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arXiv:0902.0303v2 [cond-mat.supr-con] 1 Jun 2009

L. Fanfarillo, 1 L. Benfatto, 2,1 S. Caprara, 1 C. Castellani, 1 and M. Grilli 1 1

CRS SMC, CNR-INFM and Dipartimento di Fisi a,

Università di Roma La Sapienza, Piazzale A. Moro 2, 00185, Rome, Italy 2

Centro Studi e Ri er he Enri o Fermi, via Panisperna 89/A, 00184, Rome, Italy (Dated: O tober25,2018)

Wederivetheee tivea tionforsuper ondu tingu tuationsinafour-bandmodelforpni tides, dis ussing the emergen e of a single riti al mode out of a dominant interband pairing me ha-nism. We then apply our model to al ulate the para ondu tivity in two-dimensional and lay-ered three-dimensional systems, and ompareour results with re ent resistivity measurements in SmFeAsO

0.8

F

0.2

.

PACSnumbers:74.40.+k,74.20.De,74.25.Fy

There entdis overyofsuper ondu tivityinpni tides [1℄ hasrenewed the interest in high-temperature super- ondu tivity. Pni tides share many similarities with upratesuper ondu tors,e.g.,thelayeredstru ture, the proximitytoamagneti phase[2℄,therelativelylarge ra-tiobetweenthesuper ondu ting(SC)gapandthe riti- altemperature

T

c

[3,4℄,andthesmallsuperuiddensity [2℄. However, dierently from uprates, the presen e of severalsheets oftheFermisurfa emakesthemultiband hara terofsuper ondu tivityanunavoidableingredient of any theoreti al model for pni tides. Moreover, sin e the al ulatedele tron-phonon oupling annota ount for the high values of

T

c

[5, 6℄, it has been suggested thatthepairingglueisprovidedbyspinu tuations ex- hangedbetweenele tronsindierentbands[68℄. Thus, pni tidesareexpe tedtobesomehowdierentfromother multibandsuper ondu tors(e.g.,MgB

2

)wherethemain ouplingme hanismisintraband[9℄.

Thiss enarioraisesinterestingquestionsregardingthe appropriatedes riptionofSCu tuationsinamultiband systemdominatedbyinterbandpairing. Theissueis rel-evant, be ause u tuating Cooper pairs above

T

c

on-tributetoseveralobservablequantities,su has,e.g.,the enhan ement of d ondu tivity(para ondu tivity) and ofthediamagneti responseas

T

c

isapproa hed[10℄. The natureofSCu tuationsdependsonwhetherthesystem isweaklyorstrongly oupled(andonwhetherpreformed pairsarepresentor not)and awealth ofphysi al infor-mation anbeobtainedfrom para ondu tivityand dia-magneti response,providedatheoreti alba kgroundis establishedto extra tthem.

Inthispaper,after introdu ingafour-bandmodel,as appropriate for pni tides, we dis uss the subtleties re-lated to the des ription of SC u tuationsin a system with dominant interband pairing. We then apply our resultsto omputethepara ondu tivityasso iatedwith SCu tuationsabove

T

c

. Weshowthatwheninterband pairing dominates, despite the presen e of four bands, thereareonlytwoindependentu tuatingmodes. Only oneofthemis riti alandyieldsadiverging Aslamazov-Larkin (AL) ontribution to para ondu tivity as

T

c

is

approa hed,similarlytothe aseofdominantintraband pairing [11℄. The temperature dependen e of the AL para ondu tivityisthesamederivedforordinary single-band super ondu tors [10, 12℄. Remarkably, within a BCSapproa h, we re overthe AL numeri alprefa tor, whi h is a universal oe ient in two dimensions, and depends instead on the oheren e length perpendi ular tothe planesin thethree-dimensionallayered ase. We alsondthatsubleadingtermswithrespe ttothe lead-ingAL ontribution oulddistinguishbetweendominant inter- and intra-band pairing. Within this theoreti al ba kground, we analyze re ent resistivity data in Sm-FeAsOF[4℄anddis usstheresults.

Atpresent,ARPESmeasurementsinpni tides[3℄ on-rmedthe Fermi-surfa etopologypredi tedbyLDA. It onsists of twohole-like po kets entered around the

Γ

point (labeled

α

and

β

, following Ref. [3℄), and two ele tron-like (

γ

) po kets entered around the M points of the folded Brillouin zone of the FeAs planes. Moti-vated bythemagneti hara ter of theundoped parent ompound and by the approximate nesting of the hole and the ele tron po kets with respe t to the magneti ordering waveve tor, we assume that pairing mediated by spin u tuations is ee tive only between hole and ele tron bands [8℄. Sin e the

β

band has a Fermi sur-fa e substantially largerthanthe

α

band, the

β

po ket isexpe tedtobelessnestedtothe

γ

po ket,sothatthe interband

β − γ

oupling

λ

issmallerthanthe

α − γ

ou-pling,i.e.,

η ≡ λ/Λ < 1

. Thetwoele tronpo kets have omparablesizesandforsimpli ityweassumethatthe

γ

bandsare degenerate. Therefore,the BCSHamiltonian ofourfour-bandmodelis[13℄

H =

X

i

H

i

0

+ Λ

X

q

h

Φ

γ,q

α,q

+ ηΦ

β,q

) + h.c.

i

,

(1) where

H

i

0

=

P

k

ξ

i,k

c

i,kσ

c

i,kσ

is theband Hamiltonian,

c

(+)

i,kσ

annihilates( reates)afermioninthe

i = α, β, γ

1

, γ

2

band(withthetwofolddegenerate

γ

bandslabeledas

γ

1

and

γ

2

),

ξ

i,k

isthedispersionwithrespe ttothe hemi al potential,

Φ

i,q

=

P

(2)

inthe

i

-thband,and

Φ

γ,q

≡ Φ

γ

1

,q

+ Φ

γ

2

,q

. Sin ewe as-sumedthatpairinga tsbetweenholeandele tronbands only,we anexpressthepairingtermin Eq.1bymeans of

Φ

1

≡ Φ

γ

and

Φ

2

≡ Φ

α

+ ηΦ

β

. Thus

H

I

= Λ

X

q

1

Φ

2

+h.c.) = −Λ

X

q

Φ

−Φ

+

Φ

+

),

(2) where

Φ

±

= (w

1

Φ

1

± w

2

Φ

2

)/

2

,

w

1

, w

2

are arbitrary numbers satisfying

w

1

w

2

= 1

, and for deniteness we

take

Λ > 0

, whi h isthe ase fora spin-mediated

pair-ingintera tion. FromEq.(2)oneimmediatelyseesthat when interband pairing dominates, the intera tion is a mixtureofattra tion(for

Φ

)andrepulsion(for

Φ

+

). As a onsequen e,whenweperformthestandard Hubbard-Stratonovi h (HS)de oupling of the quarti intera tion term(2)bymeansoftheHS eld

φ

,

e

±ΛΦ

Φ

=

Z

Dφ e

−|φ|

2

/Λ+

±1(Φ

φ+h.c.)

,

(3)

theHStransformationasso iatedwiththerepulsivepart ontainstheimaginaryunit. Asweshall seebelow,this would requirean imaginaryvalueof the

φ

+

HS eld at thesaddlepoint, that an beavoidedbytherotationof the pairing operators

Φ

±

dened above, via a suitable hoi eofthe

w

1,2

oe ients.

Inanalogywiththesingle-band ase[14℄,theee tive a tionfortheSC u tuationsreads

S =

X

i

X

k

X

σ

i,k

− iε

n

)c

i,kσ

c

i,kσ

+

X

q

 |φ

+

(q)|

2

Λ

− iφ

+

(q)Φ

+

(q) − iφ

+

(q)Φ

+

(q)



+

X

q

 |φ

(q)|

2

Λ

− φ

(q)Φ

(q) − φ

(q)Φ

(q)



,

where

k ≡ (k, iε

n

)

,

q ≡ (q, iω

m

)

, and

ε

n

,

ω

m

are the Matsubarafermion and boson frequen ies, respe tively. Integratingoutthefermionsweobtainthestandard on-tributiontothea tion,

P

i

Tr log A

i

kk

,wherethetra e

a tson momenta, frequen ies, and spins. The elements ofthematri es

A

i

kk

are

A

i

kk



11

= (ξ

i,k

−iε

n

kk

,

A

i

kk



22

= −(ξ

i,k

+iε

n

kk

,

with

i = α, β, γ

1

, γ

2

,

h

A

α

kk

i

12

=

w

2

2

(k − k

) − iφ

+

(k − k

)] ,

h

A

β

kk

i

12

= η

h

A

α

kk

i

12

,

h

A

γ

kk

i

12

= −

w

1

2

(k − k

) + iφ

+

(k − k

)] ,

for

γ = γ

1

, γ

2

,andthe

[A

i

kk

]

21

elements ontainthe om-plex onjugatesof theHS eldsevaluatedat

(k

− k)

.

The

q = k − k

= 0

values of theHS elds yield the SC gaps in the various bands, within the saddle-point approximation

φ

±

(q = 0) = ¯

φ

±

. However, due to the presen eoftheimaginaryunit

i

asso iatedwith theHS eld

φ

+

, in general

[A

i

kk

]

21

6= [A

i

kk

]

12

. To re over a

Hermitian

A

matrixat thesaddlepoint,theintegration ontourofthe fun tionalintegralmust bedeformed to-ward the imaginary axis of the

φ

+

eld. This an be avoided if one hooses

w

1

and

w

2

in the denition of

Φ

±

in su h awaythat

φ

¯

+

= 0

. Inour ase,the hoi e

¯

φ

+

= 0

gives

α

= −w

2

φ

¯

/

2

,

γ

= w

1

φ

¯

/

2

and

β

= η∆

α

. Hen e,theratioof thegaps inthetwohole bandsissolelydeterminedbytheratioof the ouplings,

β

/∆

α

= η

. Using

w

1

w

2

= 1

we have

φ

¯

2

= −2∆

α

γ

,

andtheequationsfor

α

and

γ

read [13℄

α

= −Λ(2Π

γ

)∆

γ

,

(4)

γ

= −Λ (Π

α

α

+ ηΠ

β

β

) ,

(5)

where

Π

i

= N

i

R

ω

0

0

dξ[tanh(E

i

/2T )]/E

i

yields the

q = 0

valueofparti le-parti lebubblewhen

T > T

c

,

N

i

is den-sityofstatesofthe

i

-thbandattheFermilevel,

ω

0

isthe ut-ofor thepairingintera tion, and

E

i

=

2

+ ∆

2

i

. Hen e,our four-bandmodel for pni tides ee tively re-du es to a two-band model, with one ele tron-like and onehole-likeee tive band. Indeed,dening

1

≡ ∆

γ

,

2

≡ ∆

α

,and

Π

1

≡ 2Π

γ

,

Π

2

≡ Π

α

+ η

2

Π

β

,Eqs.(4)-(5) re overthestandardtwo-bandexpression

1

= −ΛΠ

2

2

and

2

= −ΛΠ

1

1

. Sin e

1

/∆

2

= −w

1

/w

2

,italso

fol-lowsthat

w

2

1

/w

2

2

= Π

2

1

at

T < T

c

.

Let us now dis uss the SC u tuations for

T > T

c

, where

1,2

= 0

. Toderivetheequivalentofthestandard Gaussian Ginzburg-Landau fun tional [10℄, we expand thea tion uptotermsquadrati in theHS elds,

S

G

=

P

ικ

P

q

φ

ι

(q)L

−1

ικ

(q)φ

κ

(q)

,where

ι, κ = ±

and

L

−1

(q) =

 Λ

−1

− Π

ef f

(q)

−i Ξ

ef f

(q)

−i Ξ

ef f

(q)

Λ

−1

+ Π

ef f

(q)



,

(6) with

Π

ef f

(q) ≡

1

2

[w

2

1

Π

1

(q) + w

2

2

Π

2

(q)],

Ξ

ef f

(q) ≡

1

2

[w

2

1

Π

1

(q) − w

2

2

Π

2

(q)].

The riti al temperature is determined by the

ondi-tion

det L

−1

(q = 0) = 0

. In the BCS approximation

Π

i

(0) = N

i

ln(1.13 ω

0

/T

c

)

and, in agreement with the

Eqs.(4)-(5),weobtain

N

ef f

Λ log(1.13 ω

0

/T

c

) = 1

. Here, the parameter

N

ef f

N

1

N

2

=

p2N

γ

(N

α

+ η

2

N

β

)

playsthe role of anee tivedensity ofstates. To om-putetheu tuation ontributiontothevariousphysi al quantitiesat

T > T

c

,weevaluateEq.(6)atleadingorder in

q

(hydrodynami approximation),using thestandard expansion of the parti le-parti le bubble for a layered system,

Π

i

(q) ≈ Π

i

(0) − c

(3)

q

2

k

= q

2

x

+ q

2

y

and the

z

axisperpendi ulartothelayers. IntheBCSapproximation,e.g.,

γ

i

= πN

i

/(8T )

. Weomit thelengthier but standardBCSexpressions for

c

i,k

and

c

i,z

(see,e.g.,Ref. [10℄),thatwillnotbeexpli itlyusedin thefollowing. Bymakingfor

T > T

c

thesame hoi e pre-viouslyadoptedfor

T < T

c

,

w

2

1

Π

1

(0) = w

2

2

Π

2

(0)

,we

sim-plifythestru tureoftheu tuatingmodes. Indeed,with this hoi e, the o-diagonal termsin Eq. (6) yield on-tributionsbeyondhydrodynami sand anbenegle ted. Therefore,theleadingSCu tuationsaredes ribedby

L

−1

(q) ≈

 m

+ ν(q)

0

m

0

+

− ν(q)



,

(7) where

m

±

= Λ

−1

±

Π

1

Π

2

arethemassesofthe olle -tive modes,

ν(q) ≡ c

k

q

k

2

+ c

z

q

2

z

+ γ|ω

m

|

,

c

k

= (w

1

2

c

1,k

+

w

2

2

c

2,k

)/2

isthestinessalongthelayers,

c

z

= (w

2

1

c

1,z

+

w

2

2

c

2,z

)/2

is the stiness in the dire tion perpendi ular to thelayers, and

γ = (w

2

1

γ

1

+ w

2

2

γ

2

)/2

is thedamping oe ient. IntheBCS ase

m

±

= N

ef f

[ln(1.13 ω

0

/T

c

) ±

ln(1.13 ω

0

/T )]

and

γ = πN

ef f

/(8T )

.

Having dedu ed the hydrodynami a tion of the SC u tuations,we an al ulatethepara ondu tivityalong thelines of Ref. [15℄. Byinspe tion ofEq. (7), one an see that only the

φ

mode be omes riti al at

T

c

[i.e.,

m

(T

c

) = 0

,

m

+

(T

c

) = 2Λ

−1

℄, thus giving a diverg-ingu tuation ontributiontovariousphysi alquantities

when

T → T

c

. Theleading ontribution tothe

urrent- urrentresponsefun tion alongthelayersis

δχ(Ω

) = 4e

2

T

X

q

c

2

k

q

2

k

L

−−

(q, ω

m

)L

−−

(q, ω

m

+ Ω

),

when e the para ondu tivity

δσ

AL

= [Imδχ(Ω)/Ω]

Ω→0

isobtained,after analyti al ontinuationofthe Matsub-ara frequen y

to the real frequen y

. Therefore, thesameexpressionsknown forasingle-band super on-du torarefound,althoughwiththeee tiveparameters

m

, c

z

, γ

. Thepara ondu tivityalongthelayersis inde-pendentofthein-planestiness

c

k

,asguaranteedbythe same gauge-invarian e arguments dis ussed in Ref. [15℄ forasingle-bandsuper ondu tor. Theleading ontribu-tions to para ondu tivity along the layersin three and two dimensions(3D and 2D, respe tively) take the AL form[12℄

δσ

3D

AL

=

e

2

32~ξ

z

1

ǫ

(8)

δσ

2D

AL

=

e

2

16~d

1

ǫ

(9)

where

ξ

z

=

pc

z

is the orrelation length in the di-re tionperpendi ulartothelayers,

d

isthedistan e be-tweenlayers,and

ǫ(T ) ≡ πm

(T )/(8γT

c

)

isthe

dimen-sionlessmassofthe riti al olle tivemode. Wepointout thattheaboveexpressionsaregeneralwithina

hydrody-onanyparti ularassumptionaboutthepairingstrength [15℄. WhentheBCSexpressionforthe

Π

i

bubblesholds,

m

(T ) = N

ef f

ln(T /T

c

)

andthedimensionless mass

ap-pearinginEqs. (8)-(9)issimply

ǫ = log(T /T

c

)

.

The al ulated AL para ondu tivity expressions (8)-(9) an be now ompared with the existing results for the pni tides. As it has been widely dis ussed in the ontextof uprates[16℄, forweakly- oupledlayered ma-terials the SC u tuations usually display a 2D-3D di-mensional rossoveras

T

c

is approa hed. However,the interlayer ouplinghasadierentrelevan einthevarious familiesof uprates, with substantial 3D behavior [Eq. (8)℄ in YBa

2

Cu

3

O

6+x

samples, while more anisotropi Bi

2

Sr

2

CaCu

2

O

8

orLa

2−x

Sr

x

CuO

4

ompounds show2D u tuations [Eq. (9)℄, the 2D-3D rossover being too loseto

T

c

tobe learlyobserved. Su hasystemati sur-veyhasnotyetbeenperformedin the aseofpni tides, due also to the limited availability of lean single rys-tal. Indeed, the analysis of the 2D-3D rossovermight be biased in poly rystals by the distribution of riti al temperaturesand by themixing ofthe planarand per-pendi ulardire tions.

Havinginmindsu hlimitations,weattemptthe anal-ysis of para ondu tivity in a SmFeAsO

0.8

F

0.2

sample [4℄ with

T

c

≈ 52

K. To determine the ontribution of SC u tuations to the normal-state ondu tivity,

δσ ≡

ρ

−1

−ρ

−1

N

,weneedtoextra tthenormalstateresistivity

ρ

N

from thedata. Owing tothediverging ondu tivity, thepre ise determination of the normal-state ontribu-tionisimmaterialnear

T

c

,butbe omesrelevantforlarger valuesof

ǫ

. Wettedtheresistivityathightemperature (inthe range between

279

K,the highestavailable tem-perature,andabout

200

K) he kingthatthequalitative resultswere stable upon small variations of this range. Weusedaquadrati t

ρ

N

= a + b(T − T

0

) + c(T − T

0

)

2

,

with

T

0

= 279

K,and foundthat the resulting

para on-du tivityisroughlytwoordersofmagnitudesmallerthan the2DALresult[Eq. (9)℄. Theslopeinalog-logplotis inagreementwiththe3Dpowerlaw[Eq. (8)℄. Theplot inFig.1 learlyshowsthatthe3Dbehaviorextendsover twode adesof

ǫ

. Thettingparameters

a = 1700 µΩ

m,

b = 21.9 µΩ

m/K,and

c = 0.018 µΩ

m/K

2

wereused. The tting with Eq. (8) allows us to determine the pre isevalueof

T

c

andthe oheren elength

ξ

z

oftheSC u tuations in the dire tion perpendi ular to the FeAs layers. We nd

T

c

= 51.4

K and

ξ

z

= 19

Å. One an also see that substantial SC u tuations persist up to

ǫ ≈ (T − T

c

)/T

c

≈ 0.4

, i.e., upto

20

Kabove

T

c

. This

u tuatingregimeis thereforemu hlargerthan in on-ventional3Dsuper ondu torsand,despitethe3D behav-iorofthepara ondu tivity, allsforarelevant hara ter of the layered stru ture and for a small planar oher-en elength. Wepointoutthat at even largervaluesof

ǫ

thepara ondu tivitydrasti allydrops,inanalogywith what found in uprates (see, e.g., Ref. [16℄, and

(4)

refer-0.01

0.1

1

ε

1e-05

0.0001

0.001

0.01

0.1

∆σ

(

µΩ

cm)

-1

FIG.1:(Coloronline)Comparisonbetweentheexperimental para ondu tivityforaSmFeAsO

0.8

F

0.2

samplestudiedinRef. [4 ℄(bla k ir les)andthe2D(dashedline)and3D(solidline) expressionsofthe ALpara ondu tivity[Eqs. (8,9)℄. Forthe 3D para ondu tivity a oheren e length

ξ

z

= 19

Å has been used, while for the 2D ase the stru tural distan e between layers

d = 8.4

Åhasbeeninserted.

the ALbehaviorin multiband systemsalso depends on the role of the other (non- riti al) olle tive modes. In parti ular,it anbeshown[17℄that,whentheintraband pairingisequallydominantinallbands,the para ondu -tivity mediatedby the non riti al modes may be ome sizable,and theexperimental data should approa h the pure AL ontribution of the riti al mode from above. This is not the ase when the dominant pairing is in-terband andtherefore it isnot surprising that the data forthepni tidesampleanalyzedinthispaperalwayslay belowtheALstraightlinein Fig.1.

In on lusion, we investigated the o urren e of SC u tuationsinamultibandsystemwhereinterband pair-ingdominates, asappropriateforpni tides. In ontrast to the aseof dominantintraband me hanism (as, e.g., in MgB

2

[11℄), in the present situation the HS de ou-plingmustbea ompaniedwithaproperrotationofthe fermioneldswhi hguaranteesaHermitiansaddle-point a tionbelow

T

c

. The samerotationleadsto a straight-forwardde ouplingoftheGaussianu tuationsabove

T

c

in the hydrodynami limit. Thus, despite the apparent omplexity of the multiband stru ture in pni tides, we demonstratethattheALexpressionsfor para ondu tiv-ity stay valid notonly as far as the fun tional temper-ature dependen e is on erned, but also regarding the numeri al prefa tors. While in the BCS 2D ase the prefa tor stays universal, in the 3D ase the only dif-feren e is that a suitable redenition of the transverse oheren elengthhasto beintrodu ed. Withthis equip-ment,we onsidered theexperimental resistivitydata of the SmFeAsO

0.8

F

0.2

sample studied in Ref. [4℄, nding

above

T

c

. Re ently,anexperimental work[18℄ on u tu-ation ondu tivityin pni tides onrmed thewide u -tuating regime, even though u tuations seem to have 2D hara ter. Thus,further experimentsarein orderto onrm the nature of u tuations in pni tides, and to assesstherelevan eof Cooper-pairu tuations in these newsuper ondu tors.

A knowledgments. WewarmlythankD.Dagheroand R. Gonnelli for providing us with the data of Ref. [4℄. This work has been supported by PRIN 2007 (Proje t No. 2007FW3MJX003).

[1℄ Y.Kamihara,T.Watanabe,M. Hirano,andH.Hosono, J.Am.Chem.So .130,3296 (2008).

[2℄ See,e.g., T.Goko,etal.arXiv:0808.1425.

[3℄ H. Ding, P. Ri hard, K. Nakayama, K. Sugawara, T.Arakane,Y.Sekiba,A.Takayama,S.Souma,T.Sato, T. Takahashi, Z. Wang, X. Dai, Z. Fang, G. F. Chen, J. L. Luo and N. L. Wang, Europhys. Lett. 83, 47001 (2008).

[4℄ D.Daghero,M.Tortello,R.S.Gonnelli,V.A.Stepanov, N.D.Zhigadlo,J.Karpinski,arXiv:0812.1141.

[5℄ L. Boeri, O.V.Dolgov, andA.A. Golubev,Phys.Rev. Lett.101,026403(2008)

[6℄ I.I.Mazin,D.J.Singh,M.D.Johannes,M.H.Du,Phys. Rev.Lett.101,057003(2008);

[7℄ V.Stanev,J.Kang andZ.Tesanovi , Phys.Rev.B78, 184509 (2008);R. Sknepnek,G.Samolyuk,Y. Lee,and J.S hmalian,Phys.Rev.B79,054511(2009).

[8℄ A. V. Chubukov,D. Efremov, I. Eremin,Phys.Rev.B 78,134512 (2008); V.Cvetkovi andZ. Tesanovi , Eu-rophys.Lett.85,37002(2009).

[9℄ A.Y.Liu,I.Mazin,andJ.Kortus,Phys.Rev.Lett.87, 087005(2001).

[10℄ A. Larkin and A. Varlamov, Theory of u tuations in super ondu tors, (ClarendonPress,Oxford,2005). [11℄ A. E. Koshelev, A. A. Varlamov and V. M. Vinokur,

Phys.Rev.B72,064523(2005).

[12℄ L.G.AslamazovandA.I.Larkin,Phys.Lett.A26,238 (1968);Sov.Phys.SolidState10,875(1968).

[13℄ L. Benfatto,M. Capone, S.Caprara,C.Castellani, and C.DiCastro,Phys.Rev.B78,140502(R)(2008). [14℄ For alist ofreferen es inthesingle-band ase see, e.g.,

L. Benfatto, A. Tos hi, and S. Caprara, Phys.Rev.B. 69,184510(2004).

[15℄ S.Caprara,M.Grilli,B.Leridon,andJ.Vana ken,Phys. Rev.B79,024506(2009).

[16℄ S.Caprara,M.Grilli,B.Leridon,andJ.Lesueur,Phys. Rev.B72,107509(2005).

[17℄ L.Benfatto,S.Caprara,C.Castellani,L.Fanfarillo,and M.Grilli,unpublished.

[18℄ I.Palle hi,C.Fan iulli,M.Tropeano,A.Palenzona,M. Ferretti,A.Malagoli,A.Martinelli,I.Sheikin,M.Putti, andC.Ferdeghini,Phys.Rev.B79,104515(2009).

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