Singularities in the optical response of cuprates

6 Pages·0.17 MB·English

by  Ar. Abanov

Singularities in the optical response of cuprates.
Ar. Abanov1, Andrey V. Chubukov1, and J¨org Schmalian2
1 Department of Physics, University of Wisconsin, Madison, WI 53706
2 Department of Physics and Ames Laboratory, Iowa State University, Ames, IA 50011
(February 1, 2008)
1
0 We argue that the detailed analysis of the optical response in cuprate superconductors allows
0 one to verify the magnetic scenario of superconductivity in cuprates, as for strong coupling charge
2
carriers to antiferromagnetic spin fluctuations, the second derivative of optical conductivity should

n contain detectable singularities at 2∆+∆spin, 4∆, and 2∆+2∆spin, where ∆ is the amplitude of
a the superconducting gap, and ∆s is the resonance energy of spin fluctuations measured in neutron
J scattering. Wearguethatthereisagoodchancethatthesesingularitieshavealreadybeendetected

5 in theexperiments on optimally doped YBCO.
1
PACS numbers:71.10.Ca,74.20.Fg,74.25.-q

]
n The pairing state in cuprate superconductors is pre- tional piece due to the presence of the superconducting
o
c dominantly made out of Cooper pairs with dx2−y2- condensate. A nonzero σ1(ω) at a finite frequency is
- symmetry [1]. This salient universal property of all onlypossible iffermionshaveafinite lifetime. Morepre-
r high T materials entails constraints on the microscopic cisely,oneofthetwofermionsexitedinaprocesscausing
p c
u mechanism of superconductivity. However, it does not the AC conductivity should have a finite scattering rate,
s uniquelydetermineit,leadingtoaquestforexperiments whileanothershouldbeabletopropagate,i.e.,itsenergy
.
t whichcanidentify”fingerprints”ofaspecificmicroscopic shouldbelargerthan∆. Forclean,phonon-mediatedsu-
a
mechanism of d-wave superconductivity, a strategy sim- perconductors,there are two sources for fermionic decay
m
ilar to the one used in conventional superconductors [2]. (see Fig. 1). One is a direct four-fermion interaction,
-
d Severalresentexperimentswereinterpretedasanindi- which yields a threshold in the imaginary part of the
n rectevidencethatdx2−y2 pairingincupratesisproduced self-energy,Σ′′(ω), at ω =3∆- the minimal energy nec-
o by an exchange of collective spin fluctuations peaked at essary to pull all three fermions in the final state out of
c
or near antiferromagnetic momentum Q = (π,π) [3]. In the condensate of Cooper pairs. Another is the inter-
[
particular, the distance between the peak and the dip action between an electron and an optical phonon. It

1 in the fermionic spectralfunction, Ak(ω), in ARPES ex- yields the onset of Σ′′(ω) at ω = ∆+Ωp, where Ωp is
v
periments coincides with the frequency ∆ of the reso- the frequency ofan opticalphonon[6](for simplicity, we
0 s
nance peak measured in neutron scattering [4,5]. This is assumed that the phonon propagator has a single pole).
2
2 exactly what one should expect for fermions interacting For the values of the coupling constant used to interpret
1 with a resonating spin collective mode [4,5] (for phonon the tunneling data in strongly coupled conventional su-
0 mediated superconductors, this is known as the Holstein perconductors like Pb [9], Ω > 2∆, i.e., the onset of
p
1
effect[6]). Similarly,apeak-dipstructureoftheSIStun- conductivity is at 3∆+ ∆ = 4∆ (2∆ for dirty super-
0
neling conductance with peak-dip distance roughly con- conductors[10]), while the signaturesof phonon-assisted
/
at sistent with ∆s has been obtained in the measurements dampingonlyshowupathigherfrequencies,andareun-
m onbreak junctions by Zasadzinskiet al. for various dop- correlated with the behavior of σ1(ω) near 4∆.
ing values [7]. Carbotte et al. [8] analyzed optical con-
-
d ductivity σ(ω) in magnetically mediated d-wave super- a) b)
n
conductors and argued that ∆ can be extracted from
o s
the measurements of the second derivative of σ(ω).
c
: In this paper we reexamine the behavior of the opti-
v
cal conductivity in superconductors with quasiparticles
i
X stronglycoupledtotheir owncollectivespinmodes. Our
FIG. 1. a) The exchange diagram for boson mediated in-
r results partly agree and partly disagree with those by
teraction. The solid line stands for a propagating fermion.
a
Carbotte et al. [8] (see below). The key prediction of
The wiggled line is a phonon propagator in case of electron-
this paper, however, is novel: we argue that by measur- phononinteraction,andamagnonlineincaseofspin-fluctu-
ingthefrequencyderivativesoftheconductivity,onecan ation mediated interaction. b) The lowest order diagram for
not only verify the magnetic scenario, but, in principle, thefermionicselfenergyduetoadirectfourfermion interac-
alsoindependentlydetermineboth∆ and∆inthesame tion, also represented by a wiggly line
s
experiment.
Our argument goes as follows. For a superconductor, For spin-mediated superconductivity, the situation is
the real part of the conductivity, σ1(ω), has a δ- func- different. In the one-band model for cuprates, which we
adopt, the underlying interaction is solely a Hubbard-
1
type four-fermion interaction. Spin excitations appear the Fermi points which determine the positionof the ex-
as collective modes of fermions, and their velocity v is citonic pole in χ′′(Q,ω). Accordingly, the singularity in
s
comparable to v . For v ∼ v , the low frequency spin conductivityentirelycomesfromfermionsnearhotspots,
F s F
dynamicsisdominatedbyadecayprocessintoaparticle- and the threshold frequency 2∆+∆ involves a maxi-
s
hole pair and is purely relaxational in the normal state, mum value of the gap and the resonance spin frequency
with nearly featureless χ′′(Q,ω) [5]. (for phonon super- at momentum Q. The same argumentation implies that
conductors the relaxation is also present but is strongly 4∆ threshold also involves a maximum value of the gap.
reduced due to a smallness of the sound velocity com- Note for clarification that we are only considering here
pared to v [11]). thesingularitiesintheconductivityatω >2∆. Thereg-
F
Below Tc, fermions acquire a gap, and spin-decay be- ular partofσ1(ω) is notnecessaryconfinedto hotspots.
comes impossible for energies below 2∆. The direct In particular, for ω ≪ ∆ the optical response is dom-
four-fermion interaction process in Fig. 1b then yields inated by nodal quasiparticles for which Σ′′ is nonzero
a threshold in Σ′′ at 3∆ which gives rise to a singu- down to the lowest frequencies.
larity in the conductivity at ω = 4∆ [12]. If χ′′(Q,ω) We now proceed with the calculations. The real part
remainedfeatureless,thiswouldbetheonlyeffect. How- of the optical conductivity in a superconductor is given
ever,severalauthorshavedemonstratedthattheresidual by
attraction in a d-wave superconductor binds a particle
i
aTnhdisaeffheoclte ginivtoesarisspeintoeaxcpiteoankaint aχn′′(eQne,rωg)ya∆tsω<=2∆∆. σ1(ω)=Reω+iδ Z dθΠσ(θ,ω) (1)
s
and makes it look like the spectral function for optical
where Π (θ,ω) is the fully renormalized current-current
phonons. Accordingly,the conductivityacquiresanother σ
correlator. In Matsubara frequencies, it is given by
threshold at 2∆+∆ . Formally, this is analogous to the
s
phonon case, but in distinction to phonons, ∆s < 2∆. 1 d2k
Then 2∆+∆s < 4∆, i.e., in clean systems, the lower Πσ(iωn) ∝ β XZ (2π)2 [Gk(iωn+iωm) Gk(iωm)
threshold corresponds to the scattering by a spin exci- m
ton. Moreover, since both effects are due to the same +Fk(iωn+iωm) Fk(iωm)], (2)
underlying interaction, (the diagram in Fig. 1b is just
and the normal and anomalous Green’s functions are
thefirstterminthe seriesofgraphswhichconstitutethe
spin-mediated scattering process shown in Fig 1b) the Σk(iωm)+εk
ratio ∆s/∆ and the relative intensity of the singularities Gk(iωm)= Σ2(iω )−Φ2(iω )−ε2, (3)
k m k m k
tinioσn1i(sωa)“afitn4g∆erapnridnt2”∆o+ft∆hessapreinc-oflrurcetluaatetdio.nTmheischcoarnriesmla-. Fk(iωm)= Σ2(iω )Φ−kΦ(ω2m(i)ω )−ε2, (4)
Wewillarguethattherearestrongindicationsthatboth k m k m k
singularities have been observed in the measurements of
theopticalconductivityinYBCO [8],andtheirposition (we adsorbed a bare iωm term into Σk(iωm). In prin-
ciple, Π is modified by vertex corrections related to
σ
and relative intensity are in reasonable agreement with
dΣ/dk [11], but for spin-mediated scattering these cor-
the theory.
rections are small (see below).
Beforewe proceedwith the calculations,a commentis
As an input for the computation of Π we need
σ
in order. In the above discussion we neglected the mo-
mentum dependence of the d-wave gap. Meanwhile, the the forms of the fermionic self-energy Σk(iωm) and the
anomalous vertex Φk(iωm). We obtained these forms
computations of the optical conductivity involve averag-
in Ref. [16] by deriving and solving a set of Eliashberg
ing of the lifetime over the Fermi surface [13,14]. It is
equations within the spin-fermion model. This model
then a’priori unclear whether the angulardependence of
adequately describes the interaction between low-energy
the d-wave gap with ∆(θ) ∝ cos2θ affects the positions
fermions and their collective spin degrees of freedom
of the two thresholds in the conductivity. Carbotte et al
[5,15,16]atenergiessmallerthanE . Thefulldynamical
argued [8] that it does, and the singularity at 2∆+∆ F
(which they only considered) is determined by some avs- spinsusceptibilitypeakedat(ornear)Qmediatesdx2−y2
superconductivity. As discussed, this susceptibility is by
eraged |2∆(θ)| ≈ ∆. We argue that averaging reduces
itself affected by low-energy fermions via a decay pro-
strengths of the singularities but doesn’t shift their po-
cess into a particle and a hole, and has to be computed
sitions. Our argument is two-fold. First, we explicitly
together with the fermionic self-energy and the pairing
demonstratebelowthatthe singularityinthe conductiv-
vertex.
ity occurs at a frequency equal to the maximum value
The justification of the Eliashberg approach for
of the gap. Second, two of us and Finkel’stein argued
the spin-mediated superconductivity was discussed ear-
earlier [15] that for spin-mediated d-wave superconduc-
lier [5,15,16], and we just quote the result: at strong di-
tivity,∆(θ)isatitsmaximumathotspots(pointsatthe
mensionless spin-fermion coupling λ, vertex corrections
Fermi surface separated by Q) [15]. These are precisely
2
and v−1 dΣ/dk⊥, where k⊥ is the component of the mo- prefactor is the same for Σ′ and Φ′. Substituting these
F
mentum transverse to the Fermi surface, are small com- forms of Σ(ω) and Φ(ω) into (2) we obtain after simple
pared to dΣ/dω by logλ/λ. In what follows we will ne- algebrathat the conductivity emergesabove 2∆+∆ as
s
glect these corrections, i.e., approximate Σk(iωm) and ǫ1/2/log2ǫ, where ǫ = ω−(2∆+∆s). This singularity
Φk(iωm) by Σk(iωm)=Σ(iωm,θ) and Φ(iωm,θ). obviously causes a divergence in the derivatives of the
conductivity at ǫ=+0.
σ (ω) 1/τ In Fig.2 we show the result for the conductivity ob-
1 tained by numerically solving Eq.(2) using Σ(ω) and
Φ(ω) from Ref. [16]. We clearly see the expected thresh-
old at 2∆+∆ . The insert shows the behavior of the
ω/ω s
relaxation rate 1/τ(ω) = (4π/ω2)Re[1/σ(ω)] where ω
0 pl pl
4 isthe plasmafrequency. Observethat1/τ(ω)islinearin
ω over a rather wide frequency range. This agrees with
3
the earlier study of the normal state conductivity [18].
W(ω ) W(ω) O6.95 sample
3 O6.6 sample
2∆+∆ ω/ω
0 s 2 2 3
FIG. 2. Frequency dependence of the real part of the 1 2
optical conductivity σ1(ω) at T = 0 computed using the self 1
0
energyandthepairingvertexdeterminedfromtheEliashberg
ω(meV)
equations for λ = 1. The onset of the optical response is -1
0 50 100 150 200
ω = 2∆+∆s. The contributions from nodal regions (not
included in calculations) yield a nonzero conductivity at all
ω. They also soften the singularity at ω = 2∆+∆s, but do 1 3
not eliminate it. Insert - the behavior of the relaxation rate ω/ω
2
1/τ(ω)=(4π/ω2)Re[1/σ(ω)]. The frequency is measured in
pl
units ω¯ which sets the overall energy scale in the Eliashberg 0 1 2
solution. For λ=1, ∆=0.204ω¯ and ∆s =0.291ω¯. WF(ωIG).=3.d2 [ωRe[A1/σc(aωl)c]u]laatteTd→fr0eq.uTehnicsyqudaenpteitnydiesnaceseno-f
d2ω
As our goal is to study the singularities in σ1(ω), we sitive measure of finestructures in theoptical response. The
first perform calculations assuming that Σ and Φ are in- locationsoftheextremaare: 1–2∆+∆s,2–4∆,3–2∆+2∆s.
Dashed lines are the results are higher T. Observe that the
dependent on θ (i.e. that the superconducting gap is
maximum shifts to a lower temperature, but the minimum
flat near the hot spots), and then analyze the results for
remains at 2∆+∆s. Inset– Experimental results for W(ω)
a true d-wave gap. For a flat gap, the momentum in-
at low T from Ref. [8]. The position of the deep minimum
tegration in Eq. (2) is straightforward. Substituting k
agrees well with 2∆+∆s. The extrema at higher frequen-
integrationby integrationoverεk, and performing it, we cies are consistent with 4∆ and 2(∆+∆s) predicted by the
obtain at T =0 and ω 6=0 theory.
Π (iω )∝ dω′ dθΣ+Σ−+Φ+Φ−+D+D− (5) Wenextdemonstratethatthepositionofthesingular-
σ n Z m D+D−(D++D−) ity is not affected by the angulardependence of the gap.
Indeed, let the maximum value of the gap correspondto
Here Σ± = Σ(iω±,θ), Φ± = Φ(iω±,θ), and D± =
(Φ2± −Σ2±)1/2, where ω± = ω′ ±ω/2. The conductiv- θ = 0 and symmetry related points. At deviations from
θ =0 both ∆ and ∆ decrease. The decreaseof ∆ is ob-
ity is obtained by converting this expression to the real s
vious,thedecreaseof∆ isduetothefactthatresonance
axis [17]. The singular piece in σ1(ω) near 2∆+∆s can s
is a feedback from superconductivity, and its frequency
be obtained without a precise knowledge of Σ(ω) and
scales as (∆(θ))1/2. Since both ∆ and ∆ are maximal
Φ(ω): the only information we need is that in a d-wave s
superconductor, χ′′(Q,ω) has a δ-functional singularity at a hot spot, we can expand ω0(θ) = ∆(θ)+∆s(θ) as
at ω =∆ . This is what we found solving a set of three ω0(θ) = ω0 − aθ2, where a > 0. The singular pieces
Eliashbersg equations. Using this as an input and apply- in Σ(ω) and Φ(ω) then behave as |log(ω0 −ω −aθ2)|.
ing a spectral representation for Σ′′ and Φ′′ (which for a Substituting these forms into (2) and integrating over θ,
givenχ′′(Q,ω)aredescribedbyaconventionalsetoftwo wefindthattheconductivityitselfanditsfirstderivative
Eliashberg equations), we obtain that Σ′′(ω) and Φ′′(ω) arecontinuousatω =2∆+∆s,butthesecondderivative
of the conductivity diverges as d2σ/dω2 ∝ 1/(|ǫ|log2ǫ)
are zero up to ω = ∆+∆ , and undergo finite jumps
s
at this frequency. By Kramers-Kronig relation, Σ′ and where, we remind, ǫ = ω−(2∆+∆s). We see that the
Φ′ diverge as |log(ω −ω0)| where ω0 = ∆+∆s. The singularityisweakenedbytheangulardependenceofthe
3
gap, but it is still located at exactly 2∆+∆ . mentatω >2∆+∆ ispredominantlyanindicationthat
s s
Thesamereasoningisalsoappliedtoaregionnear4∆. themomentumdependenceofthefermionicdynamicsbe-
We found that the singularity at 4∆is also weakenedby comes irrelevant at high frequencies, and fermions from
the angular dependence of the gap, but is not shifted all over the Fermi surface behave as if they were at hot
and still should show up in the second derivative of the spots. The insert to Fig.4 shows σ−1(ω). We see that it
1
conductivity. is linear over a substantial frequency range.
We now discuss the second derivative of the conduc-
σ (ω)
tivity in more detail. In Fig.3 we present our numerical 1 1/σ
results for W(ω) = dd22ω(ωReσ−1(ω)) which effectively 1−cm 1 1
measures second derivative of conductivity (we followed −Ω
Ref[8]andusedthesameW(ω)asforphononsupercon-
ductors). We clearly see that there is a sharp maximum
0 ω/ω
inW(ω)near2∆+∆ followedbyadeepminimum. We 0
s 5
also see that W(ω) has extra extrema at 4∆ and, also, 0
5
at 2ω0 = 2∆+2∆s. The last peak is a secondary effect
due to a singularityin Σ(ω) at ω =ω0: σ1(ω) is singular
when the frequencies of bothfermions in the polarization
bubble exceed ω0. ωcm-1
The experimental result for W(ω) in YBCO is shown 0 2000 4000 6000
in the insert. We see that the theoretical and experi- FIG. 4. A comparison of the theoretical result with the
mental plots of W(ω) look rather similar, and the rel- experimental data of Puchkov et al. [19]. The substructure
ative intensities of the peaks are at least qualitatively inthetheoreticalσ1 atlowfrequenciesisanartifact. Insert-
consistentwith the theory. By the reasonswhichwe dis- thebehavior of σ1−1(ω).
playbelow,weidentify 2∆+∆ withthe deepminimum
s
in W(ω). This yields 2∆+∆ ≈ 100meV. Identifying We emphasize, however, that this linearity is only an
s
the extra extrema in the experimental W(ω) with 4∆ intermediate asymptotic. At the highest ω, our theory
and 2∆+2∆ , respectively, we obtain 4∆ ∼ 130meV, yields σ1(ω)∝ω−1/2. The lowerboundaryforω−1/2 be-
s
and 2∆ + 2∆ ∼ 150meV. We see that three sets havior decreases with increasing λ. For optimally doped
s
of data are self-consistent and yield ∆ ∼ 30meV and Bi2212,λissomewhatlargerthaninYBCO as∆≈∆s,
∆ ∼ 40 − 45meV. The value of ∆ is in good agree- andweexpectamorepronouncedω−1/2behaviorathigh
s
ment with tunneling measurements [20], and ∆ agrees frequencies. This trend is consistent with the data of
s
wellwiththeresonancefrequencyextractedfromneutron Ref. [22]. This issue, however, requires further study as
measurements [21]. We caution, however, that determi- σ1(ω) ∝ ω−1/2 at intermediate frequencies was also ob-
nation of a second derivative of a measured quantity is tained in Ref. [14] assuming a strongly momentum de-
a very subtle procedure. The good agreement between pendent scattering rate.
our theory and the experiment is promising but have to Finally, we comment on the position of the 2∆+∆s
be verifiedinfurtherexperimentalstudies. Nevertheless, peak and compare our results with those by Carbotte
ourcalculationclearlydemonstratesthepresenceandob- et al. [8]. Theoretically, at T = 0 and in clean limit,
servability of these ”higher harmonics” of the optical re- the maximum and minimum in W(ω) are at the same
sponse at 4∆ and 2∆+2∆ . frequency. We found, however, that at finite T, they
s
So far we considered only the singular part of σ1(ω). quickly move apart. We present the theoretical temper-
In Fig.4 we compare our results for σ1(ω) (ignoring the ature dependence of W(ω) in Fig 3. Carbotte et al. [8]
contributions from the nodes) directly with the experi- focused on the maximum in W(ω) and argued that it is
mental data by Puchkov et al. [19] for optimally doped locatedat∆+∆sinsteadof2∆+∆s. Wealsofoundthat
YBa2Cu3O6+δ. We used λ = 1 and the overall energy the maximum in W(ω) shifts to a lower frequency with
scale ω¯ which yield ∆ ∼ 30meV and ∆ ∼ 45meV increasing temperature, already at T where the temper-
s
as the solution of the Eliashberg set, and also ω = ature dependence of the gap may be neglected. On the
p
1.2×104cm−1,similartothatin[19]. Weseethatthefre- otherhand,theminimuminW(ω)movesverylittlewith
quency dependence of the conductivity at high frequen- increasingT andvirtuallyremainsatthesamefrequency
cies agrees well with the data. The measured conduc- asatT =0. Thisisourreasoningtousetheminimumin
tivity drops at about 100meV in rough agreement with W(ω)asamuchmorereliablefeatureforthecomparison
2∆+∆ ≈ 100meV in our theory. As in earlier stud- with experiments. This reasoning is in agreement with
s
ies [14,18], to match the magnitude of the conductivity, recentconductivity dataonoptimallydopedBi2212[22]
we had to add the constant 7×10−4Ω cm to (σ1(ω))−1. – W(ω) extracted from these data shows strong down-
Weviewthegoodagreementbetweentheoryandexperi- turn variation of the maximum in W(ω) with increasing
temperature, but the minimum in W(ω) is located at
4
around 110meV for all temperatures. P. Johnson, M. Norman, D. Pines, E. Schachinger, S.
Finally, we briefly consider whether one can extract a Shulga and J. Zasadzinski for useful conversations. We
resonance spin frequency from the measurements of the arealsothankfulto D.N.Basov,C.Homes,M.Strongin
Raman intensity in a d−wave superconductor. The Ra- and J. Tu for sharing unpublished results with us. The
man intensity is given by [23] research was supported by NSF DMR-9979749 (Ar. A
and A. Ch.) and by the Ames Laboratory, operated for
R(ω)=Im dθV2(θ)Π (ω,θ) (6) the U.S. DoE by Iowa State University under contract
Z R No. W-7405-Eng-82(J.S).
where V(θ) is Raman matrix element, and Π is same
R
bubbleasforconductivity,butwithadifferentsignofthe
anomalous FF term. The latter is a consequence of the
fact that Raman vertices are scalar and do not change
sign under k− > −k. Performing the integration over
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s
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[15] Ar.Abanov,A.Chubukov,andA.M.Finkel’stein,cond-
havesingularitiesat2∆+∆ ,4∆and2∆+2∆ ,where∆
s s
mat/9911445
is the maximum value of the d−wave gap, and ∆ <2∆
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