Table Of ContentMulti-Welled Tunneling Model for the Magnetic-Field Effect in Ultracold Glasses
Giancarlo Jug [*]
Dipartimento di Fisica e Matematica, Universit`a dell’Insubria, Via Valleggio 11, 22100 Como, Italy
CNISM – Unit`a di Ricerca di Como and INFN – Sezione di Pavia, Italy
(Dated: January 15, 2009)
Puzzling observations of both thermal and dielectric responses in multi-silicate glasses at low
temperatures T to static magnetic fields B have been reported in the last decade and call for an
extension of the standard two-level systems tunneling model. An explanation is proposed, capable
9 ofcapturingatthesametimetheT-andB-dependenceofthespecificheatCp andofthedielectric
constantǫintheseglasses. Thistheorypointstotheexistenceofanomalousmulti-welledtunneling
0
systems in the glasses – alongside the standard two-level systems – and indications are given for
0
glasses which should achievelarger electric magnetocapacitive enhancements.
2
n PACSnumbers: 61.43.Fs,77.22.Ch,77.22.-d,65.60.+a
a
J
5 The lastdecade has seenmuchrenewedinterestinthe via nuclei in the glasses carrying an electric quadrupole
1 physics of cold non-metallic glasses,materials displaying moment as well as a magnetic dipole one [6]. The nu-
some universalphysicalproperties attributedto the low- clear mechanism is supported by some features of the
]
n energyexcitationscharacterisingmostamorphoussolids. observed magnetic-field dependence of the polarization
n The two-level systems (2LS) tunneling model (TM) [1] echo (PE) experiments in the mentioned multi-silicate
-
s has been rather successful in explaining a variety of in- glasses in the millikelvin range [7]. Moreover the ampli-
i terestingphenomena dealingwiththe thermal,dielectric tude of the PE in Glycerol glass was shown to become
d
. and acoustic properties of structural glasses at tempera- strongly B-dependent only upon deuteration and thus
t
a tures T < 1 K. Limitations and failures of the 2LS TM the introduction of quadrupole-moment carrying nuclei
m (treating cooperativemotion in terms of single particles, in the glass [8]. However, though pure a-SiO2 (devoid
- mostly) on the other hand, have been also discussed [2]. of quadrupole-moment carrying nuclei) shows no PE-
d
Thesematerialsdonotpresent,normally,anyremark- amplitudemagneticfielddependence[7],Glycerolglass–
n
o able magnetic-field response phenomena other than pos- deuterated or not – shows no measurable B-dependence
c sibly a weak contribution from trace paramagnetic Fe- initsdielectricconstant[9]. Thenuclearapproachisalso
[ impurities. It came thus as a great surprise when mea- unable to account for the magnitude and features of the
1 surements showed [3] that in some multi-component sil- B-andT-dependenceofδǫ/ǫforthemulti-silicateglasses
v icate glasses (but not in pure a-SiO2) one is able to ob- [10] and the (also unusual) B- and T-dependence of the
4 serve changes in the dielectric constant ǫ(T,B) and al- heatcapacityCpofthelatter–notentirelylinkedtotheir
9 ready in magnetic fields B as weak as a few Oe. A typi- Fe impurity contents[11,12]– is notevenaddressed. To
2
cal glass showing a strong response has some 100 ppm add to the mystery, the acoustic response (also linked
2
. Fe3+ and the composition Al2O3-BaO-SiO2 (in short to the 2LS coupling to phonons) of borosilicate glasses
1
AlBaSi-O, in this paper). Measurements made on a BK7 and AF45 has been found to be independent of B
0
9 thick sol-gelfabricatedAlBaSi-O film showedchangesin [13]. Table Isummarisesthis ratherpuzzling experimen-
0 δǫ/ǫ=[ǫ(B)−ǫ(0)]/ǫ(0)oforder10−4 andcharacterised tal situation. Therefore, either the nuclear explanation
: by an enhancement peaking around 0.03 T for 10 mK is specific to the PE magnetic effect, or an entirely new
v
i <T < 200mK thenfollowedbya reduction ofǫ for B > explanation should be found for all of the observations.
X
0.1T.Afurtherenhancementwasalsoobservedatmuch
ThepurposeofthisLetteristobegintogivearationale
r higher fields (B > 10 T).
a tothe situationinTableI,aswellastostimulatefurther
Another, cleaner, multi-component silicate glass experimental and theoretical research. A novel explana-
(borosilicate BK7, with 6 ppm Fe3+) and a dirtier one tion, already shown to account for the unusual behavior
(borosilicate Duran, with 120 ppm Fe3+) have shown of C (T,B) in the multi-silicates [12], is here shown to
p
similar – but weaker – magnetic anomalies, |δǫ/ǫ| ∼ explain the behavior of the dielectric constant as well.
10−5, seemingly excluding the paramagnetic impurities This simple theory is centered on the observation, in
as their source [4]. Yet another multi-silicate glass, a- computer simulations and experiments [14], that multi-
SiO2+xCyHz wasinvestigated[5],confirmingtheunusual silicateglassespresentintheiratomicstructureboththe
findings in AlBaSi-O. A convincing explanation for the connectednetworkofSiO4-tetrahedraandacollectionof
unusual electric magnetocapacitance behavior of these pockets and channels of non-networking ions showing a
cold glasses has not yet been found. tendency to form microaggregates and to partially de-
Recently, some consensus has been gained by the idea stroy the SiO4-network. Fig. 1 shows as an example a
of a coupling of the standard 2LS to the magnetic field snapshot of a simulation of the (Na2O)-3(SiO2) glass il-
2
glass type Ref. δCp δǫ δvs δAPE and are taken, normally, to be distributed in the glass
aa--SSiiOO22+xCyHz [11],[[54]],[7] N?O YNEOS ?? N?O saoutnhiafotr∆madnidstrlnib∆ut0io(nro:uPgh2LlyS,(t∆he,∆po0)te=ntiPa¯l/b∆a0r,riPe¯r)bheainvge
a material-dependent constant. This description holds
AlBaSi-O [11],[3],[7] YES YES ? YES
for the network-forming (NF) TS. For the NM ATS in-
Duran [11],[4],[7] YES weak ? YES
stead, the simplest 3D generalization of the 2LS TM is
BK7 [11],[4],[13],[7] weak weak NO YES that of other fictitious charged particles, each moving in
AF45 [13] ? ? NO ? a multi-welled 3D potential (see Fig. 2) and coupling to
Glycerol [9],[8] ? NO ? weak the magnetic field through their orbital motion. For the
d-Glycerol [9],[8] ? NO ? YES simplest case of nw =3 potential wells, one can use:
TABLEI:Presenceofmagnetic-fieldinducedvariationsinthe E1 D0eiφ/3 D0e−iφ/3
physicalpropertiesofsomecoldglasses. Cp: heatcapacity,ǫ: H0 = D0e−iφ/3 E2 D0eiφ/3 (1)
[d?i:elnecotrinicvecsotnigsatatinotn, kvsn:owsonu]nd velocity, APE: PE amplitude D0eiφ/3 D0e−iφ/3 E3
where D0 ≃ ¯hΩe−UB/h¯Ω is some 3D barrier’s trans-
lustrating such situation. The present theory proposes parency(UB barrierheightandΩ single-wellfrequency),
thatthemagneticeffectsarisefromanomaloustunneling E1,E2,E3arethesinglewells’ground-stateenergyasym-
systems (ATS) forming in the cooling of such structure metries (E1+E2+E3 =0, say) and
within the non-networking (or network-modifying, NM)
pockets and channels, whilst the SiO4-network remains φ = 2πΦ(B)/Φ0 (2)
the nest of the ordinary non-magnetic 2LS. Φ(B) = B·S△ =BS△cosβ
is some Aharonov-Bohm phase for a “particle” carrying
charge q tracing a closed path of area S△ threaded by a
magnetic flux Φ(B) (Φ0 ≡h/|q|=ϕ0|e/q| being the ap-
propriate flux quantum (ϕ0 being the elementary one)).
V(x,y)
(b)
(c)
FIG. 1: Molecular dynamics snapshot of the structure of
sodium trisilicate at 2100 K at the density ρ = 2.2 g cm−3.
(a) Ba, Al
The blue spheres that are connected to each other represent O Si
the (NM) Na atoms. The Si-O network is drawn by (NF)
yellow (Si) and red (O) spheres that are connected to each
FIG.2: 2Drepresentationoftheplausiblesourceofmagnetic-
other bycovalent bonds(from [14], by permission).
field sensitive (anomalous) tunneling systems in (e.g.) the
AlBaSi-O glass. Thetightvitreous-SiO2 structure(a) isbro-
In order to couple the ATS to the magnetic field, a kenupbytheAl-and(large)Ba-atoms(b),thusleavingmany
simple 3D generalization of the 2LS TM is in order. As metal ions free to move as “particles” in a 3D tunneling po-
is known[1], in the TM the coldglassis thoughtto have tential characterised bynw-minima, with nw ≥3 (c).
few remaining degrees of freedom rappresented by fic-
titious “particles”, each moving quantum-mechanically In this model the choice D0 > 0 can be thought of
within a 1D double-welled potential. At low temper- as arising from the coherent tunneling motion of a small
atures only the ground states of the individual wells clusterofNM-ions;thiswilllead,asisseen,tohighvalues
|ii (i=1, 2) are relevant and in this representation the of the product |q|S△D0. For D ≡ E12+E22+E32 ≪
Hamiltonianofasingle2LSreadsH0 =−12∆σz−21∆0σx D0 and weak fields (φ ≪ 1) theplowest energy gap
(σ Pauli matrices) with ∆ the ground-state energy of the model is seen to open with increasing magnetic
µ
asymmetry between the two wells and ∆0 the barrier’s field according to the simplified expression [12] ∆E ≃
transparency. These two parameters are linked to the D2φ2+D2 containing the main physics of this model
0
real potential’s details as discussed in the TM literature p(regardless of the value of n ). One more assumption of
w
3
thistheory(seentoexplainthenearly-flatT-dependence The second term in the numerator of Eq. (7) gives rise
of C for B = 0 in some temperature range for these to a peak in δǫ/ǫ at very low B, while the first term
p
glasses [12]) is to take the parameters’ distribution uni- (present only if n > 2) gives rise to a negative contri-
w
form for U , but favoring near-degeneracy (to a degree bution to δǫ/ǫ at larger B which can win over the en-
B
fixed by a lower bound Dmin 6=0 for D) for the {Ei}: hancement term for all values of B if D0max ≫ D0min
P∗ (D0min, D0max corresponding to cutoffs in the distribu-
PATS({Ei},···;D0)= (E12+E22+E32+···)D0. (3) tDiounraonfaAnTdSBeKn7erignydebeadrrsiheorswUaBs)i.gnTifihceanotbdseerpvraetsisoionns ionf
This anomalous distribution forthe ATScanbe thought ǫ(B)forweakfields[4],thusgivingdirectevidenceforthe
of as arising from a degree of devitrification in the mate- existenceofATSwithnw >2inthemultisilicateglasses.
rial, as measured by the parameter P∗. In fact, it is Carryingoutthe averaging[...]av onegetsanalyticalex-
reported [15] that thick glass films prepared with the pressionsfor the polarizability;the uniform averageover
sol-gel technique (such as the AlBaSi-O films of the ex- orientation angles β must be performed numerically.
periments) are multiphase materials with microcrystals
Magnetocapacitance, Al O -BaO-SiO Glass "AlBaSi-O"
embedded within an amorphous glassy matrix [16, 17]. 2 3 2
Indeed, amorphous solids with the general composition
3
Al2O3-MgO-CaO-SiO2 are termed “glass ceramics” in AlBaSi-O, 118 mK
the literature [16] owing to partial devitrification occur- 2 A=20, Cp fit
best fit
ring. It seems thus reasonable to imagine that in these
1
glasses some NM-ions provide nucleation centres for the
microcrystals and that, therefore, TS presenting near- 0
0.5
symmetric wells will be favoured in the neighborhood of εε/-1
and inside such crystallites. ∆x 0
This approach has provided a good description [12] 510-2 -0.5
of the Cp(T,B) data for AlBaSi-O and Duran [11]; one -3 -1
can treat the ATS as effective 2LS having gap ∆E for Duran, 15.4 mK
“weak” fields. Within this picture, the linear-response -4 -1.5 A=4, Cp fit
best fits
quasi-staticresonantcontributionto the polarizabilityis -5 -2 BK7, 15.4 mK
∞ 0.001 0.01 0.1
dE E E -6
αRES = G { i};p tanh( )δ(E−∆E) 0.001 0.01 0.1 1
µν Z0 2E µν(cid:18) E i(cid:19) 2kBT B [T]
(4)
FIG.3: Dielectricconstant’schange(realpart)inamagnetic
where
field, measured and from the present theory, for the multi-
E nw E E component AlBaSi-O glass at T = 118 mK (driving field 15
Gµν(cid:18){Ei};pi(cid:19)= piµpiν − Ei2jpiµpjν (5) kV m−1, driving frequency 1 kHz) [inset: data and theoreti-
Xi=1 Xi,j cal calculations for the BK7 and Duran glasses at 15.4 mK].
Data from Ref. [4].
contains the single-well dipoles p = qa . This expres-
i i
sion assumes vanishing electric fields and no TS-TS in-
Fig. 3 shows the change due to a magnetic field in the
teractions, a situation which does not wholly apply to
the experiments. To keep the theory simple one can still dielectricconstantǫ(T,B)=ǫ0+α(T,B)fortheAlBaSi-
Oglasswithinthepresentoversimplifiedtheory,whichis
use Eq. (4) and the analogous one for the relaxational
used to fit the data at 118 mK [4]. It is seen that using
contribution to the polarizability. Eq. (4) must be aver-
the parameters extracted from fitting this theory [12] to
aged over the random energies’ distribution (3) ([...] ,
av the C (T,B) data [11] and a value A = 20.0 × 10−5 for
responsible for the high sensitivity to weak fields) and p
over the dipoles’ orientations and strengths ((...)). For the dimensionless prefactor A ≡ p2iP∗/(ǫ0ǫrDmin) one
a collection of ATS with n > 2 this averaging presents can reproduce all the qualitative features of the puz-
w
seriousdifficultiesandonemustresorttothedecoupling: zling B-dependence of the magnetocapacitance for this
glass. A best fit to the 118 mK experimental data yields
Gµνδ(E−∆E)≃Gµν ·δ(E−∆E), (6) the parameters reported in Table II. The T-dependence
of the calculated magnetocapacitance is also in qualita-
where [δ(E−∆E)] =g (E,B) is the fully-averaged
av ATS tive agreement with the experimental data [3, 4]. In a
density of states. To calculate G , one can envisage
µν comparison with data for the BK7 and Duran glasses
a fully isotropic distribution of planar n -polygons to
w [4] (at the much lower only available temperature of
obtain:
15.4 mK, however) the present theory reproduces the
G = 1 nw p2(nw−2)E2+D02φ2δ . (7) mainfeaturescorrectly. ThevaluesofD0min|q/e|S△ and
µν 3(cid:18)nw−1(cid:19) i E2 µν D0max|q/e|S△ are very similar to those used to fit the
4
Cp(T,B) data [12]; the small values of D0min/D0max alargermagneticresponsethanthusfarobservedshould
and of A for BK7 denote a much reduced presence of be found in the best ceramic glasses, like for instance
microcrystallitesinthematerial. ThevaluesofD are Ceran.
min
lower (due to the strong electric field applied) and more
realistic than those used in the analysis of C (T,B), in
p
The permission by its Authors to display Fig. 1 is
all cases confirming the consistency of the assumption
gratefully acknowledged.
|Ei|/D0 ≪1 in this theory.
glass A Dmin D0min|q/e|S△ D0max|q/e|S△
type [mK] [K ˚A2] [K ˚A2]
AlBaSi-O 1.42×10−4 30 9.72×104 1.69×105 [*] email: [email protected]
Duran 0.98×10−5 34 1.83×104 6.28×105 [1] W.A. Phillips, J. Low Temp. Phys. 7, 351 (1972);
BK7 0.95×10−5 24 4.16×104 1.09×106 PM.Wag..A2n5d,e1rs(o1n9,75B);.IP..HEasqlpueirnianzia(nedd.)CT.Mun.nVelainrmgaS,ysPtehmils.
in Amorphous and Crystalline Solids (Springer, Berlin,
TABLE II:Local ATS parameters extracted from this fit
1998)
[2] YuC.C.andLeggettA.J.,CommentsCond.Mat.Phys.,
Finally, the present simplified theory has focused 14, 231 (1988)
on the weak B-field regime (up to 1 T). Given the [3] P.Strehlow,C.EnssandS.Hunklinger,Phys.Rev.Lett.
large values of D0|q/e|S△ thus extracted (indicating 80, 5361 (1998); P. Strehlow, M. Wohlfahrt, A.G.M.
that small coherent clusters of some 5 to 10 NM- Jansen, R. Haueisen, G. Weiss, C. Enss and S. Hun-
klinger, Phys.Rev. Lett. 84, 1938 (2000)
ions are involved in the magnetic-sensitive tunneling
[4] M. Wohlfahrt, P. Strehlow, C. Enss and S. Hun-
[12]) one can expect the low-B expression employed klinger, Europhys. Lett. 56, 690 (2001); M.
for the gap to break down around some larger field Wohlfahrt, Ph.D. Thesis (Heidelberg 2001, www.ub.uni-
∗
B ∼ ϕ0(D0min/D0max)/(2π|q/e|S△) above which the heidelberg.de/archiv/1587)
gapgrowssub-linearlywithφ(B). Aswillbeshownelse- [5] J. Le Cochec, F. Ladieu and P. Pari, Phys. Rev. B 66,
where,this isinturnresponsibleforthe secondenhance- 064203 (2002)
ment of δǫ/ǫ observed in the experiments for B > B∗ [6] A.Wu¨rger,AFleischmannandC.Enss,Phys.Rev.Lett.
∗ 89, 237601 (2002)
(B being rather material-dependent).
[7] S. Ludwig, C. Enss, P. Strehlow, and S. Hunklinger,
Insummary,the ingredientsofthe presenttwo-species Phys.Rev.Lett.88,075501(2002);S.Ludwig,P.Nagel,
TM together with the reasonable assumption of partial S.HunklingerandC.Enss,J.LowTemp.Phys.131,89,
devitrificationinthefilms(whichcanbecheckedthrough (2003)
X-ray analysis) allows for a first good understanding of [8] P. Nagel, A. Fleischmann, S. Hunklinger and C. Enss,
the puzzle of the magnetocapacitance in the cold multi- Phys. Rev.Lett. 92, 245511 (2004)
silicate glasses. The absence of a magnetoacoustic re- [9] M. Brandt, Ph.D. Thesis (Heidelberg 2004,
http://www.ub.uni-heidelberg.de/archiv/4822)
sponse in such glasses with ATS can be understood in
[10] D. Bodea and A. Wu¨rger, J. Low Temp. Phys. 136, 39
terms of the much higher resonance frequencies of the
(2004)
NM-pockets and channels, and experiments should be
[11] L. Siebert, Ph.D. Thesis (Heidelberg 2001, www.ub.uni-
done in such conditions. As for the PE-experiments,the heidelberg.de/archiv/1601)
orbital-coupling approach with the inclusion of TS-TS [12] G. Jug, Phil. Mag. 84, 3599 (2004)
interactions has been shown to provide a partial expla- [13] M. Layer, M. Heitz, J. Classen, C. Enss and S. Hun-
nation for the B-dependence of the PE-amplitude at ul- klinger, J. Low Temp.Phys. 124, 419 (2001)
[14] A.Meyer,J.Horbach,W.Kob,F.KarglandH.Schober,
tralowtemperatures[18]. Thepresenttheory,interaction
Phys. Rev.Lett. 93, 027801 (2004)
improved, is expected to also provide an explanation for
[15] G.Schuster,G.Hechtfischer,D.BuckandW.Hoffmann,
such data, the so-calledisotope effect [8] being, possibly, Rep. Prog. Phys.57, 187 (1994) (p. 212)
fabrication- rather than isotope-related. Further experi- [16] H.BachandD.Krause,AnalysisoftheCompositionand
ments areneeded: clearly,ifthe magneticeffects aredue Structure of Glass and Glass Ceramics (Springer, New
to quadrupole moments then the response should scale York 1999)
with the quadrupole-carrying nuclear concentration. If [17] N.Pellegri,E.J.C.DawnayandE.M.Yeatman,J.Sol-Gel
Sci. and Tech. 13, 783 (1998); nanocrystals are ubiqui-
tunneling paramagneticmoments areinvolved,asis sug-
tousinsol-gelpreparedsilicateglasses,seee.g.: G.Deet
gestedbyalocalisedTMalsocapableofprovidingagood al.,J.Non-Cryst.Sol.194,225(1996);X.L.Duanetal.,
explanationfortheCp andǫdata[19],thenthemagnetic J. Cryst. Growth, 252, 311 (2003)
response should scale with the Fe-concentration. In the [18] A. Akbari and A. Langari, Phys. Rev. B 72, 024203
present approach the response scales with the NM-ions’ (2005)
concentrationandwiththedegreeofdevitrification,thus [19] A. Borisenko and A.Bakai, Physica B 388, 112 (2007)
