Fermi arc in doped high-Tc cuprates

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by  Takashi Yanagisawa

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Fermi Arc in Doped High-T Cuprates
c
Takashi Yanagisawa1, Mitake Miyazaki1,2 and Kunihiko Yamaji1
1Condensed-Matter Physics Group, Nanoelectronics Research Institute, National Institute of Advanced Industrial Science
and Technology, Tsukuba Central 2, 1-1-1 Umezono, Tsukuba 305-8568, Japan
6 2Department of Physics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-0558, Japan
0
0 (ReceivedFebruary6,2008)
2 We propose a d-density wave induced by the spin-orbit coupling in the CuO plane. The

n spectral function of high-temperature superconductors in the under doped and lightly doped
a regions is calculated in order to explain the Fermi arc spectra observed recently by angle-
J resolved photoemission spectroscopy. We take into account the tilting of CuO octahedra as

6 well as the on-site Coulomb repulsive interaction; the tilted octahedra induce the staggered
1 transferintegralbetweenpx,y orbitalsandCut2g orbitals,andbringaboutnontrivialeffectsof
spin-orbit coupling for the d electrons in the CuO plane. The spectral weight shows a peak at

] around (π/2,π/2) for light dopingandextendsaround thispoint forming an arc asthecarrier
l
e density increases, where the spectra for light doping grow continuously to be the spectra in
-
the optimally doped region. This behavior significantly agrees with that of the angle-resolved
r
t photoemission spectroscopy spectra. Furthermore, the spin-orbit term and staggered transfer
s
. effectively induceafluxstate, apseudo-gapwith time-reversal symmetrybreaking.Wehavea
t
a nodal metallic state in thelight-doping case since the pseudogap hasa dx2−y2 symmetry.
m
KEYWORDS: spectral function, high-Tc cuprate,distortion, spin-orbit coupling, stripes
-
d
n
o
In recent years, oxide high-T superconductors have spin-orbitcoupling betweent orbitalsisnotdecoupled
c 2g
c
beeninvestigatedintensively;anomalousmetallicbehav- from e networks.[8,9] For the spin-orbit coupling
[ g
iors as well as high critical temperatures T have been
1 focused in the study of high-T cuprates. Clcarifying the Hso =ξℓ·s, (1)
c
v
origin of an anomalous metal with a pseudogap is also matrix elements exist between t orbitals:
9 2g
2 a challenging problem attracting many physicists. Re- hd (r)↑|H |d (r)↑i=−(i/2)ξ, (2)
3 cently,a peak crossingthe Fermi levelinthe node direc- xz so yz
1 tion of the d-wave gap has been observed in a lightly
0 doped La Sr CuO (LSCO) by angle-resolved pho- hdyz(r)↑|Hso|dxz(r)↑i=(i/2)ξ, (3)
2−x x 4
6
toemission spectroscopy (ARPES). The spectral weight
0 hd (r)↓|H |d (r)↓i=(i/2)ξ, (4)
/ shows a peak at around (π/2,π/2) for the light-doping xz so yz
t
a case and it extends along the Fermi surface with in-
hd (r)↓|H |d (r)↓i=−(i/2)ξ. (5)
m creasingcarrierdensity.[1,2]The existenceofincommen- yz so xz
- surate correlations has also been reported by neutron- We have also matrix elements between t2g orbitals and
d scattering measurements, suggesting vertical stripes in e orbitals:
g
n
an underdoped region and diagonal stripes in a lightly
o doped region.ModulationvectorsareQ =(π±2πδ,π), hdxz(r)↓|Hso|dx2−y2(r)↑i=ξ/2, (6)
c s
v: fQocr =the(±h4oπleδ-,d0o)pi(nogr rQaste=x(≥π,0π.0±5,2wπδh)e,reQδc =ind(i0c,a±te4sπtδh)e) hdyz(r)↓|Hso|dx2−y2(r)↑i=−iξ/2, (7)
i
X approximately linear dependence δ =x.[3-5] and those for reversed spins obtained by multiplying by
r Inthe lightlydopedregionforx<0.05,the suggested −1. The matrix elements induced by the tilting are
a
modulation vector is Q = (π ±2πδ,π ±2πδ) and the
s hp (x−a/2,y)σ|H |d (r)σi=−t eiQ·r, (8)
deviation from the linear dependence exhibits δ < x.[4] x pd xz xz
From the experiments of resistivity in the lightly-doped
hp (x,y−a/2)σ|H |d (r)σi=−t eiQ·r, (9)
region of LSCO, the system holds a metallic behavior y pd yz yz
belowT ,whichmaybeduetotheformationofmetallic wherer=(x,y),Q=(π,π),aisthelatticeconstantand
N
charge stripes.[6] H denotes the hybridizationterm. The factor eiQ·r in-
pd
The purpose of this paper is to investigate the spec- ducedby the tilting ofoctahedraina staggeredmanner,
tral function of doped high-temperature superconductor which leads to the doubled unit cell. t and t are as-
xz yz
inordertoelucidatetherecentlyobservedARPESFermi sumed to be given by sinθ for the tilt angle θ. In the
arc spectra, taking into account the stripe orders sug- low-temperature tetragonal (LTT) phase, the spin-orbit
gested by neutron-scattering measurements. We empha- coupling may be significantly smaller than that in the
size the importance of distortions of CuO octahedra. low-temperature orthorhombic (LTO) phase, since the
ThetiltingofCuOoctahedrainducestransferintegrals integrals between oxygen p and Cu t orbitals remain
2g
between p orbitals and Cu t orbitals, and then the zeroalongthe tiltaxis,whereoxygenatomsnevermove:
x,y 2g
t =0 if the tilt axis is in the y direction.
yz
2 J.Phys.Soc.Jpn. FullPaper AuthorName
The dispersion in the presence of distortions for the -2.6
five-band p-d model is shown in Fig.1. The parameters
are as follows: ǫ = −2, ǫ = 0, ǫ = ǫ = −1, -2.8
dx2−y2 p dxz dyz
t = 0.2, ξ = 0.1, and t = t = 0.3 in units of
pp xz yz
t ∼ 1eV where t is the transfer between d and p -3.0
pd pd
orbitals and t is that between neighboring oxygen p
pp E
orbitals. For the parameters shown above, the splitting -3.2
k
at X = (π,0) is of the order of 10meV≈100K. Each
curve is a twofold degeneracy, i.e. Kramers degeneracy -3.4
as it should be. This structure near the Fermi energy is
well understood, using the single-band model with the -3.6
reduced Brillouin zone, as
-3.8
H = [ξ c† c +∆ c† c ], (10)
0 X k kσ kσ kσ kσ k+Qσ
kσ
-4.0
where
ξ =−2t(cos(k )+cos(k ))−4t′cos(k )cos(k ), (11) -4.2
k x y x y G X M G
and Q denotes the wave vector Q = (π,π). ∆ is a k
kσ
complex parameter satisfying
∆ =∆∗ . (12) Fig. 1. Dispersionrelationforthethree-bandd-pmodel.Thepa-
k+Qσ kσ
∆kσ is taken as[10,11] r−a1m,ettpeprs=a0r.e2,aξs=fol0lo.1w,sa:nǫddxt2x−zy=2 t=yz−=2,0.ǫ3pin=u2n,itǫsdxozft=pd,ǫdwyhzer=e
∆ =iσsinφ·(−2t)Y(k) (13) tpd is the transfer between d and p orbitals and tpp is that be-
kσ tweenneighboringoxygenporbitals.ThesplittingatX=(π,0)
with Y(k)=cos(k )−cos(k ) in order to reproduce the isoftheorderof10meV.Eachcurveisatwofolddegeneracy,i.e.,
x y
Kramersdegeneracy.
dispersion for the five-band CuO model near the Fermi
energy, where φ = λθ; θ is the tilt angle between the
Cu-O plane and the Cu-O bond, and λ is a constant
1
estimated as λ ≈ 0.2.[11] The dispersion relation of the
noninteracting part H in eq. (10) is
0
0.8
1
E = [ξ +ξ − (ξ −ξ )2+4|∆ |2]. (14)
k k k+Q q k k+Q kσ
2
At half-filling, the low-energy excitations are described
0.6
by the Dirac fermion since there is a Fermi point at
(±π/2,±π/2). A light hole doping results in a small N(w ) f = 0.03p
Fermi surface around this point with the excitation gap
near (π,0). This peculiar feature induces a pseudogap 0.4 f = 0.01p
in the density of states and we obtain a nodal metallic f = 0
state.The pseudogapinducedbythe spin-orbitcoupling
has a dx2−y2 symmetry as apparent from that of ∆kσ. 0.2
The density of states for small φ shown in Fig. 2 clearly
indicates a pseudogap structure for a small excitation
energy.
0
TheeigenstateofH0resemblesthed-densitywavepro- 0 0.2 0.4 0.6 0.8 1
posedforananomalousmetallicstateintheunderdoped w / t
high-T cuprates[12], and possesses the following order
c
parameter
Fig. 2. Densityofstatesforφ=0,0.01πand0.03π.Wesett′=0.
i∆ Y(k)=hc† c i (15)
DDW k+Qσ kσ
for a real constant∆ . Although the d-density wave
DDW linear dependence of δ on x has been explained by vari-
state appears as a solution of the mean-field equations,
ational Monte Carlo (VMC) methods[16], and further-
this state is hardly stabilized in variationalMonte Carlo
more, the saturation of incommensurability δ for x >
calculations.Thus,we are motivatedto consider the lat-
0.125is alsoconsistentwiththose obtainedbythe VMC
tice distortion, which is going to stabilize the d-density
methods.Thereisalsoatendencytowardstheformation
wave cooperating with the spin-orbit coupling.
of stripes under the lattice distortions, with vertical or
The inhomogeneous ground state under the lattice
horizontal hole-rich arrays coexisting with incommensu-
distortion in the underdoped region has been investi-
rate magnetism and superconductivity (SC).[7]
gated intensively using the two-dimensional (2D) Hub-
We are going to evaluate spectral functions in the
bard model.[13-16] Within the 2D Hubbard model, the
J.Phys.Soc.Jpn. FullPaper AuthorName 3
lightly to optimally doped regions. The tilt angle for 1 4
LSCO is estimated as 14◦-18◦ by EXAFS.[17] Thus,
3
θ is approximately 0.1π. If we use λ ∼ 0.2, we have 0.5
2
φ = λθ ≈ 0.02π. Here, we use a slightly larger value,
φ=0.05π,inactualcalculationstoobtainsufficientpre- ky/p 0 1
cision because of the numerical difficulty for the small 0
splitting.Clearly,wecanexpectthatthisdoesnotchange -0.5
the peculiar feature of the spectra. We determine the
variational parameters g and ∆Q so as to minimize the -1 -1 -0.5 0 0.5 1
ground state energy for U = 4 and the carrier density kx/p
in the range of 0 < x < 0.2 by the VMC methods. The
Hamiltonian is
Fig. 3. Contourmapofdensityofstatesforthediagonallystriped
H =H +U n n , (16) stateatdopingratex=0.03withφ/π=0.05.∆Q=0.08,α=0
0 X i↑ i↓ andt′=−0.2.
i
1 4
whereH isgivenineq.(1).Thewavefunctioniswritten
0
in a Gutzwiller form: 3
0.5
ψ =P ψ . (17) 2
G MF ky/p
P is the Gutzwiller operator 0 1
G
0
PG =Y(1−(1−g)ni↑ni↓), (18) -0.5
i
andthe mean fieldwavefunction ψMF is obtainedas an -1 -1 -0.5 0 0.5 1
eigenfunction of the Hartree-Fock Hamiltonian
kx/p
HMF =H0+X[δni−sign(σ)(−1)xi+yimi]c†iσciσ. (19)
Fig. 4. Density of states at doping rate x = 0.0612 with φ/π =
iσ
0.05. Vertical stripes with a 16-lattice periodicity are assumed.
δni and mi are expressed by the modulation vectors ∆Q=0.10,α=0andt′=−0.2.
Q and Q for the spin and charge part, respectively.[7]
s c
Equivalently, we use the form given by[7,18] 1 4

3
δni =−αXcosh((xi−xsjtr)/ξc), (20) 0.5
2
j
ky/p
and 0 1

m =∆ tanh((x −xstr)/ξ ). (21) 0
i QY i j s -0.5
j
The diagonally striped state has hole arraysin the diag-
-1
onaldirection,while the verticallystriped state has hole -1 -0.5 0 0.5 1
arraysinthedirectionparalleltothexorydirection.Al- kx/p
ternatively, it is also possible to determine the order pa-
rameter ∆ = hc† c i (ℓ =0,1,··· ,M −1) Fig. 5. Density of states at doping rate x = 0.125 with φ/π =
ℓQsσ Pk k+ℓQsσ kσ 0.05. Vertical stripes with an 8-lattice periodicity are assumed.
for δ = 1/M consistently.[19] We obtain the diagonally
striped state for x < 0.05 and the vertically striped ∆Q=0.16,α=0andt′=−0.2.
state for x > 0.05 as the ground state by the VMC 1 5
methods.[16] Among the diagonal stripes, the bond- 4
centered striped state is the most stable when we vary 0.5 3
the wave function from the site-centered stripe to the
2
bond-centeredstripe by the VMC methods. We mention ky/p
0 1
herethatthespin-orbitcouplingstabilizesthediagonally
0
striped state in the light doping region.[20]
-0.5
In the evaluations of spectral functions, we consider
the effect of P within the mean field theory since we
G -1
must consider the excited states as well as the ground -1 -0.5 0 0.5 1
state. We then evaluate the spectral weight from the kx/p
eigenvaluesE andeigenfunctions(u ) oftheHamil-
σm σm j
tonian (Hσ ) in the real-space representation, where Fig. 6. Densityofstatesfortheverticallystripedstateatdoping
MF ij
i and j are site indices. Green’s functions are ratex=0.197withφ/π=0.05.Verticalstripeswithan8-lattice
periodicityareassumed.∆Q=0.08,α=0andt′=−0.2.
(u ) (u )∗
σm i σm j
g (i,j,iω)= , (22)
σ X iω−E
σm
m
4 J.Phys.Soc.Jpn. FullPaper AuthorName
1 5 trast to a theory considering p-orbitals in apical oxygen
4 atoms. [22] We have taken into account the incommen-
0.5 3 surate structure observed by neutron-scattering exper-
2 iments. Furthermore, we can expect the following fas-
ky/p 0 1 cinating physics: pseudogap, nodal metal, time-reversal
symmetry breaking, diagonal stripes, and string-density
0
wave as a generalization of the d-density wave. In the
-0.5
half-filledcase,weobtainaFermipoint,andthus,apeak
exists near (π/2,π/2) in the density of states that ex-
-1
-1 -0.5 0 0.5 1 tends to form the Fermi arc spectra as the doping rate
kx/p increases.[2] In the low-doping case, we obtain a nodal
metallic state in the diagonally striped state, since the
Fig0..179.7wDitehnsφit/yπo=fs0t.a0t5e,s∆foQrt=he0naonrdmta′l=st−at0e.2a.tdopingratex= pwsiethudtohgeaepxphearsimaendtxs2−oyf2ressyismtimvietytr.[y6,]wThhiecharcisiscoexnpsiesctteendt
to expand with increasing temperature to form the full
Fermi surface above the splitting gap ∼100K.[1]Lastly,
1
gσ(k,iω)= N Xe−ik·(Ri−Rj)gσ(i,j,iω), (23) we comment on a possibility of superconductivity along
a ij stripes.Thecoexistenceofsuperconductivityandstripes
has been pointed out for vertical stripes by the VMC
where N is the number of atoms.The spectralfunction
a
methods.[7]Itisdifficulttohaveastablecoexistentstate
is calculated using the formula
ofSC andstripes for diagonalstripes inthe VMC meth-
1
Nσ(k,ǫ)=− Imgσ(k,ǫ+iδ). (24) ods because the SC pairs must have a dxy symmetry
π
along stripes. Thus, superconductivity is suppressed for
We show the results for x = 0.03, 0.061, 0.125 and light doping.
0.197 for the parameters obtained by the VMC meth- We are grateful to H. Eisaki for valuable discussions.
ods. The contour map of spectra for the light-doping
x = 0.03 is shown in Fig. 3 where the calculations
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J.Phys.Soc.Jpn. FullPaper AuthorName 5
054504.