The energy, momentum, and temperature dependence of the quasiparticle local density of states (LDOS) of a two-dimensional d(x2)-(y2)-wave superconductor with random disorder is investigated using the first-order T-matrix approximation. The results suggest that collective modes such as spin-charge-density waves are relevant low-energy excitations of the cuprates that contribute to the observed LDOS modulations in recent scanning tunneling microscopy studies of Bi2Sr2CaCu2Ox

Chen, C.-T.

Yeh, N.-C.

English

Caltech Authors - Main

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Collective modes and quasiparticle interference on the local density of states of
cuprate superconductors
C.-T. Chen1 and N.-C. Yeh1
1Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
(Dated: May 30, 2005)
The energy, momentum and temperature dependence of the quasiparticle local density of states
(LDOS) of a two-dimensional dx2−y2 -wave superconductor with random disorder is investigated
using the first-order T-matrix approximation. The results suggest that collective modes such as
spin/charge density waves are relevant low-energy excitations of the cuprates that contribute to the
observed LDOS modulations in recent scanning tunneling microscopy studies of Bi2Sr2CaCu2Ox.
PACS numbers: 74.50.+r, 74.62.Dh, 74.72.-h
One of the most widely debated issues in cuprate su-
perconductivity is the possibility of preformed Cooper
pairs1,2,3,4 and the origin of the pseudogap phe-
nomenon.5,6 Recent experiments have demonstrated
that the pseudogap phenomenon is unique to the
hole-doped (p-type) cuprates and is absent above Tc
in electron-doped (n-type) cuprates.7,8,9 Furthermore,
in the quasiparticle tunneling spectra of the double-
layer Bi2Sr2CaCu2O8+δ (Bi-2212)
10 and the one-layer
Bi2(Sr2−xLax)Cu2O6+δ (Bi-2201)
11 systems, it is shown
that the pseudogap can be distinguished from the super-
conducting gap: the former evolves smoothly with in-
creasing temperature whereas the latter vanishes at Tc.
These phenomena suggest that the pseudogap may be
associated with a competing order6,12 that coexists with
the superconducting phase for T < Tc and persists above
Tc until a pseudogap temperature T
∗. The competing
quantum ordered phase13,14 can be manifested in the
form of collective modes such as charge- and spin-density
waves (CDW and SDW) in the superconducting state,
as inferred from neutron scattering experiments in a va-
riety of p-type cuprates.15,16,17,18,19 However, whether
these collective modes are closely correlated with super-
conductivity remain controversial. Recent scanning tun-
neling spectroscopic studies of the Fourier transformed
(FT) quasiparticle local density of states (LDOS) of Bi-
221220,21,22 have stimulated further discussions on the
relevance of collective modes.23,24,25,26,27,28 While Bogoli-
ubov quasiparticle interference apparently plays an im-
portant role in the observed FT-LDOS in the supercon-
ducting state, certain spectral details of the LDOS can-
not be accounted for unless collective modes are consid-
ered.23,24,25 In particular, the findings of 4 high-intensity
Bragg peaks remaining above Tc in the FT-LDOS map of
Bi-221229 cannot be reconciled with quasiparticles being
the sole low-energy excitations. These new developments
motivate us to reexamine the role of collective modes in
cuprate superconductors by considering the energy (E),
momentum transfer (q) and temperature (T ) dependence
of the resulting FT-LDOS modulations.
We begin our model construction by noting that sub-
stantial quasiparticle gap inhomogeneities are observed
in the low-temperature tunneling spectroscopy of under-
and optimally doped Bi-2212 single crystals,30,31 suggest-
ing at least two types of spatially separated regions, one
with sharp quasiparticle coherence peaks at smaller en-
ergies ∆d and the other with rounded hump-like features
at larger energies ∆∗. On the other hand, low-energy
LDOS (for E < 0.5∆d) of Bi-2212 exhibit long-range
spectral homogeneity. We therefore conjecture that dy-
namic SDW or CDW coexist with cuprate superconduc-
tivity and that they are only manifested in the quasipar-
ticle LDOS when pinned by disorder. Thus, regions with
rounded hump features in the quasiparticle spectra are
manifestation of localized charge modulations due to pin-
ning of collective modes by disorder, and the wavevector
of the charge modulation is twice of that for the collinear
SDW order, as proposed in Refs. 23,24,32. In con-
trast, regions with sharp quasiparticle spectral peaks are
representative of generic Bogoliubov quasiparticle spec-
tra with a well-defined d-wave pairing order parameter
∆k ≈ ∆d cos 2θk, where ∆d is the maximum gap value
and θk is the angle between the quasiparticle wavevector
k and the antinode direction. Our model therefore as-
sumes ‘puddles’ of spatially confined ‘pseudogap regions’
with a quasiparticle scattering potential modulated at a
periodicity of 4 lattice constants along the Cu-O bonding
directions, and the spatial modulations can be of either
the ‘checkerboard’ pattern23,24 or ‘charge nematic’ with
short-range stripes.12,14 In the limit of weak perturba-
tions, we employ the first-order T-matrix approximation
and consider a (400 × 400) sample area with either 24
randomly distributed point impurities or 24 randomly
distributed puddles of charge modulations that cover ap-
proximately 6% of the sample area. For simplicity, we do
not consider the effect of disorder on either suppressing
the local pairing potential ∆d(r) or altering the nearest-
neighbor hopping coefficient (t) in the band structure of
Bi-2212, although such effects reflect the internal struc-
tures of charge modulations.25,26
Specifically, the Hamiltonian of the two-dimensional
superconductor is given by H = HBCS + Himp, where
HBCS denotes the unperturbed BCS Hamiltonian of the
d-wave superconductor, HBCS =
∑
kσ(ǫk − µ)c
†
kσckσ +∑
k ∆k[c
†
k↑c
†
−k↓+c−k↓ck↑], and Himp is the perturbation
Hamiltonian associated with impurity-induced quasipar-
2ticle scattering potential.24,26,27,33 Using the T-matrix
method, the Green’s function G associated with H is
given by G = G0 + G0TG0, where G0 is the Green’s func-
tion of HBCS and T = Himp/(1−G0Himp). The Hartree
perturbation potential for single scattering events in the
diagonal part of H and for non-interacting identical
point impurities at locations ri is Vα(q) =
∑
i Vs,me
iq·ri
for non-magnetic (Vs) and magnetic (Vm) impurities
27,
whereas that for puddles with short stripe-like modula-
tions centering at rj is
33
Vβ(q) =
∑
j
V0e
iq·rj
2 sin(qy,xRj) sin(qx,yRj)
qy,x sin(2qx,y)
, (1)
and that for checkerboard modulations is
Vγ(q) =
∑
j
V0e
iq·rj
[
2 sin(qyRj) sin(qxRj)
qy sin(2qx)
+ (qx ↔ qy)
]
(2)
Here all lengths are expressed in units of the lattice con-
stant a, Rj is the averaged radius of the j -th puddle,
and V0 denotes the magnitude of the scattering poten-
tial by pinned collective modes. For simplicity, we ne-
glect the energy dependence of Vα,β,γ and assume that
Vs, Vm and V0 are sufficiently small so that no resonance
occurs in the FT-LDOS.26 For sufficiently large scatter-
ing potentials, full T-matrix calculations become neces-
sary as in Ref. 27. However, large Vs,m would result in
strong spectral asymmetry between positive and negative
bias voltages,27 which differs from experimental observa-
tion.20,21,22 We also note that the energy dependence of
Vβ,γ reflects the spectral characteristics of the collective
modes and their interaction with quasiparticles and im-
purities. For instance, we expect Vγ ∼ ζγ
2 for pinned
SDW, where ζ is the impurity pinning strength and γ
is the coupling amplitude of quasiparticles with SDW
fluctuations.23,24 Empirically for nearly optimally doped
Bi-2212, Rj ranges from 5 ∼ 10.
31 Here we take different
values for Rj with a mean value 〈Rj〉 = 10.
Given the Hamiltonian and the scattering potentials
Vα,β,γ(q), we find that for infinite quasiparticle lifetime
and in the first-order T-matrix approximation, the FT
of the LDOS ρ(r, E) that involves elastic scattering of
quasiparticles from momentum k to k+ q is:
ρq(ω) = −
1
piN2
limδ→0
∑
k Vα,β,γ(q)×
{uk+quk (uk+quk ∓ vk+qvk)ℑ
[
1
(ω−Ek+iδ)(ω−Ek+q+iδ)
]
+uk+qvk (uk+qvk ± vk+quk)ℑ
[
1
(ω+Ek+iδ)(ω−Ek+q+iδ)
]
+vk+quk (uk+qvk ± vk+quk)ℑ
[
1
(ω−Ek+iδ)(ω+Ek+q+iδ)
]
−vk+qvk (uk+quk ∓ vk+qvk)ℑ
[
1
(ω+Ek+iδ)(ω+Ek+q+iδ)
]
}.(3)
Here N is the total number of unit cells in the sam-
ple, and ℑ[. . . ] denotes the imaginary part of the quan-
tity within the brackets, which is related to the equal-
energy quasiparticle joint density of states. The up-
per (lower) sign in the coherence factor applies to
spin-independent (spin-dependent) interactions, uk and
vk are the Bogoliubov quasiparticle coefficients, u
2
k +
v2k = 1, u
2
k = [1 + (ξk/Ek)] /2, ξk ≡ ǫk − µ, ǫk
is the tight-binding energy of the normal state of Bi-
2212 according to Norman et al.34, ǫk = t1(cos kx +
cos ky)/2 + t2 cos kx cos ky + t3(cos 2kx + cos 2ky)/2 +
t4(cos 2kx cos ky + cos kx cos 2ky)/2 + t5 cos 2kx cos 2ky,
t1−5 = −0.5951, 0.1636,−0.0519,−0.1117, 0.0510 eV, µ
is the chemical potential, and Ek =
√
ξ2k +∆
2
k.
FIG. 1: Calculated energy-dependent Fourier transform (FT)
maps of quasiparticle LDOS in the first Brillouin zone
with randomly distributed non-magnetic point defects us-
ing Eq. (3) and Vα: (a) ∆d = 40 meV and (ω/∆d) =
±0.15,±0.45,±0.75 (up and down from left to right); (b)
∆d = 20 meV and (ω/∆d) = 0.15, 0.45, 0.75 (left to right).
(c) Schematic illustration of the equal-energy contours and
representative modulation wavevectors qA, qB and qC , which
correspond to q1, q7 and q2 in Refs. 20,21.
Using Eq. (3) and Vα,β,γ(q), we obtain the energy-
dependent FT-LDOS maps in the first Brillouin zone for
non-magnetic point impurities in Fig. 1 with two differ-
ent ∆d values and for pinned SDW (with spin-dependent
coherence factor) in Fig. 2, whereas the corresponding
LDOS modulations due to Vα,β,γ(q) in real space is
shown in Figs. 3(a)-(c). For non-magnetic point impurity
scattering at T ≪ Tc, the intensities associated with qB
and qC are much stronger than those of qA, as shown in
Fig. 1 and also in Fig. 4(a). The results in Fig. 1 differ
from the STM observation20,21 that reveals comparable
3FIG. 2: Energy-dependent FT-LDOS maps with randomly
distributed pinned SDW using Eq. (3) and Vγ : (a) ∆d = 40
meV and (ω/∆d) = ±0.15,±0.45,±75, up and down from left
to right; (b) ∆d = 20 meV and (ω/∆d) = 0.15, 0.45, 0.75, from
left to right. The FT-LDOS does not exhibit discernible dif-
ferences in the spectral characteristics except the total inten-
sities if we simply replace Vγ by Vβ and assume non-magnetic
coherence factors in Eq. (3).
intensities associated with qA and qB , and weaker inten-
sities with qC . Interestingly, the intensities of qA and
qB,C become reversed if one assumes magnetic point im-
purity scattering, as illustrated in Fig. 4(b). However,
there is no evidence of magnetic scattering in the samples
used in Refs. 20,21. In contrast, the presence of pinned
collective modes, regardless of CDW or SDW, gives rise
to much stronger intensities for qA (by about two orders
of magnitude), as shown in Fig. 2. Thus, the empirical
FT-LDOS maps20,21 cannot be solely attributed to quasi-
particle scattering by non-magnetic point impurities.
FIG. 3: Real space quasiparticle LDOS for a (400 × 400)
area at T = 0 due to scattering by (a)non-magnetic point
impurities, (b) pinned CDW and (c) pinned SDW, for ∆d =
40 meV and ω = 30 meV.
The relevance of collective modes become indisputable
when we consider the temperature dependence of the FT-
LDOS. As shown in Fig. 5(a), in the limit of T → T−c ,
the q-values contribute to the FT-LDOS map become sig-
FIG. 4: Evolution of the relative intensities of FT-LDOS with
energy (ω) for qA, qB and qC as defined in Fig.1(c) and Vs, Vm
and V0 all taken to be unity: quasiparticle scattering by (a)
single non-magnetic point impurity, and (b) single magnetic
point impurity.
FIG. 5: The FT-LDOS maps at T = 0, 0.75Tc and Tc (from
left to right) for (a) point impurities Vα(q) and (b) pinned
SDW Vγ(q). We assume ∆d(T ) = ∆d(0)[1 − (T/Tc)]
1/2,
∆d(0) = 40 meV, tunneling biased voltage = 18 mV,
and Tc = 80 K. Besides temperature dependent coher-
ence factors, the thermal smearing of quasiparticle tunnel-
ing conductance (dI/dV ) is obtained by using (dI/dV ) ∝
|
∫
ρq(E)(df/dE)|(E−eV )dE|, where f(E) denotes the Fermi
function. (c) |qA|-vs.-V (biased voltage) dispersion relation
for pinned SDW at T = 0 and Tc.
nificantly extended and smeared for point-impurity scat-
tering. In contrast, pinned SDW yields strong intensi-
ties in the FT-LDOS map only at qA for T >∼ Tc, as
shown in Fig. 5(b). The overall energy dispersion due to
pinned SDW is weaker than that due to point impuri-
ties, as shown in Fig. 5(c) for |qA|-vs.-V (biased voltage)
at both T = 0 and T = Tc. In particular, we note that
the dispersion is further reduced at Tc. These findings
are consistent with recent experimental observation by
4Yazdani et al.29
The energy, momentum and temperature dependence
of our calculated FT-LDOS in Figs. 1-5 is supportive
of spatially modulated collective modes being relevant
low energy excitations in cuprates besides quasiparticles.
In particular, only pinned collective modes can account
for the observation in the FT-LDOS map above Tc. Al-
though our simplified model cannot exclude CDW, we
note that pinned CDW would have coupled directly to
the quasiparticle spectra and resulted in stripe-like peri-
odic local conductance modulation, which has not been
observed in STM studies. On the other hand, various
puzzling phenomena seem reconcilable with the SDW
scenario. For instance, the nano-scale gap variations ob-
served in Bi-221231 may be understood by noting that
the LDOS in regions with disorder-pinned SDW contains
information of disorder potential coupled with quasipar-
ticles and SDW, so that the hump-like spectral features
at ±∆∗ represent neither the SDW gap nor the super-
conducting gap ∆d, and the values of ∆
∗ vary in accor-
dance with the disorder potential. The long-range spatial
homogeneity of quasiparticle spectra in YBa2Cu3O7−δ
(YBCO)35 as opposed to the strong spatial inhomogene-
ity in Bi-2212 can also be reconciled in a similar context.
That is, SDW can be much better pinned in extreme two-
dimensional cuprates like Bi-2212 than in more three-
dimensional cuprates such as YBCO. Furthermore, SDW
can be stabilized by magnetic fields,23,24,36 which natu-
rally account for the checkerboard-like spectral modula-
tions around the vortex cores of Bi-2212.19,37 Finally, the
smooth evolution of the pseudogap phase with tempera-
ture through Tc may contribute to the anomalously large
Nernst effect under a c-axis magnetic field above Tc,
38
with spin fluctuations responsible for the excess entropy.
In summary, we employ first-order T-matrix approx-
imation to study modulations in the quasiparticle FT-
LDOS of cuprates as a function of energy, momentum and
temperature. Our results suggest that a full account for
all aspects of experimental observation below Tc must in-
clude collective modes as relevant low energy excitations
besides quasiparticles, and that only collective modes can
account for the observed FT-LDOS above Tc.
Acknowledgments
We thank Professors Subir Sachdev, Doug Scalapino
and C. S. Ting and Mr. Yuan-Yu Jau for useful dis-
cussions. This research was supported by NSF Grant
#DMR-0103045.
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Collective modes and quasiparticle interference on the local density of states of cuprate superconductors

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Collective modes and quasiparticle interference on the local density of states of cuprate superconductors

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