Physica B 280 (2000) 253}257

Theory of relaxation phenomena in glasses and dopedsemiconductors at low temperatures

Il'ya Polishchuk!,*, Alexander Burin", Yu. Kagan#, Leonid Maksimov#

!Department of General and Nuclear Physics, Russian Research Center **Kurchatov Institute++, Kurchatov Sq. 1, Moscow 123182, Russia"Department of Chemistry and Material Research Center, Northwestern University, 2145, Sheridan Road, Evanston, IL, 60208, USA

#Department of Superconductivity and Solid State Physics, Russian Research Center **Kurchatov Institute++, Kurchatov Sq. 1, Moscow 123182,Russia

Abstract

Low-temperature relaxation properties of dielectric glass and doped disordered semiconductors, also called Fermiglasses, are studied. The isomorphism in the description of both types of material is established. In particular, it is provedthat the #ip-#op relaxation mechanism, which is a characteristic of dielectric glasses, is also speci"c for Fermi glasses. Thelow-temperatures behavior for the thermal conductivity, electrical conductivity and nuclear relaxation in the glassysystems considered is investigated. The relation to experiment is analyzed, which allows us to clarify the key role of the#ip-#op processes in the formation of the universal properties of disordered dielectrics. This assertion is still a point ofmuch controversy. ( 2000 Elsevier Science B.V. All rights reserved.

Keywords: Coulomb gap; Dielectric glasses; Dipole}dipole interaction; Fermi glasses

1. Introduction

About three decades ago Zeller and Pohl [1] revealedthat amorphous insulating solids exhibit universal anom-alous low temperature behavior. In particular, the heatcapacity quasi-linear in the temperature and the thermalconductivity which varied nearly as the square of temper-ature were found below 1 K. The phenomenologicaltwo-level tunneling system (TLS) model was proposed byAnderson, Halperin and Varma, and independently byPhillips [2,3], in order to account for this behavior.Within the framework of this model every TLS is con-sidered as an isolated entity which was quite satisfactoryin explaining the experiments only above about 100 mK.The "rst evidence for the importance of the interactionbetween TLS came from the ultrasound experiments that

*Corresponding author. Present address: Department ofGeneral and Nuclear Physics, Russian Research Center `Kur-chatov Institutea, Kurchatov Sq. 1, Moscow 123182, Russia.

E-mail address: iyp@kurm.polyn.kiae.su (I. Polishchuk)

have been carried out since about 1976 [4,5]. A spectraldi!usion was shown to take place between TLS becauseof the interaction between them. It was assumed fora long time that the spectral di!usion is the only manifes-tation of the interaction between the TLS.

Later on, the measurement of the internal frictioncoe$cient [6] and investigation of dielectric loss factor[7] in di!erent amorphous solids has indicated that thelongitudinal relaxation rate behaves as q~1

1&¹ in the

millikelvin region. In addition, the echo-type experimentsrevealed that the rate of the phase memory loss(transverse relaxation) q~1

2&¹ in Suprasil I [8,9] and

orientationally disordered mixed crystal (KBr)1~x

(KCN)x

[10,11].1 It has been established that any at-tempt to interpret these experiments within theframework of the standard model of noninteractingTLS fails.

The unusual behavior of the relaxation rates in theseexperiments has been explained in our papers [12,13]. It

1Note that, if the standard single phonon relaxation of TLS isconsidered, one would expect q~1

11)&¹3 and q~1

21)&¹2.

0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 6 6 5 - 8

has been shown that the dipole}dipole interaction be-tween TLS leads to appearance of a branch of delocalizedcollective low-energy excitations. At ultra-low temper-atures, when the phonons are frozen out to a large extent,the relaxation processes are due to this collective excita-tion branch.

In Section 2 the standard model of isolated TLS isintroduced. Section 3 outlines the features of the TLSmodel that takes into account the dipole}dipole interac-tion. In particular, we describe the parameters of thecollective delocalized excitations that arise in the systembecause of this interaction. Section 4 links the model ofinteracting TLS to that of the strongly disorderedCoulomb system. For example, the system could beamorphous or doped crystalline semiconductors.The isomorphism between these models is established.Section 5 is concerned with the description of certainphenomena governed by the relaxation of the collectiveexcitation considered. In particular, we examine how thisrelaxation in#uences the thermal conductivity andnuclear relaxation. In addition, special attention is paidto the phononless electric conductivity mechanismconcerned with the spectral di!usion brought about bythe relaxation of collective excitations in the Coulombsystem.

2. The standard model of TLS

An isolated TLS is described by the standard pseudos-pin Hamiltonian [2,3]

h"!DSz!D0Sxi, (1)

and the level detuning D and transition amplitude D0

aresupposed to be distributed according to the dependence

P(D,D0) dDdD

0"

PMD0

dDdD0. (2)

In numerous physical processes, only the so-calledthermal TLS play a key role. These TLS are character-ized by the parameters of DyD

0y¹ and, correspond-

ingly, by the excitation energy E"JD2#D20K¹.

Hence the concentration of thermal TLS reads

cT"a~3(PM ¹), (3)

with a being the minimal admissible distance betweenTLS. The single-phonon relaxation rate of the thermalTLS is

q~11)

";0(¹/v)3, ;

0"

c2ov2

. (4)

Here c describes the strain TLS-phonon interaction,o and v are material density and sound velocity, respec-tively.

3. The interacting TLS and phononless relaxation

The strain interaction between TLS and phononsresults in the indirect interaction between the TLS

<K "1

2+i,j

;(Rij)Sz

iSzj, D;(R

ij)D";

0R3

ij

, (5)

where Rij

is the distance between two TLS and thecon"guration average S;(R

ij)T"0. This interaction

gives rise to a relaxation of the TLS di!erent from thephonon-assisted one (4). To clarify how this relaxationoccurs, let us consider a hopping of the excitation fromthe excited TLS with E&¹ to the other TLS with E@&¹

which was initially in the ground state. As a result of therelaxation, the "rst TLS excited before goes over into theground state, while the second TLS gains the excitation.The initial and "nal states of the pair described make upa two-level #ip-#op con"guration. Therefore the last canbe considered as a new type of two-level system with thedetuning of D

p"DE!E@D. One should emphasize that

there exist #ip-#op con"gurations for which Dp;¹.

Because of the interaction (5), two levels of the #ip-#opcon"guration are coupled by the tunneling amplitudeD0p

K;0/R3

ij. Therefore, the transition between the

levels occurs if Dp(D

0p. The pair of TLS considered is

called a resonant pair (RP).If temperature ¹'0, the excited TLS concentration

has a macroscopic value. For this reason, so does thenumber of RP. Because of the dipole}dipole interactionbetween TLS, RPs interact strongly. So the relaxation ofany RP occurs under the control of other RPs. Thus, itwould be a fallacy to assume that the RPs relax asisolated objects. It is shown in Refs. [12,13] that RPs thatbelong to the strip with D

0pK¹(PM ;

0)2 compose the

in"nite cluster, and the concentration of these RPs is

NH"a3(PM ¹)(PM ;0), (6)

The elementary excitation energy speci"c for this clusterequals DH"¹(PM ;

0)2, and the corresponding relaxation

rate for RP considered is given by the dependence

q~1H KDH . (7)

It is important that the relaxation of the RP within thein"nite cluster considered gives rise to the spectral di!u-sion in the environmental TLS. The scale of this di!usionis

C"m¹(PM ;0), (8)

m being a factor logarithmically large. Finally, note thatfor the RP concept to be valid, the coherent couplingbetween TLS within the RP should not be destroyedby phonons during the time qH , and therefore [12,13]

¹(¹0"(PM ;

0)2Jv3/;

0. (9)

254 A. Burin et al. / Physica B 280 (2000) 253}257

4. From dipole glasses to Fermi glasses through theCoulomb gap

To describe Fermi glasses, let us consider a disorderedsystem in which the electronic states are localized close tothe Fermi level. For the sake of simplicity, let the local-ization radius l of these states be of the order of theaverage distance between the localization centers. TheHamiltonian is written as [18]

H"H0#<

1#<

2, H

0"+

i

Uini#+

ij

<ijninj,

<1"+

ij

D0(R

ij)c`

jci, <

2"+

ij

;ijklc`ic`jckcl. (10)

Here H0

is the energy of the electrons in the disorderedpotential U

irandomly distributed within the energy strip

of width=, and <ij"e2/r

ijpresents the Coulomb inter-

action; D0(R

ij)ye2R

ij/l2exp(!R

ij/l) describes an elec-

tron tunneling at distance Rij. The matrix elements

;ijklye2l2/R3

ikaccount for the simultaneous tunneling of

two electrons separated by the distance Rik, both elec-

trons hopping at a distance of the order of l. The oper-ators c`

iand c

icreate and annihilate an electron at the

cite i and ni"c`

ici.

Let ei

be the energy of the charge excitation whicharises if the electron transfers from in"nity to site i andvice versa. There is an energy gap of the width of=

0in

the energy spectrum of charge excitations and the distri-bution function is [14]

g(e)"1

=Ae=

0B

2, e(=

0"

e2

lc1@2, c"

e2

l=.

(11)

Below we shall focus mainly on the temperature range of¹(=

0. Therefore the concentration of the low energy

charge excitations is small.As taken from the Fermi level, e

i'0 for the vacant site

positively charged, otherwise ej(0 for the occupied site

negatively charged at ¹"0. Transferring an electronfrom the site i to the site j requires the energy [14]

D"ej!e

i!e2/r

ij. (12)

The electron occupying either the site i or the site j, onecan treat these two sites as a kind of TLS. The distribu-tion function for these dipole depends on both thedetuning D and the size r. In the spirit of the approachconsidered brie#y in Section 3 we should concentrate"rst on the resonant thermal TLS obeying the condition¹(D

0Ke2/l exp (!r/l). This restriction is valid for

r(l ln (e2/l¹). If the temperature is not exponentiallysmall, the value of r(r

0"lc~1@2. It is shown in [14]

that, if the excitation size is shorter than r0, the low

energy distribution function is

dF(D, r)KAe

l2=B2dDd3r. (13)

Further, due to the disorder the distance between thelocalization centers is a uniformly distributed randomvalue, and one can substitute dD

0/D

0for d3r in Eq. (13)

within logarithmic accuracy. Introducing the notationsPM "(e/l2=)2 and;

0"(el)2 one can readily rewrite Eq.

(13) in the form of (2). If the dipole}dipole interaction istaken into account, the model considered is completelyisomorphic to that described in Section 3.

5. Dipole}dipole interaction in disordered dielectrics andrelated phenomena

Recently the in#uence of the dipole}dipole interactionon the low-temperature behavior of amorphous dielec-trics has been widely disputed in the literature. Below wewill describe some possible features of these substances,which should be due to the delocalized collective excita-tion described above.

5.1. Low-temperature thermal conductivity

As shown in Section 3, the thermodynamic and kineticproperties of the TLS model with dipole}dipole interac-tion should be described at the temperature below¹

0within the framework of the RP concept. There exists

an in"nite cluster of the strongly coupled RP, and theexcitations are no longer localized. Therefore, the energytransport becomes possible over this cluster at a macro-scopic distance. Let us estimate the thermal conductivitygoverned by these excitations.

The concentration of the RP considered is NH (6) and,therefore, the excitation is transferred at the distanceRH&N~1@3H during the time qH (7). Then the energydi!usion coe$cient is described as

DH"1

3

R2HqH

"

¹

+

(PM ;0)4@3

(PM ¹)2@3. (14)

The speci"c heat of the excitations concerned iscHK2(PM ¹)(PM ;

0)3. Thus one can estimate the thermal

conductivity KH using the simple kinetic theory as

KH"cHDH"2¹

3+(PM ¹)1@3(PM ;

0)13@3. (15)

The conventional low-temperature heat transport is dueto the phonons scattered by TLS. In this case

K1)"

v

2p(PM ;0)a2

0A

¹

HDB

2. (16)

A. Burin et al. / Physica B 280 (2000) 253}257 255

If the temperature is low enough, the thermal conductiv-ity is governed by the RP transport, Eq. (15). The cross-over temperature is very small for the conventionaldielectric glasses like SiO

xsince the dimensionless

parameter PM ;0(10~2. On the other hand, in the mixed

crystal the TLS density of states is three orders largerthan in the standard dielectric glasses. Thus the param-eter PM ;

0can be larger than 0.1 and the crossover point

between (15) and (16) should exceed 1 mK.

5.2. Low-temperature electric conductivity of Fermiglasses

The variable hopping length electron transport isanother class of phenomena in which the RP can playa key role. One should turn again to the model describedin Section 4.

Let us pick out a site (a), a localization centers, with acertain energy e. Next, choose a second site (b) separatedfrom the "rst by the distance R<r

0. Then, one can

neglect the correlation between the energies of the sitesconsidered. Let these sites constitute a TLS with thedetuning energy within a strip of a certain width u(e.Hence the second site should have approximately thesame energy e. Taking into account Eq. (11), one canwrite down the probability to "nd site (b) within thesphere of the radius R as

P(e,u,R)Kg(e)uR3. (17)

Let us recall that the irreversible process is inherent in theconsidered model due to the relaxation of the resonantpair (see Sections 3 and 4). Like the dielectric glass case[12,13], this relaxation gives rise to the spectral di!usionresulting in the #uctuations in u on the scale of C(R/l)"¹(PM ;

0)R/l (see (8)) during the time q

0"[¹(PM ;

0)3]~1.

If the two sites chosen above have the detuning uKC,one can calculate the minimal R for the detuning to behowever small. This is the condition necessary for theelectron to hop. It is valid with the probability of unity ifP(e,C,R)"1. The latter de"nes implicitly the hoppinglength

RC (e)"[m¹(PM ;0)g(e)/l]~1@4. (18)

In addition, the e!ective rate for the electron to hop atthe distance R depends strongly on the spectral di!usionrate CRC(e)/lq

0and is given by the expression [15]

q~1C (e)"q~10 C1!expA

D20

RC(e)C/lq0BD

K

D20l

RC (e)C. (19)

Then the di!usion coe$cient is estimated as

DC (e)"R2C(e)q~1C

"RC

;20

m¹(PM ;0)l5

expC!2RC

l D. (20)

One can now estimate the electric conductivity using theEinstein relation,

p"Pdeg(e) expA!e¹B

e2

¹

DC(e). (21)

Estimating this expression, we conclude that

p&GexpC!

¹@0

¹ D1@2

if ¹(¹A#,

expC!¹A

0¹ D

1@4if ¹A

#(¹(¹@

#.

(22)

Here ¹@0";

0/(mc1@3)2 and ¹A

0";

0/mc3, and ¹A

c"

;0c5@3 and ¹@

#";

0c.

This result should be compared with that of Shklovskiiand Efros for the phonon-induced electron hopping [14].The main conclusion of the paper [14] is that below¹@

#one should always observe the 1

2-dependence, while

above this temperature the Mott temperature 14-depend-

ence should take place. The comparison of the result ofRef. [14] with that obtained in this section leads us to theconclusion that, due to the large preexponent omitted in(22), the conductivity mechanism proposed is more e!ec-tive than the phonon assisted one at ¹(¹@

#. This means

that the Shklovskii}Efros conductivity is suppressedwithin this temperature region. One should still observethe 1

4-dependence down to the temperature ¹A

#;¹@

#. Yet,

it is due to the proposed mechanism rather than theMott one. And only below ¹A

#the 1

2-dependence can be

revealed. Again, in this case the dependence does notconcern with phonon-assisted hops. This conclusionprobably could shed some light on the fact that theCoulomb gap e!ect has not been brought out in manyexperiments.

5.3. Low-temperature nuclear relaxation

The low-temperature relaxation of nuclei having itsown quadrupole electrical moment in dielectric glasseshas a matter of investigation during last two decades. Atvery low temperature the phonon-induced nuclear relax-ation (&¹7) is inhibited by the interaction of nuclei withthe dipole moment of TLS invariably inherent in dielec-tric glasses. If a TLS experiences relaxation, it gives riseto nuclear relaxation owing to the dipole}quadrupoleinteraction between the nucleus and the TLS.

There are two ways for the TLS relaxation. In bothcases the nuclear relaxation is due to the TLS relaxation.

256 A. Burin et al. / Physica B 280 (2000) 253}257

However, in the "rst case a TLS experiences the phonon-induced relaxation (4) while in the second the TLS re-laxes only owing to the dipole}dipole interaction withthe corresponding rate (7).

Let the frequency of nucleus transition be u, the TLSrelaxation time be q, and A be the coupling constantbetween nucleus and TLS. Then, in the "rst case thenuclear relaxation rate is [16]

C1)"(PM ¹)A

A

a4uB2q~11)

&¹4, (23)

while in the second case [16]

CH"(PM ¹)(PM ;0)A

A

a4uB2q~1H &¹2. (24)

Because of the fact that parameter (PM ;0) is very small in

glasses like SiO2, the nuclear relaxation seems to be too

slow to be observed. On the other hand, in the mixedcrystals the nuclear relaxation rate can run up to theorder of 10~4}10~3 s~1. This is quite acceptable fromthe view-point of the experiment. Let us note that similarrelaxation rates have been detected in amorphous As

2S3

[17]. In addition, in this paper it has been found that thenuclear relaxation rate behaves as ¹a, 1)a)2. Thisresult probably evidences the dipole}dipole relaxation ofTLS.

6. Conclusion

Low-temperature relaxation properties of dielectricglass and doped disordered semiconductors, also calledFermi glasses, are studied. The isomorphism in the de-scription of both types of materials is established. Inparticular, it is proved that #ip-#op relaxation mecha-nism, accepted now to be common with dielectric glasses,also plays a signi"cant role in the low-temperatureproperties of the Fermi glasses. The low-temperaturesbehavior for the thermal conductivity, electric conductiv-ity and nuclear relaxation in the considered glassysystems is investigated. The relation to experiment isconsidered, which would allow one to clarify the key role

of the #ip-#op processes in the formation of universalproperties of disordered dielectrics. This assertion is stilla point of much controversy.

Acknowledgements

The research described in this publication was madepossible due to "nancial assistance from RFBR (grantNo.98-02-1629), from INTAS (grant No IR-97-1066), andRussian Eduction Ministry program for the developmentof basic natural science (grant No 97-0-14.0-80)

References

[1] R.C. Zeller, R.O. Pohl, Phys. Rev. B4 (1971) 2029.[2] P.W. Anderson, B.I. Halperin, C.M. Varma, Philos. Mag.

25 (1972) 1.[3] W.A. Phillips, J. Low Temp. Phys. 7 (1972) 351.[4] S. Hunklinger, A.K. Raychaudhari, Progr. Low Temp.

Phys. 9 (1986) 267.[5] W.A. Phillips, Rep. Prog. Phys. 50 (1987) 1657.[6] P. Esquinazi, R. Koenig, F. Pobell, Z. Phys. B87 (1992)

305.[7] D.D. Oshero!, S. Rogge, D. Natelson, Czech. J. Phys. 46

Suppl. S6 (1996) 3295.[8] L. Bernard, L. Piche, G. Schumacher, J. Jo!rin, J. Low

Temp. Phys. 45 (1979) 411.[9] G. Bayer, M.v. Schikfus, Phys. Rev. B38 (1988) 9952.

[10] M.C. Foote, B. Golding, Phys. Rev. B43 (1991) 9206.[11] C. Enss, H. Schwoerer, D. Arndt, M.v. Schikfus, Phys. Rev.

B51 (1995) 811.[12] A.L. Burin, Yu. Kagan, JETP 80 (1995) 761.[13] A.L. Burin, Yu. Kagan, L.A. Maksimov, I.Ya. Polishchuk,

Phys. Rev. Lett. 80 (1998) 2945.[14] A.L. Efros, B.I. Shklovskii, J. Phys. C: Solid State Phys.

9 (1975) L49.[15] L.A. Maksimov, A.L. Burin, I.Ya. Polishchuk, Phys. Lett.

A 204 (1995) 310.[16] Yu. Kagan, L.A. Maksimov, I.Ya. Polishchuk, Sov. Phys.

ZhETP, 1999, to be published.[17] F. Devreux, L. Malier, Phys. Rev. B 51 (1995) 11 344.[18] I.Ya. Polishchuk, L.A. Maksimov, A.L. Burin, Sov. Phys.

ZhETP 106 (1994) 1169.

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