ARTICLE IN PRESS

Physica B 403 (2008) 3942–3945

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

� Corr

Systems

E-m

journal homepage: www.elsevier.com/locate/physb

Quadrupole effect on the heat conductivity of cold glasses

Alireza Akbari a,b,�

a Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germanyb Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan, Iran

a r t i c l e i n f o

Article history:

Received 13 December 2007

Received in revised form

16 July 2008

Accepted 17 July 2008

PACS:

61.43.Fs

65.60.þa

Keywords:

Low temperature properties of glasses

Two level system

Heat conductivity

26/$ - see front matter & 2008 Elsevier B.V. A

016/j.physb.2008.07.038

esponding author at: Max Planck Institute

, 01187 Dresden, Germany.

ail address: alireza@mpipks-dresden.mpg.de

a b s t r a c t

At very low temperatures, the tunneling theory for amorphous solids predicts a thermal conductivity

k / Tp , with p ¼ 2. We have studied the effect of the nuclear quadrupole moment on the thermal

conductivity of glasses at very low temperatures. We developed a theory that couples the tunneling

motion to the nuclear quadrupoles moment in order to evaluate the thermal conductivity. Our result

suggests a cross over between two different regimes at the temperature close to the nuclear

quadrupoles energy. Below this temperature we have shown that the thermal conductivity is larger than

the standard tunneling result and therefore we have po2. However, for temperatures higher than the

nuclear quadrupoles energy, the result of standard tunneling model has been found.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

Amorphous or glassy materials differ significantly fromcrystals, especially in the low-temperature range. Below 1 K, thespecific heat Cv of dielectric glasses is much larger than incrystalline materials. Moreover, the thermal conductivity k isorders of magnitude lower than the corresponding values found intheir crystalline counterparts. Cv depends approximately linearlyand k almost quadratically on temperature [1]. The generallyaccepted basis to describe the low-temperature properties ofglasses is the phenomenological tunneling model [2,3]. To explainthese behaviors, it was considered that atoms, or groups of atoms,are tunneling between two equilibrium positions, the two minimaof a double well potential (DWP). The model is known as the two-level system (TLS). In the standard TLS (STLS) model, thesetunneling excitations are considered as independent, and somespecific assumptions are made regarding the parameters thatcharacterize them [5].

The TLS can be excited from its ground state to the upper level,therefore contributing to the heat capacity. TLSs can also scatterphonons and in this way decrease their mean free path and,correspondingly, the heat conductance.

ll rights reserved.

for the Physics of Complex

New interest in this problem was stimulated by severalexperimental results [6–9]. Until these experiments it was thegeneral belief that the dielectric properties of insulating non-magnetic glasses are independent of external magnetic fields. It isvery surprising that strong magnetic field effects were discoveredin polarization echo experiments at radio frequency and in low-frequency dielectric susceptibility measurements at very lowtemperatures [6–8]. Several generalizations of the STLS modelhave been reported after the anomalous behavior of glasses in amagnetic field. According to these solutions, the models can bedivided into ‘‘orbital’’ [10–13] and ‘‘spin’’ models (nuclearquadrupole effect) [14–17]. The ‘‘orbital’’ models can provide anexplanation for some of the magnetic field effects by consideringthe flux dependence of the tunneling splitting. Unfortunately,some assumptions have been made, which cannot be reconciledwith the standard features of the tunneling model.

A surprising outcome of these experiments is a novel isotopeeffect observed in different glasses [9]. The latter effect shows theimportant influence of the nuclear quadrupole moments on theobserved magnetic field dependence. Therefore it is very im-portant to find the effect of nuclear quadrupole moments on theresponse function of glasses. For this purpose, in this paper wehave studied the thermal properties of heat conductivity of coldglasses taking into account the quadrupole effects. In Section 2using Wurger’s formalism [15], we introduce the nuclear spins inthe frame of the TLS model. We will find the general form of theheat conductivity of cold glasses, which takes into account the

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A. Akbari / Physica B 403 (2008) 3942–3945 3943

nuclear quadrupole moment in Section 3. And finally in Section 6,we end this paper by a summary and conclusion on our results.

2. TLS coupled by a nuclear spin

The STLS can be described as a particle or a small group ofparticles moving in an effective DWP. At very low temperaturesonly the ground states of each wells are relevant. Using a pseudo-spin representation the Hamiltonian of such a TLS read as

HTLS ¼12D0sx þ

12Dsz, (1)

where D is the energy offset at the bottom of the wells, and D0 isthe tunnel matrix element. Diagonalization of this two-stateHamiltonian gives the energies

E� ¼ �12E ¼ �1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2

0 þD2q

,

where E is the energy difference between the two wells. Accordingto the randomness of the glassy structure, the energy differencebetween the two wells have a broad distribution. The energyoffset and the tunneling matrix elements are widely distributedand are independent of each other with a uniform distribution of

PðD;D0Þ ¼P0

D0, (2)

where P0 is a constant. Using the notations u ¼ D0=E;w ¼ D=E

which satisfy u2 þw2 ¼ 1, the corresponding eigenstates of thediagonal Hamiltonian are given by

jc�i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�w

2

rjLi �

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�w

2

rjRi, (3)

where jc�i and jcþi are the ground and excited state of thesystem, respectively. For the moment there is no rigorous theoryfor tunneling in glasses. It is assumed that atoms or groups ofatoms participate in one TLS. As we mentioned before, in the caseof the multi-component glasses, one or several of the tunnelingatoms carry a nuclear magnetic dipole and an electric quadrupole.When the system moves from one well to another, the atomschange their positions by a fraction of an Angstrom.

We can describe the internal motion of the nuclei by a nuclearspin I of absolute value I2

¼ _2IðI þ 1Þ. For a nucleus with spinquantum number IX1 the charge distribution rðrÞ is not isotropic.Beside the charge monopole, an electric quadrupole moment canbe defined with respect to an axis e as

Q ¼

Zd3r½3ðr � eÞ2 � r2�rðrÞ. (4)

Therefore each level of the pseudo spin projection will split toð2I þ 1Þ nuclear spin projections with the quantization axism ¼ �I; . . . ; I.

This can couple to an electric field gradient (EFG) at the nuclearposition, expressed by the curvature of the crystal field potential.The potential describing this coupling is written as [19]

VQ ¼�qQ

Ið2I � 1Þ½V11I2

1 þ V22I22 þ V33I2

3�. (5)

The bases used here ðe1; e2; e3Þ are the principal axes of the tensorVij which describes the EFG, and q is the electron charge.According to the Laplace equation the potential obeyV11 þ V22 þ V33 ¼ 0. If we define the asymmetry parameterZ ¼ ðV22 � V33Þ=V11, the quadrupole potential can be expressed as

VQ ¼ �Q ½3I21 þ ZðI

22 � I2

3Þ � I2�, (6)

where we denote by �Q ¼ �qQV11=4Ið2I � 1Þ the quadrupolecoupling constant.

Therefore we can write the quadrupole potential in terms ofthe reduced two-state coordinate:

HQ ¼ VLQ

1þ sz

2

� �þ VR

Q

1� sz

2

� �� �, (7)

where VRðLÞQ is defined in Eq. (5) for the particles in right (left) well

[16]. We can use the basis jc�ðI;m�Þi ¼ jc�i � jI;m�i which isdefined as the following [15]:

H�jc�ðI;m�Þi ¼ E�;m� jc�ðI;m�Þi, (8)

where H� ¼ HDTLS þ ððV

LQ þ VR

Q Þ=2Þ �wððVLQ � VR

Q Þ=2Þ and thereforeE�;m� ¼ �E=2þ �m� ; the corresponding eigenstates satisfy:

hI;m0�jI;m�i ¼ dm0�;m� (9)

and since Hþ and H� do not commute, their eigenstates are notgenerally orthogonal:

hI;m0�jI;m�i ¼ wm0�;m�

, (10)

where these overlaps are dependent on the angle y (here y is theangle between the axis of the nuclear quadrupole in right and leftwells [17]: deR

1; eL1).

3. Heat conductivity

The dominant effect of uniform strain field (describing theinteraction of the TLS with a phonon field) is on the energy of thetunneling state by changing the asymmetry energy. The changesin the barrier height can usually be ignored [18]. Any externalperturbation is therefore diagonal in the local representationðjLi; jRiÞwhich when transformed into the diagonal representation(jcþi; jc�i) has the form

Hint ¼D0

Esx þ

DEsz

� �ge cosðotÞ ¼ H0int cosðotÞ (11)

in the presence of a strain field x ¼ x0 cosðotÞ, where x0 and o arethe amplitude and the frequency of the strain field, respectively.The strain is given by e ¼ x0ka, and the parameter g, defined as12 ðqD=qeÞ, is equivalent to elastic dipole moment, where ka is thephonon wave-vector with polarization a. Here the tensorial natureof e has been ignored and ge is written as an average overorientations. Therefore we can easily show [4] thate ¼ ð_=2roÞ1=2ka, where r is the bulk density and _ is the Planckconstant.

Using the Fermi Golden Rule, one can obtain the contributionof a phonon with wave vector ka and polarization a to thegeneralized TLS transition probability due to phonon emission andabsorption, respectively, as

Gemam0þ�!m�

¼2p_jhcþðI;m

0þÞjH

0intjc�ðI;m�Þij

2

�nEþ;m0þ

dðEþ;m0þ � E�;m� � _oaÞ

and

Gabam��!m0þ

¼2p_jhc�ðI;m�ÞjH

0intjcþðI;m

0þÞij

2

�nE�;m� dðEþ;m0þ � E�;m� � _oaÞ,

where nE�;m�¼ e�bE�;m� =Z is the Boltzmann weight,

Z ¼P�;m�

e�bE�;m� , b ¼ 1=KBT , KB is the Boltzmann constant and

T is temperature. It must be noted here that the transition

ARTICLE IN PRESS

Temperature(K)

Σ

10-4 10-3 10-2 10-1

1

1.01

1.02

1.03

1.04

1.05

θ = 0θ

θ = 0.2θ = 0.15θ = 0.1

Fig. 1. Thermal variation of the ratio parameter S, for I ¼ 1, �Q ¼ 1 mK and the

different angles between the nuclear quadrupole moment in each TLS well.

A. Akbari / Physica B 403 (2008) 3942–39453944

between the same TLS levels are zero:

hc�ðI;m�ÞjH0intjcþðI;m

0�Þi ¼ 0�!Gm��!m0

�¼ 0.

Therefore the phonon relaxation time can be found by summing

Gabam��!mþ

�Gemam0þ�!m0�

over all spin states:

t�1a ¼

Xm� ;m0þ

2pg2aoa

rv2a

u2jwm0þ ;m�j2tm� ;m0þ ðEÞd½E� ð�m� � �m0

þþ _oaÞ�,

(12)

where va is the sound velocity. Denoting

tm� ;m0þ ðEÞ ¼ nE�;m� � nEþ;m0þ

¼ebEe�b�m� � e

�b�m0þ

ebEP

m�e�b�m� þ

Pm0þ

e�b�m0

þ

(13)

and after some calculations and averaging over TLS parameters(using Eq. (2)), it can be easily shown that

t�1a ¼

P0pg2aoa

rv2a

Xm� ;m0þ

jwm0þ;m�j2tm� ;m0þ ð�m� � �m0þ þ _oaÞ. (14)

Neglecting the phase difference between the nuclear moments inthe two wells and assuming that the EFG in both wells are the

same ðwm0þ;m�¼ dm0þ ;m� ) �m� ¼ �m0þ Þ, the famous result of the STLS

model can be found [21]:

t�1a ¼

P0pg2aoa

rv2a

tanhðb_oaÞ. (15)

The thermal conductivity kðTÞ is evaluated on the assumptionthat heat is carried by non-dispersive sound waves, therefore onecan write

kðTÞ ¼ 1

3

Xa

Z 10

lðoaÞCV ðoaÞgðoaÞva doa, (16)

where lðoaÞ ¼ va=t�1a is the phonon mean free path of angular

frequency oa, gðoaÞ ¼ o2a=2p2v3

a is the phonon frequency dis-

tribution function, and CV ðoaÞ is the heat capacity of phononwhich is given by

CV ðoaÞ ¼1

ðKBT2Þ

_oa

2

� �2

csch2 b_oa

2

� �. (17)

By defining x ¼ b_oa=2 and using the above equations the heatconductivity can be obtained,

kðTÞ ¼ SðTÞ � kTLSðTÞ, (18)

where kTLSðTÞ ¼P

a ðrva=6p_2P0g2aÞK

3BT2 is the STLS heat con-

ductivity [20], and the coefficient SðTÞ is defined by

SðTÞ ¼4

p2

Z 10

x3csch2ðxÞdxP

m0þ ;m�jwm0þ ;m�

j2tm� ;m0þ ð�m� � �m0þþ 2x=bÞ

. (19)

As the exact behavior of the heat conductivity cannot be foundanalytically, we are trying to solve Eq. (19) numerically. Assumingthat I ¼ 1 and �Q ¼ 1 mK as suggested by echo experiments, we

observed the behavior of parameter SðTÞ in terms of temperature.The results are presented in Fig. 1 for different values of

quadrupole angle (y) and by averaging over the Z parameter.It can be seen that in high-temperature regimes ð�Q5KBTÞ, this

ratio ðSÞ goes to one. As it is predictable where the nuclear parteffect can be neglected and heat conductivity behaves as the STLSmodel, decreasing temperature makes this ratio grow and will besaturated at very low temperatures.

In agreement with expectation, at zero quadrupole angle theheat conductivity is the same as the result found from the STLSmodel (please see Eq. (15) and the statements before that). At low-

temperature regimes, increasing the quadrupole angle with smallvalue cause the heat conductivity saturated value to be larger thanwhat is found from the STLS model up to 1%.

The same behavior can be found for I ¼ 32 and 2. Also it can be

shown that by changing �Q , the growing regime shows adependency on the quadrupole energy value; it means that byincreasing the magnitude of �Q , the growing regime will be shiftedto higher temperatures. It shows that there is a cross overbetween two different regimes in the temperature around thequadrupole energy value.

4. Summary and conclusion

In this paper we studied the thermal properties of heatconductivity of cold glasses taking into account the quadrupoleeffects. To describe the interaction of a TLS with nuclearquadrupole, we have used a generalization of the standard TLSHamiltonian with nuclear spin. The nuclear quadrupole of thesesystems leads to a splitting of the nuclear spin levels, which isdifferent for the ground state and the excited state. The presenceof this multi-level structure increases the heat conductivitymagnitude compared to that of simple two-level systems at lowertemperature regime (KBT�Q ).

It is shown that the heat conductivity has a cross over betweentwo different regimes at the temperature which is close to thenuclear quadrupole energy, KBT�Q . It can be observed that thereare three different regimes versus temperature.

The first regime deals with high-temperature regime: KBTb�Q

where the known standard tunneling model results have beenfound. At this regime the effect of quadrupole energy can beneglected in comparison with the TLS energy scale, therefore thenuclear spin splitting is not observable.

The second regime demonstrates the temperature aroundnuclear quadrupole energy: KBT�Q . In this regime, by decreasingthe temperature the heat conductivity increases. This is the crossover regime between the standard TLS behavior and the low-temperature regime where the nuclear quadrupole effects becomeimportant. In this area the nuclear quadrupole energy levels playan important role in the thermal behavior of the system. With

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A. Akbari / Physica B 403 (2008) 3942–3945 3945

decreasing temperature the nuclear quadrupole energy is compar-able to thermal fluctuations.

In general these sub-energy level are not the same in bothwells of the TLS. Thus their eigenstates are not orthogonal and

have to overlap with each other (jwm0þ ;m�j2adm0þ;m� and

�mþa�m� ). This effect causes the mean free path of phonons to

increase. Therefore the thermal conductivity has larger value incomparison with the simple two-level system at low-temperature

regime. This means that where KBT�Q , S ¼ 1þ �ðyÞ and heat

conductivity exponent, p, is less than two instead of the p ¼ 2which has been found for the standard TLS model.

Finally for the third regime the heat conductivity will besaturated at KBT5�Q .

To obtain a theoretical expression for this effect, one can write

wm0þ ;m�¼ dm0

þ;m� þ Bm0þ ;m�

and �mþ ¼ �omþþ gmþ

where �omþ¼ �m� ,

and move to the special limiting case where Bmþ ;m�5bgmþ, and

gmþ5�o

mþ. By a little manipulation it can be easily shown that

�ðyÞ ¼ C � ðdbgÞ2, (20)

where C ¼ 1256 ½48p2 þ p4 � 384zð3Þ� 0:428 is a numerical con-

stant; ðdbgÞ2 ¼ hb2g2i � hbgi2, hbgi ¼P

mþe�b�

omþ bgmþ

=P

mþe�b�

omþ ,

and hb2g2i ¼P

mþe�b�

omþb2g2

mþ=P

mþe�b�

omþ . It shows clearly that by

increasing the quadrupole angle the difference of sub-energy inboth wells increases which causes the SðTÞ value to increase, inagreement with numerical results.

In conclusion we believe that nuclear quadrupoles play animportant role in the nature of glasses at low temperatures. In thisrespect for solving the problem of cold glasses, it is useful to findthe effect of nuclear spin on the other response functions. As far aswe know there is no experimental result for the heat conductivity[22] in the case KBT�Q . Therefore it might be a good suggestion

for future experiments to approach lower temperatures or use theglasses with larger quadrupole energy.

Acknowledgments

I would like to express my deep gratitude to A. Langari, andD. Bodea for stimulating discussions and useful comments. I amalso grateful to A. Wurger and M. Aliee for the fruitful discussions.

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