Polaronic signatures in the optical properties of Nd$_{2-x}$Ce$_x$CuO$_4$

0.37 MB·English

by  E. Cappelluti

Polaronic signatures in the optical properties of Nd Ce CuO
2−x x 4
E. Cappelluti1,2, S. Ciuchi3,1, and S. Fratini4∗
1SMC Research Center and ISC, INFM-CNR, v. dei Taurini 19, 00185 Rome, Italy
2Dipartimento di Fisica, Universit`a “La Sapienza”, P.le A. Moro 2, 00185 Rome, Italy
3CNISM and Dipartimento di Fisica, Universit`a dell’Aquila, via Vetoio, I-67010 Coppito-L’Aquila, Italy and
4Institut N´eel - CNRS & Universit´e Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France
(Dated: February 4, 2008)
We investigate the temperature and doping dependence of the optical conductivity σ(ω) of
Nd2−xCexCuO4 in terms of magnetic/lattice polaron formation. We employ dynamical mean-field
8 theory in the context of the Holstein-t-J model where an exact analytical solution is available in
0 the limit of infinite connectivity. We show that the pseudogap features in the optical conductivity
0 of this compound can be associated to the formation of lattice polarons assisted by the magnetic
2 interaction.

n
a Understandingthephasediagramofhigh-T supercon- upon increasing the temperature; iii)the absence of any
J c
ductors still represents, twenty years after their discov- signature of the N´eel temperature in the temperature
0
ery, one of the main challenges in this field. One of evolution of the pseudogap spectral weight (defined as
3
the reasons of such difficulties relies on the fact that thespectralweightintheregionbetweentheMIRshoul-

] different interactions are simultaneously operating, in- der and a low-energy infrared cut-off ωmin 0.12 eV),
l ≃
e volving different degrees of freedom. On one hand, ev- whereas a remarkable non-monotonic kink is present at
- idences are gathering, from different experimental tech- a higher temperature T∗. The latter feature suggests a
r
t niques, about a relevant role of the electron-lattice in- prominent role of short-range correlations in governing
s
. teraction [1, 2, 3, 4, 5]. On the other hand, short- the observed spectral properties.
t
a range antiferromagnetic (AF) correlations, ruled by the A detailed theoretical study of the polaron features
m
exchange coupling J, are still present in the normal in the optical conductivity of one hole in the Holstein-
- state even when true long-range AF ordering is de- t-J model, using dynamical mean-field theory (DMFT),
d
stroyed. The interplay between these two kinds of in- has been recently reported in Ref. [11]. In this Letter,
n
o teractionshas been recentlypointed out in undoped and we apply such approach to analyze the optical data of
c weaklydoped cuprates,where angle-resolvedphotoemis- electron-dopedcupratesNd2−xCexCuO4(NCCO),where
[ sionspectroscopy(ARPES)revealedabroadhigh-energy exhaustive measurements as function of both tempera-

1 peak well distinguished from the quasi-particle states ture and doping are available [12]. We show that the
v withvanishingly smallspectralweight[6,7]. The broad- optical conductivity in these compounds can be natu-
1 nessofsuchpeakhasbeeninterpretedintermsofamulti- rally interpreted in terms of spin/lattice polarons re-
9
phonon Franck-Condon structure typical of a lattice po- sulting from the joint positive cooperation between the
6
laron,superposedonthe dynamicsofasingleholeinthe electron-phonon (el-ph) and magnetic exchange interac-
4
. t-J model [6, 7, 8, 9, 10]. The validity of this scenario tions [11, 13]. In this framework we are able to explain
1
at finite doping, and its relation with the pseudogap ob- the existence of a pseudogap in the optical spectra, as
0
served by means of different techniques is however not wellasits evolutionwithdopingandtemperature. From
8
0 clear. the comparison between theory and experiments we es-
: timate a value for the local el-ph coupling of the order
v Amongthedifferentfamiliesofcopperoxides,electron-
λ 0.7.
i
X doped cuprates represent the most suitable systems ≈We now examine the abovescenarioin the lightof the
where to investigatethe interplay betweenmagnetic and
r theoreticalDMFTresults. Weconsiderthemotionofone
a lattice polarons. This is because, unlike hole-doped sys-
electron added to the half-filled antiferromagnetic back-
tems, the long-range AF ordering extends up to quite
ground and interacting a` la Holstein with local disper-
large doping levels δ 0.14, so that the relative impor-
≈ sionless phonons. Due to the particle-hole symmetry on
tanceoflong-rangevs. short-rangemagneticcorrelations
theparentHubbard-HolsteinHamiltonian,suchproblem
can be more clearly addressed. The optical conductivity
isformallyequivalenttoconsideroneholeintheHolstein-
(OC) of the electron-doped compound Nd Ce CuO
2−x x 4 t-J model. Usingthelinearspin-waveapproximation,we
hasbeensystematicallyanalyzedasfunctionoftempera-
can thus write the Hamiltonian as [11, 13]:
tureanddopinginRef. 12. There,amid-infrared(MIR)
shoulder associatedwith a pseudogapstate has been ob- t J˜
H = h†h a +h.c. + a†a +
servedinthespectraatfinitedoping. Quiteinterestingly, 2√z X(cid:16) j i j (cid:17) 2 X i i
they observed: i) a shift of the MIR peak towards lower hiji i
frequencies upon increasing δ, corresponding to a pseu- + g h†h b +b† +ω b†b , (1)
X i i(cid:16) i i(cid:17) 0X i i
dogap closing with doping; ii) a filling of the pseudogap
i i
2
where z is the coordination number, a† and b† are the δ 0.05 0.075 0.1 0.125
creationoperatorsforbosonspindefectsandphononsre- T∗ (K) 440 340 300 200
spectively, and h† is the single spinless charge operator.
J (meV) 152 117 103 69
Alocalel-phcouplingconstantλ=g2/ω tcanbedefined
0
as the polaron energy in units of the half-bandwidth t. Lp 0.14 0.19 0.27 1.01
The parameter J˜is an effective exchange interaction. It dP−P 4.5 3.7 3.2 2.8
is linked to the microscopic J by the relation J˜= Jm,
where m is the average on-site magnetization which, in
TABLEI:Dopingevolutionofexperimentaldataandtheoret-
the z 1limit, is governedby the Curie-Weissequation ical parameters as function of doping δ. T∗ is the pseudogap
≫
m = tanh(βJm/4) with N´eel temperature TN = J/4. temperature taken from Ref. [12], J is the theoretical esti-
The one-particle Hamiltonian (1) can be thought to be mateoflowtemperatureexchangecouplingcorrespondingto
∗
representativealso for the finite doping case as far as we T , Lp is the average number of spin defects, i.e. the spin
are in the dilute limit, namely as far as the interparticle polaron size in units of the lattice constant, and dP−P is the
distance d is larger than the spin polaron size L . averageparticle-particledistancegivenbydP−P =1/√δ,also
P−P p in units of the lattice constant.
For δ <0.14 this implies L <2.7 in units of the lattice
p
consta∼nt. We shall discuss la∼ter the validity of this con-
dition. A further implication of the finite doping regime
is that the effective exchangeenergy J˜shouldbe consid- a paramagnetic analysis with J˜ = m = 0 in enforced.
eredas renormalizedby finite chargedensity effects, and We can identify thus the experimental T∗ with the N´eel
thus doping dependent J˜=J˜(δ). temperature TN of the DMFT solution of the Holstein-
t-J model, where the AF correlation length jumps from
TheHolstein-t-J Hamiltonian(1)isthebasistodiscuss
ξ = 0 for T > T to ξ = for T T and the
the physics ofa spin/lattice polaron,namely a travelling AF N AF N
∞ ≤
spin polaron size from L = for T > T to a finite
charge carrying along an on-site phonon cloud and sur- p N
∞
value for T T . Along this perspective we can deter-
rounded by a local destruction of the AF background of N
≤
mine the (doping dependent) low temperature effective
size L . Different properties of this object can be inves-
p exchange energy J˜(T = 0) = J, where m = 1, from
tigated and, as a general rule, the choice of a particular
the experimental temperature T∗ = T = J/4. The
theoretical approach depends on which property is un- N
estimates so obtained as function of δ, in good agree-
der examination and on its feasibility to investigate it.
ment with experimental estimates [19], are collected in
Quasi-particledispersion,itsrelativespectralweightand
Table I along with other characteristic quantities. The
effectivemass,forinstance,areintrinsicallyrelatedtothe
coherentmotionofthe polaronas a whole. Onthe other temperature dependence of the magnetization can thus
be evaluated from the Curie-Weiss relation. From pho-
hand, the internal local structure of the spin/lattice po-
toemission spectroscopy we take a half-bandwidth value
laron,responsible for the optical absorptionat finite fre-
t=1 eV for NCCO [20, 21, 22], and we set also ω =70
quency, is mainly related to incoherent processes which 0
meV, consistent with the energy window of the optical
can be safely caught by the dynamical mean-field ap-
phonons in undoped cuprates. With these values, the
proach, where coherent motion is disregarded[11, 13].
onlyfreeadjustableparameterresultstobetheelectron-
One of the main intriguing characteristics of the
phonon coupling constant λ. We are going to show that
spin/lattice polaron that can be captured by DMFT
the overall experimental scenario [12] is naturally repro-
within the context of the Holstein-t-J model [14] is
duced within the context of a spin/lattice polaron by
that the electron-phonon and the exchange interaction
assuming a doping-independent el-ph coupling λ = 0.7.
act in a cooperative way to reduce the local hopping
We would stress that this does not rule out a possible
amplitude, and hence to establish polaronic features
weakdependence ofλondopingwhichwouldprovidean
[8, 11, 13, 15, 16, 17, 18]. An interesting consequence
even stronger agreement with the experimental data.
of this scenario is that lattice polaron features can be
in principle switched on and off by varying the magnetic It should be stressed that an el-ph coupling λ = 0.7,
exchange coupling J˜at given el-ph coupling λ. Ouranal- at ω /t = 0.07, is not sufficient to induce lattice po-
0
ysis shows that this is precisely what occurs in NCCO. laron effects in the absence of exchange coupling since a
On the physical ground, the effective exchange coupling critical value λ = 0.91 is needed to establish a lattice
c
depends on the charge doping δ and on the tempera- polaron state at J =0 [23]. This means that lattice po-
ture T, in such a way that increasing δ and T leads to laron features in these compounds can be found only in
a reduction of J˜ and hence to an increase of the spin the presence of (short-range) AF correlations and they
polaron size. In this context we interpret the pseudogap are effectively tuned by the doping dependent exchange
temperature T∗ as the highest temperature where the coupling J(δ). In order to show and quantify this issue
spin polaron size L is smaller than the AF correlation we plot in Fig. 1 the renormalized phonon probability
p
length ξ . For T > T∗ (L > ξ ) the charge mo- distribution function P(n) [ P(n) = 1] for different
AF p AF Pn
tion does not probe anymore the AF environment and valuesofthe exchangecouplingspanningthe experimen-
3
δ = 0.05 δ = 0.75
1 T = 440 K J/t = 0.152 T = 340 K J/t = 0.117
p T* = 440 K T* = 340 K
L T = 540 K T = 540 K
0.1
P(n) 0.01 0.1 units] T = 90 K T = 90 K
J/t arb.
J/t=0.15 σω() [ T = 290 K JT/δ*t === 3000..1100 K3 T = 190 K JT/δ*t === 2000..10026 K59
T = 90 K
T = 540 K T = 540 K
J/t=0.01
T = 90 K
n = 0 1 2 ... 19 0 0.5 1 1.5 0.5 1 1.5 2
ω [eV] ω [eV]
FIG.1: Phonondistributionfunction P(n)(mainpanel) and
spinpolaronsizeLp (inset)asfunctionofexchangecoupling. FIG. 2: (color online) Temperature evolution of the optical
conductivity σ(ω) for different exchange couplings (dopings)
∗
andfor(T T ): T =90,140,190,240,290,340,390,440K).
≤
talrangeofJ (seeTableI). Themagnetic-inducedlattice Also shown is the optical conductivity at T = 540 K in the
polaron crossover occurs for J/t 0.083 (J = 83 meV normal state (dashed red line).
≃
in energy units), where the P(n) changes abruptly from
a distribution peaked at n = 0 to a multiphonon broad
structure at finite n, typical of a polaronic lattice distor- the pseudogapregime. Suchfeature representsthe natu-
tion. The strong interplay between lattice and magnetic ral counterpart, in the optical data, of the broad multi-
degrees of freedom is also witnessed by the dependence phononstructureresponsibleforthe incoherentpeakob-
of the spin polaron size L on J at T = 0, obtained by servedinARPES,andisanaturalsignatureofthe onset
p
the Hellmann-Feynman theorem[13], as shown in the in- of the lattice polaron. Consistent with this picture, the
set of Fig. 1. Here the onset of the lattice polaron for energyscale of suchbroadMIR band is reduced as pola-
J/t>0.083leadstoafurtherself-trappinglocalizationof ronicfeatures are weakenedupon doping (Fig. 2), which
the c∼harge and to an abrupt drop of L in the polaronic causes a closing of the optical pseudogap. Also interest-
p
regime. Comparing the typical values of the exchange ing is the role of temperature, also reported in Fig. 2.
energyestimatedfromtheexperimentaldatareportedin While the magnitude of the exchange coupling J rules
TableI,wecanpredictlatticepolaronfeatures,accompa- the energy scale of the MIR band [11] at low tempera-
nied by a very small amount of spin defects, for dopings ture, and hence drives the closing of the pseudogapwith
δ = 0.05,0.075,0.1 while at the larger doping δ = 0.125 δ,increasingthetemperatureforT <T∗leadstoafilling
spin/lattice polaron features are already remarkably re- ofthepseudogapitself(seeinparticulartheupperpanels
duced. Itisalsocomfortingtocheckthatforallthedop- in Fig. 2), in goodagreement with Ref. [12]. Eventually
ings considered in the present analysis δ = 0.05 0.125, for T > T∗ there is a considerable smoothing of the OC
−
bothonthe polaronicandonthe non-polaronicside,the features with respect to the spectra at T T∗.
≈
estimate of the spin polaron size Lp is always smaller Such temperature evolution can be nicely traced by
thanthe averageparticle-particledistancedP−P (seeTa- evaluating the “MIR optical spectral weight” WMIR,
ble I), confirming the validity of the present one-particle which was defined in Ref. [12] as the optical spectral
analysis in the dilute limit. weight between a low energy cut-off at ω = 0.12 eV
low
Let us now discuss the effects ofthe magnetic induced and a high-energy isosbestic point. Quite notably, the
lattice polaron formation on the optical conductivity. In temperature evolution of such spectral weight does not
Fig. 2 we plot σ(ω) evaluated for the different exchange show any signature of the AF long-range order at the
couplings corresponding to different doping levels (see N´eel temperature Texp, whereas a non-monotonic kink
N
Table I). We should stress, once more, that, due to the at a higher temperature T∗ is clearly visible. This ob-
localapproximationenforcedbythez = DMFTlimit, servation can be naturally explained within the present
∞
only incoherent contributions to the OC are here taken (local) polaronic approach, since the local polaron pro-
into account, so that no Drude-like peak is present. Dis- cesses responsible for the incoherent shoulder in Fig. 2
regardingthecoherentDrudepeak,theresemblancewith cannotdistinguishbetweenashort-rangeandlong-range
the experimentaldata of Ref. [12]is remarkable. In par- AF ordering as long as L < ξ . For the same rea-
p AF
ticularweobserveatlowtemperatureabroadincoherent sons, anomalies in W could be expected at T > T∗
MIR
shoulder in the MIR energy range which characterizes (T > TN in our analysis) where L > ξ and the∼con-
p AF
∼ ∼
4
1 integration of ∆W down to ω = 0 (black squares
MIR low
T*
in the last panel of Fig. 3).
0.5 T*
In conclusion, in this paper we have analyzed the op-
0
ticaldata ofthe electron-dopedcuprate Nd Ce CuO
s] 2−x x 4
nit -0.5 δ = 0.05 δ = 0.075 as function of doping and temperature by means of the
b. u J/t = 0.152 J/t = 0.117 DMFTresultsforoneelectronintheHolstein-t-J model.
ar
[R δ = 0.125 We haveshownthat the originof the MIR band, the op-
WMI 0.5 T* J/t = 0.069 tical pseudogap as well as its doping and temperature
∆ T* dependence can be naturally explained in terms of a po-
0
laronic scenario resulting from the joint cooperation of
-0.5 δ = 0.1 the el-ph and exchange interactions. From our analysis
J/t = 0.103
we estimate an el-ph coupling of the order of λ 0.7.
-1 ≃
0 100 200 300 400 500 100 200 300 400 500 600 Our results illustrate the relevance of the positive in-
ω [K] ω [K]
terplay between lattice and spin degrees of freedom in
electron-doped cuprates, where the AF ordering extends
over a large doping window, making these compounds
FIG. 3: Temperature dependence of ∆WMIR for different ideal systems where to investigate such effects.
exchange coupling corresponding to different doping levels.
We thank Y. Onose, Y. Tokura, A.S. Mishchenko, V.
Empty circles and black squares are obtained by integrating
Cataudellaand G.De Filippis for useful discussions. We
σ(ω) in the [0.12,1] eV and in the [0,1] eV energy window,
respectively. also thank Y. Onose and Y. Tokura for having made
available their experimental data. E.C. and S.C. ac-
knowledgealso financialsupport fromthe ResearchPro-
gram MIUR-PRIN 2005.
cept itself of a spin polaron breaks down. In order to
quantifythisissue,intheabsenceofacorrespondingisos-
bestic point in our data, we define W as the optical
MIR
spectralweightbetweenω =0.12eVandafixedhigh-
low
energy cut-off at ωhigh = 1 eV. We have checked that ∗ On leave of absenceat theICMM-CSIC, Madrid, Spain.
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−
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5
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