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Dielectric anomaly in coupled rotor systems

a r X i v :c o n d -m a t /0407050v 1 [c o n d -m a t .m e s -h a l l ] 2 J u l 2004

Hiroyuki Shima and Tsuneyoshi Nakayama

Department of Applied Physics,Hokkaido University,Sapporo 060-8628,Japan

The correlated dynamics of coupled quantum rotors carrying electric dipole moment is theoreti-cally investigated.The energy spectra of coupled rotors as a function of dipolar interaction energy is analytically solved.The calculated dielectric susceptibilities of the system show the peculiar temperature dependence di?erent from that of isolated rotors.

PACS numbers:34.10.+x,34.20.-b,77.22.-d

I.INTRODUCTION

With the advent of nanotechnologies,quantum rotors have attracted much attention in relevance to a funda-mental element of molecular scale machinery [1–3].Ar-rays of surface mounted quantum rotors with electric dipole moments are of particular interest because dipole-dipole interactions can be controlled and even designed to yield speci?c behavior,such as ferroelectricity.Ordered two-dimensional arrays of dipole rotors yield either ferro-electric or antiferroelectric ground states,depending on the lattice type,while disordered arrays are predicted to form a glass phase [4,5].

Besides technological problems,the microscopic dy-namics of quantum rotors have extensively studied from physical and chemical interest.The idea of quantum ro-tors is applicable to interstitial oxygen impurities in crys-talline germanium,where oxygen atoms are quantum-mechanically delocalized around the bond center posi-tion [6].The rotational of oxygen impurities around the Ge-Ge axis has been experimentally observed by phonon spectroscopy [7].While the rotation of oxygen impu-rities in Ge is weakly hindered by an azimuthal po-tential caused by the host lattice,several materials are known to show a free rotation of molecules.An exam-ple is ammonia groups in certain Hofmann clathrates M(NH 3)2M’(CN)4-G [8–10],usually abbreviated as M-M’-G,where M and M’are divalent metal ions and G is a guest molecule.Nearly free uniaxial quantum rotation of NH 3has been observed for the ?rst time in Ni-Ni-(C 6D 6)2by inelastic neutron scattering [8].Recently,a surprising variation of the linewidth has been observed for Ni-Ni-(C 12H 10)2[11],which has been interpreted by a novel line broadening mechanism based on rotor-rotor coupling [12].It is also known that the βphase of solid methane [13]as well as methane hydrate [14]show al-most free rotation of CH 4molecule.The linewidths of methane in clathrates show inhomogeneous broadening owing to the dipolar coupling with water molecules [15].It is therefore expected that new interesting phenomena will be found by investigating the in?uence of dipolar interaction between quantum rotors.

In the present paper,we study the correlated dynamics of coupled quantum rotors carrying electric dipole mo-ments.We give the exact solution of eigenvalue problem of interacting rotors with arbitrary con?gurations.It is

revealed that coupled rotors show a peculiar dielectric re-sponse at low temperatures,which can be interpreted by taking account of the selection rule of dipolar transition for coupled rotors.

II.

THE HAMILTONIAN

Suppose two dipole rotors q r 1and q r 2separated by the vector R .The Hamiltonian for the system is given by H =H K +W D ,where the kinetic term is

H K =?

?Θ21

4πε

|R +r 1?r 2|

|R ?r 2|

(2)

Here I is the moment of inertia for dipole rotors and εthe dielectric constant,respectively.Figure 1shows a con?guration of two dipoles rotors under consideration.We assume that rotors do not feel any potential variation along a ring of radius r .In the Jacobi coordinate,the

FIG.ro-tors radius r ,

FIG.2:Contour plot of the interaction term W D in(Θ1,Θ2) plane.Two maxima(white regions)and two minima(dark regions)are realized at positions with di?erences?Θ1≈πand?Θ2≈π.Parameter values are given in the text. vectors r1,r2and R are given by

r1=r(cosΘ1,sinΘ1cosα1,sinΘ1sinα1),

r2=r(cosΘ2cosβ?sinΘ2cosα2sinβ,

cosΘ2sinβ+sinΘ2cosα2cosβ,sinΘ2sinα2), R=R(0,0,1).(3)

A spatial pro?le of W D as a function of(Θ1,Θ2)is dis-played in Fig.2by a contour plot,in which the angles (α1,α2,β)are set as(π/4,?π/6,π/3).We should remark that two minima(dark regions)and two maxima(white ones)are located at the anti-parallel or parallel dipolar con?guration,indicating that the dipoles prefer an anti-parallel con?guration.The two minima of W D(Θ1,Θ2) arise from the dipole interaction between two rotors,i.e., the dipole interaction plays a key role for creating barri-ers and two potential minima,which strongly a?ect the energy spectra and the dielectric response of the system. Provided that the spacing R is large enough compared with the radius r,the interaction term W D can be ex-panded in terms of1/R.The lowest-order term has the form of a dipolar interaction given by

W(3)

D =

R2 .(4)

The higher-order term W(4)

D ≡W D?W(3)

is of the order

of O(r3/R4),which can be negligible for the case R?r. Actually we have con?rmed that the calculated results presented in this paper change very little by taking into account the term W(4)

III.EIGENV ALUES AND EIGENFUNCTIONS The Schr¨o dinger equation for the Hamiltonian H0=

H K+W(3)

D has analytic solutions as shown below.Trans-

forming variables toθ1=(Θ1+Θ2)/2andθ2=(Θ1?Θ2)/2,Eqs.(1)and(4)yields

H K=?

?θ21

4πεR3

i=1c i cos2(θi+γi).(6)

The parameters c i andγi(i=1,2)are functions of angles α1,α2andβde?ned in Fig.1,whose explicit forms are given by

c i=

x2i+y2i,γi=

y i ,(7) with the de?nitions

x1=sinβ(cosα1?cosα2),

x2=sinβ(cosα1+cosα2),

y1=cosβ(1?cosα1cosα2)+2sinα1sinα2,

y2=cosβ(1+cosα1cosα2)?2sinα1sinα2.(8) Consequently,we can decompose the Schr¨o dinger equa-tion H0Ψ0(θ1,θ2)=E0Ψ0(θ1,θ2)into two independent Mathieu equations.SettingΨ0(θ1,θ2)=?1(θ1)?2(θ2), we obtain

?2?i

E K

[c i E D cos2(θi+γi)?E i]?i=0,[i=1,2]

(9) where the quantities E K= 2/(2I)and E D= q2r2/(4πεR3)represent the kinetic and interaction en-ergy,respectively.The eigenvalue E of the initial Schr¨o dinger equation is expressed as the sum of E= E1+E2.Note that the periodic terms∝cos2(θi+γi) originate from two minima(or maxima)of the interaction term W D(Θ1,Θ2)shown in Fig.2[17]. Eigenfunctions of Eq.(9)are described by four types of the Mathieu functions,given by ce2n(v i,θi), se2n+1(v i,θi),ce2n+1(v i,θi)and se2n+2(v i,θi)with the de?nitions v i≡c i E D/E K and n=0,1,2···.Each of them belongs to a di?erent eigenvalue and can be ex-pressed in terms of the Fourier-cosine expansion;for in-stance,

ce2n(v i,θi)=

∞

m=0A(2n)2m(v i)cos2m(θi+γi).(10)

The coe?cients{A(2n)

}are determined by a successive relation obtained by substituting Eq.(10)into Eq.(9).

The amplitudes of{A(2n)

}rapidly decrease with increas-ing m,so that we can truncate the summation in Eq.(10) at m=20in actual calculations.

Figure3plots the calculated spectra of eigenenergies E=E1+E2as a function of E D,where E K is taken as an energy unit.The angles(α1,α2,β)are set to be (π/4,0,0)as an example.We?nd,though some levels are degenerate when E D=0,they split o?for?nite

E / E

FIG.3:The E D .Solid for the The explanation E D with a monotonous variation with increasing E D .For high-E D limit,some levels become degenerate again.It indicates that the relative motion of paired-rotors is frozen out for E D ?E K due to the strong Coulomb interaction.This behavior can be understood from the spatial pro?le of the interaction term W D (Θ1,Θ2)shown in Fig.2.With increasing E D ,the depths of two minima of W D (Θ1,Θ2)grow,and larger barrier-heights hinder the quantum transition of a particle through the bar-rier.This gives rise to localized wavefunctions around these minima.Consequently,in the limit of E D ?E K ,the amplitude of the eigenfunctions are strongly local-ized around two minima,and these two localized eigen-states are nearly degenerate.Even if the higher-order

term W (4)

D is taken into account,the energy spectra does not change much,since it only slightly disturbs the sym-metry of the depths of two minima shown in Fig. 2.When varying the angles (α1,α2,β),the curves in Fig.3slightly shift to upwards and/or downwards except for the unchanged values of

E at E D =0.

IV.

DIELECTRIC SUSCEPTIBILITIES

Let us consider the dielectric response of dipole ro-tors coupled via dipolar interaction.The real part of the frequency-dependent dielectric susceptibility is expressed as

χμμ(ω,T )=?

(E j ?E l )2?( ω)2

exp

E j

temperatures.The crossover temperature between the steady increase and the almost constant value in Fig.4is determined by the minimum-energy di?erence of eigenstates at E D =0that are allowed for dipole tran-sition ,namely,indicated as (a)in Fig.3.For the case of E D /E K >1,the strong Coulomb interaction prevents from the relative motion of rotors so that the magnitude of the susceptibility χ(T )decreases with increasing E D .It is noteworthy that,for relatively weak interaction E D /E K <0.1,a bump is appeared in the suscepti-bility at about E K /(k B T )≈5.0.The kinetic energy E K = 2/(2I )for actual rotating molecules is the order of 1meV [14],indicating that the characteristic temper-ature T ?=E K /k B ×0.2corresponding to the bump is estimated as about 1K.We made sure that the bump can be observed for any angles (α1,α2,β)when E D /E K is less than 0.1.This anomaly stems from the correlated rotation of paired-rotors via the dipolar interaction,and can be interpreted by the argument on the selection rule for dipolar transition.

To understand the origin of the bump,we decompose the total susceptibility χxx (T )give in Eq.(11)as

χxx (T )=

(j,l =j )

ηj,l (T ),

(12)

ηj,l (T )=?

E j ?E l

× exp

E j

k B T

,(13)

where

(j,l =j )is the summation over all possible com-binations of (j,l )under the condition l =j .Note the fact that only three components of ηj,l (T )are responsible for the total susceptibility (12)around the characteristic temperature T ?.We denote those components by ηa ,ηb ,and ηc ,which are characterized by the eigenfunction Ψj = θ1,θ2|E j and Ψl = θ1,θ2|E l as follows;

ηa ;Ψj =ce 0(θ1)ce 0(θ2),Ψl =ce 1(θ1)ce 1(θ2),(14)ηb ;Ψj =ce 0(θ1)ce 1(θ2),Ψl =ce 1(θ1)ce 0(θ2),(15)ηc ;

Ψj =se 1(θ1)se 2(θ2),Ψl =se 2(θ1)se 1(θ2).(16)

The alphabets subscribed on ηcorrespond to three dipo-lar transitions labeled by (a)-(c)shown in Fig.3.For example,the solid arrow of (a)in Fig.3connects the eigenstates Ψj and Ψl de?ned in Eq.(14).

For weak coupling E D ?E K ,the solution of the Mathieu equation (9)is easily solved.In the lowest order of the perturbation theory,the eigenvalues E i (i =1,2)read in

E i =

E K

2±δE b ,

E K 2

±δE c ,

5E K

εE K ·1?e ?u εE K ·ue ?u/2εE K

ue ?5u/2

Z ′(u )2

u ?u

Z ′(u )

T?=E K/k B×0.2.We should note here that,if quan-tum rotors are not interacting at all,the componentηb exactly vanished due to the degeneracy E j=E l=E K/2 (See Eq.(19))and only the componentηa is dominant for the total susceptibilityχ(T).This means that the total susceptibility is a monotonic function as the same asηa so that the bump does not emerge.The anomalous bump of the susceptibility,therefore,manifests the rele-vance of the dipolar interaction to the dielectric response of quantum rotors.

V.CONCLUSIONS

It is important to recall experiments reported in Ref.

[18],for the dielectric susceptibility of KCl crystals with Li defects.It has been found that the susceptibility does not scale linearing with the Li concentration,and even becomes smaller with increasing the concentration (≈1000ppm),where the interaction between defects be-comes relevant.In addition,a bump of the suscepti-bility is observed at about200mK for concentrations of200-1000ppm.These temperature dependences of the susceptibility together with the bumps are recovered well by our results shown in Fig.4.Noting that defects in both systems move along closed loops and correlated each other,it is natural to assume that the similar pic-ture holds.For a quantitative discussion,of course,one should take into account the e?ect of potential variation hindering the free rotation of Li+,which is caused by the Coulomb interaction between a mobile Li+ion and the host atoms K+and Cl?.The problem has been theoret-ically investigated in Ref.[19,20]based on the two-level tunneling model.

In conclusion,we have investigated the quantum dy-namics of two dipole rotors coupled via dipolar interac-tion.By solving analytically the eigenvalue problem of coupled rotors,we have demonstrated the energy spec-tra of coupled rotors as a function of dipolar interaction. The anomalous temperature dependence of dielectric sus-ceptibility is also shown.Our model is so general that it should be applicable in a variety of physical context relevant to quantum rotors.

Acknowledgments

One of the authors(H.S)was?nancially supported in part by the NOASTEC Foundation for young scientists. This work was supported in part by a Grant–in–Aid for Scienti?c Research from the Japan Ministry of Educa-tion,Science,Sports and Culture.

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