Table Of ContentNonequilibrium enhancement of high-temperature superconductivity in a 3D model of
cuprates
Zachary M. Raines and Valentin Stanev
Joint Quantum Institute and Condensed Matter Theory Center, Department of Physics,
University of Maryland, College Park, Maryland 20742-4111, USA
Victor M. Galitski
Joint Quantum Institute and Condensed Matter Theory Center, Department of Physics,
University of Maryland, College Park, Maryland 20742-4111, USA and
School of Physics, Monash University, Melbourne, Victoria 3800, Australia
5
(Dated: January 29, 2015)
1
0 Recentexperimentsinthecuprateshaveseenevidenceofatransientsuperconductingstateupon
2
optical excitation polarized along the c-axis [R. Mankowsky et al., Nature 516, 71 (2014)]. Moti-
n vatedbytheseexperimentsweproposeanextensionofthesingle-layert−J−V modelofcuprates
a to three dimensions in order to study the effects of inter-plane tunneling on the competition be-
J tween superconductivity and bond density wave order. We find that an optical pump can suppress
7 thechargeorderandsimultaneouslyenhancesuperconductivity,duetotheinherentcompetitionbe-
2 tweenthetwo. Wealsoprovideanintuitivepictureofthephysicalmechanismunderlyingthiseffect.
Furthermore, based on a simple Floquet theory we estimate the magnitude of the enhancement.
]
n
o I. INTRODUCTION planes17,18, which can naturally support the coexis-
c
tence and competition between charge ordering and
-
r Eversincethediscoveryofhigh-temperaturesupercon- superconductivity19–21. Furthermore, we focus on the
p
u ductivityincupratematerials1,ithaspresentedafunda- low energy physics of fermions near the so-called ‘hot
s mental challenge to our understanding of strongly inter- spots’, where the Fermi surface intersects the magnetic
t. acting systems. Despite the decades of strenuous efforts Brillouin zone22,23. We extend the model by introducing
a these materials still remain a profound mystery. In the an effective Hamiltonian, describing stacked planes, cou-
m
lastseveralyear,however,newexperimentalresultshave pled by a c-axis tunneling term tz. Then we investigate
- shone some light on one of the most puzzling phases of the phase diagram of this extended model by utilizing a
d
n cuprates – the pseudogap2. Clear evidence of some sort Landau expansion of the free energy. Quite surprisingly,
o ofchargeorderingcompetingwithsuperconductivityhas weobserveanon-monotonicbehaviorofthecriticaltem-
c emerged3,4. peratures of the two orders with increasing tz, and we
[ Recently, in a series of exciting papers, it has been provideanintuitivephysicalexplanationofthisinterest-
1 demonstrated that optical pumping can significantly ing feature. Finally, we consider different effects of the
v enhance the superconductivity in several hole-doped photo-excitations of the system, particularly focusing on
2 cuprates5–8 (for a brief summary see Ref. 9). In these the role of the apical oxygens, which are thought to play
2 works, a sample was subjected to a short pulse of mid- a key role in the experiments6,7,14. We find that, quite
9 infrared light, polarized along the c-axis. Reflectivity generally, there is a parameter region where an optical
6 measurements were then taken as a function of time de- pump can lead to a melting of the charge order and a
0
lay, from which the frequency dependent conductivity corresponding enhancement in superconductivity, due to
.
1 was extracted. There is an increase in it that the au- the competition between the two orders.
0 thors identify with an inhomogeneous and transient en- As a side note, let us add that the t−J −V model
5
hancement of superconductivity. Furthermore, the na- and its variants tend to have as their leading instabil-
1
: ture of this enhancement seems to be different from the ity a (Q,Q) type ordering vector19,20,24, with the (Q,0)
v already well known effect due to quasiparticle photo- ordering vector seen in experiment4 as a sub-leading in-
i
X excitation7,10–13. Several works have investigated these stability (withboth ordershavingpredominantly d-wave
experiments and proposed an increase of inter-layer cou- symmetry). While several extensions of the model have
r
a pling as one of the dominant effects14,15. However, there been proposed as a way to stabilize the experimentally
is another possible effect to consider, since resonant x- observed order21,25,26, they introduce additional, and for
ray spectroscopy seems to suggest that the charge order our purposes unnecessary, complications. In order to
is suppressed in the same region of the phase diagram keep the model simple, we restrict our attention to the
where superconductivity is enhanced in the optical exci- instabilityatwavevector(Q,Q). Thephysicalcontentof
tation experiments16. A natural question to ask, then, our results lead us to expect that the qualitative behav-
ishowcoexistingsuperconductivityandchargeorderbe- ior of the effects would be similar for the experimentally
have under such perturbation. relevant (Q,0) charge order.
To address this problem we start from the t − The structureof the paperis as follows. First, we pro-
J − V model of the quasi two dimensional CuO videashortdescriptionofthetwo-dimensionalt−J−V
2
2
modelandtheassociatedLandautheoryinSectionII.In
Section III we extend this model to include c-axis tun-
neling, and explore the effects of t on the phase dia-
z
4
gram. Time-dependent perturbation of t is introduced
z
inSectionIV,andthehigh-frequencylimitisstudied. In
Section V we summarise and discuss our findings.
1
II. TWO-DIMENSIONAL MODEL
2 2
In order to study the interplay of bond density wave
(BDW) and superconducting orders we employ a 2D
t − J − V model of a CuO plane17–19,21. It provides
2
a natural platform for exploring the general features of 3
the interaction and the coupling between these two or-
derswithinasinglecopperoxideplane. TheHamiltonian
is FIG. 1. Hotspots exist where the Fermi surface intersects
the magnetic brillioun zone boundary. The charge ordering
H =(cid:88)t c† c + 1 (cid:88)JS(cid:126) ·S(cid:126) + 1 (cid:88)Vn n , (1) vectors are given by the separations between hotspots 1 and
ij σ,i σ,j 2 i j 2 i j
2 accross the Brillouin zone border and the rotated vector
i,j (cid:104)i,j(cid:105) (cid:104)i,j(cid:105)
(Q,−Q).
where V and J are nearest neighbor interactions,
n = (cid:80) c† c is the charge density, and S(cid:126) =
i σ i,σ i,σ i
1(cid:80) c† (cid:126)σ c isthesitespindensity,withiandjbeing d-waveformfactor,asthiswasfoundtobethestrongest
2 σ i,a ab i,b mean-field instability19,23. If we further enforce time re-
site indices, and σ, a, and b are spin indices. The term
versal symmetry this allows us to concentrate our atten-
t contains nearest, next to nearest, and next to next
ij tion to four hotspots. Having restricted our fermions to
to nearest hopping27. V describes the nearest-neighbor
live within a range Λ of the hotspots we obtain an effec-
tail of the Coulomb repulsion, which tends to suppress
tive Hamiltonian
the d-wave superconductivity and enhance the BDW or-
der. J is the usual nearest neighbor anti-ferromagnetic (cid:88)
H = ξ c† c
exchange interaction. i,(cid:126)k i,(cid:126)k,σ i,(cid:126)k,σ
Of course, the pure t−J model has been extensively (cid:126)k,i
used in the studies of cuprates as an effective one-band +gabcd(cid:88)(cid:104)c† c c† c
description of the CuO planes28. It naturally leads to 1,(cid:126)k,a 2,(cid:126)k,d 4,p(cid:126),c 3,p(cid:126),b
a d-wave superconducti2vity as its dominant instability. (cid:126)k,p(cid:126)
(cid:105)
In order to have a region where superconductivity and −c† c† c c , (2)
1,(cid:126)k,a 2,−(cid:126)k,c 4,−p(cid:126),d 3,p(cid:126),b
BDW order coexist the model can be extended by the
introduction of V, which suppresses superconductivity where
andbooststhechargeorder–thisistherationalebehind
the t−J −V model. (cid:88)···(cid:39)(cid:90) Λ (cid:90) Λ d2k ···,
We now construct a low-energy effective model by re- −Λ −Λ (2π)2
(cid:126)k
strictingourattentiontofermionslivingwithinalimited
region surrounding ‘hotspots’ – points where the Fermi the interaction is
surface intersects the magnetic Brillioun zone bound-
1
ary, as depicted in Fig. 1 (such models have been re- gabcd =−4JK(cid:126)(cid:126)σab·(cid:126)σcd−VK(cid:126)δabδcd, (3)
cently introduced and used in a number of studies of
charge order in cuprates19,20,29,30). These points are (cid:126)k is now the deviation from the hotspot, a−d are the
of special interest because the Fermi surface is nested electron spin indices, and i is now a hotspot index.
with wave-vector K(cid:126) = (π,π), and given the importance At this point we undertake a mean field decomposi-
of anti-ferromagnetic spin fluctuations to pairing in the tionoftheinteraction(four-fermion)termsoftheHamil-
cuprates19,31,32,weexpectthatthemostrelevantinterac- tonian simultaneously in the BDW and superconducting
tions will be those with exchanged momentum K(cid:126). Close channels,wheread-waveformfactorisassumedforboth
tothehotspotswecanreplacetheinteractionsJ ,V with orders, by defining
q(cid:126) q(cid:126)
constants J ,V , and restrict the two unconstrained
fermion momK(cid:126)entK(cid:126)a (cid:126)k −(cid:126)k(cid:48) = (cid:126)q to lie in hot regions sep- iσayb∆¯ =− gs(cid:88)(cid:104)c†1,(cid:126)k,ac†2,−(cid:126)k,b(cid:105)
arated by K(cid:126). (cid:126)k (4)
(cid:88)
Furthermore, we only consider a charge ordering in- = g (cid:104)c† c† (cid:105),
s 3,(cid:126)k,a 4,−(cid:126)k,b
stability with a diagonal (Q,Q) ordering vector and a
(cid:126)k
3
and with w,u ,u > 0. The exact expressions for the
BDW SC
coefficients are given in Table I. It is in fact possible to
(cid:88) (cid:88)
φδ =−g (cid:104)c† c (cid:105)=g (cid:104)c† c (cid:105), (5) solve the mean field problem exactly via a sequence of
ab c 1,(cid:126)k,a 2,(cid:126)k,b c 3,(cid:126)k,a 4,(cid:126)k,b
(cid:126)k (cid:126)k Bogoliubov transformations. This method breaks down,
however, once we introduce c-axis hopping, and so, in
where anticipation of this extension of the model, we have cho-
seninsteadtoworkwithaLandauexpansion, whichwill
1 (cid:88)
g = gabcd(iσy )(iσy )=3J −4V carry over to the more complicated case.
s 2 ac db
a,b,c,d
(6)
1 (cid:88) Term Expression
gc = 2a,b,c,dgabcdδadδbd =3J +4V, ΠΠSC 2(cid:82)(cid:82)(cid:126)kn(1ξ−2,2Tnξ)(1−ξ1n,(Tξ)1,T)
achreartgheecheffanecnteilvserceospuepcltinivgeslyi,natnhde(cid:104)s·u··p(cid:105)erincodnicdautcetsinagthaenrd- uuSBCDW (cid:82) 1 (cid:104)(cid:82)n(cid:126)k(cid:48)(2ξξ112,(cid:104)T(cid:126)kn)(cid:48)+(ξ,nT(cid:48)ξ()1ξ−+,ξ2T1−)2+2nξ(12ξ,nT()ξ(cid:105)2,T)−n(ξ1,T)(cid:105)
mal average. The order ∆ is the projection of uni- wBDW (cid:126)k (ξ1−ξ22)(cid:82)2 11 (cid:104)n(cid:48)(ξ ,2T)+ 1−2n(ξ1,ξT1)−(cid:105)ξ2
form d-wave superconductivity onto the hotspots, and (cid:126)k ξ1(ξ1−ξ2) 1 2ξ1
φ describes a d-wave charge order, that lives on the
TABLEI.MicroscopicexpressionsforcoefficientsoftheLan-
bonds between copper sites, and has modulation vec-
dau theory. n is the Fermi function and n(cid:48) the derivative of
tor Q(cid:126) given by the separation between hotspots. It is
the Fermi function with respect to energy.
clear that V enhances superconductivity and simultane-
ously suppresses charge order. Because of the d-wave
form factor of the order parameters, two of the remain-
ing hotspots become redundant and at the mean field
level the behavior of the system may be described by a 3.0
4×4 Hamiltonian in hotspot-Nambu space. Defining a φ>0
(cid:16) (cid:17)T ∆>0
Nambu spinor ψ = c ,c ,c† ,c† the
(cid:126)k 1,(cid:126)k,↑ 2,(cid:126)k,↑ 2,−(cid:126)k,↓ 1,−(cid:126)k,↓
Hamiltonian takes the form
2.5
H=(cid:88)ψ†Hˆ ((cid:126)k)ψ + 2 |∆|2+ 2 |φ|2, (7)
(cid:126)k MF (cid:126)k g g
s c
(cid:126)k V)
me2.0
where (T
ξ ((cid:126)k) φ¯ ∆ 0
1
HˆMF((cid:126)k)= ∆φ¯ ξ20((cid:126)k) −ξ0((cid:126)k) −∆φ¯ , (8) 1.5
1
0 ∆¯ −φ −ξ ((cid:126)k)
2
1.0
and ∆ and φ are the superconductivity and BDW order 0 50 100 150 200
parameters respectively. V (meV)
Bothorderparametersaregenerallycomplexnumbers,
whichwecanwriteas∆=|∆|eiθ∆ andφ=|φ|eiθφ. How- FIG. 2. (Color online) Phase diagram of the 2D model27, for
ever, since we can always remove the complex phases fixed J. The regions are from left to right, SC, BDW + SC,
(at the mean-field level) via a gauge transformation33 andBDW,withthewhiteregionatthetopbeingthenormal
c1 → c1e−i(θ∆−θφ)/2, c2 → c2e−i(θφ+θ∆)/2, we will con- pTh=ase0.aNsothteetLhaantdtahuettheemorpyerisatnuortevdaoliedsinnottheixstleimndit.down to
sider only real and non-negative values for φ and ∆ in
our analysis.
From Equation 7 we can readily derive a Landau In what follows we hold J fixed and use V to adjust
free energy for the ∆ and φ orders. Evaluating f = the splitting between the superconducting and BDW in-
f +1 (cid:104)H −H (cid:105) ,usingtheabovedecouplingand stabilities. Doing so, we obtain a phase diagram such
exMpFandNing to fouMrtFhMorFder in the order parameters, we as that in Figure 2. In it we see the following easy-to-
obtain understand behavior: when V is zero or very small the
d wave superconductivity is the only relevant instability
(cid:20) 2 (cid:21) (cid:20) 2 (cid:21) of the model. With V increasing, superconductivity is
f = g −ΠBDW φ2+ g −ΠSC ∆2 suppressed and eventually BDW appears as the leading
c s
order. Notably,intheregionwheretheseinstabilitiesare
+u φ4+u ∆4+wφ2∆2, (9)
BDW SC comparablethereisa(rathernarrow)coexistencephase.
4
Thesefeaturesareconsistentwithpreviousstudiesofthe within the same plane. As a consequence of this, the or-
model19. der parameters do not depend on k and the vector Q(cid:126)
z
We expect this phase diagram to be strictly valid only which separates fermions contributing to pairing in the
close to T and in the region in which the critical tem- charge channel cannot change with k . Because of this
c z
peratures of both orders are comparable. However, com- restriction, we find that while at k =0 superconductiv-
z
paring the Landau expansion with the exact numerical ity and BDW pair the same points in the 2D Brillioun
mean-fieldsolutionshowsthatevenforintermediatetem- zone, this is no longer true for k (cid:54)= 0, t (cid:54)= 0. The
z z
peratures there is only a small, purely quantitative cor- particleandholebeingpairedintheBDWchannelmust
rection to the shape of the coexistence region. remain separated by Q(cid:126) = (Q,Q) even when the Fermi
surface is no longer nested with this vector, as can be
seen in Figure 3. Superconductivity on the other hand
III. EXTENSION TO STACKED PLANES continues to pair(cid:126)k and −(cid:126)k for all k . As a consequence,
z
one has to be careful to define the hot regions properly
Motivatedbytheexperimentswhichhaveusedoptical inthiscase. Thereareseveralwaysonemightdoso, but
excitation polarized along the c-axis to create transient fortunately, as shown in the appendix, they all lead to
states of enhanced superconductivity6–8, we seek to ex- the same qualitative behavior.
tendthepurelytwo-dimensionalmodelfromtheprevious That being the case, we implement the following pro-
sectiontoincludecouplingbetweenplanes. Thesimplest cedure. At each k there is a region centered on where
z
modelthatencapsulatesthisbehaviorcanberepresented the 2D Fermi surface intersects the 2D magnetic Bril-
by the following Hamiltonian: lioun zone. φ now pairs quasi-particles separated by the
fixed charge ordering vector and is only non-zero when
(cid:88)
H = H +H . (10) the momenta of both fall within a hot region. With this
3D l T
procedure in place, the free energy takes the same form
l
as Equation 9 but with the coefficients now being the
Here l is a layer index, and Hl is the single-layer Hamil- three dimensional integrals shown in Table III. Here, we
tonian considered in Equation 1. There are Nz copies of havemadethesimplifyingassumptionsthat∆andφare
those, which are coupled via a c-axis tunneling term not modulated along the c-axis. As can readily be seen,
the expressions in Table III reduce to those in Table I in
HT = (cid:88) c(cid:126)†k(cid:107),kzσtz((cid:126)k(cid:107),kz)c(cid:126)k(cid:107),kz,σ, (11) the limit of no c-axis hopping.
(cid:126)k(cid:107),kz
where c-axis momentum k is conjugate to the plane in-
z
dex l.
In what follows we will consider and compare three 1 kz=0
different forms for the tunneling t in Eq. 11. The exact
expressions for each tunneling typze are presented in Ta- 1 kz=π
bleII. TypeAtunneling(nearestneighborhoppingalong
the c-axis) we introduce mainly for its simplicity, type B Q
comes from a one-band tight binding fit to band struc-
ture calculations for LSCO34, and type C was proposed
as an approximate tunneling form for several families of 2
cuprate superconductor35. Despite the significant differ-
ences between these tunneling forms, it turns out that 2
the effects we obtain are not specific to any of them, but
are in all cases qualitatively similar.
Type c-axis tunneling FIG.3. c-axishoppingcausesawarpingoftheFermisurface
BA34 −2tzcos(cid:0)k2z(cid:1)(cosk−x2−tzccoosskkyz)2cos(cid:0)k2x(cid:1)cos(cid:16)k2y(cid:17) wlehadicshtocaausdeessttrhuecthioontspofotQ(cid:126)s ntoesmtinogveawasayafrfuonmctkiozn=o0f.kz and
C35 −2t cos(k )(cosk −cosk )2
z z x y
Using each of the three tunneling forms, we calculated
TABLE II. c-axis tunneling elements used in the calculation the state which minimized the free energy for a range
of T, V and t . The (Q,Q) order remains the leading
z
We now retrace the same steps as in section II. The chargeinstabilityatthequadraticlevel,soweagainonly
derivation proceeds similarly, but there are some sub- decouple in this channel and the d-wave superconduct-
tleties that need to be considered first. ing channel. The phase diagram for tunneling type B is
Due to the model containing solely in-plane interac- presented on Fig. 4 (the other two types lead to quali-
tions, we only need to consider pairing of quasi-particles tatively similar diagrams). We start (for t = 0) with
z
5
Term Expression
Π (cid:82) 1−2n(ξ1,T)
ΠSC 2(cid:82)(cid:48) n(cid:126)k(ξ1−Q,ξT1)−n(ξ1,T)
uBDW (cid:82) 1 (cid:126)k(cid:104)n(cid:48)(ξ,ξT1−)ξ+1−1Q−2n(ξ,T)(cid:105)
SC (cid:126)k 2ξ2 2ξ
u (cid:82)(cid:48) 1 (cid:104)n(cid:48)(ξ ,T)+n(cid:48)(ξ ,T)+2n(ξ1−Q,T)−n(ξ1,T)(cid:105)
wBDW (cid:126)k (ξ1−ξ1−Q2)(cid:82)2(cid:48) 11 (cid:104)n(cid:48)(1ξ−,QT)+ 1−2n(ξ1,ξT1)−(cid:105)ξ1−Q
(cid:126)k ξ1(ξ1−ξ1−Q) 1 2ξ1
TABLE III. Microscopic expressions for the coefficients of the Landau theory. ξ is the energy at a point at the point
1−Q
(cid:126)k(cid:48) =(cid:126)k+(cid:126)k −Q(cid:126). Primed integration indicates a restriction of the integral to regions where both (cid:126)k and k(cid:126)(cid:48) lie within a hot
HS
region. The k integration is from −π to π.
z
the coexistence case in which BDW is the leading insta- fect on superconductivity through the biquadratic term
bility. As we can see, for small but finite t the charge in the free energy.
z
orderisfurtherenhancedattheexpenseofsuperconduc- Thenon-monotonicbehaviorofthechargesusceptibil-
tivity. However, once t becomes sufficiently large, the ity Π can be understood as a competition of two
z BDW
tendency reverses, and superconductivity is boosted by effects. First, it is well known within this model that
the increase of three-dimensionality, until it becomes the in the absence of Fermi surface curvature, the diagonal
leading instability. BDWorderexhibitsaBCStypetransitioncharacterized
by a logarithmic divergence. Curvature however cuts off
the divergence of the logarithm and weakens the charge
instability19,23,36. Specifically, we can write the disper-
4.0 sions as
φ>0
∆>0 ξ =ξ +ξ (12)
1 + −
3.5
ξ =ξ −ξ (13)
2 + −
In that case the charge susceptibility becomes
3.0
(cid:90) sinhξ− coshξ−
V) Π = 2T 2T (14)
(meT 2.5 BDW (cid:126)k ξ−[sinh2 2ξ+T +cosh2 2ξ−T]
inthe2Dlimit. Onecanseeclearlythatinthelimitthat
2.0
ξ → 0, the absence of curvature, the charge (particle-
+
hole) instability becomes as strong as the superconduct-
ing (particle-particle) one, and for any non-zero ξ the
1.5 +
charge susceptibility is weaker compared to that of the
BCS case.
1.00 10 20 30 40 50 As the c-axis hopping tz increases the 2D curvature
decreases. This is because the k = 0 component for
t (meV) z
z all tunneling types is of the form −t η((cid:126)k )c† c , with
z (cid:107) (cid:126)k(cid:107) (cid:126)k(cid:107)
FIG. 4. (Color online) Phase diagram of a system of stacked η >0. Nearthehotspotsthisactsasaneffectiveupward
planesatafixedcouplingstrengthfortunnelingtypeBinTa- shiftinthechemicalpotential,whichmovesthehotspots
ble II. Types A and C exhibit qualitatively similar behavior, such that ξ is decreased relative to ξ . As a result,
+ −
but with different scales on the tz axis. The inset indicates for increasing tz, the kz = 0 Fermi surfaces changes in
where the starting point at tz = 0 lies in V −T plane of suchawayastoeffectivelyenhancethechargeinstability.
Figure 2 (V =110meV).
Fundamentally, this is a consequence of the effect of the
tunneling term on the properties of the Fermi surface of
To understand this peculiar shape of the phase dia- each plane.
gram,welookatthequadraticcoefficientsinthefreeen- In opposition to the aforementioned effect, increasing
ergy. The behavior of the superconducting coefficients, t will lead to a progressive destruction of nesting away
z
Π and u , can be explained entirely by the depen- from k = 0, as depicted in Figure 3. For t = 0, the
SC SC z z
denceofthedensityofstatesandbandwidth,ρ(ξ,t )and Fermi surface is nested at the hotspots for all k . How-
z z
W(t ). As can be seen in Figure 5, these lead to only a ever, as t increases the Fermi surface warps more with
z z
small effect on the superconducting susceptibility. Thus, k , decreasing the portion of the phase space for which
z
thefeaturesvisibleinFigure4aremostlydeterminedby there is approximate nesting, and thus weakening the
the behavior of the charge instability and its indirect ef- charge susceptibility.
6
For small t the warping of the Fermi surface is small, enhancements of t will generally lead to suppression of
z z
andsothe2Ddecreaseofcurvatureisnearlythesamefor charge order and enhancement of superconductivity.
a wide range of k values, leading to an overall strength- It should be noted that the effects demonstrated here
z
ening of the charge instability. However, as t increases are obtain within the grand canonical ensemble at fixed
z
Fermi surface warping along the c-axis becomes more chemical potential. In anticipation of section IV the ex-
pronounced. Asaconsequence,theavailablephasespace perimentalprogramweenvisioncomprisesplacingasam-
for charge ordering is substantially reduced, while at the ple in contact with its environment, which functions as
same time the significance of the decreased curvature is aparticlebathmaintainingµ,andthenapplyingpertur-
lessened away from k = 0, together eventually caus- bations that will lead to a change of t in the effective
z z
ing a weakening of the charge ordering instability. The Hamiltonian. This, however, means that the volume en-
non-monotonic shape of the phase diagram can conse- closedbytheFermisurfaceisnotconstantandtherefore
quently be understood as demonstrating a crossover be- averageparticlenumberisnotconserved. Ifonewerein-
tween regimes in which the 2D and the 3D effects of t stead to consider a system where such a constraint were
z
dominate. important this might change the behavior of the phase
diagram at small t , where the effects are largely gov-
z
erned by the position of the k =0 Fermi surface. How-
1.020 z
ever, larger t should still lead to Fermi surface warping
z
and thus the enhancement of superconductivity in that
regime should remain. Therefore, while the 2D effects
that are important predominantly at small t could be
z
washed away, the suppression of charge order (via de-
1.000 struction of nesting), and thus the enhancement of su-
perconductivity, appears much more robust.
Π (t )/Π (0)
SC z SC
Π (t )/Π (0)
CDW z CDW
0.985 IV. ENHANCEMENT OF
0 50 SUPERCONDUCTIVITY VIA PERIODIC
tz (meV) MODULATION
(a)QuadraticSusceptibilities
Very recently, it has been argued that in YBCO the c-
axisvibrations,inducedbytheexternalfield,canchange
1.2
the equilibrium lattice structure14. The result is a tran-
sientshiftoftheCuO planes–theintra-bilayerdistance
2
increases, while the inter-bilayer one decreases. In our
model this would lead to effectively bringing the layers
1.0 closer. Intuitively, it is clear that this should lead to en-
hancementoftheinter-planecouplingt . Inthistherole
z
uSC(tz)/uSC(0) of the apical oxygen seems quite significant; previous ex-
u (t )/u (0) perimentalworkshavefoundthatthereisarangeofdop-
CDW z CDW
w(t )/w(0) ings for which the inter-layer hopping exhibits a roughly
z
0.8 exponential dependence on the hole-doping37 while the
0 50 bond length between the plane copper and the apical
tz (meV) oxygen exhibits a roughly linear dependence on doping
(b)QuarticTerms through the same region38. A natural interpretation is
that the hopping exhibits an exponential dependence on
FIG. 5. (Color online) Quadratic susceptibilities and quar- an effective barrier width d:
tic coefficients of the Landau free energy as a function of t ,
z
scaledbytheirvalueattz =0,fortypeBtunneling(atfixed t =Ae−αd, (15)
z
temperature). TypesAandCagainexhibitsimilarbehavior.
Note that there is no significance to the crossing of lines for where d is approximately linear in the distance between
different coefficients, as the leading instability will be deter- the plane and the apical oxygen37,39. Thus, decreasing
mined by gΠ.
theinterlayerdistanceleadstoanexponential increaseof
the c-axis tunneling.
Interestingly, the value of t for which superconduc- There is a second, more subtle, way in which driving
z
tivity reaches a minimum is of similar magnitude to the the apical oxygens can enhance the interlayer tunneling.
strengthofc-axishoppingforvariousfamiliesofcuprates Let us consider a vibration of these ions without change
obtained as a fit to band structure34. Given this, if we of their equilibrium positions. Then, in line with the
imagine a system where t is near or greater than the above reasoning, we can model the effect of harmonic
z
point of minimum superconducting T in Figure 4, then oscillations of the apical oxygen on t as oscillations of
c z
7
the effective barrier width, leading to a time dependent V. DISCUSSION AND CONCLUSION
hopping element
The primary motivation for this investigation came
t (t)=Ae−αd(t) =t exp[−αd cos(Ωt)], (16) from the recent experiments on transient enhancement
z z0 1
of superconductivity in the cuprates via mid-infrared
optical excitations6–8,14. To model these experiments
where d(t)=d0+d1cos(Ωt), and tz0 =Ae−αd0. we considered an extension of the t−J −V model of
cupratestothreedimensions,andtheeffectsofthisthree-
Let us consider the high frequency limit. Then, we
dimensionalityonthecompetitionbetweensuperconduc-
expect that the quasiparticles will see an effective time
tivity and bond density wave orders. We showed that
averagedHamiltonian. Whileinexperimentthefrequen-
for our extended model, increased inter-plane tunneling
cies are not extremely high, we consider this limit as
leads to a suppression of charge ordering, and a coincid-
a particularly simple case, from which we may extract
ing enhancement of superconductivity due to the inher-
relevant qualitative trends. More formally, we can ob-
ent competition between the two orders. The evolution
tain a Floquet Hamiltonian related to the time depen-
of charge order takes place in two steps, corresponding
denthoppings. Thestroboscopicdynamicsofthesystem
to the regions where 2D and 3D effects of increased in-
willbegovernedbythisFloquetHamiltonian, whichcan
terlayer coupling, respectively dominate. The primary
be obtained as series in 1/Ω via a Magnus expansion of
the time evolution operator40. For a high frequency os- effect of interest is that the charge instability is sensi-
tive to the c-axis curvature of the Fermi surface, which
cillation, we keep only the first term of this expansion,
destroys nesting at the charge ordering vector. This ef-
which is just the time-dependent Hamiltonian averaged
fectisgenericacrossseveraltunnelingformsproposedfor
over one period. In this case the Floquet Hamiltonian
various cuprate materials.
is the original Hamiltonian (Equation 10), but with the
modification Theseresultsprovideaphysicalpictureexplainingthe
enhancementofsuperconductivitybythedecreaseofthe
inter-bilayer distance, caused by optical excitation. We
(cid:90) 2Ωπ dt further showed that periodic oscillations of the apical
t →(cid:104)t (t)(cid:105)=Ω t (t)=t I (αd ), (17)
z z 2π z z0 0 1 oxygens(identifiedintheexperimentsasimportant)can
0
also lead to an effective increase of the inter-layer cou-
pling. Boththeseeffectsindirectlypromotesuperconduc-
whereI isthemodifiedBesselfunctionofthefirstkind.
0 tivityviaasuppressionofthecompetingchargeordering.
I is bounded below by 1, and increases monotonically
0 Webelievethatthemechanismpresentedherecouldplay
withthemagnitudeofd . Therefore,withinthisapprox-
1 a significant part in the observed enhancement of super-
imation,anyoscillationoftheapicaloxygensunavoidably
conductivity and could be useful in pursuing new ways
leads to enhancement of the effective c-axis tunneling t .
z to raise T . Other mechanisms have been proposed with
This can be easily understood from the exponential de- c
regard to these experiments: the suppression of phase
pendence of the tunneling on the apical oxygen position:
fluctuations15,42 as well as the usual enhancement of su-
inthetailoftheexponent,astrongerenhancementisob-
perconductivity due to microwave stimulation10 could
tained from decreasing the argument than the suppres-
certainly play a complementary role.
sion when increasing it at π/Ω time later. Thus, the
tunneling amplitude is, on average, enhanced. Note that At the end, let us also note that this work considers
this observation is rather general, and likely applicable onlythemeanfieldbehaviorofsuchasystem. Asiswell
well beyond the region of validity of the high-frequency known, fluctuations play an extremely important role in
approximation used above. the superconducting transition of cuprates43–45. Never-
theless, we expect that the effects discussed here on a
ByusingdatafromRefs.37and38wecanobtainanes-
mean field level will remain important in a more com-
timatetothemagnitudeofboththeoscillatoryenhance-
pletedescriptionofthesystem(enteringthroughtherel-
ment and the slower, quasi-static, effect41. Assuming an
evantenergyscales, forexample). Themeanfieldtheory
oscillation distance of d ∼ 2.2pm, as is observed in the
1 presented in this paper is only the first necessary step in
experiments discussed in Ref. 14, we find approximately
the study of these effects, and including fluctuations is
a 10−15% enhancement of t in the steady state aver-
z an important direction for future work.
age. Thisinturnmayleadtoafewpercentuptoaround
a 15% enhancement in the superconducting T depend-
c
ing where in the non-monotonic structure of Fig. 4 the
sample is before perturbation. A stronger effect is the
ACKNOWLEDGMENTS
shift of the equilibrium position of the apical oxygen. A
quasi-static shift of the apical oxygen position by 2.4pm
(again see Ref. 14) can enhance T by up to about 60%. We are grateful to J.D. Sau for enlightening dis-
c
These effects provide a qualitative picture of a possible cussions. This work was supported by DOE-BES
mechanism underlying the explanation given in Ref. 14. DESC0001911 and Simons Foundation.
8
Appendix: Low energy model away from k =0 grationsoverthesamehotregionsasthesuperconducting
z
terms, but enforce that pairing occur between particles
separated by the fixed ordering vector Q(cid:126). We label this
the moving approximation.
A final approach, and the one we used in this work, is
to take a similar tact as the moving approximation, but
to further restrict the integration such that both of the
paired particles lie in a hot region. We will call this the
truncated approximation. Ifwearecommittedtotheno-
tion that only fermions near the hotspots are important,
thisseemsthemostnaturaloftheapproximations,aswe
Q Unmoving
Moving
Truncated
W
D
C
Π
FIG. 6. Different choices of hot regions for charge integrals.
Thegraydashedlinesdenotestheunmoving hotregions. The
black dashed lines represent the moving hot regions, where 0 50
t (meV)
one must be careful to avoid double counting. Finally the z
gray-filled areas represent the truncated hot regions.
FIG.7. (Coloronline)Quadraticsusceptibilitiesasafunction
ofc-axishoppingfordifferentchoicesofintegrationregionin
As discussed in section III there are subtleties associ-
integrals involving charge order. The qualitative behavior is
ated with defining the hotspot model for stacked planes
the same for all three schemes.
with inter-plane hopping. Specifically, one has to define
theintegralsinvolvingchargeordersuchthatthesepara-
tion between paired particles remains constant with kz, are then only counting the fermions that live within the
despite the fact that the Fermi surface changes shape. true hot regions. The coefficients Π , u , and
BDW BDW
We considered three different methods of handling this w are shown for a range of t in Figure 7. As can be
z
issue. Each is depicted in Figure 6. In all cases the coef- seen, while the results differ numerically, the qualitative
ficients ΠSC and uSC are unchanged, so we only need to behavior is the same. Since the hotspot model is itself a
decidehowtoimplementtheotherthree: ΠBDW,uBDW qualitative model, we chose to adopt the truncated ap-
and w. proximation as it seemed the most in line with the spirit
The first is to define two different hot regions: one re- of the model.
mains bound to the Fermi surface and is associated with While these issues make the 3D model slightly more
superconductivity, the other corresponds to the kz = 0 complicated, the qualitative behavior of the system ap-
hotregionsforallkz andisassociatedwithchargeorder. pears to be fairly robust to their method of resolution.
Allintegralsinvolvingchargeorderaredoneovertheun- This suggests that the extended model, like its simpler
moving hotspots. We label this procedure the unmoving progenitor,canbeusedasatooltouncovergeneralphys-
approximation. ical mechanisms underlying the complicated behavior of
Another approach is to perform the charge order inte- various lattice models.
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27 In this work we used t = 430meV, t = −0.32t , t =
1 2 1 3
