Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity (Review Article)
A term first coined by Mott back in 1968 a “pseudogap” is the depletion of the electronic density of states at the Fermi level, and pseudogaps have been observed in many systems. However, since the discovery of the hightemperature superconductors (HTSC) in 1986, the central role attributed to the ps...
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nasplib_isofts_kiev_ua-123456789-1220742025-02-23T20:00:42Z Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity (Review Article) Kordyuk, A.A. К 70-летию со дня рождения В. М. Локтева A term first coined by Mott back in 1968 a “pseudogap” is the depletion of the electronic density of states at the Fermi level, and pseudogaps have been observed in many systems. However, since the discovery of the hightemperature superconductors (HTSC) in 1986, the central role attributed to the pseudogap in these systems has meant that by many researchers now associate the term pseudogap exclusively with the HTSC phenomenon. Recently, the problem has got a lot of new attention with the rediscovery of two distinct energy scales (“two-gap scenario”) and charge density waves patterns in the cuprates. Despite many excellent reviews on the pseudogap phenomenon in HTSC, published from its very discovery up to now, the mechanism of the pseudogap and its relation to superconductivity are still open questions. The present review represents a contribution dealing with the pseudogap, focusing on results from angle resolved photoemission spectroscopy (ARPES) and ends up with the conclusion that the pseudogap in cuprates is a complex phenomenon which includes at least three different “intertwined” orders: spin and charge density waves and preformed pairs, which appears in different parts of the phase diagram. The density waves in cuprates are competing to superconductivity for the electronic states but, on the other hand, should drive the electronic structure to vicinity of Lifshitz transition, that could be a key similarity between the superconducting cuprates and iron-based superconductors. One may also note that since the pseudogap in cuprates has multiple origins there is no need to recoin the term suggested by Mott. I acknowledge discussions with L. Alff, A. Bianconi, S.V. Borisenko, B. Büchner, A.V. Chubukov, T. Dahm, I. Eremin, D.V. Evtushinsky, J. Fink, A.M. Gabovich, M.S. Golden, M. Grilli, D.S. Inosov, A.L. Kasatkin, T.K. Kim, Yu.V. Kopaev, M.M. Korshunov, S.A. Kuzmichev, V.V. M. Loktev, I.A. Nekrasov, S.G. Ovchinnikov, N.M. Plakida, M.V. Sadovskii, A.V. Semenov, S.G. Sharapov, D.J. Scalapino, A.L. Solovjov, M.A. Tanatar, T. Valla, C.M. Varma, A.N. Yaresko, and V.B. Zabolotnyy. The work was supported by the National Academy of Sciences of Ukraine (project 73-02-14) and the State Fund for Fundamental Research (project F50/052). 2015 Article Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity (Review Article) / A.A. Kordyuk // Физика низких температур. — 2015. — Т. 41, № 5. — С. 417-444. — Бібліогр.: 318 назв. — англ. 0132-6414 PACS: 74.20.–z, 74.25.Jb, 74.70.Xa, 79.60.–i https://nasplib.isofts.kiev.ua/handle/123456789/122074 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 70-летию со дня рождения В. М. Локтева К 70-летию со дня рождения В. М. Локтева |
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К 70-летию со дня рождения В. М. Локтева К 70-летию со дня рождения В. М. Локтева Kordyuk, A.A. Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity (Review Article) Физика низких температур |
| description |
A term first coined by Mott back in 1968 a “pseudogap” is the depletion of the electronic density of states at the Fermi level, and pseudogaps have been observed in many systems. However, since the discovery of the hightemperature superconductors (HTSC) in 1986, the central role attributed to the pseudogap in these systems has meant that by many researchers now associate the term pseudogap exclusively with the HTSC phenomenon. Recently, the problem has got a lot of new attention with the rediscovery of two distinct energy scales (“two-gap scenario”) and charge density waves patterns in the cuprates. Despite many excellent reviews on the pseudogap phenomenon in HTSC, published from its very discovery up to now, the mechanism of the pseudogap and its relation to superconductivity are still open questions. The present review represents a contribution dealing with the pseudogap, focusing on results from angle resolved photoemission spectroscopy (ARPES) and ends up with the conclusion that the pseudogap in cuprates is a complex phenomenon which includes at least three different “intertwined” orders: spin and charge density waves and preformed pairs, which appears in different parts of the phase diagram. The density waves in cuprates are competing to superconductivity for the electronic states but, on the other hand, should drive the electronic structure to vicinity of Lifshitz transition, that could be a key similarity between the superconducting cuprates and iron-based superconductors. One may also note that since the pseudogap in cuprates has multiple origins there is no need to recoin the term suggested by Mott. |
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| author |
Kordyuk, A.A. |
| author_facet |
Kordyuk, A.A. |
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Kordyuk, A.A. |
| title |
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity (Review Article) |
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Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity (Review Article) |
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Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity (Review Article) |
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Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity (Review Article) |
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Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity (Review Article) |
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pseudogap from arpes experiment: three gaps in cuprates and topological superconductivity (review article) |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 70-летию со дня рождения В. М. Локтева |
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| citation_txt |
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity (Review Article) / A.A. Kordyuk // Физика низких температур. — 2015. — Т. 41, № 5. — С. 417-444. — Бібліогр.: 318 назв. — англ. |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5, pp. 417–444
Pseudogap from ARPES experiment: three gaps
in cuprates and topological superconductivity
(Review Article)
A.A. Kordyuk
Institute of Metal Physics of the National Academy of Sciences of Ukraine, Kyiv 03142, Ukraine
E-mail: kordyuk@gmail.com
Received December 24, 2014, published online March 23, 2015
A term first coined by Mott back in 1968 a “pseudogap” is the depletion of the electronic density of states at
the Fermi level, and pseudogaps have been observed in many systems. However, since the discovery of the high-
temperature superconductors (HTSC) in 1986, the central role attributed to the pseudogap in these systems has
meant that by many researchers now associate the term pseudogap exclusively with the HTSC phenomenon. Re-
cently, the problem has got a lot of new attention with the rediscovery of two distinct energy scales (“two-gap
scenario”) and charge density waves patterns in the cuprates. Despite many excellent reviews on the pseudogap
phenomenon in HTSC, published from its very discovery up to now, the mechanism of the pseudogap and its re-
lation to superconductivity are still open questions. The present review represents a contribution dealing with
the pseudogap, focusing on results from angle resolved photoemission spectroscopy (ARPES) and ends up with
the conclusion that the pseudogap in cuprates is a complex phenomenon which includes at least three different
“intertwined” orders: spin and charge density waves and preformed pairs, which appears in different parts of the
phase diagram. The density waves in cuprates are competing to superconductivity for the electronic states but,
on the other hand, should drive the electronic structure to vicinity of Lifshitz transition, that could be a key simi-
larity between the superconducting cuprates and iron-based superconductors. One may also note that since
the pseudogap in cuprates has multiple origins there is no need to recoin the term suggested by Mott.
PACS: 74.20.–z Theories and models of superconducting state;
74.25.Jb Electronic structure;
74.70.Xa Pnictides and chalcogenides;
79.60.–i Photoemission and photoelectron spectra.
Keywords: pseudogap, superconductivity, electronic ordering, Fermi surface topological transition, angle resolved
photoemission spectroscopy.
Contents
1. Introduction ................................................................................................................................... 418
2. Theories of pseudogap .................................................................................................................. 419
3. Pseudogap in experiments ............................................................................................................. 421
4. Pseudogap in Cu-SC and transition metal dichalcogenides ........................................................... 427
4.1. Measuring gaps in ARPES .................................................................................................... 428
4.2. Two gaps in Cu-SC ............................................................................................................... 430
4.3. Charge density wave gaps in transition metal dichalcogenides ............................................. 431
4.4. Charge density wave in cuprates ........................................................................................... 432
4.5. Van Hove singularities nesting and Mott gap in transition metal dichalcogenides ............... 433
4.6. Three gaps in Cu-SC ............................................................................................................. 434
4.7. Two sides of the phase diagram ............................................................................................ 435
5. Pseudogap in Fe-SC ...................................................................................................................... 435
6. Pseudogap and superconductivity ................................................................................................. 436
7. Conclusions ................................................................................................................................... 437
References ......................................................................................................................................... 437
© A.A. Kordyuk, 2015
A.A. Kordyuk
1. Introduction
The term pseudogap was suggested by Nevill Mott in
1968 [1] to name a minimum in the electronic density of
states (DOS) of liquid mercury at the Fermi level. Later he
had shown that when this pseudogap is deep enough the
one-electron states become localized [2].
Next, the term pseudogap was narrowed to “fluctuating
band gap”, the gap formed by fluctuating charge density
wave (CDW) at a Peierls transition [3] in quasi-one-
dimensional (1D) metals [4–7], as shown in Fig. 1.
In fact, the systems with fluctuating CDW can be de-
scribed similarly to disordered systems without long-range
order [8], so, the pseudogap should not be necessarily re-
lated with low dimensionality. Indeed, in quasi-two-dimen-
sional (2D) metals with CDW ordering, such as transition
metal dichalcogenides (TMD) [9] (see also recent review
[10]), the fluctuation effects are considered negligible but a
partial gap, which can be called “pseudogap” according to
the Mott’s definition, appears in a number of CDW phases.
Two such kinds of pseudogaps have been discussed: tradi-
tional Peierls gap but smeared out due to incommensura-
bility [11,12] (or, may be, short-range-order CDW fluctua-
tions [13] as in “nearly commensurate” [14,15] or “quasi-
commensurate” [16] CDW state); and a “correlation gap”
of Mott–Hubbard insulating phase in a commensurate
CDW state [16–18].
Curiously, except the study of fluctuating effects in 1D
CDW compounds, the pseudogap phenomena in 2D CDW
systems, despite a variety of the aforementioned possibi-
lities, had not earned so much attention [9,19] as it had
done later in the field of high-temperature superconductors
(HTSC) [20–28] for which it is often considered unique [27].
On one hand, the discovery of the superconducting cup-
rates (Cu-SC) slowed down noticeably the study of the
CDW-materials. On the other hand, the role of the pseu-
dogap in HTSC might be greatly exaggerated — partly due
to real complexity of the phenomenon but partly because
a lot of people struggling to find the mechanism of high-
temperature superconductivity needed a “guilty” why that
has appeared to be so hard. In this sense, the well-turned
definition of the pseudogap in cuprates as “a not-under-
stood suppression of excited states” was given by Robert
Laughlin in early years of HTSC era [29].
Nevertheless, the pseudogap phenomenon in cuprates
has stimulated appearance of many fascinating theories
(some of which will be briefly overviewed in Sec. 2), and
has been extended to a number of other materials, for ex-
ample, A15 superconductors [30], manganites [31,32],
Kondo insulators [33,34], thin films of conventional super-
conductors [35] and nanoislands [36,37], Co–Fe-based half
metals [38], ultracold Fermi gases [39].
In many of those systems the pseudogap phenomenon is
discussed as a pseudogap phase on the phase diagrams of
temperature vs charge carrier concentration (also called
“doping”) or vs pressure, where the pseudogap phase neigh-
bors both the density wave and superconducting phases.
Fig. 1. The electronic density of states normalized to the metallic
density of state, plotted versus ω/kTc, for various temperatures.
The T/Tc = 0 curve is the mean-field result. After [5].
Fig. 2. (Color online) Examples of the phase diagrams of quasi-2D metals in which the charge or spin ordering compete or coexist with
superconductivity and a pseudogap phase: a transition metal dichalcogenide [40] (a), a high-Tc cuprate [41] (b), and an iron-based su-
perconductor [42] (c).
418 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity
Figure 2 shows three recent examples of such phase dia-
grams for a transition metal dichalcogenide [40] (a), a high-Tc
cuprate [41] (b), and an iron-based superconductor [42] (c).
In present review we mostly discuss these three families
of quasi-2D superconductors from an empirical point of
view, focusing on results from the angle resolved photo-
emission spectroscopy (ARPES), which is the most direct
tool to access the electronic density of states at the Fermi
level [43–45]. We end up with the conclusion that the
pseudogap in cuprates is a complex phenomenon which
includes at least three different “intertwined” orders: spin
and charge density waves (similar to the 2D CDW com-
pounds) and preformed pairs, which appears in different
parts of the phase diagram. The density waves in cuprates
are competing to superconductivity for the electronic states
but, on the other hand, should drive the electronic structure
to vicinity of Lifshitz topological transition, the proximity
to which is shown to correlate to cT maximum in all the
iron-based superconductors (Fe-SC) [46].
The paper is organized as following. Section 2 gives a
short overview of selected theories of the pseudogap in
cuprates. The manifestations of the pseudogap in different
experiments are briefly discussed in Sec. 3. Then, in the
rest of the paper, the focus is made on ARPES results,
starting from a short introduction to ARPES data analysis
and gap extraction methods (Sec. 4.1), the pseudogap phe-
nomenon is considered in HTSC cuprates and CDW bear-
ing TDM in Sec. 4. The growing evidence for the pseudo-
gap in Fe-SC are reviewed in Sec. 5. Possible relation of
the pseudogap to superconductivity is discussed in Sec. 6.
2. Theories of pseudogap
The theories of the pseudogap in cuprates are reviewed
in a number of papers [21–26,47–50] and textbooks [28,51].
Most of these theories can be classified by their predictions
about a crossover line, T*, which borders the pseudogap
phase from a normal metal (or a “strange metal”) on T–x
phase diagram (see Fig. 3). Here I briefly recall some of
the most discussed models.
Diagram (a) is for the models which consider the
pseudogap phase as a precursor to the superconducting
state, the preformed pairs scenarios [21,52].
The fluctuations in bulk clean superconductors are ex-
tremely small. It is evident from very sharp transitions of
thermal and electrical properties and has been shown theo-
retically by Levanyuk and Ginzburg back in 1960 [52]. The
corresponding Ginzburg number 4= / ( / )c c FGi T T T Eδ
10–12–10–14, where Tδ is the range of temperatures in
which the fluctuation corrections are relevant and FE is the
Fermi energy. In thin dirty superconducting films the fluctu-
ations should be increased drastically [53]: = /c FGi T E for
clean 2D superconductor and 1/ FGi E−τ for dirty 2D
superconductor [52], where 1−τ is the quasiparticle scatter-
ing rate at .FE Thus, the width of the superconducting tran-
sition became experimentally measurable, but still 1.Gi
The said behavior was deduced for the conventional su-
perconductors to which the mean-field BCS theory or the
Ginzburg–Landau model of the second-order phase transi-
tion is applicable. In superconductors with very small cor-
relation length ξ ( 1)Fkξ the Bose–Einstein condensa-
tion (BEC) of local pairs takes place at cT while the
formation of singlet electron pairs (that could be bipola-
rons [54]) is assumed at some higher temperature [28].
Therefore, soon after discovery of HTSC, when it became
clear that these materials are quasi-2D and dirty, with ex-
tremely small ξ , the superconductive fluctuations was the
first scenario for the pseudogap [55]. Moreover, in strictly
2D systems, the phase fluctuations of the order parameter
destroy the long-range order at finite temperature and only
the Berezinskii–Kosterlitz–Thouless (BKT) superconduct-
ing instability may occur [21,56].
The “phase fluctuation” scenario [57,58] stems from the
empirical “Uemura relation”, that cT is proportional to the
zero-temperature superfluid density (0)sn (or “phase stiff-
ness”) [59,60]. It was suggested that HTSC with low su-
perconducting carrier density are characterized by a rela-
tively small phase stiffness for the superconducting order
parameter and by poor screening, both of which imply
a significantly larger role for phase fluctuations. So, the pseu-
Fig. 3. (Color online) Three theoretical idealizations for the interplay of pseudogap (PG) and superconductivity (SC) in the temperature-
doping phase diagram of the HTSCs. Tc, T *, and Tcoh temperatures represent the phase transition to the SC state and crossovers to the PG
and a coherent states, respectively.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 419
A.A. Kordyuk
dogap state is a region where the phase coherence is de-
stroyed, but the amplitude of the order parameter remains
finite. Two crossover lines in Fig. 3(a), T* and Tcoh, border
the regions where pairs are formed and become coherent,
respectively, while superconductivity appears only under
both lines [58].
One should note that calculated T*(x) for either phase
fluctuation [51] or BKT model [21,61] show decrease with
lowering the charge carrier density, as in Fig. 4. Also,
while the experimental T*(x) dependence looks universal
for all the hole doped cuprates, the fluctuation effects
should be very sensitive to dimensionality and therefore
different for different families. For example, the striking
difference in the shape of the specific heat anomaly at cT
is observed for quasi-2D Bi2Sr2CaCu2O8+x (BSCCO or
Bi-2212), where it follows the BEC phase transition, and
for more 3D YBa2Cu3O7–δ (YBCO), with classical BCS
jump [62]. Another problem of BEC models is that the
Bose quasiparticles have no Fermi surface, while it is
clearly observed by ARPES [21,28].
Nevertheless, recently, the “checkerboard” pattern ob-
served in experiments [63–65] has been explained by the
model in which CDW is induced by superconducting fluc-
tuations [66].
The spin singlet scenario [47,67] leads to the same
phase diagram: the spin singlets play the role of preformed
pairs, i.e., the pseudogap state is a liquid of spins without
long-range order (the original RVB idea of Anderson [68])
and superconductivity occurs below two crossover lines
due to spin-charge recombination. Similar considerations
occur also for the SO(5) model [69] which attempts to uni-
fy antiferromagnetism and superconductivity. An impor-
tant aspect of these scenarios is the general doping depend-
ence of T*. Since the energy gain associated with spin
singlet formation is the superexchange energy, J, the T*
line is proportional to ,J tx− where t is the hopping energy
of the doped hole [26,70].
Diagram (b) in Fig. 3 is for scenarios in which another
order with a quantum critical point (QCP) interplays with
superconductivity. In QCP theories [71–73], the transition
between the ordered and disordered quantum phases trans-
forms in a region of critical fluctuations which can mediate
singular interactions between the quasiparticles, providing
at the same time a strong pairing mechanism [74]. As for
the nature of QCP, various proposals have been discussed.
In Ref. 75, in which CDW and QCP were put together
for the first time, it had been proposed that in the presence
of the long-range Coulomb forces a uniform Fermi liquid
can be made unstable by a moderate electron–phonon cou-
pling (Hubbard–Holstein model) giving rise to incommen-
surate CDW in the form of “frustrated phase separation”,
and the related QCP around optimal doping. Within this
scenario, the static CDW compete (and kill) superconduc-
tivity like in some 1/8 doping systems, but, as long as
CDW fluctuations stay dynamic, they can mediate super-
conductivity, and even the d-wave pairing can arise from
CDW fluctuations without any spin interaction [76,77]. It
was noted that CDW may also evolve into a spin-charge
separation deeper in the charge-ordered phase as a conse-
quence of modulation of charge density, anharmonic ef-
fects [78], closer proximity to the antiferromagnetic (AFM)
phase, pinning, and so on.
The QCP determined by magnetic interaction [79] leads
to the spin-fluctuation scenarios. In the spin-fermion mo-
del, the pseudogap phase reflects the onset of strong AFM
spin correlations, a spin-liquid without long-range order
[80–83]. The full analysis of the normal state properties of
the spin-fermion model near the antiferromagnetic instabil-
ity in two dimensions was given in [84]. Recently, it has
been shown [50,85] that within this model, a magnetically
mediated interaction, which is known to give rise to d-wave
superconductivity and charge order with momentum along
zone diagonal [189], also gives rise to the charge density
wave with a “d-symmetry form factor” consistent with re-
cent experiments [87].
The antiferromagnetic scenario within the Hubbard mo-
del was also considered in the two-particle self-consistent
approach [48,88] and studied within a generalized dynam-
ical mean-field theory [89,90].
Several exotic scenarios of symmetry breaking, in
which T * would be a true phase line, had been also sug-
gested. For example, the orbital current state proposed by
Varma [91] and the “flux-density wave” [92] or “d-density
wave” current state [93].
Diagram (c) in Fig. 3 is a result of similar competition
between superconductivity and another ordering which
does not require the QCP for its understanding. These
could be either a spin-charge separation, predicted [94–96]
and found [97] long ago in some families of cuprates and
known as “stripes”, or “ordinary” (Peierls type) CDW or
spin density wave (SDW) [98–100], like in the transition
metal dichalcogenides [9,10]. The former can be responsi-
ble for the pseudogap in one-electron spectrum either due
to density wave [101] or by causing an electronic nematic
order (quantum liquid-crystal) [102]. Broken rotational
symmetry in the pseudogap phase of cuprates is really ob-
served [103]. And nematic order becomes very fashionable
today [101,104].
Fig. 4. Effect of thermal and quantum phase fluctuations (left),
and of dimensional crossover (right) on the critical temperature
for phase coherence Tc. Adapted from [61].
420 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity
A driving force for the Peierls type ordering is peculiarity
of the electronic band structure: either the Fermi surface
nesting [11,13] or nesting of Van Hove singularities (VHs)
[105–107]. Nowadays, the FS nesting is considered respon-
sible for CDW and pseudogap not only in cuprates and tran-
sition metal dichalcogenides but also in a number of other
low-dimensional metals such as manganites [108,109], bina-
ry and ternary molybdenum oxides [110], Bi-dichalcoge-
nide layered superconductors [111,112], etc. The competi-
tion between density wave and superconductivity is usually
considered in frame of the Bilbro–McMillan relation [113],
according to which SC∆ and 2 2
SC DW∆ + ∆ increase essen-
tially identically with falling temperature, so, the density
wave is suppressed by superconductivity and can be sup-
pressed completely, as shown in Fig. 3(c). Recently it has
been shown that for QCP models this relation will lead to
max 2 max 2( ( )/ ) ( ( )/ ) = 1DW DW c cT x T T x T+ [114].
Many other possible reasons for pseudogap formation
have been suggested, such as, for example, an intrinsic
inhomogeneity [115], d-wave-type Fermi surface defor-
mations (Pomeranchuk instability) [116], or interaction
with diatomic negative U centers [117], but it is hardly
possible even to mention all of them here.
To conclude, there are many theories for the pseudogap
phenomenon in HTSC and, may be consequently, there is
no consensus on its origin. On the other hand, it seems that
the main problem of the acceptance of these theories, until
recently, was a general expectation that they should de-
scribe the whole pseudogap region on the phase diagram
and all its experimental manifestations, briefly considered
in the following section. Nowadays, there is growing evi-
dence that the cuprates do indeed provide a complicated
background for theorists revealing simultaneously a bunch
of different phenomena: stripes, CDW, SDW, electronic
fluctuations and localization. Thus, it seems that at least
several of those models are related to reality of HTSC.
3. Pseudogap in experiments
Opening of a gap or just a depletion of the electronic
density of states at the Fermi level can hardly be missing
by a number of experimental probes. Indeed, any transition
to one of possible CDW states in, for example, transition
metal dichalcogenides, left signatures in temperature de-
pendences of different experimental parameters: heat capa-
city, resistivity, magnetic susceptibility, etc. Those signa-
tures were usually accompanied by change of the diffraction
patterns, so, the character of the symmetry change was
more or less clear [9].
In cuprates, the pseudogap was observed in many ex-
periments as something that starts to happen above cT [20],
while any indication of new order could not be found by
diffraction techniques. Then a depletion of the spectral
weight was observed directly by ARPES [118,119] and tun-
neling spectroscopy [120,121], and some kind of CDW/SDW,
a spin-charge separation in form of “stripes”, was found in
some HTSC compounds [97]. Nowadays, there are many
experimental evidences for CDW in almost all families of
cuprates, but the nature of the pseudogap remains puzzling.
In this section, before turning to the ARPES results, we
briefly consider experimental manifestations of the pseudo-
gap in cuprates by other experimental probes: spectroscopic
methods such as nuclear magnetic resonance (NMR), infra-
red optical conductivity (IR), Raman scattering (RS), and
tunneling spectroscopies (except STM/STS these are intrin-
sic tunneling, superconductor/insulator/normal-metal (SIN)
and superconductor/insulator/superconductor (SIS) tunnel-
ing, and Andreev reflection tunneling (AR)), and inelastic
neutron scattering (INS), as well as traditional thermody-
namic/transport probes such as heat conductivity and resis-
tivity measurements (or “dc conductivity”). ARPES and
tunneling measure directly the density of single electronic
states while other spectroscopies as well as thermodynam-
ic/transport probe the two-particle spectrum.
NMR. The pseudogap in cuprates was first detected by
NMR [122,123], which measures the Knight shift, ,sK and
spin-lattice relaxation rate, 11/ .T The Knight shift is a meas-
ure of the polarization of electrons by the applied magnetic
field and is proportional to the real part of the paramagnetic
(Pauli) susceptibility, ( = 0, ),′χ ωq that, in the Fermi liquid
model is proportional to the density of states at the Fermi
level and should be independent on T. The spin-lattice relaxa-
tion rate is related to the imaginary part of susceptibility, such
as 2
11/ | ( ) | ( , )/ ,T T F ′′χ ω ω∑q q q
where ( )F q is the form
factor for the particular nuclear site — by probing various
nuclei in the unit cell one can probe different parts of mo-
mentum space [20,28]. In Fig. 5 the T-dependent Knight
shift (a) and the spin-lattice relaxation rate (b) are shown for
underdoped, optimally doped, and overdoped BSCCO [124].
The suppression of both quantities starts below T* but no
additional anomaly is seen at cT , that has been considered
in support of the preformed pairs scenario [26].
Specific heat. If a gap, which lowers the kinetic energy
of electrons, opens (or starts to develop) at T*, one should
see a peculiarity in any thermodynamic/transport quantity
at T* rather than at cT when the energy of the electrons
does not change. Indeed, the specific heat jump at cT fades
out with underdoping, see Fig. 5(c),(d), but usually there is
no jump at T* (though some measurements reveal a weak
bump [125]). In general, the specific heat data have fre-
quently been cited in support of diagram (b) of Fig. 3 [26]
since the determined T* line cuts through the cT dome
[126,127]. It is also consistent with the sharp decrease of
the specific heat jump or the superconducting condensation
energy 0U , defined as the entropy difference integrated
from = 0T to ,cT which is a constant 2
0 = 0.24BCS
n cU Tγ
for a BCS superconductor with d-wave pairing [28] (see
Fig. 5(e)). Based on those NMR and heat capacity data, it
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 421
A.A. Kordyuk
has been concluded [127] that the pseudogap and super-
conductivity are “two gaps”, independent and competing.
So, smooth evolution of tunneling spectra from the pseu-
dogap into superconductivity does not necessarily imply
the pseudogap is a short-range pairing state with the same
mean-field gap energy as superconductivity [128,129].
One should note that the interpretation of specific heat
measurements is tricky because at transition temperatures
the phonon contribution in cuprates is typically a hundred
times stronger than the electronic one [20] and differential
techniques should be used.
Transport properties. After discovery of a new super-
conductor, its transport properties, i.e., dc conductivity, Hall
effect, thermal conductivity and thermopower, are the first
quantities to study. Any phase transition which affects the
electronic density of states at the Fermi level should be seen
as a peculiarity on temperature dependences of transport
properties. Though, it is often hard to say which peculiarity
to expect. For example, the dc conductivity depends on both
charge carrier concentration n (or density of states at ,FE
(0))N and scattering time τ (in simplest Drude model,
2= / ).n e mσ τ If, due to CDW, a full gap opens, it is a
Peierls type of metal–insulator transition and resistivity
changes from metallic to insulating ( / < 0).d dTρ If a partial
gap opens, than n decreases but τ increases due to less space
for electron to scatter. So, depending on Fermi surface ge-
ometry, the resistivity (see Fig. 6(a) [9]) can show steps as
for 1T-TaS2, which has several subsequent transitions to an
incommensurate at 550 K, quasicommensurate (or “nearly
commensurate” [14,130]) at 350 K and commensurate CDW
at 180 K [131], or kinks as for 2H-TaSe2 with transitions
to an incommensurate at 122 K and commensurate CDW
a 90 K.
In cuprates, the transition to the pseudogap state is less
pronounced in resistivity (see Fig. 6(b)–(d) [132]) but still
detectable and heavily discussed. Soon after discovery of
HTSC, a peculiar feature of cuprates, a quasilinear depend-
ence of resistivity over a wide temperature range has been
found [133]. It means that the experimental magnitude of the
resistivity in cuprates at high temperatures is much larger
than the Ioffe–Regel limit considered within the convention-
al semiclassical transport theory based on the Boltzmann
equation [28]. This linear region on the phase diagram has
inspired appearance of many new HTSC theories modeling
this “strange metal” behavior, such as fluctuating staggered
currents [134] or the “marginal” Fermi liquid (MFL) model
[91]. On the other hand, it has been shown [135] that within
the t J− model the saturation resistivity should be much
larger than the Ioffe–Regel limit, so, the absence of satura-
tion of resistivity at high temperatures is expected for
strongly correlated systems.
The linear resistivity is observed only in a narrow re-
gion of temperatures near the optimal doping, as has been
shown [132] by mapping of the in-plane resistivity curva-
ture 2 2( / )abd dTρ of the La2–xSrxCuO4 (LSCO), YBCO,
and Bi2Sr2–zLazCuO6+δ (BSLCO) crystals (see Fig. 6).
The pseudogap temperature, determined on these maps as
the inflection point of the resistivity 2 2( / = 0),abd dTρ
Fig. 5. (Color online) The T-dependent Knight shift (a) and the spin-lattice relaxation rate (b) are shown for underdoped, optimally
doped, and overdoped BSCCO [124]. (c) Temperature dependence of the Sommerfeld constant for Y0.8Ca0.2Ba2Cu3O6+x, labels
show [126]. (d) A sketch to indicate the transition temperatures Tc, T * and a crossover to superconducting fluctuations, Tf, for a optimal-
ly doped and two underdoped samples. (e) The doping dependence of the gap energy Eg, of the condensation energy U0, and of Tc, the
SC gap determined from heat capacity is shown on (f) [127].
422 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity
decreases linearly with doping and terminates near the op-
timal value = 0.16.p Below T* the curvature is positive
until the superconducting fluctuations make it negative
again.
The idea of two pseudogaps has been confirmed by
measurements of the c-axis resistivity and magnetoresis-
tance [136]: while T* increases with decreasing hole doping
and is field-insensitive, a field-sensitive gap is found at low-
er temperature, which scales with ,cT and may be considered
therefore as a precursor to superconductivity. By applying
magnetic field to Y1–xCaxBa2(Cu1–yZny)3O7–δ thin films
and changing the Zn concentration to suppress both the su-
perconductivity and superconducting fluctuations, it has
been shown that the pseudogap region persists below cT on
the overdoped side and T* extrapolates to zero at about
0.19 holes concentration [137].
Nernst effect. The Nernst effect is considered as one of
the most convincing evidences for the existence of the pre-
formed pairs [20,28]. The Nernst effect in solids is the de-
tection of an electric field E perpendicular to orthogonally
applied temperature gradient T∇ and magnetic field H
[138]. The Nernst signal, defined as ( , ) = / ,Ne H T E T∇ is
generally much larger in ferromagnets and superconduc-
tors than in nonmagnetic normal metals. In the supercon-
ducting state, the Nernst signal is the sum of the vortex and
quasiparticle terms, = ,qp
N N Ne e e+v which can be distin-
guished with proper analysis, measuring the thermopower,
Hall angle, and resistivity in addition to the Nernst effect
[139]. In Fig. 7 the onset of Nev is defined by temperature
onsetT on the phase diagrams of LSCO and Bi-2212 (num-
bers on the contour curves indicate the value of the vortex
Nernst coefficient 0= /Ne Hν µv in nV/KT). The observation
of a large vortex Nernst signal in an extended region above
cT in hole-doped cuprates provides evidence that vortex
excitations survive there [138,139]. The results support the
preformed pairs scenario and suggest that superfluidity
vanishes because long-range phase coherence is destroyed
by thermally created vortices (in zero field). Interestingly,
in electron-doped cuprates (e.g., NCCO) where the PG is
believed absent the vortex Nernst signal is also absent. So,
the comparison of Nernst effect in hole and electron-doped
cuprates shows that the “thermally created vortices” are
not generic to any highly anisotropic layered superconduc-
tor but may be related to the physics of the pseudogap state
in hole-doped cuprates [138]. The vortex Nernst signal
above cT is analogous to an excess current observed in
Fig. 6. (Color online) Resistivity over phase transitions: (a) for selected transition metal dichalcogenides [9]; (b)–(d) for high-Tc
cuprates. Resistivity curvature maps for LSCO (e) and BSLCO (f) [132].
Fig. 7. (Color online) The phase diagrams of LSCO (left) and
BSCCO (right) showing the Nernst region between Tc and Tonset
(numbers on the contour curves indicate the value of the Nernst
coefficient). The Tonset-curves peak near x = 0.10. The dashed lines
are T * estimated from heat-capacity measurements. After [138].
180
160
140
120
100
80
60
40
20
0 0
T,
K
T,
K
La Sr CuO2– 4x x
0.1 0.1 0.2 0.2 0.3 0.3
Tonset Tonset
200
150
100
50
Bi-2212
Sr content x Hole density ρ
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 423
A.A. Kordyuk
the same temperature range in the Andreev contacts [140]
that also indicates the presence of Cooper pairs.
Optics. Like transport measurements, optical studies
of electronic spectra [141,142] provide information on
the spectrum of collective electron–hole pair excitations,
where a transition takes place from an initial state to a dif-
ferent final state. The difference is in final states. While in
transport techniques the initial and final states have the
same energy, in optics they hold the same momentum.
Both the first- and second-order processes of light scatter-
ing are used. In the former, the light excites bosonic de-
grees of freedom: phonons, electron–hole pairs, spin waves
or other electronic density fluctuations. These are studied
by infrared and optic absorption. The second-order pro-
cesses when a photon absorbed and reemitted are used in
the Raman scattering.
The absorption spectroscopy methods measure reflec-
tance on single crystals or transmission in thin-films, that
allows one to study the complex dielectric function
1 2( ) ( ) ( )iε ω = ε ω + ε ω in the long-wave limit ( = 0),q from
which the dynamical complex conductivity ( ) =σ ω
1 2( ) ( )i= σ ω + σ ω can be derived: 1 24 = ,πσ ωε 24 =πσ
1(1 )= ω − ε [28,142]. The real part of conductivity, 1( ),σ ω is
proportional to the joint density of states (Kubo–Greenwood
formula) and determines absorption of radiation at the fre-
quency ω. The real part of the inverse conductivity 1( )−σ ω
is proportional to the quasiparticle scattering rate 1−τ while
its imaginary part is proportional to mass renormalization
* / = 1 ( ).m m + λ ω The Kramers–Kronig (KK) relations allow
one to calculate both the real and the imaginary parts of
( )ε ω or ( )σ ω from the raw experimental data. In the ellipso-
metric technique [143], the real and imaginary parts of ( )ε ω
can be measured independently.
Simple Drude model predicts that reflectance decreases
monotonically with frequency. In HTSC, a structure in the
form of a “kink” was found. In underdoped materials, this
kink starts to develop already in the normal state at tem-
peratures similar to T∗ derived from other experiments
and, therefore, was interpreted as a manifestation of the
pseudogap. The corresponding changes in the optical con-
ductivity appears as a depletion of the spectral weight in
the range 300–700 cm–1 (about 40–90 meV) [144,145], as
one can see in Fig. 8(a)–(c) for YBCO. Since ( )σ ω just
above this range looks not changing with temperature, it
has been concluded that the gapped spectral weight is
shifted to lower frequencies, resulting in a narrowing of the
Drude peak [141]. The measurements over much wider
frequency range, as one can see in Fig. 8(d),(e) [146–148],
shows that much higher energies could be involved. Simi-
lar depletion by the pseudogap is observed for the derived
from conductivity scattering rate, as shown in Fig. 8(f)–(h)
for BSCCO [149].
Naturally, the origin of the pseudogap has been ad-
dressed in many optical studies. Most of that ideas can be
found in the topical reviews [20,141,142], which, neverthe-
less, ended with the conclusions that there is no unified
Fig. 8. (Color online) Optical spectroscopy data which show pseudogap in different HTSC. (a)–(c) Inplane conductivity of YBCO for va-
rious temperatures in both the normal and superconducting states [145]: (a) optimally doped with Tc = 93 K at T = 120, 100, 90 (dashed),
70, 20 K (from top to bottom); (b) underdoped 82 K at T = 150, 120, 90, 80 (dashed), 70, 20 K; (c) underdoped 56 K T = 200, 150, 120,
100, 80, 60 (dashed), 50, 20 K. (d), (e) Conductivity of electron-doped NCCO [147] and hole-doped LSCO [146] (symbols) in wider
frequency range (1 eV ≈ 8066 cm–1) compared to model calculations (solid lines) [148]. (f)–(h) Doping and temperature dependence of
the scattering rate of BSCCO [149].
424 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity
view on the nature of the pseudogap state. That was also
noted on controversy between optical experiments and
ARPES about coherence state [142]: from ARPES point of
view, it is set only below ,cT but infrared methods provide
evidence for coherence below the spin-gap temperature
> .s cT T Also, an important role of magnetic correlations in
the pseudogap state has been found by optical study of
(Sm,Nd)Ba2{Cu1–y(Ni,Zn)y}3O7–δ with magnetic (Ni) and
nonmagnetic (Zn) impurities [150]. The broadband infra-
red ellipsometry measurements of the c-axis conductivity
of underdoped RBa2Cu3O7–δ (R = Y, Nd, and La) have
separated energy scales due to the pseudogap and the su-
perconducting gap and provided evidence that these gaps
do not share the same electronic states [151].
Raman scattering, like optical absorption, measures
a two-particle excitation spectrum providing direct insight
into the total energy needed to break up a two-particle
bound state. In metals, the Raman effect is difficult to ob-
serve because of a small penetration depth and limited en-
ergy range [20]. The signal is often riding on a high back-
ground, which might result in a considerable data scat-
tering, and the nodal results need a numerical analysis [27].
But its big advantage, compared to the infrared spectrosco-
py, is that the symmetry selection rules enable to measure
some momentum dependence of the spectrum [160]. For
cuprates there are two useful momentum averages: B1g
symmetry, that is peaked at (π,0), and B2g symmetry, peak-
ed at (π/2,π/2). Figure 9 shows typical Raman spectra for
HgBa2CuO4–δ (Hg-1201) for these two symmetries [152].
One can see that the peaks in these two symmetries depend
on doping in opposite directions.
These two energy scales are plotted on the phase dia-
gram in Fig. 10 taken from Ref. 152, which has reanimated
the interest to the “two gaps” scenario discussed earlier
[128,129]. Very similar diagrams have been suggested in
Refs. 27 and 160. It has been noted that the B1g peak coin-
cides with the pseudogap values, 2 PG∆ derived from other
experiments, while the B2g peak follows the superconduct-
ing gap 2 = 8 .B ck T∆
Resonant inelastic x-ray scattering (RIXS) [161,162]
is similar to Raman spectroscopy but has the additional
advantage of full-momentum-space resolution. Despite re-
markable progress of this new spectroscopic technique in
the past decade [163], the results of this experiment are not
fully understood [162]. Nevertheless, many exciting RIXS
measurements have already been reported, and it is gener-
ally believed that RIXS can be an extremely powerful tool
to probe the interplay between charge, spin, orbital, and
lattice degrees of freedom. In particular, it has been shown
that RIXS is a suitable probe across all energy scales, in-
cluding pseudogap, charge-transfer gap, and Mott gap in
cuprates [164]. Recent RIXS experiments [165–168] to-
gether with x-ray diffraction [169] has revealed CDW or-
dering in cuprates.
Fig. 9. (Color online) Raman spectra for HgBa2CuO4+δ (Hg-1201)
for B2g (left) and B1g (right) symmetries. The arrows indicate
the position of the superconducting peak maxima. Ov.: overdoped;
Opt.: optimally doped; Und.: underdoped. After [152].
NR
B2g B1g
ANR
Opt. Opt.
95 K 95 K
Und. Und.
89 K 89 K
Und.
63 K
Ov.Ov.
92 K92 K
Und.
63 K
( ,0)( )π π π, ( ,0)( )π π π,
Γ Γ
T T<< cT T<< c
T T> cT T> c
H
ol
e
do
pi
ng
χ
(ω
)
’’
, a
rb
. u
ni
ts
0 0 200 200 400 400 600 600 800 800
Raman shift , cmω –1 Raman shift , cmω –1
417 cm–1
660 cm–1
400 cm–1
380 cm–1
505 cm–1515 cm–1
230 cm–1
Fig. 10. (Color online) Antinodal and nodal peak energies nor-
malized to max
cT for Hg-1201 [152], Bi-2212 [153,154], Y-123
[154] and LSCO [154]). The ratios max2 / cT∆ determined by
ARPES [155–157] and tunnelling spectroscopy [158,159] are
shown for comparison. After [152].
Nodal B2g
Antinodal B1g
m x4 /T Tc c = 4(1 – 82.6 (0.16 ) )2 – p
T T << c
10
5
0 0.05 0.10 0.15 0.20 0.25
Raman B1g B2g
Hg-1201
Bi-2212
Bi-2212
Bi-2212
Y-123
Y-123
LSCO
Tunnelling
ARPES
Doping p
ω
ω
A
N
N
,
/T
cm
ax
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 425
A.A. Kordyuk
Inelastic neutron scattering (INS) method works simi-
lar to RIXS but with neutrons instead of photons. Due to
large penetration depth it is the most “bulk” among the
spectroscopies considered here but requires very large sin-
gle crystals and has mainly been done on YBCO, LSCO,
and HgBa2CuO4+δ (Hg-1201). Like optical methods, INS
measures except phonons the two-particle (electron–hole)
excitations but with spin flip and with momentum resolu-
tion — in joint momentum-energy space. The most promi-
nent feature seeing by INS in cuprates is a “spin reso-
nance” [170] that is peaked at the antiferromagnetic wave-
vector and at energy about 40 meV. The resonance is a part
of a “hourglass shape” spin excitation spectrum [171,172]
which became incommensurate above and below the reso-
nance energy but never extends to zero energy being limit-
ed at low energies by the so-called spin-gap [173]. In the
normal state both YBCO and LSCO show a much weaker
spectrum, which is centered around = ( , )π πQ and is broad-
er in momentum than in the superconducting state. In the
pseudogap state, some intermediate picture is observed,
with a gradually sharpening response at the antiferromag-
netic wavevector, which has been considered as a precur-
sor of the magnetic resonance mode that starts to develop
below T* [173,174]. Other authors believe that there is no
justification for a separation of the normal state spin excita-
tions spectrum into resonant and nonresonant parts [175].
For the scope of this review, it is important to mention
the role of INS in discovery [97] and study [101,176] of
incommensurate SDW and CDW, called “stripes”, in the
hole-doped cuprates. As mentioned, the pseudogap can be
a consequence of fluctuating stripes [101] or an electronic
nematic order [102].
Commensurate AFM ordering has been observed in the
superconducting YBCO by elastic neutron scattering [177].
More recently, the polarized neutron diffraction experi-
ments on YBCO [178] and Hg-1201 [179] have shown an
existence of a magnetic order below T* consistent with the
circulating orbital currents and QCP scenario. This has
been further supported by INS observation of a 5256 meV
collective magnetic mode appearing below the same tem-
perature [180]. The idea of the intra-unit-cell magnetic
order has been also supported by recent polarized elastic
neutron scattering experiments on BSCCO [181] which
raise important questions concerning the range of the mag-
netic correlations and the role of disorder around optimal
doping.
Tunneling spectroscopies, like ARPES, measure the
single-particle density of states. So, it is the most direct
probe to see the pseudogap in Mott’s definition [1]. There
are a number of different tunneling probes: intrinsic tunnel-
ing spectroscopy [182,183], Andreev reflection tunneling
(ART) [184,185], superconductor/insulator/superconductor
(SIS) [186] tunneling (in fact, both ART and SIS probe the
two-particle DOS) and superconductor/insulator/normal
metal (SIN) [120,187], as well as scanning-tunneling mi-
croscopy/spectroscopy (STM/STS) [121,188]. The latter
provides sub-atomic the spatial resolution and, with the Fou-
rier transformation [189,190], an access to the momentum
space [191–193].
The most convincing tunneling results showing that the
superconducting and pseudogaps represent different coex-
isting phenomena were obtained by intrinsic tunneling from
one- and two-layers BSCCO [182,183,194,195]. The data
for T* presented in Fig. 11 have been obtained by SIS tun-
Fig. 11. (Color online) Pseudogap in tunneling spectroscopy on BSCCO. (a) SIN tunneling spectra of optimally doped sample (Tc =
= 85–90 K) [120], note that zero bias tunneling conductance G(0) does not saturates at T * ≈ 150 K (b). (c) STM spectra for underdoped
sample (83 K) [121], the depletion of the density of states at Fermi level is seen to persist in the normal state, the size of the pseudogap
looks independent on temperature. (d)–(f) Inhomogeneity of the pseudogap: (d) each curve is STM spectrum integrated over many tip
positions with the same gap value, (e) characteristic spectra from the two regions ∆ < 65 and ∆ 65 meV [65]. (f) 180 Å square maps
of gaps (defined as half the distance between the edges of the gap) and corresponding histograms of the superconducting gap (left) and
pseudogap (right) for underdoped sample (15 K) [198].
0 meV 0 meV40 meV 40 meV
(f)
(e)(d)(c)
∆ < 65 meV
∆ < 65 meV
–400–100
–200 –100 100
400100
200
00
0
Sample bias, mV Sample bias, mV
46.4 K85.0 K
102.0 K
117.0 K
4.2 K
4.2 K
63.3 K
77.0 K
91.0 K
88.9 K
21.5 K
80.9 K
59.0 K
109.0 K
175.0 K
151.0 K
194.8 K
76.0 K
70.0 K
98.4 K
84.0 K
46.0 K
123.0 K
182.0 K
166.6 K
202.2 K
293.2 K
dI
dV/
, a
rb
. u
ni
ts
dI
dV/
, G
Ω
–1
VSample, mV
1.5
1.0
0.5
0.10
0.08
0.06
0.04
0.02
G
(0
),
S
0 50 100 150 200 250 300
T, K
(b)
(a)
Jun.23
Jun.5: G(0)X30
Voltage, mV
Jun.23
–150 –100 –50 0 50 100 150
1.4
1.2
1.0
0.8
0.6
0.4
G
V
T
(
,
)/G
V(
,1
43
K
)
426 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity
neling on break junctions [158] and SIN point-contact tun-
neling [159] also support the two-gaps scenario. Andreev
reflection is expected to be similar to SIN and STM, but
appears to be sensitive to the superconducting energy scale
only, that may be because the tunneling mechanisms are
actually different [27].
STM, despite more complicated theoretical justification
[196], has appeared to be extremely useful for study the
pseudogap phenomenon in cuprates allowing one to ex-
plore spatial inhomogeneity [65,197,198] and detect new
orderings. In superconducting state, STM/STS reveals in-
tense and sharp peaks at the superconducting gap edges
which smoothly transform to broad maxima at the pseudo-
gap energy above ,cT as one can see in Fig. 11 from the re-
sult of early SIN tunneling (a), (b) [120] and STM (c) [121]
experiments. The depletion of the density of states at Fermi
level is seen to persist in the normal state, even above T*,
at which it evolves more rapidly. The visual smoothness of
the gap transition over cT may suggest a common origin of
the gaps [121]. On the other hand, the size of the pseudo-
gap looks independent on temperature, that makes it mark-
edly different from superconducting gap (see discussion in
Ref. 128). Also, the studies of the normalized differential
conductance [198] have shown a coexistence of a sharp
homogeneous superconducting gap superimposed on a large
but inhomogeneous pseudogap, see Fig. 11(f).
The much weaker inhomogeneity observed at low ener-
gies in the Fourier transform maps of the STM spectra
shows two type of modulations. The first one is due to the
quasiparticle interference [191–193] on the d-wave gapped
electronic structure [189]. It allows to recover the momen-
tum dependence of the superconducting gap [65,190]. The
second one is a nondispersive modulation at higher ener-
gies, which can be related to the incoherent pseudogap states
at the antinodes [64]. They could be related to a short-
range local charge ordering with periods close to four lat-
tice spacing in the form of the square “checkerboard”
[63,199,200] or unidirectional domains [201]. These two
modulations coexist in the superconducting state but com-
pete with each other for the electronic states.
4. Pseudogap in Cu-SC and transition metal
dichalcogenides
ARPES is the most direct tool to measure the one parti-
cle spectrum with momentum resolution [43–45]. Natural-
ly, it has been successfully used to show that both super-
conducting gap and pseudogap are anisotropic: absent
Fig. 12. (Color online) Pseudogap anisotropy by ARPES. (a) Energy distribution curves (EDCs) from the antinodal region of
underdoped (U) and overdoped (O) BSCCO in the normal (N) and superconducting (SC) states [118]. (b) Hole pockets around (π/2, π/2)
as one of explanations of the gapped sections of Fermi surface [202]. (c) Midpoints of the leading edge, the “leading edge gap” (LEG),
of the EDCs of underdoped BSCCO vs. temperature, which inspired the “Fermi arc” idea, sketched in panel (d): d-wave node below Tc
becomes a gapless arc above Tc which expands with increasing temperature to form the full Fermi surface at T * [156]. More accurate
set of EDCs along the Fermi surface from node (N) to antinode (A) for optimally doped two-layer BSCCO (Tc = 90 K) at T = 140 K (e)
contrasted to heavily overdoped Bi-2201 (Tc = 0) at T = 40 K (f) [156].
200 100 0
Binding energy, meV
In
te
ns
ity
, a
rb
. u
ni
ts SC-U
SC-O
N-U
N-O
(a) (c) (e)
(f)
(d)
(b)
M
id
po
in
t,
m
eV
20
10
0
–10
0 50 100 150
a
a
b
b
UD 77 K
Γ M
M
M M
M
M
M
M
M
T, K
N
or
m
al
iz
ed
in
te
ns
ity
A
N
Bi2212 optimal = 90 KTc
Bi2201 overdoped = 0 KTc
–0.3 –0.2 –0.1 0
Energy, eV
1.0
0.8
0.6
0.4
0.2
0
N
A
0 0.4 0.8
kx, /aπ
k y,
/aπ
N
or
m
al
iz
ed
in
te
ns
ity
–0.3 –0.2 –0.1 0
Energy, eV
Y Y Y
Γ Γ
(0,0)
(0, )π
( ,0)π
( , )π π
Slightly
overdoped- = 85 KTc
Under-
doped- ~ 67 KTc
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 427
A.A. Kordyuk
along the nodal direction and maximal at the antinodal
region, and doping dependent: vanishing with overdoping,
but the pseudogap is vanishing earlier [118,119]. More-
over, while the superconducting gap follows a d-wave like
dependence being zero only at the nodes, the pseudogap
behaves more unusually, leaving non-gapped sections of
the Fermi surface around the nodes [202] later called
“Fermi arcs” [156]. It was also suggested [156] that “Fermi
arc” gradually changes its length from zero at cT to the full
Fermi surface at T*, as shown in Fig. 12. Panel (e) shows
that the pseudogap increases gradually from the node and
stays constant in the whole antinodal region for optimally
doped two-layer BSCCO that is in contrast to heavily
overdoped ( cT = 0) one-layer Bi-2201 (f) [156].
4.1. Measuring gaps in ARPES
Despite the clear evidences for the pseudogap anisotro-
py, the determination of the momentum resolved gap value
in ARPES is far from being straightforward [203]. First,
one should distinguish a gap from a number of possible
artifacts. Second challenge is to derive the gap value ∆ that
can be compared to other experiments and theoretical mo-
dels. Among possible artifacts in cuprates: charging by pho-
tocurrent, superstructure [204], misalignment [203], bilayer
splitting [155], matrix elements [205], photoemission back-
ground [206], and Van Hove singularity [207]. Most of
them, if known, can be taken into account due to improved
accuracy of the state-of-the-art ARPES technique [45].
If the gap model in known, as in the case of BCS-like
superconducting gap or CDW gap, the best way to derive
the gap value from experimental spectrum is to fit it to the
model. And it seems that the most accurate method to ex-
tract the value of the BCS-like gap from ARPES spectra is
fitting of a partial DOS (the momentum integrated EDCs
along a cut perpendicular to the Fermi surface) to the for-
mula derived by Evtushinsky [208]:
IEDC( ) = ( , ) Re ,if T R
E ω
′′ ω+ Σ
ω ω ⊗
(1)
which coincides with the Dynes function [209] multiplied
by the Fermi function and convolved with the energy reso-
lution function .Rω Here 2 2= ( ) ,''E iω+ Σ −∆k ′′Σ is the
imaginary part of the self-energy, and ∆k is the momen-
tum-dependent superconducting gap. This formula is ob-
tained in approximation of linear bare electron dispersion,
but there is also useful analytical solution for a shallow
parabolic band [208]. A similar method of gap extraction is
widely used in angle-integrated photoemission spectrosco-
py [210]. For our case it could be useful if the pseudogap
in cuprates is due to either preformed pairs or Peierls-like
density waves.
If the model behind the gap is not known, other empiri-
cal methods could be used. The most straightforward one
is to measure the peak position of the gapped EDC and
assume that ∆ is the distance to 0.FE ≡ It works well for
momentum integrated spectra with BCS-like gap if such a
“coherence peak” is well defined, that is usually not the
case for the pseudogap in cuprates. Moreover, looking for
a gap in momentum resolved ARPES spectrum, one deals
with the kF-EDC (EDC taken at Fermi momentum), which
never peaks at .FE In a normal non-gapped state this EDC
is a symmetrical spectral function ( ) = ( ),A Aω −ω which
width is twice of the scattering rate 1(0, ) = ,T h −′′Σ τ multi-
plied by the Fermi function: ( , ) = ( , ) ( , ).I T A T f Tω ω ω So,
its peak position is temperature dependent, as one can see
in Fig. 13(a) [203].
Two procedures have been suggested to work around
this problem, the symmetrization [156] and division by
Fermi function [211]. If at Fk the gaped spectral function
obeys a particle-hole symmetry ( ) = ( ),A Aω −ω both pro-
cedures should lead to the same result: ( ) ( ) =I Iω + −ω
( )/ ( ) = ( ).I f A= ω ω ω This, however, does not help much to
determine small gaps, when < ( ) :′′∆ Σ ∆ In this case two
peaks below and above Fermi level are just not resolved
and the symmetrized EDC is peaked at .FE Thus, after
symmetrization procedure, a smooth evolution of the gap
with either temperature or momentum will look like a
sharp gap opening at ( , ) = ( = , , ).k T k T′′∆ Σ ω ∆
The position (binding energy) of the midpoint of the
leading edge of EDC is called the “leading edge shift” or
“leading edge gap” (LEG) [81,119]. Naturally, it is sensi-
tive to the gap size but also depends on a number of pa-
rameters [203], as one can see in Fig. 13: quasiparticle
scattering rate and temperature, momentum and energy
Fig. 13. “Leading edge gap” (LEG) in non-gapped ARPES spec-
tra. (a) Leading edge midpoint of kF-EDC depends on tempera-
ture, momentum (b) and energy (d) resolutions. False fast “open-
ing” of the gap can be seen for EDCs slightly away from kF (c).
After [203].
(a)
(c) (d)
(b)
1.0
0.5
0N
or
m
al
iz
ed
in
te
ns
ity
LE
G
, m
eV
LE
G
, m
eV
LE
G
, m
eV
–5
0
–5
–5
–10
–10
–10
–15
–15 –15
–20 –20
–25
0.04 0 –0.04
Binding energy, eV
10 K
300 K
Rk = 0.070 Å–1
0
0 0
100
100 100
200
200 200
300
300300
T, K
T, KT, K
0.050
0.030
0.015
0.007
0.001
Rω = 10 meV
20
30
40
50
k k – = 0.010 F – Å–1
–0.008...
0.010
428 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity
resolutions, displacement from ,Fk etc. At 150 K, for ex-
ample, for standard experimental resolutions (thicker mid-
dle curve on panels (b)–(d)) LEG is 10 meV above ,FE so,
one can roughly say that LEG would be at FE if the
pseudogap is about 10 meV.
The “Fermi arcs” story is illustrative in this respect. Ini-
tially, the “gapless arc” was defined as a set of Fermi mo-
menta for which the leading edge midpoint is above FE
(LEG < 0 in binding energy) [156]. Negative LEG is
equivalent to a peak in the spectral function at ,FE so, the
symmetrization procedure has been used instead of LEG in a
number of detailed study of Fermi arcs evolution with tem-
perature, see [212], for example. The observed dependence
of the length of the arcs with temperature is consistent with
temperature dependence of the kF-EDC width, as explained
above, or, in more theoretical language, as a consequence of
inelastic scattering in a phase-disordered d-wave supercon-
ductor [213]. Thus, comparing a number of proposed mod-
els for the Fermi arcs, authors of Ref. 214 have concluded
that the best one to model the ARPES data is a d-wave ener-
gy gap with a lifetime broadening whose temperature de-
pendence is suggestive of fluctuating pairs.
Nevertheless, the question is not closed and Fermi arcs
remain enigmatic. The initially proposed scenario of hole
pockets [202] is still considered. And while authors of [215]
report on coexistence of both the Fermi arcs and hole pock-
ets, the authors of Ref. 216 insist that the Fermi arcs are illu-
sion made by fully enclosed hole pockets with vanishingly
small spectral weight at the magnetic zone boundary.
Interestingly that in view of “two gaps” scenario
[128,129], now widely accepted [27,152,211,217–223],
the Fermi arcs are natural signature of a competing to super-
conductivity order which is peaked at the antinodal region.
To finish with LEG method one should admit that it is
less susceptible, in spite of Fig. 13(c), to sudden artificial
changes of the derived gap values and the real gap opening
can be detected in LEG(T). Also, LEG is a good quantity
for the ARPES map of gaps [155]. Moreover, the LEG
method works much better if applied to the momentum in-
tegrated spectrum (aforementioned partial DOS), since in-
tegration along a cut perpendicular to the Fermi surface
removes the problem of Fk determination error and, flat-
tening the spectrum, place the leading edge midpoint of
non-gapped spectra at the Fermi level [219,224]. In the same
way the symmetrization of this partial DOS has much more
sense than of single EDC and can be effectively used for
visualization of the gap. So, both the LEG and symmetriz-
ation methods applied to partial DOS are simple but most
robust procedures of gap detection, but to determine the
gap value one shout fit it to the appropriate model, such as
Eq. (1), for example.
All the said about the gap evaluation from ARPES is
valid for a deep band, if it is much deeper than the gap.
The Van Hove singularities (VHs) nearby the Fermi level
complicate the situation [207,225]. Typical set of EDCs
from the antinodal region in superconducting state is shown
in Fig. 14(a) [157]. In a wide doping range around optimal
doping the spectra have so-called “peak-dip-hump” line
shape [226] that has been considered [157] as a conse-
quence of interaction with the spin-fluctuations resonance
seen by inelastic neutron scattering [170]. The “supercon-
ducting peak” which dominates the overdoped spectra van-
ishes with underdoping evolving into a kink, which can be
defined as the second derivative maximum of the spectra,
as shown in panel (d) [217]. The energies of all the fea-
tures, “peak”, “dip”, and “hump”, scale similarly, increas-
ing with underdoping (see Fig. 14(b),(c)), but it is the su-
perconducting peak position that fits the pseudogap values
derived from other experiments [27,152], as has been
shown earlier in Fig. 10.
Fig. 14. (Color online) Doping dependence of the pseudogap from ARPES. (a)–(c) EDCs from the antinodal region of BSCCO samples
of different doping levels and two energy scales derived from them: positions of “peak” and “hump” [157]. (d) The symmetrized spectra
of three underdoped BSCCO samples on which the “superconducting peak” is evolving into a kink, which can be defined as the second
derivative maximum of the spectra; inset shows the absence of temperature dependence of these spectra for the UD 30 K sample taken
at 10 K (blue) and 50 K (red). (e) Doping dependence of the peak position for these three spectra (black symbols) and for three other
momenta at the Fermi surface closer to the node, as marked in the inset [217].
In
te
ns
ity
, a
rb
. u
ni
t s
Binding energy, eV
0.6 0.4 0.2 0 –0.2 –0.2 –0.1 0 0.1 0.2
O72K
O87K
U89K
U83K
U75K
U55K
U15K
(a) (b)
x0
100
Tc
Tc
Tc= 50 K
Tc= 40 K
Tc= 30 K
T c
, K
T*
SC
State
H
um
p/
pe
ak 5
4
3
2
(c)
(d) (e)
1.2
0.8
0.4
0
En
er
gy
, 1
0
K3
E n
er
gy
, 1
0
K3
peak
( ,0) pumpπ
3
2
1
0
0.04 0.12 0.20
x E E – , eVF
100
80
60
40
20
0
Pe
ak
p
os
iti
on
r e
la
tiv
e
to
n
o d
e,
m
eV
250
200
150
100
50
0
Hole concentration
0 0.05 0.10 0.15 0.20
Intermediate
region
kF around
( ,0)π
Ph
o t
oe
le
ct
ro
n
in
te
n s
ity
, a
rb
. u
n i
ts
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 429
A.A. Kordyuk
Later it has been shown [207] that the “peak-dip-hump”
structure is completely due to the bi-layer splitting (the peak
and hump correspond to the VHs’s of the antibonding and
bonding bands, respectively) at the overdoped side, and
only with underdoping the (π,0)-spectra become affected
by both the superconducting gap and the spin-fluctuations
resonance [225]: the latter contributes to the dip while the
peak, being sandwiched between the gap and the reso-
nance, becomes narrower and finally looses its spectral
weight. One can mention here that besides the spin-fluctua-
tions also the low-energy CDW modes can contribute to
the peak-dip-hump structure [227].
The asymmetric STM spectra also can be naturally ex-
plained by the bi-layer split VHs [228]. So, the doping de-
pendence of the gaps derived from ( ,0)π ARPES spectra
and from tunneling in the superconducting state should be
taken with caution. On the other hand, the asymmetry of
SIN tunneling spectra can be due to a contribution to the
Green function (and tunnel current) that represents the
electron–hole pairing and is proportional to the CDW order
parameter depending on its phase [229,230] (as shown in
the earlier work [231]). Also, there are reports that the bi-
layer splitting may be vanishing with underdoping [232],
but the most careful spectra for underdoped one-layer Bi-
compound [223] do not show the “peak-dip-hump” line
shape.
Despite all the mentioned complications, one can make
the following conclusions. (1) Maximal (for given sample)
pseudogap value exhibits similar doping dependence as
the temperature at which it starts to develop, i.e.
( ) ( ).x T x∗ ∗∆ (2) This dependence is essentially different
from the dependence of the superconducting transition
temperature: ( ) ( ).cT x T x∗ But the relation between cT
and SC∆ remained controversial since different techniques
gave different ( )SC x∆ dependences. One can say that this
controversy is now resolved [27].
4.2. Two gaps in Cu-SC
The idea that the pseudogap and superconducting gap
are two distinct gaps [128,129] rather than one is a precur-
sor of another has become started to find wide acceptance
when a number of evidences for different doping depend-
ence of the gaps measured in different experiments has
reached some critical value (see Fig. 10 and Refs. 27, 152)
and, that may be more important, when those different de-
pendence have been observed in one experiment, first in
Raman [152] and then in ARPES [211,217,218]. It has
been shown that in superconducting state the gap measured
around the node does not increase with underdoping as the
antinodal gap but scales with .cT Studying the evolution of
the spectral weight of some portions of ARPES spectra
(the weight under the “coherent peak” and the weight de-
pleted by the pseudogap) it has been concluded that the
pseudogap state competes with the superconductivity [220].
These results are summarized in Fig. 15.
Another difference between the pseudogap and super-
conducting gap has come from STM: the superconducting
gap is homogeneous while the pseudogap is not [198].
One may conclude that the pseudogap which opens at
T* and the superconducting gap have different and com-
peting mechanisms and that T* is not the temperature of
the preformed pairs: .SC c pT T T∗ ∗∆ ∆ This does
neither exclude an existence of the preformed pairs nor
uncover the T* origin. To do the latter, one should find
the pseudogap features peculiar for a certain mechanism.
And probably this could be done empirically, comparing
the pseudogap in cuprates to known cases.
Indeed, it has been found [219] that from ARPES point
of view, the pseudogap in BSCCO is remarkably similar to
the incommensurate CDW gap in another quasi-2D metal,
the transition metal dichalcogenide 2H-TaSe2. Figure 16
shows evolution of the gap with temperature as a tempera-
Fig. 15. (Color online) The gaps above and below Tc (a)–(c) and the coherent and pseudogap spectral weights (d)–(f) over the Fermi
surface for different doping levels of Bi-2201. After [220].
430 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity
ture map (a) and as the position of the leading edge (b).
The temperature dependence of LEG in an underdoped Tb-
BSCCO with cT = 77 K and T* = 170 K looks identical to
the same quantity (c) measured in 2H-TaSe2 with the tran-
sitions to the commensurate and incommensurate CDW
phases at ICCT = 90 K and NICT = 122 K, respectively
[219,233]. Note, that if one plots the peak position from
panel (a), it would increase above cT having a local maxi-
mum at about 120 K. Such a behavior has been considered
as the most convincing evidence for the existence of two
distinct gaps [223].
So, the incommensurate CDW or other density wave
could be the main reason for the pseudogap below T*, but
the spectroscopic consequences of it are not trivial and
even difficult to calculate from the first principles [234]. In
this case, one may try to use TMD as model systems to
compare in details the charge ordering gaps to the pseudo-
gap in cuprates.
4.3. Charge density wave gaps in transition metal
dichalcogenides
Quasi-2D transition metal dichalcogenides in which
a number of CDW phases are realized [9] can be useful
model systems to study the spectroscopic manifestations of
those phases and their relation to the electronic structure.
In general, the quasi-2D electronic systems have a weaker
tendency towards the formation of CDW and SDW insta-
bilities than quasi-1D metals because the Fermi surfaces
in 2D can be only partially nested and therefore partially
gapped, so the system may be metallic even in the CDW
state. The 2D character and the existence of an anisotropic
gap make these systems similar to the HTSC cuprates [235],
especially taking into account similarity between T* and
cT lines in cuprates and CDWT and cT lines in the T-doping
and T-pressure phase diagrams of dichalcogenides [236],
see Fig. 2. For topical review on the origin of charge den-
sity waves in layered transition metal dichalcogenides see
Ref. 10.
2H-TaSe2 [11] and 2H-NbSe2 [12,237,238] seem to be
perfect model systems to understand the effect of different
CDW on electronic density of states and ARPES spectra.
Figure 17 shows the Fermi surface of 2H-TaSe2 [11,239],
a compound in which there are two phase transitions into
the states with incommensurate (122 K) and commensurate
3 3× (90 K) CDW. It is the first transition at which a jump
in the heat capacity and a kink in the resistance are ob-
served, while the second transition has almost no effect on
these properties [9]. From ARPES point of view the situa-
tion is opposite. The Fermi surface (shown in the upper left
panel) remains virtually unchanged up to 90 K, and a new
order appears just below the commensurate transition. The
explanation for this dichotomy comes from the behavior of
the spectral weight near the Fermi level on the Fermi sur-
face sheet centered around K-points. Below 122 K the spec-
tral weight starts to decrease sharply, that is the pseudogap
opening (see the cross-section 5–6). When passing through
90 K, the pseudogap is transformed into a band gap in
the new Brillouin zone, but this transition is not accompa-
nied by such a gain in kinetic energy.
It is a good example when both the commensurate and
incommensurate CDW are driven by the Fermi surface nest-
ing, that, as the name implies, is a measure of coincidence
of the Fermi surface parts shifted by a “nesting” vector.
Numerically, the nesting vectors can be found by autocor-
relation of the measured Fermi surface [11], or, more physi-
cally, from peaks of the imaginary part of electronic sus-
ceptibility [240,241]. Interestingly, there is opinion that the
Fermi surface nesting is a misconception since it is very
sensitive to the Fermi surface geometry while the calcula-
tions show that the Fermi surfaces almost never nest at
Fig. 16. (Color online) Nonmonotonic pseudogap in cuprates. (a) The temperature map which consists of a number of momentum inte-
grated energy distribution curves (EDCs) measured at different temperatures at a “hot spot”. The gap is seen as a shift of the leading
edge midpoint (LEM) which corresponds to white color close to the Fermi level. (b) The position of LEM as function of temperature for
an underdoped Tb-BSCCO with Tc = 77 K and T * = 170 K is remarkably similar to the pseudogap in a transition metal dichalcogenide
2H-TaSe2 (c) with the transitions to the commensurate and incommensurate CDW phases at TICC = 90 K and TNIC = 122 K, respective-
ly. After [219,233].
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 431
A.A. Kordyuk
the right CDW vectors [242]. The mentioned ARPES stud-
ies have shown that the nesting, which, of course, is better to
discuss in terms of peaks in electron susceptibility, is indeed
very sensitive to the Fermi surface geometry [11]. That is
why the nesting vectors coincide with CDW vectors when
derived from the experimental band structures rather than
from the calculated ones. In fact, the incommensurate
CDW in 2H-TaSe2 has appeared to be more complex at
some temperature range, consisting of one commensurate
and two incommensurate wave vectors [243,244].
The ARPES data on 2H-TaSe2 and other TMDs prove
empirically that the formation of the incommensurate charge
density wave, which can be described within the scenario
based on short-range-order CDW fluctuations [13], leads to
depletion of the spectral weight at the Fermi level, while
the transition from incommensurate to commensurate order
leads rather to a redistribution of the spectral weight in mo-
mentum. This is consistent with the sign changing Hall
coefficient in this compound [239]. So, the incommensu-
rate gap in dichalcogenides looks very similar to the pseu-
dogap in cuprates [219].
Among other types of CDW, which are observed in
2H-TMDs and could be similar to CDW in cuprates, I would
mention the striped incommensurate CDW [243,245] and
nearly commensurate CDW observed in 2H-NbSe2 by
STM [15]. The latter is established in nanoscale regions in
the vicinity of defects at temperatures that are several times
the bulk transition temperature .CDWT
Other analogies may be found between cuprates and
1T-TDMs in VHs nesting and correlation gap, as discussed
in Sec. 4.5.
4.4. Charge density wave in cuprates
Until recently, CDW in cuprates remained almost pure-
ly theoretical idea, but now one may say that that was due
to the dynamical nature of CDW fluctuations [246,247].
Last years of experimental studies added much to the ev-
idence concerning CDW in cuprates [26,248,249]. Initially,
a copper-oxygen bond-oriented “checkerboard” pattern
has been observed by STM in vortex cores in BSCCO [63].
The proposed explanation was a spin density wave local-
ized surrounding each vortex core, but similar pattern had
been observed also in zero field [199]. In BSCCO above cT
there is energy-independent incommensurate periodicity in
the pseudogap state close to 1/4 [64] or 1/4.5 [65], if meas-
ured deep in superconducting state.
Transport measurements for LSCO also find a tendency
towards charge ordering at particular rational hole-doping
fractions of 1/16, 3/32, 1/8, and 3/16 at which resistivity is
peaked [250]. The charge ordering, in terms of Cooper
pairs density waves (PDW), was expected to be particular-
ly pronounced near certain “magic” doping levels, where
the charge modulation is commensurate with the underly-
ing lattice [251,252].
Raman at higher frequencies on LSCO [253] has shown
that the spin fluctuations are present even in overdoped
Fig. 17. (Color online) Evolution of the Fermi surface (upper row) [239] and underlying electronic structure (lower row) of 2H-TaSe2
with temperature. Fermi surface changes topology at 90 K (transition to the commensurate CDW state) while the pseudogap opens on
some parts of the Fermi surface at 122 K (incommensurate CDW transition). After [11].
k y
, 1
/Å
kx, 1/Å kx, 1/Å
0
0
0
0.1
0.1
–0.1
–0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
–0.4–0.4 –0.1 –0.1 0 00.4 0.4–0.2 0.1 0.10.2 0.2
B
in
di
ng
e
ne
rg
y,
e
V
Binding energy, eV
Binding energy, eV
0 0.5 1.0 1.5
1.0 1.0
0.5 0.5
0 0
–0.5 –0.5
–1.0 –1.0
180 K
–2 –2 –1 –1 0 0 1 1
max
min
30 K
M
om
en
tu
m
, 1
/Å
Momentum, 1/Å
Momentum, 1/Å
0
–0.5
–1.0
no
rm
al
In
te
ns
ity
, a
rb
. u
ni
ts
In
te
ns
ity
, a
rb
. u
ni
ts
0.3
0.3
0.2
0.2
0
0
0.1
0.1
–0.1
–0.1
107 K
290 K 1
2
34
56
min max
432 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity
samples, but their strength tends to decrease substantially
upon overdoping, while the charge-ordering fluctuations
increase and reach a maximum intensity around x ≈ 0.19.
Recent neutron and x-ray scattering experiments on un-
derdoped Bi2Sr2–xLaxCuO6+δ [254] point to a surface-en-
hanced incipient CDW instability, driven by Fermi surface
nesting.
Hard x-ray diffraction measurements [41] on LSCO of
three compositions (x = 0.11, 0.12, 0.13) revealed CDW
order with onset temperatures in the range 51–80 K and
ordering wave vectors close to (0.23, 0, 0.5). On entering
the superconducting state the CDW is suppressed, demon-
strating the strong competition between the charge order
and superconductivity. CDW order coexists with incom-
mensurate magnetic order and the wave vector of CDW is
twice of the wave vector of SDW. This fluctuating CDW
order is strongly coupled to, and competes with, supercon-
ductivity, as demonstrated by the observed nonmonotonic
temperature dependence of the scattering intensity and
the correlation length [169,255].
In many studies the break of four-fold rotational sym-
metry have detected in the pseudogap state, pointing to
stripe or nematic order [101,104]. For example, the uni-di-
rectional stripes within the checkerboard has been detected
by STM [200]. A large in-plane anisotropy of the Nernst
effect has been observed in YBCO [103]. The anisotropy,
as reported, sets in precisely at T* throughout the doping
phase diagram.
So, nowadays there are enough evidences for the CDW
ordering in cuprates. These waves are generally consistent
with the idea of Fermi surface nesting, thus should gap the
straight sections of the Fermi surface, but it is unlikely that
they can be responsible alone for the whole pseudogap
state bordered by T*(x). Then other possible constituents
of the pseudogap are SDW due to VHs nesting, AFM or-
der, and Mott gap, each one or all together.
4.5. Van Hove singularities nesting and Mott gap
in transition metal dichalcogenides
Let us first consider Van Hove singularity driven CDW
in 1T-TMD’s. Some of those compounds are known as
“excitonic insulators” [10]. The driving force for new or-
dering is a win of electron kinetic energy that happens
when two VHs’s of opposite character (e.g., top and bot-
tom of different bands) residing near the Fermi level are
folded to the same momentum, as shown in Fig. 18(b).
Among a few known examples is 1T-TiSe2 [256]. It shows
large band renormalizations at high-symmetry points of the
Brillouin zone and a very large transfer of spectral weight
to backfolded bands.
Another example of VHs nesting has been found re-
cently in 5d transition metal compound IrTe2 [257]. It has
been shown that the band related to the saddle points at the
Fermi level is strongly reconstructed below transition tem-
perature, removing VHs from FE and the wavevector be-
tween the adjacent saddle points is consistent with the in-
plane structural modulation vector.
Partial gaps have been reported for other 1T-compounds:
1T-VSe2 [258] and classical 1T-TaS2, where CDW is called
“quasicommensurate” [16,131] or “nearly commensurate”
[14,130,259,260] (domain-like discommensurate [14], i.e.,
commensurate domains separated by discommensurate areas
[130]). In case of 1T-TaS2, the commensurate CDW phase
has been discussed in relation to Mott transition [16,260]. It
has been suggested that the Mott phase melts into a textured
CDW and superconductivity develops within the CDW
state, and survives to very high pressures [260]. This com-
pound becomes superconducting when subjected to external
pressure [260] or chemical doping or Fe [261]. 1T-TaS2
with Cu intercalation reveals a disorder-induced metallic
state; a non-Fermi liquid with a pseudogap that persists at
finite temperatures [262]. A Mott transition has been found
also at the surface of 1T-TaSe2 [18,263].
Fig. 18. (Color online) Time-domain classification of CDW insulators. (a) Mott insulator. (b) Excitonic insulator. (c) Peierls insulator.
(d) Corresponding timescales of the responses to impulsive near-infrared excitation and their assignment to elementary model-specific
processes. After [264].
0
x
0
x
E
x
N E( )
E
F
E
F
E
F
E k( ) E k( )
kk
a
(a)
(d)
(b) (c)
π/a π/a–π/a
Pump pulse
exitation
Electron hopping Buildup of
screening Amplitude mode
oscillation
0 fs 1 fs 10 fs 100 fs 1,000 fs
∆
∆ ∆
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 433
A.A. Kordyuk
The assignment of the partial gap observed by ARPES
to Peierls or Mott type could be controversial [16,259], but
it seems that the time resolved ARPES, measuring the
melting times of electronic order parameters, can help to
resolve this controversy [264]. A time-domain classifica-
tion of charge-density-wave insulators is shown in Fig. 18
[264]: the Mott insulator collapses due to an ultrafast rear-
rangement of the electronic states on the elementary time-
scale of electron hopping, the excitonic insulator breaks
down because the Coulomb attraction causing electrons
and holes to form excitons is screened by the added free
carriers, and the Peierls insulator melts with atomic rear-
rangement. In particular, it has been proved that Rb inter-
calated 1T-TaS2 is a Peierls insulator while the 1T-TiSe2 is
an excitonic insulator.
While the mechanism of the Mott transition in TND is
under active consideration now [10,264], one may think
about it in terms of critical depth of the pseudogap derived
by Mott in 1969 for liquid metals [2]. One can also expect
that flattening of the band leads to localization of the band
forming electrons.
4.6. Three gaps in Cu-SC
From incommensurate CDW one may expect a transfer
of the spectral weight from the pseudogap to other momen-
ta while the Mott transition involves the weight transfer to
higher binding energies above 1–2 eV [148]. So, in order
to distinguish between different mechanisms of pseudogap
formation, careful temperature dependence of ARPES
spectra in the whole Brillouin zone is required, that is a lot
of experimental work still to be done.
As an example, Fig. 19 shows the same “hot-spot” EDC
as in Fig. 16 but not normalized. One can see that from 160
to 120 K the spectral weight disappears. It may be trans-
ferred from around FE either to much higher energies or to
other momenta. In the superconducting state the spectral
weigh recovers in “coherence peak”. It has been shown
while ago [265] that the weight to the peak is transferred
from other momenta and higher binding energy (up to
0.3 eV), so, one may assume that both the incommensurate
CDW and localization do affect the “hot-spot” spectrum,
but more temperature dependent studies are clearly needed.
The AFM ( , )π π interaction in cuprates is certainly a
strong one, taking into account its persistence on electron-
doped side of the phase diagram (see Fig. 20) and the en-
ergy transfer involved at Mott transition [148]. Based on
comparison with TMD, one may speculate that the Mott
transition in cuprates occurs due to commensurate SDW
gap development (as in the spin-fermion model [84], for
example) for which the reason is VHs at ( ,0).π Also, due to
interaction of two extended saddle points with opposite
curvatures, the resulting band flattening is expected. One
should note that some evidence for incommensurate SDW
has been obtained in neutron experiments on YBCO [266].
In Refs. 221, A222 it has been shown that temperature
evolution of antinodal ARPES spectrum for Bi-2201 is
mostly consistent with a commensurate ( , )π π density-
wave order, but not with the preformed pairs scenario.
On the other hand, some evidence for the preformed pairs
in the underdoped Bi-2212 with cT = 65 K has been found
looking for the particle–hole symmetry in the pseudogap
state [267]. A d-wave symmetry of the pseudogap has been
observed in nonsuperconducting La2–xBaxCuO4 (LBCO)
(x = 1/8) and concluded that the Cooper pairs form spin-
charge-ordered structures instead of becoming superconduct-
ing [224]. Finally, evidence for the preformed pairs state
have been found in accurate ARPES experiments by Kamin-
ski [223].
Fig. 19. (Color online) Temperature evolution of the hot spot EDC for underdoped BSCCO (77 K). The transition temperatures on
the phase diagram (center) correspond to marked changes in EDC evolution as it can be seen from the temperature map (left): at T * ,
the pseudogap starts to increase rapidly, the spectral weight starts to decrease; at Tp, the spectral weight starts to increase; at Tc, the su-
perconducting gap opens, the spectral weight continues to increase up to TSC. The examples of non-normalized EDC’s at 160, 120, and
30 K (right) illustrate the spectral weight evolution. Adopted from [219].
200
150
100
50
T,
K
–0.2 0
Energy, eV Hole doping
T *
Tp
Tc
TSC
–0.2 –0.1 0 0.1
Energy, eV
In
te
nc
ity
A
B
C
160 K
120 K
30 K
434 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity
To conclude, now it seems evident that at least three
mechanisms form the pseudogap in the hole doped cup-
rates: the preformed pairing, the incommensurate CDW
due to nesting of the straight parallel Fermi surface sec-
tions around ( ,0),π and the ( , )π π SDW which is dominant
constituent of the pseudogap assosiated with T* and is
either causing or caused by the Mott localization. These
phases occupy different parts of the phase diagram, as
shown in Fig. 21, and gap different parts of the Fermi sur-
face [222,223] competing for it.
4.7. Two sides of the phase diagram
It is believed that electron- and hole-doped cuprates rep-
resent the Slater and Mott pictures, respectively [148,268].
Although the electron-doped cuprates share the same lay-
ered structure based on CuO2 planes, their phase diagram
differs essentially. In Nd2–xCexCuO4–y (NCCO), for example,
the 3D antiferromagnetic state extends up to x = 0.15, and
the superconducting region is confined to a narrow doping
range (0.15–0.17) neighboring the AFM state. On the other
hand, the superconducting dome of another electron-doped
compound, La2–xCexCuO4–y, is in a similar position as for
the hole-doped LSCO [269]. So, one may conclude that
universality of the phase diagram at the electron-doped
side is still an open question.
The presence of the pseudogap phase at the electron-
doped side is also controversial, but in any case it is not so
extended as on the hole side. Some experiments show ex-
istence of a pseudogap when superconductivity is sup-
pressed by magnetic field [138,270], that excludes precur-
sor of superconductivity as its origin.
The magnetic excitations are present in both hole- and
electron-doped cuprates been even stronger in the latter
[271], but it does not correspond to a higher superconduct-
ing transition temperature. Thus, it is important to identify
which factors, the magnetic excitations, the underlying
Fermi surface topology, or additional effects, are not opti-
mized here.
ARPES confirms that the Brillouin zone is magnetic,
i.e., there is clear observations of a gap along the magnetic
zone boundary [272–274]. Figure 20 show a fragmented
Fermi surface, which suggests that the large Fermi surface
is gapped by into electron and hole pockets [272–274], and
“shadow” and main bands are split along the magnetic BZ
boundary [273]. This can be described by the generalized
dynamical mean-field theory with the k-dependent self-
energy (LDA + DMFT + )Σk [274]. Similar s-wave-like
dependence of the pseudogap has been recently suggested
based on the analysis of Raman spectra and for hole-doped
BSCCO [275].
One may conclude that the electron–hole asymmetry of
the phase diagram of cuprates is a piece of pseudogap puz-
zle that should be addressed by any consistent model.
5. Pseudogap in Fe-SC
In the iron-based superconductors, the pseudogap is
hardly seen by ARPES [46]. It has been reported in several
early studies on polycrystalline samples [276–278] and
later on Ba1–xKxFe2As2 (BKFA) single crystals [279], but
that observations are not supported by a majority of
ARPES [208,280–285] and STM [286,287] experiments.
It is surprising because from a nearly perfect Fermi sur-
face nesting one would expect the pseudogap due to in-
commensurate ordering like in transition metal dichal-
cogenides and cuprates. The absence of the pseudogap in
Fig. 20. ARPES evidence for AFM ordering in superconducting electron-doped cuprates: fragmented Fermi surfaces of
Nd1.87Ce0.13CuO4 (a) [272] and Sm1.86Ce0.14CuO4 (b),(c) and split “shadow” and main bands along the magnetic zone boundary (d) [273].
Fig. 21. Compiled phase diagram of HTSC cuprates. Insets show
a sketch of the AFM split conducting band along the magnetic
zone boundary illustrating the idea of “topological superconduc-
tivity”.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 435
A.A. Kordyuk
ARPES spectra may be just a consequence of low spectral
weight modulation by the magnetic ordering that may
question its importance for superconductivity, discussed in
previous section. Also, the band gap due to antiferromag-
netic order, even commensurate, is small and partial, it
opens the gap on Fermi surface parts but not even along
each direction [288].
Meanwhile, a growing evidence for pseudogap comes
from other experiments [46]. NMR on some of 1111 com-
pounds and Ba(Fe1–xCox)2As2 (BFCA) [289] and nuclear
spin-lattice relaxation rate on Ca(Fe1–xCox)2As2 [290] re-
veal a pseudogap-like gradual decrease of 1( 1 )T T − below
some temperature above cT as function of doping, similarly
to the spin-gap behavior in cuprates.
The interplane resistivity data for BFCA over a broad
doping range also shows a clear correlation with the NMR
Knight shift, assigned to the formation of the pseudogap
[291]. In SmFeAsO1–x, the pseudogap was determined
from resistivity measurements [292,293]. The evidence for
the superconducting pairs in the normal state (up to tem-
perature 1.3 )cT T≈ has been obtained using point-contact
spectroscopy on BFCA film [294].
The optical spectroscopies reveal the presence of the
low- and high-energy pseudogaps in the Ba122 [295] and
FeSe [296]. The former shares striking similarities with the
infrared pseudogap in YBCO while the later is similar to
features in an electron-doped NCCO. Recently a pseu-
dogap-like feature has been observed in LiFeAs above cT
up to 40 K by ultrafast optical spectroscopy [297].
In magnetic torque measurements of the isovalent-dop-
ing system BaFe2(As1–xPx)2 (BFAP), electronic nematicity
has been observed above the structural and superconduct-
ing transitions [298]. It has been supported by recent
ARPES study of the same compound [299] in which a
composition-dependent pseudogap formation has been re-
ported. The pseudogap develops a dome on the phase dia-
gram very similar to cuprates and is accompanied by
inequivalent energy shifts in the Fe /zx yz orbitals, which
are thus responsible for breaking the fourfold rotational
symmetry.
The pseudogap related to the fourfold symmetry break-
ing and electronic nematic fluctuations has been observed
by a time-resolved optical study for electron-doped BFCA
[300] and near optimally doped Sm(Fe,Co)AsO [301]. The
observed anisotropy persists into the superconducting state,
that indicates that the superconductivity is coexisting with
nematicity and the pseudogap in these compounds.
Very recently, the pseudogap-like behavior has been
found in the novel iron-based superconductor with a triclinic
crystal structure (CaFe1–xPtxAs)10Pt3As8 ( cT = 13 K), con-
taining platinum–arsenide intermediary layers, studied by
µSR, INS, and NMR [302]. Authors have found two su-
perconducting gaps like in other Fe-SCs, but smaller, about
2 and 0.3 meV, and also an unusual peak in the spin-ex-
citation spectrum around 7 meV, which disappears only
above T* = 45 K. A suppression of the spin-lattice relaxa-
tion rate observed by NMR immediately below this tem-
perature indicates that T* could mark the onset of a pseu-
dogap, which is likely associated with the emergence of
preformed Cooper pairs.
To conclude, there is much less consensus about the
pseudogap in the iron-based superconductors than in cup-
rates. The fact that in contrast to cuprates the pseudogap in
Fe-SC is not easily seen by ARPES says for its more so-
phisticated appearance in multiband superconductors. Thus,
at the moment, unlike the CDW bearing dichalcogenides,
the ferro-pnictides and ferro-chalcogenides can hardly pro-
vide deeper incite into pseudogap origins. On the other
hand, due to their multiband electronic structure, studying
these materials may shed some light on the interplay of the
pseudogap and superconductivity.
6. Pseudogap and superconductivity
Density wave (SDW or CDW) in cuprates, like CDW in
TMD, competes with superconductivity for the phase
space and is generally expected to suppress .cT Though the
interaction of two orders can be more complex [50,255].
At this point, I would like to recall the idea of CDW-in-
duced superconductivity [303–305], in which the super-
conducting transition temperature can be increased when
one of CDW-induced peaks in the density of states (due to
new VHs) is shifted to the Fermi level. This idea was criti-
cized since it looks unlikely that a self-consistent solution
of both orders caused by the same mechanism (competing
for the same electronic states) could lead to such situation.
On the other hand, if the density wave has different origin,
one can imagine the situations when such an enhancement
would be possible.
For example, if spin and charge degrees of freedom are
decoupled [306], the AFM ordering can enhance the elec-
tronic density of states at certain momenta. The VHs nest-
ing scenario in cuprates [107] is different by origin but
should have the same consequences. The situation when
the upper split band at ( ,0)π is just touching the Fermi
level, as shown in the right inset in Fig. 21, should be fa-
vorable for both ( , )π π density wave and superconductivi-
ty. The pessimistic view on such a scenario says that such
an increase of DOS in 2D system would not enough to
explain HTSC, especially taking into account finite scatter-
ing rate [24,107].
The new experience with the iron-based superconductors
may help to understand the superconducting mechanism
in both Fe-SC and Cu-SC. It has been found [46,307] that
the Fermi surface of every optimally doped Fe-SC com-
pound (the compounds with highest )cT has the Van Hove
singularities of the Fe /3 xz yzd bands in the vicinity to
the Fermi level. The ARPES data for new Fe-SC com-
pounds received thereafter, such as Ca1–xNaxFe2As2 [308],
Rb–Fe–Se (“245” family) [309], and Ca–Pt–Fe–As [310]
436 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity
completely support this observation. This suggests that
the proximity to an electronic topological transition, known
as Lifshitz transition, for one of the multiple Fermi surfac-
es makes the superconductivity dome at the phase diagram
of Fe-SCs [46]. It seems that new Bi-dichalcogenide lay-
ered superconductors follow the same empirical rule:
LaO0.54F0.46BiS2 at optimal doping has the Fermi surface
in close proximity to the topological change [311]. The
high-Tc superconductivity driven by “shape-resonance pair-
ing” in a multiband system in the proximity of a Lifshitz
topological transition [312–314] is one of possible models
to explain the observed correlation.
With the discussed ( , )π π density wave taken into ac-
count, the high-Tc cuprates may share the same “topologi-
cal” mechanism. If the superconducting dome at the hole
side is made by shallow electron pockets around ( ,0)π , the
dome at the electron side is made by the hole pockets
around ( /2, /2),π π as shown in the left inset in Fig. 21.
The role of Lifshitz transition can be twofold here: shaping
the spectrum of magnetic fluctuations [315] and formation
of critically slow quasiparticles [316]. Earlier, an enhan-
cement of superconductivity due to proximity to Lifshitz
transition has been discussed in connection to the ( ,0)π
saddle point [317] (see also [318] and references therein),
but the main objection against the relevance of this scenar-
io for the cuprates was that for optimal doping the saddle
point is essentially below the Fermi level.
7. Conclusions
The present review represents a contribution dealing
with the pseudogap, focusing on ARPES results. Based on
the available data, it is tempting to conclude that the pse-
udogap in cuprates is a complex phenomenon which in-
cludes different combinations of density waves (CDW with
Fermi surface nesting vector and SDW with AFM vector)
and preformed pairs in different parts of the phase dia-
gram. Although the density waves are generally competing
to superconductivity, the ( , )π π SDW, the main constituent
of the pseudogap phase, may be responsible for a “topolog-
ical” mechanism of superconducting pairing, that may be
similar for high-Tc cuprates, iron-based superconductors,
and even superconducting transition metal dichalcoge-
nides.
Acknowledgments
I acknowledge discussions with L. Alff, A. Bianconi,
S.V. Borisenko, B. Büchner, A.V. Chubukov, T. Dahm,
I. Eremin, D.V. Evtushinsky, J. Fink, A.M. Gabovich, M.S.
Golden, M. Grilli, D.S. Inosov, A.L. Kasatkin, T.K. Kim,
Yu.V. Kopaev, M.M. Korshunov, S.A. Kuzmichev, V.V.
M. Loktev, I.A. Nekrasov, S.G. Ovchinnikov, N.M. Plakida,
M.V. Sadovskii, A.V. Semenov, S.G. Sharapov, D.J. Scala-
pino, A.L. Solovjov, M.A. Tanatar, T. Valla, C.M. Varma,
A.N. Yaresko, and V.B. Zabolotnyy. The work was sup-
ported by the National Academy of Sciences of Ukraine
(project 73-02-14) and the State Fund for Fundamental
Research (project F50/052).
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1. Introduction
2. Theories of pseudogap
3. Pseudogap in experiments
4. Pseudogap in Cu-SC and transition metal dichalcogenides
4.1. Measuring gaps in ARPES
4.2. Two gaps in Cu-SC
4.3. Charge density wave gaps in transition metal dichalcogenides
4.4. Charge density wave in cuprates
4.5. Van Hove singularities nesting and Mott gap in transition metal dichalcogenides
4.6. Three gaps in Cu-SC
4.7. Two sides of the phase diagram
5. Pseudogap in Fe-SC
6. Pseudogap and superconductivity
7. Conclusions
References
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