# The role of the pseudogap in cuprate superconductors demonstrated by the
Hall effect

Júlia C. Anjos, Hércules Santana and E. V. L. de Mello  Instituto de Física,
Universidade Federal Fluminense, 24210-346 Niterói, RJ, Brazil
<EMAIL_ADDRESS>
###### Abstract

Cuprate high-temperature superconductors are known to have a normal-state
pseudogap but, after many years of intense research, its relation to the
superconductivity is still a mystery. Similarly, the in-plane Hall coefficient
$R_{\rm H}$ has a large temperature variation caused by changes in the carrier
density which is also a long-standing puzzle. We approach both problem by a
kinetic theory of phase separation that reproduces the charge density wave by
a free energy array of potential wells. The charge modulation favors local
hole-hole attraction yielding local superconducting pairs and a transition by
long-range phase order at $T_{\rm c}$. As the temperature increases the
modulations decrease and the charge density increases by thermal activation.
This approach reproduces the $R_{\rm H}$ of La2-xSrxCuO4 compounds of
$x=0.0-0.25$ and clarifies the role of the pseudogap and the disorder in
cuprates.

The underlying physics of the pseudogap has long been probed by many different
experiments and techniquesTimusk and Statt (1999) but it is still shrouded in
mysteryM. R. Norman and Kallin (2005). It is observed in both electron and
hole-doped cuprate, but remains unresolved despite many years of intense
studiesKordyuk (2015). More concerning still, there is not a agreement on the
precise phase boundaries and phenomenology of this normal state gap. It is
agreed that it represents a partial gap in the electronic density of
statesHüfner  _et al._ (2008) and is seen in the antinodal region of
$k$-spaceWise  _et al._ (2008). Once, it was believed to be solely associated
with the under-doped side of the phase diagram but now it is described as a
line falling to zero either at a critical doping in the overdoped regionBadoux
_et al._ (2016); Hüfner  _et al._ (2008) or at the end of the superconducting
domeHüfner  _et al._ (2008); Hashimoto  _et al._ (2014).

On the other hand, the in-plane Hall coefficient $R_{\rm H}=1/ne$, where $e$
is the electron charge and $n$ the itinerant carrier density, probes the
charge dynamics of metals and semiconductorsOng (1990). In metals described by
the Fermi liquid model, $R_{H}$ is constant and negative because the
conduction band density of electrons. In cuprate high-temperature
superconductors (HTS), it is positive due to the hole carriers and has a large
variation with the temperature. Comparison between powder and single crystals
data demonstrated that the magnitude and temperature dependence of $R_{\rm H}$
is dominated by the in-plane contributionHwang  _et al._ (1994) but after all
these years there is not an explanation why $n$ varies with the temperature.

We propose here that $R_{\rm H}$ is influenced by the planar charge
instabilities that is observed in different forms like stripesTranquada  _et
al._ (1995), checkerboardHanaguri  _et al._ (2004) or puddlesLang  _et al._
(2002). At first, this was a phenomenon attributed solely to weakly doped
compounds around $p=1/8$ per Cu atom, where the charge density wave (CDW)
signal is generally more intenseGhiringhelli  _et al._ (2012); Hücker  _et
al._ (2014); Blanco-Canosa  _et al._ (2014). However, a variety of
complementary experimental probes have detected charge instability in all
hole-doped HTS familiesComin and Damascelli (2016) as well as in Nd-based
electron-dopedda Silva Neto  _et al._ (2018). More recently, CDW has been
detected in overdoped La2-xSrxCuO4 (LSCO) up to at least $p\equiv x=0.21$Wu
_et al._ (2017); Chen  _et al._ (2019); Fei  _et al._ (2019); Miao  _et al._
(2021) and possibly up to $p=0.27$ in the form of puddlesLi  _et al._ (2022);
Tromp  _et al._ (2023). This overall presence of various forms of
incommensurate charge order (CO) suggests that they may be more intimately
related with the superconducting phase than initially thought.

Accordingly, we have developed a model which the pseudogap is a mesoscopic
electronic phase separation that leads to CO in HTS is precursor of the
superconductivityde Mello (2012); deMello and Sonier (2014); de Mello and
Sonier (2017); de Mello (2021); Santana and de Mello (2022, 2023). We study
this phenomenon by the time-dependent non-linear Cahn–Hilliard (CH)
differential equation used to describe the order-disorder transition of binary
alloys formation. The method minimizes a Ginzburg-Landau (GL) free energy
given by the (phase separation) order parameter power expansionde Mello
(2012); deMello and Sonier (2014). The GL potential ($V_{\rm GL}$) forms an
array of wells hosting alternate high and low hole densities domains of
wavelength $\lambda_{\rm CO}$ size (see Fig.1). The low density regions favor
antiferromagnetic (AF) fluctuations that are reminiscent of the Mott AF ground
state of low doping compounds and whose existence was central to several
magnetic mediated pairing theoriesM. R. Norman and Kallin (2005); Hüfner  _et
al._ (2008). When $T$ approaches $T^{*}(p)$, the $V_{\rm GL}$ amplitude or
modulation decreases untightening the CO, what makes, for instance, the
softening of the x-ray peaksHücker  _et al._ (2014); Blanco-Canosa  _et al._
(2014) and the angle-resolved photoemission spectroscopy (ARPES)Hashimoto  _et
al._ (2014); Kanigel  _et al._ (2008). We take this effect in $R_{H}(p,T)$ by
calculating the flow of holes tunnelling through the nanoscopic barriers shown
in Fig. 1 and insets.

The presence of $T^{*}(p)$ and the similarities between CO in hole- and
electron-doped phase diagramsda Silva Neto  _et al._ (2018) suggest the
existence of CDW modulations also around the half-filled ($p=0$ or $n=1$). In
other words, even the insulator compounds with $n\approx 1$ posses the $V_{\rm
GL}$ or charge modulations. Indeed, the variations of $R_{H}(p,T)$ at high
temperaturesOno  _et al._ (2007) in compounds with very low $p$ support this
statement. Therefore, in samples with larger doping, the additional $p$ holes
tend to fill in alternating high and low-density domains. Then, in a typical
compound with $n=1+p$ itinerant carriers, the half-filled $n=1$ are associated
with the $V_{\rm GL}$ free energy potential and $p$ become localized in the
CDW domains in agreement with a coexisting two-gap or two-particle scenario
verified by different experimentsHüfner  _et al._ (2008); Wise  _et al._
(2008); Yoshida  _et al._ (2012); Kondo  _et al._ (2015).

In this scenario, the tunnelling process through the array of $V_{\rm GL}$ and
the two well-known energies of pseudogap and superconducting gapHüfner  _et
al._ (2008); Wise  _et al._ (2008); Yoshida  _et al._ (2012); Kondo  _et al._
(2015) are sufficient to interpret the dynamics of the holes as a function of
temperature. Expressing these two energies as proportional to $T^{*}(p)$ and
the onset of pair formation $T_{c}^{\rm max}(p)$, we reproduce the main
aspects of the temperature and doping dependence of $R_{\rm H}(p,T)$ of
LSCOHwang  _et al._ (1994); Ando  _et al._ (2004); Tsukada and Ono (2006); Ono
_et al._ (2007). These Hall effect calculations with these two aspects of the
charges provide knowledge to understand the basic building blocks of HTS,
notably the role of the pseudogap.

The calculations start with the time-dependent CH equation which reproduces
the CDW charge instability. It is written in terms of the local phase
separation order parameter $u(r_{i},t)=(p(r_{i},t)-p)/p$, where $p(r_{i},t)$
is the local charge or hole density at a unit cell position $i$ in the CuO
plane and $p$ is the average doping levelde Mello and Sonier (2017); de Mello
(2021, 2020a, 2020b); Santana and de Mello (2022). The simulation evolves to
total phase separation, but it is stopped when the charges reach a
configuration close to a given CO with the $\lambda_{\rm CO}$ of a sample. The
CH equation is based on minimizing the GL free energy:de Mello and Kasal
(2012); deMello and Sonier (2014); de Mello and Sonier (2017):

$f(u)={{\frac{1}{2}\varepsilon|\nabla u|^{2}+V_{\rm GL}(u,T)}},$ (1)

where $\varepsilon$ is the parameter that controls the charge modulations
$\lambda_{\rm CO}$ and ${V_{\rm GL}}(u,T)=-A^{2}u^{2}/2+B^{2}u^{4}/4+...]$ is
a double-well potential. The charge oscillations appear below the phase
separation temperature $T_{\rm PS}$, $B=1$ and $A^{2}=\alpha(p)[T_{\rm
PS}-T]$. In Fig. 1 we show a typical low temperature $V_{\rm GL}(u(r_{i},T))$
simulation and its valleys which tend to bound the carriers in alternating
high and low densities CDW domains. We show in the inset (a), the $V_{\rm
GL}(u(r_{i},T)$ amplitude along the $x$-direction that diminishes when the
temperature raises towards $T_{\rm PS}\geq T^{*}$ where the system becomes
homogeneous. The size of the barriers between the two (high and low charge
density) phases is proportional to $A^{4}/B^{2}$ or $\langle V_{\rm
GL}(p,T)\rangle=\alpha(p)^{2}[T^{*}-T]^{2}$ , for $T\leq T^{*}$.

Thus, at low temperatures, ${V_{\rm GL}}$ produces alternating high and low
charge domains and such inhomogeneous carrier distribution induces ions‘
fluctuations that may lead to hole-hole attractionde Mello (2020a, 2021). Such
indirect interaction yields local superconducting pairs proportional to the
barrier size $\langle V_{\rm GL}(p,T)\rangle$ what is in agreement with the
HTS coherence lengths being smaller than the CO wavelengthdeMello and Sonier
(2014); de Mello and Sonier (2017); Santana and de Mello (2022), i.e.,
$\xi_{\rm sc}\leq\lambda_{\rm CO}$. On the other side, when the temperature
increases, $V_{\rm GL}(p,T)$ diminishes as shown in inset (a) and the carriers
become uniform which leads to the temperature effect on $R_{\rm H}(p,T)$ that
we will detail below.

Figure 1:  Low-temperature CH simulations of the planar GL free energy phase
separation potential with its valleys and barriers that originate the CDW
structure. Inset (a) shows the $V_{\rm GL}$ profile along the $x$-direction
for increasing $T$ up to near $T^{*}$. In the inset (b) we describe the
simplified model to the $T=0$ K potential barriers to the transmission
coefficient used to calculate the number of carriers with the temperature.

We mentioned before that the $p=0$ Hall measurementsOno  _et al._ (2007) also
display an increase in the number of hole carriers with the temperature
similar to most of the compounds, up to the overdoped side. This behavior
indicates that possibly all the samples undergo a continuous phase separation
below $T\approx T^{*}$. At low temperatures, ${V_{\rm GL}(p,T)}$ has a series
of “ice cream cones” as shown in Fig. 1 which slowly “melts” when the
temperature raises. For $p\geq 0.0$, at low temperatures, the $p$ doping holes
tend to become localized or even trapped in the $n=1$ CDW domains, where they
may also form local pairsde Mello (2020a, b); Santana and de Mello (2022).
According to this scenario, we distinguish three independent temperature
dependence to $R_{\rm H}(p,T)$;

First, the two sources of carriers mentioned above:
i- The $p$ holes of a $n=1+p$ compound become localized in the CDW alternating
charge domains and, eventually, at low temperatures they form local
superconducting order parameter that establish local amplitudes and the
superconducting state by phase coherencede Mello (2012). The energy of these
pairs are not scaled by $T_{\rm c}$ but by $T_{\rm c}^{\rm max}$, that is, the
onset of experimental superconducting pairs or precursor pairs detected by
many experiments like ARPESKanigel  _et al._ (2008) and STMGomes  _et al._
(2007). Below $T_{\rm c}^{\rm max}$ the localized superconducting pairs form
an array of mesoscopic Josephson junctions, and $R_{\rm H}$ is small. As the
temperatures increases above $T_{\rm c}$ but below $T^{*}$, these pairs
eventually brake free and $R_{\rm H}$ has a maximum near $T_{\rm c}^{\rm
max}$. The number of unbound holes is obtained by thermal excitation;
$n_{1}(p,T)=pB_{\rm c}(1+exp(-T_{\rm c}^{\rm max}/T))$, where $B_{\rm c}=2$ is
the number of CuO planes in the unit cell of LSCOShibauchi  _et al._ (1994).

ii- The $R_{\rm H}(p,T)$ variation with the temperatureOno  _et al._ (2007)
indicates that the CDW free energy modulations are already present at half-
filled ($n=1.0$ and $p=0$). Therefore we assume that such modulations are
formed by the half-filled holes and not by the hole density ($p$) from doping.
Again, with the rise in temperature, the modulations of ${V_{\rm GL}}$
decrease as shown in the inset (a) of Fig. 1 because part of the these $n=1.0$
holes diffuse through the system. Since the CDW sets in at $T^{*}$ this
process is scaled by the pseudogap temperature and the temperature dependence
of the number density of such carriers is given by thermal activation;
$n_{2}(p,T)=nB_{\rm c}(1+exp(-T^{*}(p)/T))$, where $n=1$ is the same to all
compounds from $p=0$ to $p=0.25$.

iii- Furthermore, all carriers have to move through the background of the
$V_{\rm GL}(p,T)$ modulations that are large at low doping and low
temperatures but decrease when the system is warmed, as shown in the inset (a)
of Fig. 1. This is taken into account by the tunnelling current probability
between the charge domains which is not easy to calculate exactly, and we
adopted a simplified model based on one-dimensional quantum mechanics
transmission coefficientGasiorowicz (1974). In this way, the tunnelling
expression through rectangular barriers like those shown in the inset (b) of
Fig. 1 with width of $\lambda_{\rm CO}/2$ and height proportional to $\langle
V_{\rm GL}(p,T)\rangle$ is given by.

$T_{\rm trans}^{2}=n(p,T)e^{-[{\gamma\langle V_{\rm
GL}\rangle(1-(T/T^{*}))^{2}}]^{1/2}}$ (2)

where $n(p,T)=n_{1}(p,T)+n_{2}(p,T)$ is the total number density of holes,
$n_{1}(p,T)$ and $n_{2}(p,T)$ are given just above,
$\gamma=2m\hbar^{2}/\lambda^{2}_{\rm CO}$ and $m$ is the electron mass. The
values of $\langle V_{\rm GL}\rangle$ have been calculated before in Refs. de
Mello, 2021; Santana and de Mello, 2022 and the ones used in the calculations
are listed in Table 1. Notice that $\gamma\langle V_{\rm GL}\rangle$ is
adimensional and a measurement of the CDW rougheness.

Now we use the above calculated $n(p,T)$ to reproduce the temperature
dependence of the number of holes that is the main and most puzzling
ingredient of $R_{\rm H}(p,T)$. These very simple expressions are robust
enough to describe all $p\leq 0.25$ experimental valuesHwang  _et al._ (1994);
Ando  _et al._ (2004); Tsukada and Ono (2006); Ono  _et al._ (2007) using only
their values of $T^{*}(p)$ and $T_{\rm c}^{\rm max}(p)$. For $0.0<p\leq 0.05$
we keep the $n(p,T)$ constant for $T\leq T_{\rm AF}=300$ K, the Neèl
temperature, because the antiferromagnetic phase. The low density AF clusters
act like as if the carriers are frozen what yields a plateauOno  _et al._
(2007) in $R_{\rm H}(p,T)$.

The values of $\langle V_{\rm GL}\rangle^{\prime}$
$=2m\hbar^{2}/\lambda^{2}_{\rm CO}\langle V_{\rm GL}\rangle$ = $\gamma\langle
V_{\rm GL}\rangle$ for under and overdoped LSCO are obtained from the tables
in the supplemental information of Ref. de Mello, 2021 and $\lambda^{2}_{\rm
CO}(p)$ from Ref. Comin and Damascelli, 2016. For instance, the optimal doped
$p=0.16$ hasde Mello (2021) $\langle V_{\rm GL}\rangle=0.234$ meV and
$\lambda_{\rm CO}=3.9a_{0}$ with $a_{0}=3.78\AA$ and $\sqrt{\gamma\langle
V_{\rm GL}\rangle}=1.833$, and therefore we use in Eq. 2 the value 1.80 to
$p=0.15$. $\langle V_{\rm GL}\rangle^{\prime}$ to other $p$’s are either
calculated in this way or, like the small doping compounds that we do not have
all the data, are extrapolated. We list these results in the last column of
the Table 1 below.

The $R_{\rm H}(p,T)$ results from the above analysis are shown in Fig. 2 with
the experimental data of Ref. Ono  _et al._ , 2007 and the agreement is very
good from $p=0.0$ to 0.23. Only $p=0.25$ diverges from our calculations above
$T=100$ K. The decrease of the experimental $R_{\rm H}(p,T)$ is most likely
due to the negative Hall coefficient of out-of planes electrons of strong
overdoped samples that was reported by Ref. Tsukada and Ono, 2006. The
presence of electrons may be connected with the decrease of the rougheness
parameter below 1.0 shown in Table 1.

Figure 2: The $R_{\rm H}(p,T)$ experimental points from Ref. Ono  _et al._ , 2007 and $p=0.21$ from Ref. Ando  _et al._ , 2004 with the theoretical model derived in the text (the continuous lines). The agreement is good from $p=0.0$ to 0.23. The case of $p=0.25$ is the only one yielding much larger values above $T=100$ K, what is most likely due to the negative $R_{\rm H}(p,T)$ of out-of-plane electrons in strong overdoped samples according to Ref. Tsukada and Ono, 2006.  Table 1:  Properties of LSCO used in the calculations. The first column is hole doping density per unit cell. Second is the $T^{*}$ and third is $T_{\rm c}$ that does not enter in the calculations but is a good reference and the last column is the renormalized free energy barrier $\sqrt{(}\langle V_{\rm GL}\rangle^{\prime})$ or rougheness factor explained in the text.  Sample |  $T^{*}$ (K) |  $T_{\rm c}$ (K) |  $T_{\rm c}^{\rm max}$ (K) | $\sqrt{(}\langle V_{\rm GL}\rangle^{\prime})$
---|---|---|---|---
p = 0.00 | 1800 | 0.0 | 0.0 | 9.0
p = 0.01 | 1700 | 0.0 | 10.0 | 6.0
p = 0.05 | 1350 | 1.0 | 70.0 | 4.0
p = 0.08 | 1300 | 20.0 | 280.0 | 3.0
p = 0.12 | 900 | 34.0 | 200.0 | 2.3
p = 0.15 | 850 | 39.0 | 160.0 | 1.8
p = 0.18 | 650 | 36.0 | 160.0 | 1.1
p = 0.21 | 350 | 24.0 | 110.0 | 0.6
p = 0.23 | 140 | 18.0 | 70.0 | 0.4
p = 0.25 | 80 | 12.0 | 40.0 | 0.2

In conclusion, the pseudogap‘s role in cuprate superconductors as a mediator
of phase separation and the two-particle and two-gap scenarioHüfner  _et al._
(2008) are identified by the $R_{\rm H}(p,T)$ calculations. The model based on
phase-ordering kineticsde Mello (2021) reveal the nature of the two types of
carriers in a typical compound of $n=1+p$ hole doping: The half-filled charges
($n=1$) of a single band are responsible by the phase separation free energy
shown in Fig. 1 and scaled by the temperature $T^{*}(p)$. The other $p$ holes
of a $n=1+p$ compound fill in the alternating CDW domains breaking the uniform
charge distribution and inducing ion-mediated pair interaction. The energy of
these pairs are scaled by the onset temperature of superconducting pair
formationGomes  _et al._ (2007); Kanigel  _et al._ (2008) $T_{\rm c}^{\rm
max}(p)$, that is just a few times $T_{\rm c}$ (see Table 1). For low doping
the physical properties like superfluid density or quantum oscillationsBadoux
_et al._ (2016); de Mello (2020b) are dominated by the thermal activated $n=p$
holes because $T_{\rm c}^{\rm max}(p)\ll T^{*}(p)$. Increasing doping, above
$p\approx 0.19$ the two energies or temperatures converge to the same order of
magnitudeChen  _et al._ (2019) (see Table 1) and the carrier concentration
detected in experiments crosses over from $p$ to $1+p$Badoux  _et al._ (2016);
Li  _et al._ (2022); Tromp  _et al._ (2023); de Mello (2020b). Taking into
account the above temperature dependent contributions we reproduced all the
LSCO $R_{\rm H}(p,T)$ measurements.

We acknowledge partial support from the Brazilian agencies CNPq and by
Fundação Carlos Chagas Filho de Amparo Pesquisa do Estado do Rio de Janeiro
(FAPERJ), Projects No. E-26/010.001497/2019 and No. E-26/211.270/2021.

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