Pairing Theory of Striped Superconductivity

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by  Florian Loder

Pairing Theory of Striped Superconductivity
Florian Loder, Siegfried Graser, Arno P. Kampf, and Thilo Kopp
Center for Electronic Correlations and Magnetism,
Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany
(Dated: January 19, 2011)
Stripedhigh-T superconductorssuchasLa Nd Sr CuO andLa Ba CuO nearx=1/8
c 2−y−x y x 4 2−x x 4
show afascinatingcompetitionbetweenspinandchargeorderandsuperconductivity. A theory for
thesesystemsthereforehastocaptureboththespincorrelationsofanantiferromagnetandthepair
correlationsofasuperconductor. ForthispurposewepresenthereaneffectiveHartree-Focktheory
1
incorporating both electron pairing with finite center-of-mass momentum and antiferromagnetism.
1
We show that this theory reproduces the key experimental features such as the formation of the
0
antiferromagneticstripepatternsat7/8bandfillingorthequasione-dimensionalelectronicstructure
2
observed by photoemission spectroscopy.
n
a
J
Unidirectional charge- and spin-density modulations densitywavestatewithunidirectionalchargemodulation

8
werepredicted[1]fordopedtransitionmetaloxideseven were either explored with respect to symmetry aspects
1
before their experimental discovery in layered nickelates and the nature of defects [12, 14] or its spectral proper-

] [2] and the rare-earth doped cuprate La2−xSrxCuO4 [3] ties [15].
n and eventually in La Ba CuO [4]. Stripe patterns Although striped SC states were encountered before
2−x x 4
o
emerge as a compromise between correlation driven an- [16–18], a simple microscopic model Hamiltonian which
c
- tiferromagnetism and an optimized kinetic energy gain supports a superconducting PDW groundstate with spin
r for mobile charge carriers [5]. Charge and spin stripe andchargestripeshasbeenlackingsofar. Hereweelab-
p
u textures were indeed obtained in various approximate orate on the existence of these solutions in an isotropic
s model analyses of correlated electron systems, but it has 2D pairing Hamiltonian, characterize their real- and
.
t remainedunresolvedwhichmodelsystemssustainstable momentum-space properties, and relate them to existing
a
groundstate solutions with stripes and superconductiv- experimental data.
m
ity. Here we report a pairing theory for the coexistence Our model is a tight-binding Hamiltonian H = H +
0
d- of charge and spin stripes with d-wave superconductiv- HI, where H0 = −(cid:80)i,j(cid:80)stijc†iscjs describes the hop-
n ity that results from an extension of the BCS theory of ping motion of free electrons on a square lattice. The
o superconductivity with an attractive pairing interaction operator c (c† ) annihilates (creates) an electron on
c js js
for tight-binding electrons moving on a square lattice. lattice site j with spin s =↑,↓; t are hopping ma-
[ ij
Charge and spin densities and the local pairing ampli- trix elements with amplitude t between nearest-neighbo,

1 tudes adjust spatially in a stripe pattern with transverse and t(cid:48) between next-nearest neighbor sites. Here we use
v
signchangefortheantiferromagnetic(AF)orderparame- t(cid:48) =−0.4t for all calculations. The BCS-type attractive
2
ter. Hoppinganisotropyweakensorevendestroyssuper- interaction
0
4 conductivity, as observed in the low-temperature tetrag- V (cid:88)
3 onal phase of cuprate superconductors [6–9]. HI =−2 c†isc†j−scj−scis (1)
.
1 Transportexperimentsinthehigh-temperaturesuper- (cid:104)i,j(cid:105),s
0 conductor La2−xBaxCuO4 for x = 1/8 uncovered a se- is restricted to nearest-neighbor sites; V >0 is the pair-
1
quence of thermal phase transitions [6, 7, 10]. Charge- ing interaction strength. In the complete mean-field de-
1
: and spin-stripe order emerges sequentially upon cooling coupling scheme
v before two dimensional (2D) superconducting (SC) fluc-
i 1 (cid:88)(cid:104)
X tuations set in which ultimately lead to 3D supercon- H −→ ∆∗ c c +∆ c† c† −Vn c† c
I 2 ji j↓ i↑ ij i↑ j↓ j↓ i↑ i↑
ductivity below 4K. These measurements provided com-
r (cid:104)i,j(cid:105) (2)
a pelling evidence for what has since been called a striped ∆∗ ∆ (cid:105)
superconductor. The subsequently developed theory for −Vn c† c + ji ij +Vn n
j↑ i↓ i↓ V i↑ j↓
the striped superconductor introduced the concept of a
pair density wave (PDW) in which the order parameter weintroducethebondorderparameter∆ij =−V(cid:104)cj↓ci↑(cid:105)
for the pairing of electrons in a superconductor is spa- forsuperconductivityandthelocalspinresolveddensities
tiallymodulatedwithrespecttothecenterofmasscoor- nis = (cid:104)c†iscis(cid:105). Using a Bogoliubov-de Gennes transfor-
dinate of the electron pair [11, 12]. This implies that mation, the model is solved self-consistently at an elec-
Cooper pairs with finite momenta ±q form accompa- tron density 7/8 (for details on the formalism see e.g.
nied by a charge-density modulation with wavenumber Ref. 19). The terms Vn c† c are typically not ac-
j,−s is is
2q [13]. The phenomenological characteristics of a pair counted for in the standard BCS theory. However, for
2
0.2
x - bonds
Δij0
y - bonds
-0.2
4 8 12 16
(a) (b) (c)
-0.87 -0.84 0 0.84 0.87 0.63 0.95 1.00
FIG.1: Real-spacecharacterizationofthestripedSCstate. (a)Themagnetizationm =n −n exhibitsAFstripesseparated
i i↑ i↓
by non-magnetic anti-phase domain walls. (b) The charge density n = n +n reaches nearly one electron per site inside
i i↑ i↓
theAFstripeswhichareseparatedbysinglelinesnearquarterfilling,resultinginanoverallmeanchargedensity7/8,i.e. hole
doping1/8. (c)TheSCbondorderparameteronthehorizontal(purple)andontheverticalbonds(blue). Itislargestonthe
vertical bonds along the line of non-magnetic sites. Results were obtained for V =2t on a 16×12 lattice.
a nearest-neighbor pairing interaction they are a strong ment. While stripe formation minimizes the magnetic
source for antiferromagnetism, since they provide an en- energy, kinetic energy is gained by transverse fluctua-
ergy gain −V for each AF bond, but only −V/4 for a tions. This is the origin of the sign change in the AF
bond between two non spin-polarized sites. This is the order between neighboring stripes.
drivingforcefortheformationofAFstripesinourmodel. The unidirectional character of the SC order param-
There are two qualitatively different regimes of inter- eter is evident from its considerably smaller values on
action strengths: for weak V, below a critical interac- the bonds perpendicular to the stripe orientation (see
tion strength V ≈ 0.9t, the only solution of the self- Fig. 1c). The SC order parameter acquires its maximum
c1
consistency equations is a homogeneous SC phase with valueonthebondswhichconnecttothesitesinthenon-
d-wave symmetry and without antiferromagnetism. As magnetic channels. The sign change between the SC or-
V is increased beyond V , there is a sharp crossover der parameters on the horizontal and the vertical bonds
c1
into a regime where antiferromagnetism is the dominant connected to the same site verifies the d-wave character
order, superconductivity is suppressed and eventually of the SC order parameter.
disappears above a second critical interaction strength Thegroundstatesolutionwithstripedsuperconductiv-
V ≈ 3t. The characteristics of this latter regime are ity is not unique with respect to a sign change of ∆
c2 ij
best illustrated for strong interactions V >V when H between neighboring hole-rich channels. A solution de-
c2 I
favors energetically a homogeneous AF phase for a half- generate to the one shown in Fig. 1 exists without this
filled band. For a filling ρ=7/8 instead a configuration sign change. Since in our model analysis all physical
ispreferredwiththree-legged,half-filledspinladdersand quantities depend on ∆2 only, these two variants of the
ij
non-magnetic lines at an average density ρ = 1/2 in be- striped superconductor have the same energy, provided
tween. This regularly striped solution which is unique ∆ vanishes at the center of the AF stripes. For inter-
ij
for the filling ρ = 7/8 was indeed inferred from elastic action strengths close to V , where the AF order weak-
c1
neutronscatteringdataforthelow-temperaturephaseof ens and ∆ becomes finite also within the AF stripes,
ij
La Nd Sr CuO [3]. this degeneracy is lifted and the state without the sign
1.48 0.4 0.12 4
If the interaction strength V is reduced below V , su- change is favored. A similar conclusion was reached
c2
perconductivity emerges and resides predominantly on within a renormalized mean-field theory for a general-
the quarter-filled channels in between the AF stripes. ized t-J model by Yang et al., if the hopping amplitudes
Figure 1 displays the self-consistently determined mag- are anisotropic [18].
netization m = n −n (a), the charge density n = A qualitative difference of the two striped SC states
i i↑ i↓ i
n + n (b), and the SC order parameter (c) of the with and without sign change of the order parameter
i↑ i↓
striped SC state for V = 2t on a 16×12 lattice. The concerns the center-of-mass momenta q of the electron
resultspresentedherearestablegroundstatesolutionsir- pairs. In the former state all pairs have either momenta
respective of the system size, provided that the selected q =±π/4a or q =±3π/4a with q =0, corresponding
x x y
geometry is commensurate with the wavelength of the totheperiodicityofeightlatticesites. Thestatewithout
stripes. Along the AF stripes the magnetic energy gain sign change has in addition a finite q=0-component for
is maximized by a nearly perfect antiparallel spin align- pairswithvanishingtotalmomentum,i.e. (cid:104)c c (cid:105)=(cid:54) 0.
k↑ −k↓
3
(a) momentum distribution (b) spectral density
0.13 0.97 1.81 0 0.3 0.61
central AF site
π π border AF site
metallic site
0 0
-π -π
-4 -2 0 2 4 6
-π 0 π -π 0 π
(c) pair density (d) spin density
0 0.039 0.078 0.67 1.15 1.63
FIG. 3: Local density of states (LDOS). The colors corre-
π π
spondtothethreedistinctsitesofthestripedsuperconductor:
sites with minimum charge density (dark blue), sites in the
center (blue) and on the edge (light blue) of the AF stripes.
0 0
n(k)inFig.2aisduetoafiniteenergygapinthedensity
of states (see Fig. 3). A remarkably similar momentum
distribution has indeed been measured by Zhou et al.
for the rare-earth doped cuprate La Nd Sr CuO
1.28 0.6 0.12 4
-π -π with static stripe order but no superconductivity [20].
-π 0 π -π 0 π
For a comparison with the measured spectral weight we
FIG. 2: Momentum-space characterization of the striped su- display the integrated spectral function
perconductor (a), The momentum distribution n(k) is a su-
(cid:90) µ 1 (cid:90) µ
perposition of incoherent states well below the Fermi energy,
A(k,ω)dω =− ImG(k,k,ω)dω (3)
and a partially filled quasi one-dimensional band (horizontal π
µ−ωc µ−ωc
bar). (b), The integrated spectral weight (see Eq. (3)) cap-
tures only the conducting states originating from the motion in Fig. 2b. The lower energy cut-off at ω = −t re-
c
alongthestripes. (c),ThepairdensityP(k)showsthedistri- stricts the spectral weight to the contributions of the
butionoftheSCpairsinmomentumspace. (d),TheFourier non-magnetic channels (c.f. Fig. 3). The chemical po-
transformedspindensityρ (k)islargestnear(0,±π)forver-
S tential µ corresponds to 7/8 filling. The quasiparticle
tical stripes. Results were obtained for V = 2t on a 16×12
excitations near the Fermi level occupy the horizontal
lattice using 9×9 supercells.
bar in momentum space with a strongly reduced spec-
tral weight in the center of the BZ as observed in the
measurements of Ref. 20. This “breach” in the spectral
Thefinitecenter-of-massmomentacoexistingwithq=0
function is not captured by the physics of isolated spin
arestabilizedonlybytheAFstripesandvanishtogether
ladders and indicates that the conducting states are not
with stripe order, thereby recovering the homogeneous
decoupled from the magnetic stripes.
d-wavesuperconductor. Thestatewithsignchangehow-
SimilarlytoRef.15wedefinetheelectron-pairdensity
ever remains striped for sufficiently large V even in the P(k) for singlet pairing as P2(k) = (cid:80) (cid:104)c c (cid:105)2.
q −k+q↓ k↑
absence of AF order and realizes the pure PDW state
P(k) serves as a measure at which momenta in the BZ
discussed in Ref. 13.
electronpairspredominantlyforminthesuperconductor.
InFig.2thestripedsuperconductorischaracterizedin For the striped superconductor discussed above electron
momentum space. The calculations were performed on a pairs have the finite center of mass momenta ±q where
16×12latticewith9×9supercellstoensuresatisfactory q =0 and q =π/4 or 3π/4 according to a stripe wave-
y x
momentumresolution. Themomentumdistributionn(k) length of 8 lattice constants. The pair density is ex-
in Fig. 2a clearly exhibits the unidirectional character of pectedtobelargestneartheFermisurfaceofthenormal
the striped system with a horizontal bar of high occupa- conducting system as is indeed verified in Fig. 2c. In a
tionprobabilityandadiffuseregionaroundtheBrillouin similarwaywealsotranslatethespin-stripepatterninto
zone (BZ) center. This diffuse background traces the momentumspaceρ (k)=(cid:80) (cid:80) (cid:104)sc† c (cid:105).Asshown
S q s k+qs ks
original 2D Fermi surface of the uncorrelated electrons. in Fig. 2d, ρ (k) is strongest near (0,±π) for vertically
S
Intheabsenceofsuperconductivitythestripeorderleads orientedAFstripes,i.e.,inthoseregionsoftheBZwhere
to a sharp Fermi surface with occupied states for mo- no electron pairs form.
mentakwithk ≤π/4. Theabsenceofdiscontinuitesin The absence of a discontinuity in n(k) is tied to the
y
4
dashed lines in Fig. 4. Since ∆ is a measure of T , the
c
resultsfor∆candirectlybecomparedtotheT datafor
c
La Nd Sr CuO [8, 9] in the inset of Fig. 4. There
2−y−x y x 4
t /t = 1
0.2 x y isalmostnosuppressionofTc fortheisotropiccompound
] with y = 0, but a complete destruction of superconduc-
x
[t tx/ty = 1 tivity around x=1/8 in the anisotropic Nd-doped com-
Δ ij 30 pounds with y =0.2 and y =0.4.
xij0.1 K]20 The presented pairing model for the coexistence of SC
ma T [c andAFstripeorderreproducesthemostprominentprop-
10
erties of striped high-T cuprates remarkably well. Al-
c
t /t = 0.83 0 though it was shown before that the pure PDW can be
0 x y 0.05 0.1 0.15 0.2 0.25
the groundstate of a pairing Hamiltonian in the absence
0.125 0.15 0.175
x of magnetism [13], the issue concerning the sign change
of the SC order parameter will only be resolved by a
FIG. 4: Doping dependence of the superconducting order
phase sensitive extension of the present calculation, e.g.
parameter. ThepurpleandbluecurvesforthemaximumSC
aJosephsoncouplingtermwhichhastobeidentifiedbe-
orderparametermax ∆ correspondtoV =2tforisotropic
ij ij
t = t and anisotropic hopping t /t = 0.83, respectively. yond the Hartree-Fock decoupling scheme [11]. Defects
x y x y
The grey line is the result for t = t with a reduced V = may certainly affect the stability of the stripe state con-
x y
1.8t. The dashed lines indicate an extrapolation to values of siderably; the pure PDW is indeed supposed to be frag-
x=1wherethenumericalproceduredoesnotconvergetothe ile with respect to impurities. [11] As we have verified,
discussedsolution. Inset: MeasureddopingdependenceofT
c the inclusion of potential scatterers in our model shows
in La Nd Sr CuO for y = 0.2 (Ref. 8) (red), y = 0.4
2−y−x y x 4 that superconductivity is weakened and eventually van-
(dark blue), and y=0 (grey) (Ref. 9).
ishes. The AF stripe order, however, is affected only
little. Moreover, impurities can act as pinning forces for
fluctuating stripes and thereby support the formation of
openingofafullgapintheDOS.ThelocalDOSinFig.3 static stripe order.
shows a large energy gap in the center of the AF stripes The authors gratefully acknowledge discussions with
which is reduced on the edge of the stripe. But also on Brian M. Andersen, Raymond Fr´esard, Peter Hirschfeld,
the non-magnetic sites, where the SC order parameter and Steve Kivelson. This work was supported by the
is strongest, there exists a small gap, because the quasi DFG (TRR 80).
one-dimensionalityadmixesasignificantextendeds-wave
component. This is in contrast to the rather 2D pure
PDW state without antiferromagnetism, where the local
DOS is gapless. [12, 13, 15]
[1] J. Zaanen, and O. Gunnarsson, Phys. Rev. B 40, 7391
The special affinity to stripe formation at the den- (1989).
sity ρ = 7/8 in the cuprates is evident from the varia- [2] J.M.Tranquadaetal.,Phys.Rev.Lett.73,1003(1994).
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c
[4] M. Fujita et al., Phys. Rev. B 70, 104517 1 (2004).
The striped compounds La Nd Sr CuO [8, 9] and
2−y−x y x 4 [5] S. A. Kivelson et al., Rev. Mod. Phys. 75, 1201 (2003).
La Ba CuO [6, 7] show a sharp dip in T (x) for
15/8 1/8 4 c [6] J.M.Tranquadaet al.,Phys.Rev.B78,174529(2008).
hole doping x = 1−ρ = 1/8. The observed reduction
[7] M.Hu¨ckeretal.,PreprintarXiv:1005.5191,unpublished.
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79, 064515 (2009).
the doping dependence of the maximum SC order pa-
[12] E. Berg et al., New J. Phys. 11, 115004 (2009).
rameter ∆ = max ∆ shown in Fig. 4. If we simulate
ij ij [13] F. Loder, A. P. Kampf, and T. Kopp, Phys. Rev. B 81,
thelatticeanisotropyfromtheoctahedraltiltintheLTT
020511(R) (2010).
phase by introducing an anisotropy in the hopping am- [14] D. Agterberg, and H. Tsunetsugu, Nat. Phys. 4, 639
plitudes t (cid:54)= t , superconductivity is weakened and the (2008).
x y
minimum at x = 1/8 develops into a sharp dip, which [15] S. Baruch, and D. Orgad. Phys. Rev. B 77, 174502
reaches ∆ = 0 for t /t (cid:46) 0.83. For weaker pairing in- (2008),
x y
[16] M.Raczkowskietal.,Phys.Rev.B76,140505(R)(2007).
teraction strengths the dip at x = 1/8 develops also for
[17] B. M. Andersen, and P. Hedeg˚ard, Phys. Rev. Lett. 95,
isotropichoppingt =t . Whenxisdecreased,theelec-
x y 037002 (2005).
trondensityintheSCstripesincreases, andsequentially
[18] K.-Y. Yang et al., New J. Phys. 11, 055053 (2009).
atspecificvaluesofxtheconductingstripesturnantifer- [19] M. Schmid et al., New J. Phys. 12, 053043 (2010).
romagnetic one by one. This process is indicated by the [20] X. J. Zhou et al., Science 286, 268 (1999).