Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity

3.8 MB·English

by  A. A. Kordyuk

Pseudogap from ARPES experiment: three gaps in cuprates
and topological superconductivity
A. A. Kordyuk
Institute of Metal Physics of National Academy of Sciences of Ukraine, 03142 Kyiv, Ukraine
AtermfirstcoinedbyMottbackin1968a‘pseudogap’isthedepletionoftheelectronicdensityof
statesat theFermi level, and pseudogapshavebeenobservedinmanysystems. However, since the
discoveryofthehightemperaturesuperconductors(HTSC)in1986,thecentralroleattributedtothe
pseudogapinthesesystemshasmeantthatbymanyresearchersnowassociatethetermpseudogap
exclusively with the HTSC phenomenon. Recently, the problem has got a lot of new attention
5
with the rediscovery of two distinct energy scales (’two-gap scenario’) and charge density waves
1
patterns in the cuprates. Despite many excellent reviews on the pseudogap phenomenon in HTSC,
0
published from its very discovery up to now, the mechanism of the pseudogap and its relation to
2
superconductivityarestillopenquestions. Thepresentreviewrepresentsacontributiondealingwith

b the pseudogap, focusing on results from angle resolved photoemission spectroscopy (ARPES) and
e endsupwiththeconclusionthatthepseudogapincupratesisacomplexphenomenonwhichincludes
F
at least three different ‘intertwined’ orders: spin and charge density waves and preformed pairs,

1 whichappearsindifferentpartsofthephasediagram. Thedensitywavesincupratesarecompeting
2 to superconductivity for the electronic states but, on the other hand, should drive the electronic
structuretovicinityofLifshitztransition,thatcouldbeakeysimilaritybetweenthesuperconducting

] cuprates and iron based superconductors. One may also note that since the pseudogap in cuprates
n
has multiple origins there is no need to recoin the term suggested by Mott.
o
c
PACSnumbers: 74.20.-z,74.25.Jb,74.70.Xa,79.60.i
-
r
p
u Contents
s
.
t
a I. Introduction 1
m
- II. Theories of pseudogap 3
d
n
III. Pseudogap in experiments 5
o
c
[ IV. Pseudogap in Cu-SC and TMD 12
A. Measuring gaps in ARPES 12
2
B. Two gaps in Cu-SC 15
v
C. CDW gaps in TMD 16
4
5 D. CDW in cuprates 17
1 E. VHs nesting and Mott gap in TMD 18
4 F. Three gaps in Cu-SC 18
0
G. Two sides of the phase diagram 19
.
1 FIG. 1: The electronic density of states normalized to the
0 V. Pseudogap in Fe-SC 20 metallic density of state, plotted versus ω/kT , for various
c
5 temperatures. The T/T = 0 curve is the mean-field result.
c
1 VI. Pseudogap and superconductivity 21 After [5].
:
v
i VII. Conclusions 21
X
r Acknowledgments 22 he had shown that when this pseudogap is deep enough
a
the one-electron states become localized [2].
References 22 Next, the term pseudogap was narrowed to ‘fluctuat-
ing band gap’, thegapformedbyfluctuatingchargeden-
sity wave (CDW) at a Peierls transition [3] in quasi-one-
I. INTRODUCTION dimensional (1D) metals [4–7], as shown in Fig.1.
In fact, the systems with fluctuating CDW can be
The term pseudogap was suggested by Nevill Mott in described similarly to disordered systems without long-
1968 [1] to name a minimum in the electronic density of range order [8], so, the pseudogap should not be neces-
states (DOS) of liquid mercury at the Fermi level. Later sarily related with low dimensionality. Indeed, in quasi-
2
FIG. 2: 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-T cuprate [41] (b), and an iron
c
based superconductor [42] (c).
two-dimensional (2D) metals with CDW ordering, such is discussed as a pseudogap phase on the phase dia-
as transition metal dichalcogenides (TMD) [9] (see also grams of temperature vs charge carrier concentration
recent review [10]), the fluctuation effects are considered (also called ‘doping’) or vs pressure, where the pseudo-
negligible but a partial gap, which can be called ‘pseu- gap phase neighbors both the density wave and super-
dogap’ according to the Mott’s definition, appears in a conducting phases. Fig.2 shows three recent examples
number of CDW phases. Two such kinds of pseudogaps of such phase diagrams for a transition metal dichalco-
havebeendiscussed: traditionalPeierlsgapbutsmeared genide [40] (a), a high-T cuprate [41] (b), and an iron
c
outduetoincommensurability [11,12](or,maybe,short- based superconductor [42] (c).
range-orderCDWfluctuations[13]asin‘nearlycommen-
surate’[14,15]or‘quasicommensurate’[16]CDWstate); Inpresentreviewwemostlydiscussthesethreefamilies
anda‘correlationgap’ofMott-Hubbardinsulatingphase of quasi 2D superconductors from an empirical point of
in a commensurate CDW state [16–18]. view, focusing on results from the angle resolved pho-
Curiously, except the study of fluctuating effects in toemission spectroscopy (ARPES), which is the most
1D CDW compounds, the pseudogap phenomena in 2D direct tool to access the electronic density of states at
CDW systems, despite a variety of the aforementioned the Fermi level [43–45]. We end up with the conclusion
possibilities, had not earned so much attention [9, 19] as thatthepseudogapincupratesisacomplexphenomenon
it had done later in the field of high temperature super- which includes at least three different ‘intertwined’ or-
conductors (HTSC) [20–28] for which it is often consid- ders: spin and charge density waves (similar to the 2D
ered unique [27]. On one hand, the discovery of the su- CDWcompounds)andpreformedpairs,whichappearsin
perconductingcuprates(Cu-SC)sloweddownnoticeably different parts of the phase diagram. The density waves
the study of the CDW-materials. On the other hand, in cuprates are competing to superconductivity for the
the role of the pseudogap in HTSC might be greatly electronic states but, on the other hand, should drive
exaggerated—partly due to real complexity of the phe- the electronic structure to vicinity of Lifshitz topological
nomenonbutpartlybecausealotofpeoplestrugglingto transition, the proximity to which is shown to correlate
findthemechanismofhightemperaturesuperconductiv- to T maximum in all the iron based superconductors
c
ityneededa‘guilty’whythathasappearedtobesohard. (Fe-SC) [46].
Inthissense, thewell-turneddefinitionofthepseudogap
in cuprates as “a not-understood suppression of excited The paper is organized as following. Sec.II gives a
states” was given by Robert Laughlin in early years of short overview of selected theories of the pseudogap in
HTSC era [29]. cuprates. The manifestations of the pseudogap in differ-
Nevertheless, the pseudogap phenomenon in cuprates entexperimentsarebrieflydiscussedinSec.III.Then,in
has stimulated appearance of many fascinating theories the rest of the paper, the focus is made on ARPES re-
(some of which will be briefly overviewed in Sec.II), and sults, starting from a short introduction to ARPES data
has been extended to a number of other materials, for analysis and gap extraction methods (Sec.IVA), the
example, A15 superconductors [30], manganites [31, 32], pseudogap phenomenon is considered in HTSC cuprates
Kondo insulators [33, 34], thin films of conventional su- and CDW bearing TDM in Sec.IV. The growing evi-
perconductors[35]andnanoislands[36,37],Co-Fe-based denceforthepseudogapinFe-SCarereviewedinSec.V.
half metals [38], ultracold Fermi gases [39]. Possible relation of the pseudogap to superconductivity
In many of those systems the pseudogap phenomenon is discussed in Sec.VI.
3
(a) (b) (c)
ure T* Tcoh T* T T*
t
a coh
r T
e PG PG c PG
p
m T T
e c c
T
SC SC SC
QCP
Charge carriers concentration
FIG. 3: Three theoretical idealizations for the interplay of pseudogap (PG) and superconductivity (SC) in the temperature-
doping phase diagram of the HTSCs. T , T*, and T temperatures represent the phase transition to the SC state and
c coh
crossovers to the PG and a coherent states, respectively.
II. THEORIES OF PSEUDOGAP tureandonlytheBerezinskii-Kosterlitz-Thouless(BKT)
superconducting instability may occur [21, 56].
Thetheoriesofthepseudogapincupratesarereviewed The‘phasefluctuation’scenario[57,58]stemsfromthe
in a number of papers [21, 22, 24, 26, 47–50] and text- empirical ‘Uemura relation’, that Tc is proportional to
books [28, 51]. Most of these theories can be classified the zero-temperature superfluid density ns(0) (or ‘phase
by their predictions about a crossover line, T*, which stiffness’) [59, 60]. It was suggested that HTSC with low
borders the pseudogap phase from a normal metal (or superconducting carrier density are characterized by a
a ‘strange metal’) on T −x phase diagram (see Fig.3). relatively small phase stiffness for the superconducting
Here I briefly recall some of the most discussed models. order parameter and by poor screening, both of which
imply a significantly larger role for phase fluctuations.
Diagram(a)isforthemodelswhichconsiderthepseu-
So, the pseudogap state is a region where the phase co-
dogapphaseasaprecursortothesuperconductingstate,
herence is destroyed, but the amplitude of the order pa-
the preformed pairs scenarios [21, 52].
rameter remains finite. Two crossover lines in Fig.3 (a),
Thefluctuationsinbulkcleansuperconductorsareex-
T* and T , border the regions where pairs are formed
tremely small. It is evident from very sharp transitions coh
and become coherent, respectively, while superconduc-
of thermal and electrical properties and has been shown
tivity appears only under both lines [58].
theoretically by Levanyuk and Ginzburg back in 1960
[52]. ThecorrespondingGinzburgnumberGi=δT/T ∼ OneshouldnotethatcalculatedT*(x)foreitherphase
c
(T /E )4 ∼10−12−10−14,whereδT istherangeoftem- fluctuation[51]orBKTmodel[21,61]showdecreasewith
c F
peratures in which the fluctuation corrections are rele- lowering the charge carrier density, as in Fig.4. Also,
vantandE istheFermienergy. Inthindirtysupercon- whiletheexperimentalT*(x)dependencelooksuniversal
F
ductingfilmsthefluctuationsshouldbeincreaseddrasti- for all the hole doped cuprates, the fluctuation effects
cally [53]: Gi=T /E for clean 2D superconductor and should be very sensitive to dimensionality and therefore
c F
Gi ∼ τ−1/E for dirty 2D superconductor [52], where different for different families. For example, the striking
F
τ−1 is the quasiparticle scattering rate at E . Thus, the difference in the shape of the specific heat anomaly at
F
width of the superconducting transition became experi- Tc is observed for quasi 2D Bi2Sr2CaCu2O8+x (BSCCO
mentally measurable, but still Gi(cid:28)1. or Bi-2212), where it follows the BEC phase transition,
and for more 3D YBa Cu O (YBCO), with classical
Thesaidbehaviorwasdeducedfortheconventionalsu- 2 3 7−δ
BCS jump [62]. Another problem of BEC models is that
perconductorstowhichthemean-fieldBCStheoryorthe
the Bose quasiparticles have no Fermi surface, while it is
Ginzburg-Landau model of the second-order phase tran-
clearly observed by ARPES [21, 28].
sition is applicable. In superconductors with very small
correlationlengthξ(ξk (cid:28)1)theBose-Einsteinconden- Nevertheless, recently, the ‘checkerboard’ pattern ob-
F
sation (BEC) of local pairs takes place at T while the served in experiments [63–65] has been explained by the
c
formation of singlet electron pairs (that could be bipo- modelinwhichCDWisinducedbysuperconductingfluc-
larons [54]) is assumed at some higher temperature [28]. tuations [66].
Therefore, soon after discovery of HTSC, when it be- The spin singlet scenario [47, 67] leads to the same
came clear that these materials are quasi-2D and dirty, phase diagram: the spin singlets play the role of pre-
withextremelysmallξ, thesuperconductivefluctuations formed pairs, i.e.the pseudogap state is a liquid of spins
was the first scenario for the pseudogap [55]. Moreover, without long range order (the original RVB idea of An-
instrictly2Dsystems,thephasefluctuationsoftheorder derson [68]) and superconductivity occurs below two
parameterdestroythelong-rangeorderatfinitetempera- crossover lines due to spin-charge recombination. Sim-
4
FIG.4: Effectofthermalandquantumphasefluctuations(left),andofdimensionalcrossover(right)onthecriticaltemperature
for phase coherence T . Adapted from [61].
c
ilar considerations occur also for the SO(5) model [69] given in [84]. Recently, it has been shown [50, 85] that
which attempts to unify antiferromagnetism and super- within this model, a magnetically mediated interaction,
conductivity. An important aspect of these scenarios is which is known to give rise to d-wave superconductivity
the general doping dependence of T*. Since the energy and charge order with momentum along zone diagonal
gainassociatedwithspinsingletformationisthesuperex- [86], also gives rise to the charge density wave with a
change energy, J, the T* line is proportional to J −tx, ‘d-symmetry form factor’ consistent with recent experi-
where t is the hopping energy of the doped hole [26, 70]. ments [87].
Diagram (b) in Fig.3 is for scenarios in which an- The anti-ferromagnetic scenario within the Hubbard
other order with a quantum critical point (QCP) in- model was also considered in the two-particle self-
terplays with superconductivity. In QCP theories [71– consistent approach [48, 88] and studied within a gen-
73], the transition between the ordered and disordered eralized dynamical mean-field theory [89, 90].
quantum phases transforms in a region of critical fluctu- Several exotic scenarios of symmetry breaking, in
ations which can mediate singular interactions between which T* would be a true phase line, had been also
the quasiparticles, providing at the same time a strong suggested. For example, the orbital current state pro-
pairing mechanism [74]. As for the nature of QCP, vari- posed by Varma [91] and the ‘flux-density wave’ [92] or
ous proposals have been discussed. ‘d-density wave’ current state [93].
InRef.75,inwhichCDWandQCPwereputtogether Diagram (c)in Fig.3 isa resultof similar competition
for the first time, it had been proposed that in the pres- between superconductivity and another ordering which
ence of the long-range Coulomb forces a uniform Fermi does not require the QCP for its understanding. These
liquid can be made unstable by a moderate electron- could be either a spin-charge separation, predicted [94–
phononcoupling(Hubbard-Holsteinmodel)givingriseto 96] and found [97] long ago in some families of cuprates
incommensurate CDW in the form of “frustrated phase andknownas‘stripes’,or‘ordinary’(Peierlstype)CDW
separation”, and the related QCP around optimal dop- or spin density wave (SDW) [98–100], like in the tran-
ing. Within this scenario, the static CDW compete (and sition metal dichalcogenides [9, 10]. The former can be
kill) superconductivity like in some 1/8 doping systems, responsible for the pseudogap in one-electron spectrum
but,aslongasCDWfluctuationsstaydynamic,theycan either due to density wave fluctuations [101] or by caus-
mediate superconductivity, and even the d-wave pairing ing an electronic nematic order (quantum liquid-crystal)
can arise from CDW fluctuations without any spin in- [102]. Broken rotational symmetry in the pseudogap
teraction [76, 77]. It was noted that CDW may also phase of cuprates is really observed [103]. And nematic
evolveintoaspin-chargeseparationdeeperinthecharge- order becomes very fashionable today [101, 104].
ordered phase as a consequence of modulation of charge
A driving force for the Peierls type ordering is pecu-
density, anharmonic effects [78], closer proximity to the
liarity of the electronic band structure: either the Fermi
anti-ferromagnetic (AFM) phase, pinning, and so on.
surface nesting [11, 13] or nesting of Van Hove singular-
The QCP determined by magnetic interaction [79] ities (VHs) [105–107]. Nowadays, the FS nesting is con-
leads to the spin-fluctuation scenarios. In the spin- sidered responsible for CDW and pseudogap not only in
fermion model, the pseudogap phase reflects the on- cuprates and transition metal dichalcogenides but also
set of strong AFM spin correlations, a spin-liquid with- in a number of other low dimensional metals such as
out long-range order [80–83]. The full analysis of the manganites [108, 109], binary and ternary molybdenum
normal state properties of the spin-fermion model near oxides [110], Bi-dichalcogenide layered superconductors
the anti-ferromagnetic instability in two dimensions was [111, 112], etc. The competition between density wave
5
and superconductivity is usually considered in frame of (IR), Raman scattering (RS), and tunneling spectro-
the Bilbro-McMillan relation [113], according to which scopies (except STM/STS these are intrinsic tunnel-
(cid:112)
∆ and ∆2 +∆2 increase essentially identically ing, superconductor/insulator/normal-metal (SIN) and
SC SC DW
with falling temperature, so, the density wave is sup- superconductor/insulator/superconductor (SIS) tunnel-
pressedbysuperconductivityandcanbesuppressedcom- ing, and Andreev reflection tunneling (AR)), and in-
pletely, as shown in Fig.3 (c). Recently it has been elastic neutron scattering (INS), as well as traditional
shown that for QCP models this relation will lead to thermodynamic/transport probes such as heat conduc-
(T (x)/Tmax)2+(T (x)/Tmax)2 =1 [114]. tivity and resistivity measurements (or ‘dc conductiv-
DW DW c c
Many other possible reasons for pseudogap formation ity’). ARPESandtunnelingmeasuredirectlythedensity
have been suggested, such as, for example, an intrinsic of single electronic states while other spectroscopies as
inhomogeneity[115],d-wave-typeFermisurfacedeforma- well as thermodynamic/transport probe the two-particle
tions(Pomeranchukinstability)[116],orinteractionwith spectrum.
diatomic negative U centers [117], but it is hardly possi- NMR. The pseudogap in cuprates was first detected
ble even to mention all of them here. byNMR[122,123],whichmeasurestheKnightshift,K ,
s
Toconclude,therearemanytheoriesforthepseudogap and spin-lattice relaxation rate, 1/T . The Knight shift
1
phenomenon in HTSC and, may be consequently, there is a measure of the polarization of electrons by the ap-
isnoconsensusonitsorigin. Ontheotherhand,itseems plied magnetic field and is proportional to the real part
that the main problem of the acceptance of these theo- of the paramagnetic (Pauli) susceptibility, χ(cid:48)(q = 0,ω),
ries, until recently, was a general expectation that they that, in the Fermi liquid model is proportional to the
shoulddescribethewholepseudogapregiononthephase density of states at the Fermi level and should be in-
diagram and all its experimental manifestations, briefly dependent on T. The spin-lattice relaxation rate is re-
considered in the following section. Nowadays, there is lated to the imaginary part of susceptibility, such as
growing evidence that the cuprates do indeed provide a 1/T T ∼(cid:80) |F(q)|2χ(cid:48)(cid:48)(q,ω)/ω, where F(q) is the form
1 q
complicated background for theorists revealing simulta- factor for the particular nuclear site—by probing vari-
neously a bunch of different phenomena: stripes, CDW, ous nuclei in the unit cell one can probe different parts
SDW, electronic fluctuations and localization. Thus, it of momentum space [20, 28]. In Fig.5 the T-dependent
seems that at least several of those models are related to Knight shift (a) and the spin-lattice relaxation rate (b)
reality of HTSC. are shown for underdoped, optimally doped, and over-
doped BSCCO [124]. The suppression of both quantities
starts below T* but no additional anomaly is seen at
III. PSEUDOGAP IN EXPERIMENTS T ,thathasbeenconsideredinsupportofthepreformed
c
pairs scenario [26].
Opening of a gap or just a depletion of the electronic Specific heat. If a gap, which lowers the kinetic en-
density of states at the Fermi level can hardly be miss- ergy of electrons, opens (or starts to develop) at T*, one
ing by a number of experimental probes. Indeed, any shouldseeapeculiarityinanythermodynamic/transport
transition to one of possible CDW states in, for exam- quantity at T* rather than at T when the energy of
c
ple, transition metal dichalcogenides, left signatures in the electrons does not change. Indeed, the specific heat
temperature dependences of different experimental pa- jump at T fades out with underdoping, see Fig.5 (c,d),
c
rameters: heat capacity, resistivity, magnetic suscepti- but usually there is no jump at T* (though some mea-
bility, etc. Those signatures were usually accompanied surements reveal a weak bump [125]). In general, the
bychangeofthediffractionpatterns,so,thecharacterof specific heat data have frequently been cited in support
the symmetry change was more or less clear [9]. ofdiagram(b)ofFig.3[26]sincethedeterminedT*line
In cuprates, the pseudogap was observed in many ex- cutsthroughtheT dome[126,127]. Itisalsoconsistent
c
periments as something that starts to happen above T with the sharp decrease of the specific heat jump or the
c
[20],whileanyindicationofnewordercouldnotbefound superconducting condensation energy U , defined as the
0
by diffraction techniques. Then a depletion of the spec- entropy difference integrated from T = 0 to T , which
c
tral weight was observed directly by ARPES [118, 119] is a constant UBCS = 0.24γ T2 for a BCS supercon-
0 n c
and tunneling spectroscopy [120, 121], and some kind of ductor with d-wave pairing [28] (see Fig.5 (e)). Based
CDW/SDW,aspin-chargeseparationinformof‘stripes’, on those NMR and heat capacity data, it has been con-
was found in some HTSC compounds [97]. Nowadays, cluded [127] that the pseudogap and superconductivity
there are many experimental evidences for CDW in al- are ‘two gaps’, independent and competing. So, smooth
most all families of cuprates, but the nature of the pseu- evolution of tunneling spectra from the pseudogap into
dogap remains puzzling. superconductivitydoesnotnecessarilyimplythepseudo-
In this section, before turning to the ARPES re- gap is a short-range pairing state with the same mean-
sults, we briefly consider experimental manifestations field gap energy as superconductivity [128, 129]. One
of the pseudogap in cuprates by other experimental should note that the interpretation of specific heat mea-
probes: spectroscopic methods such as nuclear mag- surements is tricky because at transition temperatures
netic resonance (NMR), infrared optical conductivity the phonon contribution in cuprates is typically a hun-
6
FIG.5: TheT-dependentKnightshift(a)andthespin-latticerelaxationrate(b)areshownforunderdoped,optimallydoped,
andoverdopedBSCCO[124]. (c)TemperaturedependenceoftheSommerfeldconstantforY Ca Ba Cu O ,labelsshow
0.8 0.2 2 3 6+x
x [126]. (d) A sketch to indicate the transition temperatures T , T* and a crossover to superconducting fluctuations, T , for a
c f
optimally doped and two underdoped samples. (e) The doping dependence of the gap energy E , of the condensation energy
g
U , and of T , the SC gap determined from heat capacity is shown on (f) [127].
0 c
dred times stronger than the electronic one [20] and dif- still detectable and heavily discussed. Soon after dis-
ferential techniques should be used. covery of HTSC, a peculiar feature of cuprates, a quasi-
Transport properties. After discovery of a new su- linear dependence of resistivity over a wide temperature
perconductor,itstransportproperties,i.e.,dcconductiv- range has been found [133]. It means that the experi-
ity, Hall effect, thermal conductivity and thermopower, mental magnitude of the resistivity in cuprates at high
are the first quantities to study. Any phase transition temperatures is much larger than the Ioffe-Regel limit
whichaffectstheelectronicdensityofstatesattheFermi considered within the conventional semiclassical trans-
level should be seen as a peculiarity on temperature de- port theory based on the Boltzmann equation [28]. This
pendences of transport properties. Though, it is often linear region on the phase diagram has inspired appear-
hard to say which peculiarity to expect. For example, anceofmanynewHTSCtheoriesmodelingthis“strange
the dc conductivity depends on both charge carrier con- metal” behavior, such as fluctuating staggered currents
centration n (or density of states at E , N(0)) and scat- [134] or the “marginal” Fermi-liquid (MFL) model [91].
F
teringtimeτ (insimplestDrudemodel,σ =nτe2/m). If, On the other hand, it has been shown [135] that within
duetoCDW,afullgapopens,itisaPeierlstypeofmetal- thet−J modelthesaturationresistivityshouldbemuch
insulator transition and resistivity changes from metal- larger than the Ioffe-Regel limit, so, the absence of satu-
lic to insulating (dρ/dT < 0). If a partial gap opens, ration of resistivity at high temperatures is expected for
than n decreases but τ increases due to less space for strongly correlated systems.
electron to scatter. So, depending on Fermi surface ge- The linear resistivity is observed only in a narrow re-
ometry, the resistivity (see Fig.6 (a) [9]) can show steps gion of temperatures near the optimal doping, as has
as for 1T-TaS2, which has several subsequent transitions been shown [132] by mapping of the in-plane resistiv-
to an incommensurate at 550 K, quasicommensurate (or itycurvature(d2ρ /dT2)oftheLa Sr CuO (LSCO),
ab 2−x x 4
‘nearly commensurate’ [14, 130]) at 350 K and commen- YBCO, and Bi Sr La CuO (BSLCO) crystals (see
2 2−z z 6+δ
surateCDWat180K[131],orkinksasfor2H-TaSe2with Fig.6). The pseudogap temperature, determined on
transitionstoanincommensurateat122Kandcommen- these maps as the inflection point of the resistivity
surate CDW at 90 K. (d2ρ /dT2 =0), decreases linearly with doping and ter-
ab
In cuprates, the transition to the pseudogap state is minates near the optimal value p = 0.16. Below T* the
less pronounced in resistivity (see Fig.6 (b-d) [132]) but curvature is positive until the superconducting fluctua-
7
FIG. 6: Resistivity over phase transitions: (a) for selected transition metal dichalcogenides [9]; (b-d) for high-T cuprates.
c
Resistivity curvature maps for LSCO (e) and BSLCO (f) [132].
tions make it negative again. contourcurvesindicatethevalueofthevortexNernstco-
The idea of two pseudogaps has been confirmed efficient ν = ev /µ H in nV/KT). The observation of a
N 0
by measurements of the c-axis resistivity and magne- largevortexNernstsignalinanextendedregionaboveT
c
toresistance [136]: while T* increases with decreas- in hole-doped cuprates provides evidence that vortex ex-
ing hole doping and is field-insensitive, a field-sensitive citationssurvivethere[138,139]. Theresultssupportthe
gap is found at lower temperature, which scales with preformed pairs scenario and suggest that superfluidity
T , and may be considered therefore as a precursor vanishesbecauselong-rangephasecoherenceisdestroyed
c
to superconductivity. By applying magnetic field to by thermally created vortices (in zero field). Interest-
Y Ca Ba (Cu Zn ) O thin films and changing ingly,inelectrondopedcuprates(e.g. NCCO)wherethe
1−x x 2 1−y y 3 7−δ
the Zn concentration to suppress both the supercon- PG is believed absent the vortex Nernst signal is also
ductivity and superconducting fluctuations, it has been absent. So, the comparison of Nernst effect in hole and
shown that the pseudogap region persists below T on electron-doped cuprates shows that the ‘thermally cre-
c
the overdoped side and T* extrapolates to zero at about ated vortices’ are not generic to any highly anisotropic
0.19 holes concentration [137]. layeredsuperconductorbutmayberelatedtothephysics
Nernst effect. The Nernst effect is considered as ofthepseudogapstateinhole-dopedcuprates[138]. The
one of the most convincing evidences for the existence vortex Nernst signal above T is analogous to an excess
c
of the preformed pairs [20, 28]. The Nernst effect in current observed in the same temperature range in the
solids is the detection of an electric field E perpendic- Andreev contacts [140] that also indicates the presence
ular to orthogonally applied temperature gradient ∇T of Cooper pairs.
and magnetic field H [138]. The Nernst signal, defined Optics. Like transport measurements, optical studies
as e (H,T) = E/∇T, is generally much larger in ferro- of electronic spectra [141, 142] provide information on
N
magnets and superconductors than in nonmagnetic nor- the spectrum of collective electron-hole pair excitations,
mal metals. In the superconducting state, the Nernst where a transition takes place from an initial state to
signal is the sum of the vortex and quasiparticle terms, a different final state. The difference is in final states.
e = ev +eqp, which can be distinguished with proper While in transport techniques the initial and final states
N N N
analysis,measuringthethermopower,Hallangle,andre- have the same energy, in optics they hold the same mo-
sistivity in addition to the Nernst effect [139]. In Fig.7 mentum. Both the first- and second-order processes of
the onset of ev is defined by temperature T on the light scattering are used. In the former, the light ex-
N onset
phase diagrams of LSCO and Bi-2212 (numbers on the cites bosonic degrees of freedom: phonons, electron-hole
8
FIG. 7: The phase diagrams of LSCO (left) and BSCCO (right) showing the Nernst region between T and T (numbers
c onset
onthecontourcurvesindicatethevalueoftheNernstcoefficient). TheT -curvespeaknearx=0.10. Thedashedlinesare
onset
T* estimated from heat-capacity measurements. After [138].
pairs,spinwavesorotherelectronicdensityfluctuations. spectral weight is shifted to lower frequencies, resulting
These are studied by infrared and optic absorption. in a narrowing of the Drude peak [141]. The measure-
Thesecond-orderprocesseswhenaphotonabsorbedand mentsovermuchwiderfrequencyrange,asonecanseein
reemitted are used in the Raman scattering. Fig.8 (d,e) [146–148], shows that much higher energies
could be involved. Similar depletion by the pseudogap
The absorption spectroscopy methods measure re-
is observed for the derived from conductivity scattering
flectance on single crystals or transmission in thin-films,
rate, as shown in Fig.8 (f-h) for BSCCO [149].
that allows one to study the complex dielectric function
(cid:15)(ω) = (cid:15) (ω) + i(cid:15) (ω) in the long-wave limit (q = 0), Naturally, the origin of the pseudogap has been ad-
1 2
from which the dynamical complex conductivity σ(ω)= dressed in many optical studies. Most of that ideas
σ (ω) + iσ (ω) can be derived: 4πσ = ω(cid:15) , 4πσ = can be found in the topical reviews [20, 141, 142],
1 2 1 2 2
ω(1−(cid:15) ) [28, 142]. The real part of conductivity, σ (ω), which, nevertheless, ended with the conclusions that
1 1
is proportional to the joint density of states (Kubo- there is no unified view on the nature of the pseudo-
Greenwood formula) and determines absorption of radi- gap state. That was also noted on controversy be-
ation at the frequency ω. The real part of the inverse tween optical experiments and ARPES about coher-
conductivity σ−1(ω) is proportional to the quasiparticle ence state [142]: from ARPES point of view, it is set
scattering rate τ−1 while its imaginary part is propor- only below T , but infrared methods provide evidence
c
tional to mass renormalization m∗/m = 1+λ(ω). The for coherence below the spin-gap temperature T > T .
s c
Kramers-Kronig (KK) relations allow one to calculate Also, an important role of magnetic correlations in the
both the real and the imaginary parts of ε(ω) or σ(ω) pseudogap state has been found by optical study of
from the raw experimental data. In the ellipsometric (Sm,Nd)Ba {Cu (Ni,Zn) } O with magnetic (Ni)
2 1−y y 3 7−δ
technique[143], therealandimaginarypartsofε(ω)can and nonmagnetic (Zn) impurities [150]. The broadband
be measured independently. infraredellipsometrymeasurementsofthec-axisconduc-
tivity of underdoped RBa Cu O (R = Y, Nd, and
SimpleDrudemodelpredictsthatreflectancedecreases 2 3 7−δ
La) have separated energy scales due to the pseudogap
monotonically with frequency. In HTSC, a structure in
andthesuperconductinggapandprovidedevidencethat
the form of a ‘kink’ was found. In underdoped mate-
these gaps do not share the same electronic states [151].
rials, this kink starts to develop already in the normal
state at temperatures similar to T* derived from other Raman scattering,likeopticalabsorption,measures
experiments and, therefore, was interpreted as a mani- a two-particle excitation spectrum providing direct in-
festation of the pseudogap. The corresponding changes sight into the total energy needed to break up a two-
in the optical conductivity appears as a depletion of the particle bound state. In metals, the Raman effect is dif-
spectral weight in the range 300-700 cm−1 (about 40-90 ficult to observe because of a small penetration depth
meV) [144, 145], as one can see in Fig.8 for YBCO (a- and limited energy range [20]. The signal is often riding
c). Since σ(ω) just above this range looks not changing on a high background, which might result in a consid-
withtemperature,ithasbeenconcludedthatthegapped erable data scattering, and the nodal results need a nu-
9
FIG. 8: Optical spectroscopy data which show pseudogap in different HTSC. (a-c) Inplane conductivity of YBCO for various
temperatures in both the normal and superconducting states [145]: (a) optimally doped with T = 93 K at T = 120, 100, 90
c
(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].
mericalanalysis[27]. Butitsbigadvantage,comparedto dom. In particular, it has been shown that RIXS is a
theinfraredspectroscopy, isthatthesymmetryselection suitableprobeacrossallenergyscales, including pseudo-
rules enable to measure some momentum dependence of gap,charge-transfergap,andMottgapincuprates[164].
the spectrum [160]. For cuprates there are two useful Recent RIXS experiments [165–168] together with X-ray
momentum averages: B symmetry, that is peaked at diffraction [169] has revealed CDW ordering in cuprates.
1g
(π,0), and B symmetry, peaked at (π/2,π/2). Fig.9
2g
Inelastic neutron scattering (INS) method works
shows typical Raman spectra for HgBa CuO (Hg-
2 4+δ
similar to RIXS but with neutrons instead of photons.
1201) for these two symmetries [152]. One can see that
Due to large penetration depth it is the most ‘bulk’
the peaks in these two symmetries depend on doping in
among the spectroscopies considered here but requires
opposite directions.
very large single crystals and has mainly been done on
These two energy scales are plotted on the phase di-
YBCO, LSCO, and HgBa CuO (Hg-1201). Like op-
2 4+δ
agram in Fig.10 taken from Ref.152, which has reani-
tical methods, INS measures except phonons the two-
mated the interest to the ‘two gaps’ scenario discussed
particle(electron-hole)excitationsbutwithspinflipand
earlier [128, 129]. Very similar diagrams have been sug-
with momentum resolution—in joint momentum-energy
gested in Refs.160 and 27. It has been noted that the
space. The most prominent feature seeing by INS in
B peakcoincideswiththepseudogapvalues,2∆ de-
1g PG cuprates is a ‘spin resonance’ [170] that is peaked at
rivedfromotherexperiments,whiletheB peakfollows
2g theantiferromagneticwavevectorandatenergyabout40
the superconducting gap 2∆=8k T .
B c meV. The resonance is a part of a ‘hourglass shape’ spin
Resonant inelastic x-ray scattering (RIXS) [161, excitation spectrum [171, 172] which became incommen-
162] is similar to Raman spectroscopy but has the addi- surate above and below the resonance energy but never
tionaladvantageoffull-momentum-spaceresolution. De- extends to zero energy being limited at low energies by
spiteremarkableprogressofthisnew spectroscopic tech- the so-called spin-gap [173]. In the normal state both
nique in the past decade [163], the results of this ex- YBCO and LSCO show a much weaker spectrum, which
periment are not fully understood [162]. Nevertheless, is centered around Q=(π,π) and is broader in momen-
many exciting RIXS measurements have already been tum than in the superconducting state. In the pseudo-
reported, and it is generally believed that RIXS can be gap state, some intermediate picture is observed, with a
an extremely powerful tool to probe the interplay be- gradually sharpening response at the antiferromagnetic
tween charge, spin, orbital, and lattice degrees of free- wavevector, which has been considered as a precursor of
10
5π,<,0U5π,<πU
5π,<,0U5π,<πU
NR ANR
Γ Γ
B B
2g 1g
Ov7 400,cm–1 Ov7 417,cm–1
92,K 92,K
T<<T T<<T
c c
T>T T>T
c c
Opt7 515,cm–1 Opt7 505,cm–1
95,K 95,K
H
ole,d
op
Und7 380,cm–1 Und7 660,cm–1 ing
89,K 89,K
nitsU
u
b7,
5U5arω U63n,dK7 230,cm–1 U63n,dK7 FIG. 10: Antinodal and nodal peak energies normalized to
Tmax for Hg-1201 [152], Bi-2212 [153, 154], Y-123 [154] and
c
LSCO [154]). The ratios 2∆/Tmax determined by ARPES
c
[155–157] and tunnelling spectroscopy [158, 159] are shown
for comparison. After [152].
0 200 400 600 800 0 200 400 600 800
Raman,shiftω5cm–1U
FIG. 9: Raman spectra for HgBa CuO (Hg-1201) for B
2 4+δ 2g optimal doping.
(left)andB (right)symmetries. Thearrowsindicatethepo-
1g
sitionofthesuperconductingpeakmaxima. Ov.: overdoped; Tunneling spectroscopies, like ARPES, measure
Opt.: optimally doped; Und.: underdoped. After [152]. 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 tunneling spectroscopy [182, 183], Andreev
the magnetic resonance mode that starts to develop be- reflection tunneling (ART) [184, 185], superconduc-
low T* [173, 174]. Other authors believe that there is no tor/insulator/superconductor (SIS) [186] tunneling (in
justification for a separation of the normal state spin ex- fact,bothARTandSISprobethetwo-particleDOS)and
citations spectrum into resonant and non-resonant parts superconductor/insulator/normalmetal(SIN)[120,187],
[175]. as well as scanning-tunneling microscopy/spectroscopy
Forthescopeofthisreview,itisimportanttomention (STM/STS) [121, 188]. The latter provides sub-atomic
the role of INS in discovery [97] and study [101, 176] the spatial resolution and, with the Fourier transforma-
of incommensurate SDW and CDW, called ‘stripes’, in tion [189, 190], an access to the momentum space [191–
the hole doped cuprates. As mentioned, the pseudogap 193].
can be a consequence of fluctuating stripes [101] or an The most convincing tunneling results showing that
electronic nematic order [102]. the superconducting and pseudogaps represent different
Commensurate AFM ordering has been observed in coexisting phenomena were obtained by intrinsic tunnel-
thesuperconductingYBCObyelasticneutronscattering ingfromone-andtwo-layersBSCCO[182,183,194,195].
[177]. Morerecently,thepolarizedneutrondiffractionex- The data for T* presented in Fig.11 have been obtained
perimentsonYBCO[178]andHg-1201[179]haveshown by SIS tunneling on break junctions [158] and SIN point
anexistenceofamagneticorderbelowT*consistentwith contact tunneling [159] also support the two-gaps sce-
the circulating orbital currents and QCP scenario. This nario. Andreev reflection is expected to be similar to
hasbeenfurthersupportedby INSobservationof a5256 SIN and STM, but appears to be sensitive to the super-
meVcollectivemagneticmodeappearingbelowthesame conducting energy scale only, that may be because the
temperature [180]. The idea of the intra-unit-cell mag- tunneling mechanisms are actually different [27].
netic order has been also supported by recent polarized STM, despite more complicated theoretical justifica-
elastic neutron scattering experiments on BSCCO [181] tion [196], has appeared to be extremely useful for study
which raise important questions concerning the range of the pseudogap phenomenon in cuprates allowing one to
themagneticcorrelationsandtheroleofdisorderaround explore spatial inhomogeneity [65, 197, 198] and detect