Pseudogap-induced anisotropic suppression of electronic Raman response in cuprate superconductors

0.23 MB·

by  Pengfei Jing

Pseudogap-induced anisotropic suppression of electronic Raman response in cuprate
superconductors
Pengfei Jing and Yiqun Liu
Department of Physics, Beijing Normal University, Beijing 100875, China
Huaisong Zhao
College of Physics, Qingdao University, Qingdao 266071, China
7
1
0 Lu¨lin Kuang
2 Sugon National Research Center for High-performance Computing Engineering Technology, Beijing 100093, China

r
a Shiping Feng∗
M
Department of Physics, Beijing Normal University, Beijing 100875, China

7 It has become clear that the anomalous properties of cuprate superconductors are intimately
2
related to theformation of a pseudogap. Within theframework of thekinetic-energy-drivensuper-

conducting mechanism, the effect of the pseudogap on the electronic Raman response of cuprate
]
n superconductors in the superconducting-state is studied by taking into account the interplay be-
o tweenthesuperconductinggap andpseudogap. Itis shown thatthelow-energy spectra almost rise
c as the cube of energy in the B1g channel and linearly with energy in the B2g channel. However,
- the pseudogap is strongly anisotropic in momentum space, where the magnitude of the pseudogap
r
around the nodes is smaller than that around the antinodes, which leads to that the low-energy
p
u spectral weight of the B1g spectrum is suppressed heavily by the pseudogap, while the pseudogap
s has a more modest effect on the electronic Raman response in theB2g orientation.
.
t
a PACSnumbers: 74.72.Kf,74.25.nd,74.72.Gh,74.20.Mn
m
-
d It is now generally accepted that the unusual proper- scattering geometry. In particular, the earliest evi-
n ties of cuprate superconductors in the underdoped and dence of the pseudogap effect manifested itself as a
o optimally doped regimes are heavily influenced by the suppression of the SC-state ERR around the antinodal
c pseudogap1–3. This pseudogap exists above the super- regime was observed on YBa Cu O in the under-
[ 2 3 7−x
conducing (SC) transitiontemperature T but below the doped regime14. Later, these early observations have
c
3 pseudogapcrossovertemperatureT∗,andmanifestsitself been further confirmed15–18, where a heavy loss in the
v as a loss of the spectral weight of the low-energy quasi- spectralweightwasfoundintheB spectrum,andthen
1g
3 particle excitations1–3. The origin of the pseudogap in the overall low-energy spectral weight in the B spec-
9 1g
cuprate superconductors remains a mystery second only trum is suppressed heavily by the pseudogap, however,
6
0 to that of superconductivity itself, and it is widely be- only a light loss of the spectral weight was detected in
0 lieved that by the investigation of the former one might theB2g spectrum,indicatingastrongmomentumdepen-
. uncover essential insights for the understanding of the dence of the pseudogap in cuprate superconductors. In
1
latter. thiscase,theelectronicRamanscatteringthereforegives
0
7 In order to fully understand the nature of the pseu- the important information for the nature of the momen-
1 dogapstate incuprate superconductors,the two-particle tumdependenceofthepseudogapwhichisnotaccessible
: Ramanscatteringexperiments that canprobe the quasi- via the conductivity measurement.
v
particle dynamics on different regions of the Brillouin Theoretically, ERR of cuprate superconductors in the
i
X zone (BZ) have provided useful information for clar- SC-state has been studied extensively3,4,19–25. In par-
r ifying the nature of the pseudogap phase of cuprate ticular, the ERR spectra in both the B and B ori-
1g 2g
a
superconductors3,4. Inastrikingcontrasttotheinfrared entations have been extracted to fit the electron-boson
measurements of the conductivity spectrum2, the elec- spectral density, and the results provide information on
tronicRamanscatteringcanbe usedtogivedirectlytwo the angular variation of the electron self-energy and the
spectra B and B , each representing a different av- correspondingspectraldensityaroundEFS20. Moreover,
1g 2g
erage over the electron Fermi surface (EFS)3,4. After ithasbeenshownthatERRinthe B symmetryallows
1g
intensive investigations over more than two decades, it one to distinguish between phonon-mediated and mag-
has been shown the electronic Raman response (ERR) netically mediated d-wavesuperconductivity21. Further-
spectrum in the SC-state is rather universal within the more,themainfeaturesobservedinERRhavebeenanal-
wholecupratesuperconductors3–18,wherethebroadcon- ysedbytheself-energyeffectintheproximitytoad-wave
tinuum scattering is depressed at low energies and the flux-phaseorderinstability25. However,tothebestofour
spectral weight lost is transferred to higher energies, knowledge,the anisotropicsuppressionofthe ERRspec-
forming a broad peak whose position depends on the trumby the momentum dependence ofthe pseudogapin
2
cuprate superconductors has not been treated starting terms of the single-electron Green’s function G(k,ω) in
from a microscopic theory. In this paper, we study the the Nambu representation as,
pseudogap-induced anisotropic suppression of the ERR
1
spectrum in cuprate superconductors in the SC-state by χ˜(q,iq )= γα γα
taking into account the interplay between the pseudo- m N Xk 1k+21q 2k+21q
gap and SC gap. Within the framework of the kinetic- 1
energy-drivenSCmechanism26–28,wecalculatethe ERR × Tr[G(k+q,iωn+iqm)τ3G(k,iωn)τ3]. (5)
β
function in terms of the Raman density-density correla- Xiωn
tion function, and then the obtained results show that
Itshouldbe emphasizedthatin the obtaining aboveRa-
the low-energy B response depends linearly on energy
2g mandensity-density correlationfunction, the vertex cor-
ω,whilethelow-energyB spectrumvariesasω3. How-
1g rection has been ignored, since it has been shown that
ever, our results also show that the pseudogap on EFS
the vertex correction in the pseudogap phase is negligi-
is strong momentum dependent, with the actual mini-
bly small29,30. Eq. (5) alsoindicates that for the evalua-
mumthatdoesnotappeararoundthenodes,butlocates
tion of the electronic Raman density-density correlation
exactly at the hot spots. In particular, the magnitude
function in the SC-state, we firstly need to obtain the
of the pseudogap around the nodes is smaller than that
single-electron Green’s function.
around the antinodes. This special structure of the mo-
The t-J model on a square-lattice is widely accepted
mentum dependence of the pseudogap therefore leads to
for the description of the essential physics of cuprate
that the low-energyspectral weight of the B spectrum
1g superconductors31,32. This t-J model is defined as,
is suppressed heavily by the pseudogap, while the pseu-
dthoegaBphaosriaenmtaotrieonm.odesteffectonthe ERRspectrumin H = − tll′Cl†σCl′σ+µ Cl†σClσ +J Sl·Sl′,(6)
2g Xll′σ Xlσ <Xll′>
ERR is manifested itself by the dynamical Raman re-
sponse function S˜(ω), which can be obtained directly supplementedbyanimportanton-sitelocalconstraintto
from the imaginary part of the Raman density-density avoid the double occupancy, C† C ≤ 1, where C†
σ lσ lσ lσ
correlation function χ˜(q,ω) as4, and Clσ are the electron operPators that respectively cre-
ateandannihilateelectronswithspinσonthelatticesite
S˜(ω)=−1[1+n (ω)]Imχ˜(q∼0,ω), (1) l,Slisaspinoperatorlocatedonthelatticesitel,µisthe
B
π chemicalpotential,J isthemagneticinteractionbetween
the nearest-neighbor (NN) sites on a square-lattice, and
where nB(ω) is the boson distribution function, while the summation < ll′ > is carried over the NN bonds.
the Raman density-density correlation function χ˜(q,ω)
For the hopping tll′, we take only t for the NN hopping
is defined as,
amplitude and −t′ for the next NN hopping amplitude,
respectively. In this paper, these parameters are chosen
χ˜(q,τ −τ′)=−hTρ (q,τ)ρ (−q,τ′)i, (2)
γ γ ast/J =2.5andt′/t=0.3. Theeffectofthestrongelec-
tron correlation in the t-J model (6) manifests itself by
where the Raman density operator in the Nambu repre-
the no-double electron occupancy local constraint33–37.
sentation can be expressed as,
Ithasbeenshownthatthisno-doubleelectronoccupancy
local constraint can be treated properly in actual cal-
ρ (q)= γα C† τ C , (3)
γ k+q k+q 3 k culations within the fermion-spin theory28,38, where the
Xk 2 constrainedelectronis decoupledas a chargecarrierand
with the Paulimatrix τ , andthe bareRamanvertexγα a localized spin, i.e., the electron operators Cl↑ and Cl↓
that has been classified3by the representations B1g ankd are decoupled as Cl↑ = h†l↑Sl− and Cl↓ = h†l↓Sl+, respec-
B of the point group D as4, tively, with the charge carrier h = e−iΦlσh that rep-
2g 4h lσ l
resents the charge degree of freedom together with some
γB1g = 1b [cos(k )−cos(k )], (4a) effects of spin configuration rearrangements due to the
k 4 ωi,ωs x y presence of the doped charge carrier itself, while the lo-
γkB2g = b′ωi,ωssin(kx)sin(ky), (4b) calized spin Sl describes the spin degree of freedom. In
this fermion-spin representation, the original t-J model
respectively, where as a qualitative discussion, the mag- (6) can be rewritten as,
nitude of the energy dependence of the prefactors b and
b′ can be rescaled to units. In cuprate superconductors, H = tll′(h†l↑hl′↑Sl−Sl+′ +h†l↓hl′↓Sl+Sl−′)
theB spectrumsamplestheantinodalregion,theFermi Xll′
1g
momentumontheBZboundary,whiletheB2g spectrum − µ h†lσhlσ +Jeff Sl·Sl′, (7)
samples the nodalregion,therefore ERRprobes comple- Xlσ Xhll′i
mentary regimes of EFS4. Substituting the Raman den-
sityoperator(3)intoEq. (2),theRamandensity-density Following this t-J model in the fermion-spin represen-
correlationfunctionχ˜(q,ω)thereforecanberewrittenin tation (7), the kinetic-energy-driven SC mechanism has
3
been developed26–28, where the interaction between the Recently, we39 have developed a full charge-spin recom-
charge carriers and spin directly from the kinetic energy binationschemetofullyrecombineachargecarrieranda
by the exchange of spin excitations generates the SC- localized spin into a constrained electron, where the ob-
stateintheparticle-particlechannelandpseudogapstate tained electronpropagatorcan give a consistent descrip-
in the particle-hole channel, therefore there is an inter- tion of the nature of EFS in cuprate superconductors.
playbetweentheSCgapandpseudogapinthewholeSC In the following discussions, we reproduce only the main
dome27. details in the calculations of the single-electron Green’s
In the framework of the charge-spin separation, the function of the t-J model under the fermion-spin repre-
single-electron Green’s function is obtained in terms of sentation in the SC-state. In Ref. 39, the single-electron
the charge-spinrecombination33–37. However, how a mi- diagonal and off-diagonal Green’s functions of the t-J
croscopictheorybasedonthecharge-spinseparationthat model in the SC-sate have been obtained in terms of the
can give a consistent description of EFS in terms of the full charge-spin recombination scheme as,
charge-spinrecombinationisaverydifficultproblem34,35.
ω+ε +Σ (k,−ω)
G(k,ω) = k 1 , (8a)
[ω−ε −Σ (k,ω)][ω+ε +Σ (k,−ω)]−∆¯2(k)
k 1 k 1
∆¯(k)
ℑ†(k,ω) = − , (8b)
[ω−ε −Σ (k,ω)][ω+ε +Σ (k,−ω)]−∆¯2(k)
k 1 k 1
where the bare electron excitation spectrum ε = a = (E2 − ε2 )/(E2 − E2 ), however, the
k 2k 2k 0k 1k 2k
−Ztγ +Zt′γ′ +µ, with γ = (cosk +cosk )/2, γ′ = SC quasiparticle spectrum has been divided
k k k x y k
cosk cosk , Z is the number of the NN or next NN sites into two branches, E = [Φ +Φ ]/2 and
x y 1k 1k 2k
on a square lattice, and the d-wave SC gap ∆¯(k) = E = [Φ −Φ ]/2, respecptively, by the pseu-
2k 1k 2k
Σ2(k,ω = 0) = ∆¯γk(d), with γk(d) = (coskx −cosky)/2, dogap, wphere Φ1k = ε2k + ε20k + 2∆¯2PG(k) + ∆¯2(k),
while the electron self-energies Σ1(k,ω) in the particle- Φ = (ε2 −ε2 )b +4∆¯2 (k)b +∆¯4(k),
holechannelandΣ2(k,ω)intheparticle-particlechannel b2k = ε2 − qε2 k+ 2∆¯0k2(k1)k, andPGb 2k= (ε −
duetotheinteractionbetweenelectronsbytheexchange 1k k 0k 2k k
ε )2 + ∆¯2(k), while the coherence factors,
ofspinexcitationsareevaluatedintermsofthespinbub- 0k
U2 = [a (1 + ε /E ) − a (1 + ε /E )]/2,
ble, and have been given explicitly in Ref. 39. 1k 1k k 1k 3k 0k 1k
In the previous discussions39, we on the other hand V12k = [a1k(1 − εk/E1k) − a3k(1 − ε0k/E1k)]/2,
U2 = −[a (1 + ε /E ) − a (1 + ε /E )]/2, and
have shown that the momentum dependence of the 2k 2k k 2k 3k 0k 2k
V2 = −[a (1 − ε /E )− a (1 − ε /E )]/2, with
pseudogap is directly related to the electron self-energy 2k 2k k 2k 3k 0k 2k
a =∆¯2 (k)/(E2 −E2 ).
Σ1(k,ω) in the particle-hole channel as, 3k PG 1k 2k
Substituting the Green’s function (10) into Eqs. (5)
Σ (k,ω)≈ [∆¯PG(k)]2, (9) and (1), the ERR function S˜(ω) is therefore obtained
1 ω+ε explicitly as,
0k
where ε0k = L(2e)(k)/L(1e)(k), ∆¯PG(k) is so-called as S˜(ω) = [1+n (ω)] 2 γα γα {L(1)(k)
the pseudogap, and can be obtained as, ∆¯ (k) = B N 1k 2k µν
PG Xkµν
L(2e)(k)/qL(1e)(k), while L(1e)(k) = −Σ1o(k,ω = 0) and × [A(µ1ν)(k)δ(ω+Eνk−Eµk)
L(e)(k) = Σ (k,ω = 0) are evaluated directly from
2 1 − A(2)(k)δ(ω−E +E )]
Σ (k,ω) given explicitly in Ref. 39. µν νk µk
1
With the help of the self-energy Σ1(k,ω) in Eq. (9), + L(µ2ν)(k)[A(µ3ν)(k)δ(ω−Eνk−Eµk)
we therefore obtain the single-electron diagonal and off-
− A(4)(k)δ(ω+E +E )]}, (11)
diagonal Green’s functions in Eq. (8) explicitly as, µν νk µk
G(k,ω) = Uν2k + Vν2k , (10a) where L(µ1ν)(k) = nF(Eνk) − nF(Eµk), L(µ2ν)(k) = 1 −
(cid:18)ω−E ω+E (cid:19) n (E ) − n (E ), A(1)(k) = U2 U2 − Ξ (k),
νX=1,2 νk νk F νk F µk µν µk νk µν
ℑ†(k,ω) = νX=1,2aν2kE∆¯ν(kk)(cid:18)ω+1Eνk − ω−1Eνk(cid:19),(10b) aAΞµ(µ2νν)a((kk)),∆¯2A=((µk4ν))V/(k(µ24k)EV=ν2kE−Vµ2kΞ)U.µνν2k(k−),ΞAµν(µ3(ν)k()k,)an=d ΞUµµ2νk(Vkν2)k −=
µk νk µk νk
where a = (E2 − ε2 )/(E2 − E2 ), We are now ready to discuss ERR of cuprate super-
1k 1k 0k 1k 2k
4
conductors in the SC-state. In the following discussions,
we only focus on the low-energy features related to the
anisotropicsuppressionofthelow-energyspectralweight
by the momentum dependence of the pseudogap. As a
comparison of the results between the cases of the pres-
enceandabsenceofthe pseudogap,wefirstlydiscussthe
case of the absence of the pseudogap, i.e., ∆¯ = 0. In
PG
this case, the ERR function S˜(ω) in Eq. (11) is reduced
as,
1 ∆¯2(k)
S˜(0)(q,ω)=[1+n (ω)] γα γα
B 2N Xk 1k 2k Ek(0)2
1
× th βE(0) [δ(ω−2E(0))−δ(ω+2E(0))], (12)
(cid:18)2 k (cid:19) k k
with E(0) = ε2+|∆¯(k)|2. We have performed a cal-
k k
q
culation for the ERR function (12) in both B and B
1g 2g
orientations, and the results of the B and B spectra
1g 2g
(dash-dotted line) at doping δ = 0.15 with temperature
T = 0.002J are plotted in Fig. 1a and Fig. 1b, respec-
tively,wherebothB andB spectraarecharacterized
1g 2g
clearly by the presence of the pair-breaking peaks, how-
ever,the peakinthe B spectrumislocatedaroundthe
1g
energyω =2∆¯,whiletheintensityoftheB spectrumis
2g
weakerthantheB oneandthepeakpositionislocated FIG.1: (a)B1g and(b)B2g spectra(solidline)asafunction
1g
at a lower energy3–13. of energy at δ = 0.15 with T = 0.002J for t/J = 2.5 and
′
t/t = 0.3. The dotted lines are a cubic and a linear fits
However, when the pseudogap effect is included in
for the low-energy B1g and B2g spectra, respectively. For
terms of the electron self-energy Σ (k,ω), the spectral
1 comparison, the corresponding results of the B1g and B2g
weight of the ERR spectrum is suppressed. To see this spectra in the case of the absence of the pseudogap (dash-
point clearly,we also plot the results of the B1g and B2g dotted lines) are also shown in (a) and (b),respectively.
spectra (solidline) incase of the presence ofthe pseudo-
gap ∆¯ obtained from the Eq. (11) at δ = 0.15 with
PG
T = 0.002J in Fig. 1a and Fig. 1b, respectively. In
of calculations for the B spectrum at different doping
comparisonwiththecorrespondingresultsintheabsence 2g
levels, and the results show that the B spectrum has a
of the pseudogap (dash-dotted line), we therefore find 2g
domelike shape of the doping dependence, and actually
thatalthoughthelow-energyspectralweightoftheERR
scales with T throughout the doping range.
spectrum in the both B and B symmetries has been c
1g 2g
redistributed, the low-energy spectral weight in the B The polesofthe electronGreen’sfunction (8a)atzero
1g
channelissuppressedheavilybythepseudogap,whilethe energy determine directly EFS in momentum space,and
spectral weight in the B channel is reduced lightly, in the everything on the other hand happens at EFS. An
2g
qualitative agreement with the experimental data14–18. explanation of the anisotropic suppressions of ERR in
In particular, we have also calculated the energy depen- cuprate superconductors in the SC-state can be found
dence of the ERR functions up to higher energies, and from the electron self-energy Σ (k,ω) in Eq. (9) in the
1
the results show that the redistribution of the spectral particle-hole channel, where the momentum dependence
weightinducedbythepseudogapleadstoatransferofthe ofthepseudogapisrelatedexplicitlytotheelectronscat-
missing low-energy spectral weight to the higher-energy tering as ImΣ (k,ω) ≈ 2π[∆¯ (k)]2δ(ω+ε ), also re-
1 PG 0k
region8,14–18. Moreover, a numerical fit has been made flecting a fact that the product of [∆¯ (k )]2 and the
PG F
to the low-energy data, and the results show that in the delta function δ(ε ) has the same momentum depen-
0kF
depleted low-energy region of the B spectrum, the in- dence on EFS as that of |ImΣ (k )|. To show the mo-
2g 1 F
tensity riseslinearlywithenergyω,while the low-energy mentumdependenceofthepseudogaponEFSclearly,we
spectrumvaries asω3 in the B channel(see the dotted plot the angular dependence of the |ImΣ (k,0)| on EFS
1g 1
line inFig. 1),whicharealsoconsistentwith the experi- atδ =0.15withT =0.002J inFig. 2,where|ImΣ (k,0)|
1
mental data3–18. On the other hand, these results are in [then the pseudogap ∆¯ (k )] is strongly anisotropic
PG F
a striking contrast to the conventional superconductors, in momentum space. The most striking feature is that
wheretheERRspectruminthelow-energyregimeforan the actualminimum of|ImΣ (k,0)| [then the pseudogap
1
isotropics-wavegapischaracterizedbytheexponentially ∆¯ (k )] does not appear around the node, but locates
PG F
activated behavior. Furthermore, we have made a series exactly at the hot spot k , and then the charge-order
HS
5
state is driven by the Fermi-arc instability, with a char- In conclusion, within the framework of the kinetic-
acteristic wave vector corresponding to the hot spots of energy-drivenSCmechanism,wehavestudiedpseudogap
EFS40–46. This pseudogap opening around the node is effect on the ERR spectrum in cuprate superconductors
also consistent with the experimental observations18,47. in the SC-state by taking into account the interplay be-
On the other hand, the magnitude of |ImΣ (k,0)| [then tweenthe pseudogapandSC gap. Our results showthat
1
the pseudogap ∆¯ (k )] still exhibits the largest value the low-energyB spectrum depends linearly onenergy
PG F 2g
aroundtheantinode,andthenitdecreaseswiththemove ω,whilethelow-energyB spectrumdisplaysacubeen-
1g
of the momentum away from the antinode. In particu- ergy dependence. In particular, the pseudogap is strong
lar, the magnitude of |ImΣ (k,0)| [then the pseudogap momentumdependent,wherethemagnitudeofthepseu-
1
∆¯ (k )] around the node is smaller than that around dogap around the nodes is smaller than that around the
PG F
the antinode. This special momentum dependence of antinodes. Thisspecialstructureofthepseudogapthere-
|ImΣ (k,0)| [then the pseudogap ∆¯ (k )] therefore fore leads to that the low-energy spectral weight in the
1 PG F
suppresses heavily the low-energy spectral weight of B B orientation is suppressed heavily by the pseudogap,
1g 1g
spectrum, but has a more modest effect on ERR in the while the pseudogap has a more modest effect on ERR
B channel. in the B channel. Our theoretical results are in quali-
2g 2g
tative agreement with the experimental data.
Acknowledgements
The authors would like to thank Deheng Gao and
Yinping Mou for helpful discussions. PJ, YL, LK,
and SF are supported by the National Key Research
and Development Program of China under Grant No.
FIG. 2: The angular dependence of the pseudogap on the 2016YFA0300304, and National Natural Science Foun-
electronFermisurfaceatδ=0.15withT =0.002J fort/J = dationofChina(NSFC)underGrantNo. 11574032,and
2.5 and t′/t=0.3. HZ is supported by NSFC under Grant No. 11547034.
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