# Carrier density crossover and quasiparticle mass enhancement in a doped 5$d$
Mott insulator

Yu-Te<EMAIL_ADDRESS>High Field Magnet Laboratory (HFML-EMFL) and
Institute for Molecules and Materials, Radboud University, Toernooiveld 7,
6525 ED Nijmegen, Netherlands  Center for Theory and Computation, National
Tsing Hua University, 101, Section 2, Kuang-Fu Road, Hsinchu 300044, Taiwan
Andreas Rydh  Department of Physics, Stockholm University, SE-106 91
Stockholm, Sweden  Maarten Berben  High Field Magnet Laboratory (HFML-EMFL)
and Institute for Molecules and Materials, Radboud University, Toernooiveld 7,
6525 ED Nijmegen, Netherlands  Caitlin Duffy  High Field Magnet Laboratory
(HFML-EMFL) and Institute for Molecules and Materials, Radboud University,
Toernooiveld 7, 6525 ED Nijmegen, Netherlands  Alberto de la Torre  Department
of Physics, Northeastern University, Boston, MA 02115, United States  Robin S.
Perry  London Centre for Nanotechnology and Dept. of Physics and Astronomy,
University College London, London WC1E 6BT, United Kingdom  ISIS Neutron and
Muon Source, Rutherford Appleton Laboratory, Harwell OX11 0QX, United Kingdom
Nigel E<EMAIL_ADDRESS>High Field Magnet Laboratory (HFML-
EMFL) and Institute for Molecules and Materials, Radboud University,
Toernooiveld 7, 6525 ED Nijmegen, Netherlands  H. H. Wills Physics Laboratory,
University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom

High-temperature superconductivity in cuprates emerges upon doping the parent
Mott insulator. Robust signatures of the low-doped electronic state include a
Hall carrier density that initially tracks the number of doped holes and the
emergence of an anisotropic pseudogap; the latter characterised by
disconnected Fermi arcs, closure at a critical doping level $p^{*}\approx
0.19$, and, in some cases, a strongly enhanced carrier effective mass. In
Sr2IrO4, a spin-orbit-coupled Mott insulator often regarded as a 5$d$ analogue
of the cuprates, surface probes have revealed the emergence of an anisotropic
pseudogap and Fermi arcs under electron doping, though neither the
corresponding $p^{*}$ nor bulk signatures of pseudogap closing have as yet
been observed. Here, we report electrical transport and specific heat
measurements on Sr2-xLaxIrO4 over an extended doping range 0 $\leq x\leq$
0.20. The effective carrier density $n_{\rm H}$ at low temperatures exhibits a
crossover from $n_{\rm H}\approx x$ to $n_{\rm H}\approx 1+x$ near $x$ = 0.16,
accompanied by a five-orders-of-magnitude increase in conductivity and a six-
fold enhancement in the electronic specific heat. These striking parallels in
the bulk pseudogap phenomenology, coupled with the absence of
superconductivity in electron-doped Sr2IrO4, disfavour the pseudogap as a
state of precursor pairing and thereby narrow the search for the key
ingredient underpinning the formation of the superconducting condensate in
doped Mott insulators.

There are two principal pathways along which a system of itinerant electrons
approaches a Mott state [1]. When the interactions are predominantly local, a
paramagnetic Fermi liquid survives up until the metal-insulator transition,
albeit with a divergent quasiparticle mass. This type of transition is
observed, for instance, in Sr-doped LaTiO3 [2]. The alternative transition to
the Mott state is characterised by a vanishing carrier number and a non-
diverging quasiparticle mass. This scenario is believed to be realised in the
high-$T_{\rm c}$ cuprates [3, 4], though there the oxygen 2$p$ states render
the system a charge-transfer insulator. Moreover, the electronic states in
cuprates are $k$-space differentiated such that mobile carriers are found only
near the zone diagonals, due to the presence of an anisotropic pseudogap (PG)
that removes low-energy spectral weight preferentially near the zone edges and
breaks the underlying Fermi surface into a series of disconnected ‘Fermi arcs’
[5].

Since its discovery [6], the precise phase extent and phenomenology of the PG
has been a subject of intense scrutiny and debate, though a consensus is
beginning to emerge – that of an anisotropic, states-non-conserving gap whose
amplitude diminishes linearly with hole doping ($p$) and collapses at a
critical doping $p^{\ast}$ deep within the superconducting (SC) phase boundary
[7, 8]. Understanding the origin of PG formation is widely regarded as a key
step in the elucidation not only of the Mott transition in cuprates, but also
of the pairing mechanism for high-$T_{\rm c}$ superconductivity. Indeed,
experimental and theoretical studies have long proposed that the PG is a
signature of precursor pairing [9, 10]. Other models incorporating non-local
interactions, such as cluster dynamical mean field theory [11] or the dual
fermion approach [12], attribute both PG formation and electron pairing to the
presence of short-range spin correlations that persist beyond the doping level
at which antiferromagnetic order is destroyed. Recent Hall effect [13, 14] and
specific heat [15] measurements have found evidence for an abrupt change in
the carrier density ($n$) and electronic specific heat coefficient ($\gamma$)
near $p^{*}$, prompting suggestions that the PG may terminate at a quantum
critical point (QCP). Although other transport and thermodynamic properties
are yet to be captured within such a scenario [16], this prospect has
nonetheless motivated the discovery of a multiplicity of competing orders
within the PG regime and more broadly, the search for related phenomena in
layered materials that exhibit similar attributes to the cuprates. In this
regard, the single-layer iridate Sr2IrO4 has proven to be an excellent case-
study.

Sr2IrO4 is isostructural to the archetypal cuprate La2CuO4 (albeit with a
distorted array of IrO6 octahedra) and features a single half-filled
electronic band with $J_{\rm eff}~{}=~{}1/2$ relevant for its low-energy
excitations [17]. The undoped compound is an antiferromagnetic Mott insulator
due to the strong spin-orbit coupling of Ir 5$d$ orbitals which cooperates
with the moderate Coulomb interaction to realise a Mott state. Upon $\approx
4\%$ electron doping, antiferromagnetism is destroyed [18] though short-range
spin correlations persist to much higher doping concentrations [19, 20, 21,
22]. Surface-sensitive probes have revealed a PG state with striking
similarities to that found in the cuprates, including a suppression of
spectral weight around the zone edges and a $d$-wave energy gap by angle-
resolved photoemission spectroscopy (ARPES) [23, 24, 25], and a V-shaped PG
coexisting a U-shaped Mott gap by scanning tunneling microscopy [26]. As a
result, Sr2IrO4 has been considered as a close 5$d$ analogue of the
high-$T_{\rm c}$ cuprates and several theoretical studies [27, 28, 29] have
predicted unconventional superconductivity under sufficient electron doping.
As of now, however, superconductivity has not been confirmed. Moreover, a
systematic investigation of the transport and thermodynamic characteristics of
electron-doped Sr2IrO4, essential for understanding the nature of PG in this
5$d$ Mott insulator, has been lacking.

Here, we report transport and specific heat measurements on a series of
Sr2-xLaxIrO4 single crystals covering a wide range of La-doping ($0\leq x\leq
0.20$). The evolution of the electronic ground state with carrier doping
reveals striking parallels in the iridates and cuprates, which allows us to
identify the critical doping level at which the PG closes in Sr2-xLaxIrO4. The
absence of superconductivity in the iridates not only enables its electronic
ground state to be studied down to low temperatures and magnetic field
strengths, it also emphasises the disparate nature of the PG and SC ground
states.

### Electrical transport

Figure 1 shows the Hall resistivity $\rho_{\rm yx}$ measured on an extensive
series of Sr2-xLaxIrO4 crystals. The substitution of Sr2+ by La3+ introduces
one additional electron per unit formula. As such, the La-content $x$
corresponds to the nominal electron doping on the Ir4+ ion. We find $\rho_{\rm
yx}$ to be negative and to vary linearly with magnetic field ($H$) even in
field strengths up to 35 T (see Fig. S1), consistent with the expectation that
the doped electrons are solely responsible for the charge transport in
Sr2-xLaxIrO4. For a material with a single type of charge carrier and an
electron mean-free-path $\ell$ that is isotropic in $k$-space, the magnitude
of the Hall coefficient $R_{\rm H}$ is set by the Hall carrier density $n_{\rm
H}$:

$R_{\rm H}=\frac{\rho_{\rm yx}}{\mu_{0}H}=\frac{1}{n_{\rm H}e},$ (1)

where $e$ is the elementary charge. At a fixed low temperature $T=30~{}$K ,
$|\rho_{\rm yx}|$ and $|R_{\rm H}|$ are found to decrease by more than two
orders of magnitude as $x$ increases from $x$ = 0 to $x\approx$ 0.20 (Fig.
1a-c). Given the lack of any non-linearity in $\rho_{\rm yx}(H)$ that might
indicate the presence of two carrier types (i.e. electrons and holes), this
marked decrease in $|R_{\rm H}|$ most naturally reflects a substantial
increase in the number of itinerant carriers upon electron doping. In
addition, we observe a qualitative change in the temperature dependence of
$R_{\rm H}(T)$ as $x$ increases beyond a threshold value $x_{\rm c}\approx
0.16$. For $x<0.14$, $|R_{\rm H}(T)|$ increases monotonically as $T$
decreases, presumably due to the presence of an energy gap at the Fermi level
$E_{\rm F}$ and a loss of itinerant carriers as thermal fluctuations are
reduced. In contrast, for $x>0.17$, $|R_{\rm H}|(T)$ exhibits a maximum at
$T\approx$ 50 K, below which a modest reduction in $|R_{\rm H}|$ occurs. The
marked increase in $n_{\rm H}$, coupled with the emergence of a non-monotonic
form for $R_{\rm H}(T)$, suggests a recovery of the spectral weight at $E_{\rm
F}$ and a marked change in the electronic ground state in Sr2-xLaxIrO4 beyond
$x=x_{\rm c}$. At the same time, we find a sharp increase in the longitudinal
conductivity $\sigma_{\rm xx}$ at low $T$ as $x\rightarrow 0.14$ (Fig. 2),
reaching a plateau that corresponds to the conductance quantum per IrO2 plane,
$\frac{h}{e^{2}}\frac{1}{d}$. The subsequent dip in carrier mobility $\mu_{\rm
H}$ as $x$ increases beyond 0.17 likely reflects the enhanced scattering as
more carriers become delocalised. Intriguingly, as later shown in Fig. 4, the
effective zero-temperature carrier density per unit cell volume $|n_{\rm
H}|V_{\rm uc}(0)$ closely tracks $n=x$ up to $x\approx 0.13$, above which
$|n_{\rm H}|V_{\rm uc}$ crosses over to approaching $n=1+x$ at $x\approx
0.20$, the maximum La-content that can be currently achieved in bulk crystals.
This $n=x$ to $n=1+x$ evolution of the effective carrier density in electron-
doped Sr2IrO4 across $x_{\rm c}=$ 0.16 is our first key experimental finding.

Figure 1: Hall effect in Sr2-xLaxIrO4. a-c, Magnetic-field dependent Hall
resistivity $\rho_{\rm yx}(H)$ measured at specified temperatures for (a) $x$
= 0, (b) $x$ = 0.122, and (c) $x$ = 0.196. A negative $\rho_{\rm yx}$ that
varies linearly with applied magnetic field is found for all $x$ and $T$. Note
$|\rho_{\rm yx}|$ decreases by orders of magnitude as $x$ increases,
reflecting a large increase in the effective Hall carrier density $|n_{\rm
H}|$ induced by La doping. The high noise level for $x=0$ at low $T$ is caused
by noise in the longitudinal component of the measured signals in the highly
insulating state. d, Absolute Hall coefficient $|R_{\rm H}|$ as a function of
La-doping $x$ and temperature $T$ plotted on a log-linear scale. Dashed lines
are linear extrapolations of the $R_{\rm H}(T)$ data to the zero-temperature
axis. Semi-filled triangle marks the corresponding $R_{\rm H}$ for an
isotropic, fully delocalised $J_{\rm eff}=1/2$ band at half filling.  Figure
2:  Longitudinal resistivity and mobility of Sr2-xLaxIrO4. a, $\rho_{\rm
xx}(T)$ for representative $x$ showing a rapid suppression of low-$T$
$\rho_{\rm xx}$ with electron doping. For $x=0.035$ and $0.10$, $\rho_{\rm
xx}$ data is unavailable due to the small sample size ($<400~{}\mu$m). Dashed
lines are fits made to the data below 50 K using a variable range hopping
model in three dimensions: $\rho=\rho_{0}$ exp$(T_{0}/T)^{1/4}$. b, Carrier
mobility calculated using $\mu_{\rm H}=R_{\rm H}/\rho_{\rm xx}$ for the same
samples shown in (a). Grey shaded region is the temperature range in which
$R_{\rm H}$ is not available. Open symbols at $T=0$ are calculated using the
extrapolated $R_{\rm H}(T=0)$ and fitted $\rho_{\rm xx}$ at 2 K. c,
Conductivity $\sigma_{\rm xx}$ at 2 K for all $x$ studied. Horizontal dashed
line corresponds to the conductance quantum per IrO2 plane,
$\frac{h}{e^{2}}\frac{1}{d}$. d, $\mu_{\rm H}$ in the $T=0$ limit. Both
$\sigma_{\rm xx}$(2 K) and $\mu_{\rm H}$(0) show a rapid initial increase as
$x\rightarrow 0.14$, above which a moderate variation is induced by increasing
$x$.  Figure 3: Low-temperature specific heat of Sr2-xLaxIrO4. a, Specific
heat measured in zero applied magnetic field, plotted as $C/T$ versus $T^{2}$,
between 1.5 and 20 K. A smooth, monotonic addenda background is subtracted
from the data shown. The slopes $d(C/T)/d(T^{2})$ between 10 and 20 K are
similar for all $x$, indicating a phonon contribution that varies little with
La doping. b, Low-temperature region ($T\leq$ 5 K) for the data shown in (a)
with a vertical shift of 10 mJ mol-1 K-2 applied to each $x$ for clarity. Grey
dashed lines are fits made to the data between 2 and 5 K using
$C/T=\gamma+\beta T^{2}$ and the vertical shadings correspond to the magnitude
of zero-temperature intercepts $\gamma=C/T(T=0)$ for each $x$. A sizeable
enhancement in $\gamma$ at intermediate La-doping $0.122\leq x\leq 0.174$ is
observed.

### Specific heat

Measurements of the specific heat at low $T$ allow us to directly probe the
electronic density of states (DOS) at $E_{\rm F}$ and track of its evolution
as a function of $x$. Figure 3 shows the specific heat $C$ of Sr2-xLaxIrO4
below 20 K, plotted as $C/T$ versus $T^{2}$, measured on small flakes of the
same crystals used for the Hall measurements. Between 10 and 20 K (i.e. 100
K${}^{2}\leq T^{2}\leq$ 400 K2), $C/T$ varies approximately linearly with
$T^{2}$ with a slope $\beta\approx 0.43$ mJ mol-1 K-2, consistent with
previous reports on polycrystalline Sr2IrO4 [30, 31] and corresponding to a
Debye temperature $\Theta_{\rm D}\approx$ 320 K. To quantitatively extract the
electronic specific heat, we fit the $C/T$ data between 2 and 5 K (the lowest
$T$ window in which an approximately linear-in-$T^{2}$ behaviour is observed)
using:

$\frac{C}{T}=\gamma+\beta T^{2},$ (2)

where $\gamma$ and $\beta$ correspond to the electronic and phononic
contributions, respectively. For very lightly doped Sr2-xLaxIrO4 with $x$ =
0.012 and 0.036, a small $\gamma=0.65-1.4$ mJ mol-1 K-2 is found (see Fig.
3b), comparable to that observed in pristine Sr2IrO4 ($\gamma=1.5~{}\pm~{}0.5$
mJ mol-1 K-2; ref.[30, 31]). As $x$ increases, $\gamma$ shows a substantial
enhancement followed by a sharp drop at $x\approx 0.20$. The largest $\gamma$
is found for $x$ = 0.138 with $\gamma=9.5\pm 1.4$ mJ mol-1 K-2, approximately
one order of magnitude larger than found in the low-doped material. We note
that the nuclear contribution is negligibly small above 1 K (see Fig. S3). We
also note a similar enhancement in the measured $C/T$ at $T=$ 2 K and 1 K
(Fig. S4), for which the phononic contribution is expected to decrease very
sharply, implying that our estimate for the doping dependence of $\gamma$ is
robust in the $T=0$ limit. This large enhancement of low-$T$ specific heat
centred around $x_{\rm c}=0.16$, coinciding with the marked increase in
carrier density as summarised in Fig. 4, is our second key finding.

For a Q2D metal, $\gamma$ is linked to the quasiparticle effective mass
$m^{*}$ via:

$\gamma=\frac{\pi k_{\rm B}^{2}a^{2}N_{\rm A}}{3\hbar^{2}N_{\rm
uc}}\Sigma_{i}m^{*}_{i},$ (3)

where $a$ is the in-plane lattice parameter, $N_{\rm A}$ the Avogadro’s
number, $N_{\rm uc}$ the number of formula units in a unit cell, and $i$
denotes the distinct Fermi pocket. An enhancement in $\gamma$ thus indicates a
mass enhancement of the itinerant carriers, which can originate from the
underlying band structure (e.g. on approach to a van Hove singularity (vHs))
or many-body interactions (e.g. critical fluctuations near a quantum critical
point (QCP)). In quantum critical metals, the mass enhancement near a putative
QCP typically manifests as a pronounced peak in $\gamma$ accompanied by a
logarithmic divergence at low $T$ (i.e. $C/T~{}\propto$ ln(1/$T$) as
$T\rightarrow 0$) [32]. The absence of such a divergent $C/T(T)$ in
Sr2-xLaxIrO4 (Fig. S3), however, combined with the lack of long-range
electronic order near $x$ = 0.16 (see below), appears difficult to reconcile
with a scenario involving conventional quantum criticality.

Figure 4: Doping evolution of the Hall number and specific heat in the $T=0$
limit. a, Absolute Hall number multiplied by the unit cell volume $|n_{\rm
H}|V_{\rm uc}$(0), a measure of the effective carrier number per IrO2 plane
estimated at 0 K, as a function of $x$. Each data point is an average of the
experimental $n_{\rm H}$ at the lowest measured temperature (20 K for
$x>0.06$; 30 K for $x<0.06$) and the extrapolated $n_{\rm H}$ at 0 K (Fig.
1d). The lower and upper dotted lines correspond to $n=x$ and $n=1+x$,
respectively. Errors bars in $n_{\rm H}$ reflect the compound uncertainty in
sample thickness and 0-K extrapolation (i.e. $dn_{\rm H}(0)/{n_{\rm
H}(0)}=\sqrt{(dt/t)^{2}+(dn_{\rm H0})/dn_{\rm H0})^{2}}$ where $t$ is the
sample thickness and $n_{\rm H0}$ is the fitted 0-K intercept of $n_{\rm H}$).
Errors in $x$ are given by the standard uncertainty of EDX measurements. b,
Electronic specific heat coefficient $\gamma$ estimated from the extrapolated
$C/T(T=0)$ values. Data at $x=0$ is reproduced from previous reports [30, 31].
Dashed lines are guides to the eye. A marked enhancement in $\gamma$ is
observed, centred around $x=0.16$, coinciding with the drastic increase in
$|n_{\rm H}|$. Three regimes of distinct electronic ground state are
identified: a Mott insulator (MI) for $0\leq x\lesssim 0.085$, a pseudogap
state (PG) for $0.085\lesssim x\lesssim 0.17$, and a disordered metal (DM) for
$x\gtrsim 0.17$.

### Possible origins of the observed crossovers

As $x\rightarrow x_{\rm c}$ from the parent Mott state, we find that the
effective zero-temperature carrier density $n_{\rm H}(0)$ shows a gradual
crossover from $n=x$ to $n=1+x$, accompanied by the conductivity per IrO2
plane ($\sigma_{\rm p})$ approaching the conductance quantum and a six-fold
enhancement in electronic specific heat $\gamma$. Beyond $x_{\rm c}$, $\gamma$
is found to drop markedly in a manner that is reminiscent of what is found in
hole-doped La1.6-xNd0.4SrxCuO4 beyond $p^{\ast}\approx 0.23$ (ref. [15, 14]);
likewise, the evolution of transport coefficients in Sr2-xLaxIrO4 is
qualitatively similar to that reported in the hole-doped cuprates [3, 33, 13,
14, 34]. At low doping ($x\lesssim 0.13$), $|n_{\rm H}|(x)\approx x$, implying
that only the doped electrons become mobile, consistent with previous ARPES
studies in lightly electron-doped Sr2IrO4 showing a small spectral weight at
$E_{\rm F}$ [25, 35]. A similar relation ($n_{\rm H}(x)\approx p$ determined
at $T$ = 50 K) is found in La2-xSrxCuO4 (LSCO) below $x$ = 0.08 [3, 13]. In
YBa2Cu3O7-δ [13] and Nd-doped LSCO [14], a sharp crossover from $n_{\rm
H}\approx p$ to $n_{\rm H}\approx 1+p$ is observed in the vicinity of $p^{*}$,
suggesting that the hole initially localised in the parent state is recovered
as soon as the pseudogap vanishes. Such a drastic change in $n_{\rm H}$ may
signify a Fermi surface reconstruction (FSR) driven by an
electronic/structural order parameter with sufficiently long-ranged
correlations. In Nd-LSCO, for example, this is likely to be spin stripe
correlations [36]. In Sr2-xLaxIrO4, the only long-range electronic order known
to exist is the $(\pi,\pi)$-antiferromagnetic (AFM) order, which shares the
same reduced Brillouin zone boundary caused by the IrO6 octahedra rotations.
The resultant reconstructed Fermi surface of Sr2-xLaxIrO4 is expected to
comprise both electron-like pockets around ($\pi$/2, $\pi$/2) nodal region and
hole-like pockets around the (0, $\pi$) antinodal region [25]. Consequently,
the measured Hall signals should include contributions from both Fermi
pockets, which would typically give rise to a non-linearity of $\rho_{\rm
yx}(H)$ at sufficiently high $H$ and a $R_{\rm H}(T=0)$ value that markedly
deviates from expectations for a single conducting channel, both of which are
not observed (see Fig. S1). Moreover, the AFM order is known to vanish at a
doping level ($x\leq 0.04$) far below $x_{\rm c}$, while the lattice
distortion does not fundamentally alter the underlying electronic structure
(except leading to a back-folded shadow band that is photon energy-dependent
[23, 25, 37]). Furthermore, the structural distortion is not suppressed by La-
doping [21] thus cannot lead to a profound change in electronic structure
across the doping series investigated. It is therefore unlikely that FSR
driven by long-range order can explain the drop in $|n_{\rm H}|$ below $x_{\rm
c}\approx$ 0.16 in Sr2-xLaxIrO4.

Models of FSR in the absence of long-range order within the PG regime have
been proposed for the cuprates [38, 39, 40], and numerical calculations [41,
42] based on the model proposed by Yang, Rice, and Zhang (YRZ) [38] show that
both $n_{\rm H}$ and $\gamma$ can exhibit a marked enhancement near $p^{*}$
due to the appearance of the reconstructed electron-pocket. While it is not
yet established whether the YRZ model is applicable to Sr2IrO4, evidence for
the persistence of short-range AFM fluctuations up to 20% electron doping has
been reported [20]. A recent dynamical mean-field study of Sr2IrO4[43] found
that the electronic ground state evolves with electron doping through several
distinct regimes: a Mott-insulating state at low dopings, a PG and a
differentiated regime with suppressed spectral weight near the zone boundary
at intermediate dopings, and a uniform metallic regime in which the electronic
states at $E_{\rm F}$ become increasingly more coherent with increasing
doping. Such a scenario is consistent with the $n\approx x$ to $n\approx 1+x$
crossover, though it is not yet clear how $\gamma$ and $m^{*}$ would evolve
within such a model.

The PG in cuprates has also been interpreted as a spatially inhomogeneous,
doping-dependent localisation gap $\Delta$ that at low doping localises
exactly one hole per CuO2 unit [44]. In this picture, the $n_{\rm H}\approx p$
to $n_{\rm H}\approx 1+p$ crossover is a gradual one, as observed in certain
cuprate families[34], due to its broad distribution in energy and real-space
heterogeneity and that the holes originally localised in the parent Mott state
only become fully delocalised at the edge of the SC dome where the PG is
assumed to close. While this picture can account for a gradual crossover in
$n_{\rm H}$ with doping, the link between the crossover and PG closing is not
yet proven [34, 45].

An alternative driver for the enhanced $\gamma$ in Sr2-xLaxIrO4 is a vHs
crossing at which the Q2D Fermi surface undergoes a Lifshitz transition from
electron-like to hole-like. In such a scenario, $n_{\rm H}$ (in the
isotropic-$\ell$ limit) is expected to change sign. Were $\ell$ to be
anisotropic, however, $n_{\rm H}$ may no longer correspond to the carrier
concentration or its character [46, 47] (see Methods). In analogy with the
cuprates, the observation of Fermi arcs in the PG regime suggests a ‘nodal-
antinodal dichotomy’ that manifests itself at higher doping with $k$-dependent
scattering and a departure from the isotropic-$\ell$ limit. In this
circumstance, one expects the regions with hole-like curvature near the
antinodes to have a shorter $\ell$ than the electron-like sections close to
the nodes. Consequently, the Hall conductivity $\sigma_{\rm xy}$ will continue
to be dominated by those regions of the Fermi surface with electron-like
curvature and $R_{\rm H}$ will remain negative. The hole-like regions will
nevertheless contribute progressively to the total conductivity $\sigma_{\rm
xy}$ and $\sigma_{\rm xx}$, resulting in a reduction in $|R_{\rm H}|$ and a
corresponding growth in $|n_{\rm H}|$. Thus, the enhancements in $|n_{\rm
H}|$, $\sigma_{\rm xx}$, and $\gamma$ may be consistent with a scenario based
on a vHs crossing near $x_{\rm c}\approx$ 0.16.

While the coincidence of the vHs crossing and PG closure appears to be well-
established for LSCO [48], it is not the case in Sr2-xLaxIrO4. Previous band
structure calculations find that the vHs in Sr2IrO4 lies at approximately 0.12
eV above the chemical potential (irrespective of the inclusion of lattice
distortions), corresponding to an electron doping exceeding 20% [28, 49, 50].
This conclusion is confirmed by our tight-binding calculations that locates
the vHs at $x\approx$ 0.29 (see Supplementary Materials). All these
calculations appear to place the vHs crossing in Sr2-xLaxIrO4 well beyond
$x_{\rm c}$, though future ARPES experiments on more highly doped crystals are
required to confirm its location and clarify its impact on the thermodynamic
properties. Irrespective of the vHs location, without PG closure and a
contribution from the hole-like regions to $\sigma_{\rm xy}$, the measured
$n_{\rm H}$ would not approach $n=1+x$. Moreover, the $n_{\rm H}$ crossover to
$n=x$ would not occur without PG opening as the $\ell$-anisotropy required to
reproduce the crossover is unphysically large (see Supplemental Materials).
Hence, we can conclude that the drop in $n_{\rm H}$ from $n=1+x$ below $x_{\rm
c}$ is associated with a loss of itinerant carriers upon PG opening. This
observation not only identifies the location of PG closure, it also places
Sr2-xLaxIrO4 alongside the cuprates in the second category of doped Mott
insulators with the electronic specific-heat coefficient diminishing upon
approaching the Mott state, and implies that non-local interactions must play
a prominent role in the physics of the iridates.

Despite the striking similarities in the normal-state properties of
Sr2-xLaxIrO4 and the cuprates, to date no robust signatures of
superconductivity have been reported in the former. In this regard, it is
instructive to compare the characteristic energy scales in Sr2IrO4 to those in
isostructural La2CuO4. We summarise in Table S2 the leading terms associated
with carrier hopping ($t$), spin correlations ($J$), and charge correlations
($U$) of the two compounds. (Note for Sr2IrO4, $J$ characterises the Hund’s
coupling which promotes spin exchange [51].) As the 5$d$ orbitals are more
spatially extended, $t$ is larger in Sr2IrO4 while $J$ and $U$ are smaller. It
has been pointed out previously[51] that the phase space available for stable
superconductivity reduces with increasing $J/U$ while it expands with
increasing $U/t$, with a suggested critical value of $U/t>$ 6 for the
realisation of superconductivity. Though $J/U$ is similarly small in both
oxides, $U/t$ is substantially reduced for Sr2IrO4 and very close to (or
slightly below) the critical value. It is possible that a distinctly
favourable energy landscape for superconductivity is only realised in the
cuprates, which marks the key difference between the two oxide families,
despite sharing the same pseudogap phenomenology. A more mundane reason for
the absence of superconductivity is the higher level of disorder reflected in
the low-$T$ resistivity upturns (at sufficiently high dopings). In underdoped
LSCO ($x$ = 0.06), the in-plane resistivity undergoes a minimum [3] with
$\rho_{ab}~{}\approx$ 600 $\mu\Omega$ cm at 50 K. By contrast, in Sr2-xLaxIrO4
($x\approx$ 0.17, i.e. with a significantly higher carrier density),
$\rho_{ab}\approx$ 650 $\mu\Omega$ cm at 50 K, implying a lower overall
carrier mobility. The growth of cleaner iridates, were it ever to be possible,
would obviously help to address this point. The PG, meanwhile, is seen to be
robust to the presence of disorder.

In summary, transport and thermodynamic measurements on a series of
Sr2-xLaxIrO4 crystals reveal two defining bulk signatures of the fate of
carriers doped into a 5$d$ Mott insulator Sr2IrO4: the density of the
itinerant carriers undergoes a dramatic reduction upon the removal of electron
dopants at $x_{\rm c}\approx 0.16$ while the electronic specific-heat
coefficient is strongly enhanced upon approaching $x_{\rm c}$. Our findings
indicate that the low-$T$ electronic ground state of Sr2-xLaxIrO4 undergoes a
profound transformation with electron doping and, aided by numerical
calculations, we conclude that 1) the loss of itinerant carriers signifies the
pseudogap opening near $x_{\rm c}$; 2) the concomitant enhancement in
quasiparticle effective mass is not associated with an identifiable long-range
order parameter or a QCP; 3) a bulk pseudogapped state does not promote
superconductivity in a doped 5d Mott insulator. The striking similarity
between the PG phenomenology of the hole-doped cuprates and electron-doped
Sr2IrO4 thus suggests that PG formation is generic to doped Q2D
antiferromagnetic Mott insulators but is not of itself a precursor to
superconductivity.

## Methods

### Single crystal growth

Crystals of Sr2-xLaxIrO4 are grown using a flux method with an off-
stoichiometric mixture of SrCO3:IrO2:La2O3 and anhydrous SrCl2 flux in a small
Pt crucible (30 ml) with a loose lid. For $x<0.1$, starting materials with a
typical ratio of SrCO3:IrO2:La2O3:SrCl2 = $1.9-2x:1.0:x/2:9.0$ are heated to
1245 ∘C, held for 12 hours, and cooled to 1100 ∘C at a rate $\approx$ 8
∘C/hour then quenched to room temperature. For $x>0.1$, starting materials
with a typical ratio of SrCO3:IrO2:La2O3:SrCl2 = $1.5:1.0:18:9.0$ are heated
to 1350 ∘C and held for 30 hours then quenched. The resulting square platelet
crystals have a typical lateral size of 0.2–2.0 mm and are separated from the
flux by rinsing with de-ionised water. La content $x$, which corresponds to
the nominal electron doping on the Ir site, is determined for each measured
crystal using energy-dispersive X-ray (EDX) spectroscopy on cleaved surfaces
using a JEOL JSM-6610LV SEM from Oxford Instrument EDS, with a typical
uncertainty in $x$ of 0.01–0.02. It is worth noting that the reproducibility
between growth runs was poor with identical starting constituents producing
different results; a 50% variation in EDX-measured $x$ values is often found
across each batch compared to the nominal $x$.

### Magnetotransport measurements

Platelets of Sr2-xLaxIrO4 with a typical dimensions of $(600\times 250\times
75)$ $\mu$m3 are selected for electrical transport measurements. Gold wires of
25 or 50 $\mu$m diameter are attached to the crystals using DuPont 6838 silver
paint, with the crystal sides carefully painted to eliminate the $c$-axis
contribution, and annealed in flowing oxygen at 450 ∘C for 10 minutes to
achieve sub-Ohm contact resistances. Measurements in magnetic fields up to 8
T, applied along the $c$-axis, are performed using a Cryogen-Free Measurement
System by Cryogenic Ltd and measurements up to 35 T are performed using a
Florida-Bitter magnet coupled to a 4He-flow cryostat at the High Field Magnet
Laboratory at Radboud University. Longitudinal resistivity measurements were
performed down to $\approx$ 3 K (except for $x$ = 0.012 down to 20 K) and Hall
measurements down to 20 K. We note that small temperature fluctuations at
$T<20$ K, where all the samples show a resistive upturn with a sizeable
$|d\rho_{\rm xx}/dT|$ (see Fig. 2), lead to increased noise levels in the
measured the Hall signals (due to imperfect contact alignment) and prevent a
reliable extraction of the Hall resistivity below 20 K. The noise in our
$\rho_{\rm yx}$ measurements is at a constant level $\sim 0.02~{}\mu\Omega$ cm
for $x>0.10$. For $x\leq 0.10$, the noise level is highly dependent on $x$ and
$T$ due to the divergent longitudinal component, and reaches 2$~{}\mu\Omega$
cm at 30 K for $x=0$. Nonetheless, the percentage uncertainty in $R_{\rm H}$
at 8 T is less than 5% for all $x$ and $T$, and accounted for by the error
bars of the corresponding $n_{\rm H}(0)$.

### Specific heat measurements

The present series of Sr2-xLaxIrO4 crystals has a low mass ($\lesssim$ 1 mg)
which limits the feasibility of standard specific heat methods. In order to
perform measurements on miniature crystals, we adopt a Si-N membrane-based
nanocalorimetry method which has been successfully employed on materials with
sub-$\mu$g masses [52]. Samples from the same crystals used for the transport
measurements were cleaved to yield flakes with clean surfaces with 50–100
$\mu$m in lateral dimensions and 10–30 $\mu$m in thickness. Measurements were
performed in a Bluefors dilution refrigerator with a differential calorimeter
operating using the ac steady-state method, with automatic adjustments of
thermometer bias, ac heaters, and temperature oscillation frequency. Samples
were mounted with a very small amount of Apiezon N grease, measured separately
before sample mounting. The NiCrSiOx thin-film cermet calorimeter thermometers
[53] were operated at the $n^{\rm th}$ harmonic ($n$ between 3 and 8) of the
ac heater, proving the temperature oscillation response at the $n+2$ harmonic.
Absolute temperature control was obtained through a local offset heater on the
calorimeter membrane, with the dilution refrigerator sitting at base
temperature. Reference cell thermometer was used only as a thermometer bridge,
providing a clean, high-resolution differential measurement of the temperature
oscillation.

### Numerical calculations

We adopt the tight-binding model developed in ref.[28] to parameterise the
Fermi surface of Sr2IrO4. The doping of electrons is modelled by shifting the
chemical potential to match the enclosed Fermi surface area with the expected
value (i.e. corresponding Luttinger count). We find the van Hove singularity
(vHs) lies at $n\approx 1.29$ (i.e. 29% electron doping), consistent with a
previous report which found the vHs locates above 20% electron doping [49].
For the electronic specific heat $\gamma$, we used the electronic density of
states $D$ derived from the tight-binding Fermi surface parametrisation:

$\gamma=\frac{\pi^{2}}{3}k_{\rm B}^{2}D,$ (4)

$D=\frac{1}{2\pi^{2}d}\int^{2\pi}_{0}\frac{k_{\rm F}}{\hbar v_{\rm
F}}d\theta,$ (5)

where $k_{\rm F}(v_{\rm F})$ are the Fermi wavevector (velocity), $d$ is the
distance between two IrO2 planes, and $\theta$ is the azimuthal angle around
the 2D Fermi surface. For longitudinal and Hall resistivities, we used the
Boltzmann transport equations generalised to a 2D system with anisotropic
scatterings:

$\sigma_{ij}=\frac{e^{2}}{2\pi^{2}\hbar
d}\int^{2\pi}_{0}\tau\frac{\mathbf{v_{i}}\cdot\mathbf{v_{j}}}{|v_{\rm
F}|\cos\phi}k_{\rm F}d\theta,$ (6)

where $\phi$ is the angle between $v_{\rm F}$ and $k_{\rm F}$.

## Data availability

The source data used to create the main figures presented are provided with
this paper. Any additional data are available from the corresponding authors
upon request.

## Code availability

The codes associated with the band structure and transport coefficient
calculations that support this study are available from the corresponding
authors upon request.

## Acknowledgements

We thank F. Baumberger, A. Georges, and A. Tamai for fruitful discussions. We
would like to thank R. D. H. Hinlopen for assistance with numerical
calculations and G. Stenning and D. Nye for help on the PPMS and Smartlab
instruments in the Materials Characterisation Laboratory at the ISIS Neutron
and Muon Source. We acknowledge the support of HFML-RU/NWO-I, member of the
European Magnetic Field Laboratory (EMFL). This work was supported by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement no. 835279-Catch-22), by
the Swedish Research Council D. Nr. 2021-04360, and by the UK Engineering and
Physical Sciences Research Council grants EP/N034694/1 and EP/V02986X/1.

## Author contributions

YTH designed the project. RSP and AT grew and characterised the crystals. YTH,
MB, CD performed the magnetotransport measurements. AR performed the specific
heat measurements. YTH performed the numerical calculations. YTH and NEH wrote
the manuscript with inputs from all authors.

## Competing interests

The authors declare no competing interests.

## References

  * [1] M. Imada  “Two types of Mott transitions”  In _J. Phys. Soc. Japan_ 62, 1993, pp. 1105–1108
  * [2] Y. Tokura et al.  “Filling dependence of electronic properties on the verge of metal–Mott-insulator transition in Sr1-xLaxTiO3”  In _Physical Review Letters_ 70, 1993, pp. 2126–2129
  * [3] Y. Ando et al.  “Evolution of the Hall coefficient and the peculiar electronic structure of the cuprate superconductors”  In _Physical Review Letters_ 92, 2004, pp. 197001
  * [4] W.. Padilla et al.  “Constant effective mass across the phase diagram of high-$T_{c}$ cuprates”  In _Physical Review B_ 72, 2005, pp. 060511
  * [5] M.. Norman et al.  “Destruction of the Fermi surface in underdoped high-$T_{c}$ superconductors”  In _Nature_ 392, 1998, pp. 157–160
  * [6] H. Alloul, T. Ohno and P. Mendels  “89Y NMR evidence for a Fermi-liquid behavior in YBa2Cu3O6+x”  In _Physical Review Letters_ 63, 1989, pp. 1700–1703
  * [7] B. Keimer et al.  “From quantum matter to high-temperature superconductivity in copper oxides”  In _Nature_ 518, 2015, pp. 179–186
  * [8] J.. Tallon, J. Storey, J.. Cooper and J.. Loram  “Locating the pseudogap closing point in cuprate superconductors: Absence of entrant or reentrant behavior”  In _Physical Review B_ 101, 2018, pp. 174512
  * [9] V.. Emery and S.. Kivelson  “Importance of phase fuctuations in superconductors with small superfuid density”  In _Nature_ 374, 1995, pp. 434–437
  * [10] P. Zhou et al.  “Electron pairing in the pseudogap state revealed by shot noise in copper oxide junctions”  In _Nature_ 572, 2019, pp. 493
  * [11] H. Park, K. Haule and G. Kotliar  “Cluster dynamical mean field theory of the Mott transition”  In _Physical Review Letters_ 101, 2008, pp. 186403
  * [12] A.. Rubtsov, M.. Katsnelson and A.. Lichtenstein  “Dual fermion approach to nonlocal correlations in the Hubbard model”  In _Physical Review B_ 77, 2008, pp. 033101
  * [13] S. Badoux et al.  “Change of carrier density at the pseudogap critical point of a cuprate superconductor”  In _Nature_ 531, 2016, pp. 210–214
  * [14] C. Collignon et al.  “Fermi-surface transformation across the pseudogap critical point of the cuprate superconductor La1.6-xNd0.4SrxCuO4”  In _Physical Review B_ 95, 2017, pp. 224517
  * [15] B. Michon et al.  “Thermodynamic signatures of quantum criticality in cuprate superconductors”  In _Nature_ 567, 2019, pp. 218–222
  * [16] N.. Hussey, J. Buhot and S. Licciardello  “A tale of two metals: constrasting criticalities in the pnictides and hole-doped cuprates”  In _Reports on Progress in Physics_ 81, 2018, pp. 052501
  * [17] J. Bertinshaw, Y.. Kim, G. Khaliullin and B.. Kim  “Square lattice iridates”  In _Annual Review of Condensed Matter Physics_ 10, 2019, pp. 315–336
  * [18] X. Chen et al.  “Influence of electron doping on the ground state of (Sr1-xLax)2IrO4”  In _Physical Review B_ 92, 2015, pp. 075215
  * [19] H. Gretarsson et al.  “Persistent paramagnons deep in the metallic phase of Sr2-xLaxIrO4”  In _Physical Review Letters_ 117, 2016, pp. 107001
  * [20] D. Pincini et al.  “Anisotropic exchange and spin-wave damping in pure and electron-doped Sr2IrO4”  In _Physical Review B_ 96, 2017, pp. 075162
  * [21] K. Horigane et al.  “Magnetic phase diagram of Sr2-xLaxIrO4 synthesized by mechanical alloying”  In _Physical Review B_ 7, 2018, pp. 064425
  * [22] S.. Rodan et al.  “Quantum critical nature of the short-range magnetic order in Sr2-xLaxIrO4”  In _Physical Review B_ 98, 2018, pp. 214412
  * [23] Y.. Kim et al.  “Fermi arcs in a doped pseudospin-1/2 Heisenberg antiferromagnet”  In _Science_ 345, 2014, pp. 187–190
  * [24] Y.. Kim, N.. Sung, J.. Denlinger and B.. Kim  “Observation of a $d$-wave gap in electron-doped Sr2IrO4”  In _Nature Physics_ 12, 2015, pp. 37–41
  * [25] A. Torre et al.  “Collapse of the Mott gap and emergence of a nodal liquid in lightly doped Sr2IrO4”  In _Physical Review Letters_ 115, 2015, pp. 176402
  * [26] I. Battisti et al.  “Universality of pseudogap and emergent order in lightly doped Mott insulators”  In _Nature Physics_ 13, 2017, pp. 21–25
  * [27] F. Wang and T. Senthil  “Twisted Hubbard model for Sr2IrO4: magnetism and possible high-temperature superconductivity”  In _Physical Review Letters_ 106, 2011, pp. 136402
  * [28] H. Watanabe, T. Shirakawa and S. Yunoki  “Monte Carlo study of an unconventional superconducting phase in iridate oxide $J_{\rm eff}=1/2$ Mott insulators induced by carrier doping”  In _Physical Review Letters_ 110, 2013, pp. 027002
  * [29] Z.. Meng, Y.. Kim and H.-Y. Kee  “Odd-parity triplet superconducting phase in multiorbital materials with a strong spin-orbit coupling: application to doped Sr2IrO4”  In _Physical Review Letters_ 113, 2014, pp. 177003
  * [30] S.. Carter et al.  “Mechanism for the metal-insulator transition in Sr2Ir1-xRuxO4”  In _Physical Review B_ 51, 1995, pp. 17184
  * [31] N.. Kini et al.  “Transport and thermal properties of weakly ferromagnetic Sr2IrO4”  In _Journal of Physics: Condensed Matter_ 18, 2006, pp. 8205–8216
  * [32] Löhneysen, H. v. and Rosch, A. and Vojta, M. and Woelfle, P.  “Fermi-liquid instabilities at magnetic quantum phase transitions”  In _Review of Modern Physics_ 79, 2007, pp. 1015
  * [33] F. Laliberté et al.  “Origin of the metal-to-insulator crossover in cuprate superconductors”  In _Preprints_ , 2016, pp. arXiv:1606.04491
  * [34] C. Putzke et al.  “Reduced Hall carrier density in the overdoped strange metal regime of cuprate superconductor”  In _Nature Physics_ 17, 2021, pp. 826–831
  * [35] V. Brouet et al.  “Transfer of spectral weight across the gap of Sr2IrO4 induced by La doping”  In _Physical Review B_ 92, 2015, pp. 081117(R)
  * [36] Q. Ma et al.  “Magnetic field tuning of parallel spin stripe order and fluctuations near the pseudogap quantum critical point in La1.36Nd0.4Sr0.24CuO4”  In _Physical Review B_ 106, 2022, pp. 214427
  * [37] S. Peng et al.  “Electronic nature of the pseudogap in electron-doped Sr2IrO4”  In _npj Quantum Materials_ 7, 2022, pp. 58
  * [38] K.-Y. Yang, T.. Rice and F.-C. Zhang  “Phenomenological theory of the pseudogap state”  In _Physical Review B_ 73, 2006, pp. 174501
  * [39] S. Chatterjee and S. Sachdev  “Fractionalized Fermi liquid with bosonic chargons as a candidate for the pseudogap metal”  In _Physical Review B_ 94, 2016, pp. 205117
  * [40] C. Morice, X. Montiel and C. Pepin  “Evolution of Hall resistivity and spectral function with doping in the SU(2) theory of cuprates”  In _Physical Review B_ 96, 2017, pp. 134511
  * [41] J.. Storey  “Hall effect and Fermi surface reconstruction via electron pockets in the high-$T_{\rm c}$ cuprates”  In _Europhysics Letters_ 113, 2016, pp. 27003
  * [42] S. Verret et al.  “Phenomenological theories of the low-temperature pseudogap: Hall number, specific heat, and Seebeck coefficient”  In _Physical Review B_ 96, 2017, pp. 125139
  * [43] A. Moutenet, A. Georges and M. Ferrero  “Pseudogap and electronic structure of electron-doped Sr2IrO4”  In _Physical Review B_ 97, 2018, pp. 115109
  * [44] D. Pelc et al.  “Unusual behavior of cuprates explained by heterogeneous charge localization”  In _Science Advances_ 5, 2019, pp. eaau4538
  * [45] M. Berben et al.  “Superconducting dome and pseudogap endpoint in Bi2201”  In _Physical Review Materials_ 6, 2022, pp. 044804
  * [46] N.. Ong  “Geometric interpretation of the weak-field Hall conductivity in two-dimensional metals with abitrary Fermi surface”  In _Physical Review B_ 43, 1991, pp. 193
  * [47] N.. Hussey  “The normal state scattering rate of high-$T_{c}$ cuprates”  In _The European Physical Journal B_ 31, 2003, pp. 495
  * [48] Y Zhong et al.  “Differentiated roles of Lifshitz transitions of thermodynamics and superconductivity in La2-xSrxCuO4”  In _Proceedings of the National Academy of Sciences_ 119, 2022, pp. 2204630119
  * [49] Y. Yang et al.  “Superconductivity in doped Sr2IrO4: A functional renormalization group study”  In _Physical Review B_ 89, 2014, pp. 094518
  * [50] C. Lane et al.  “First-pinciples calculation of spin and orbital contributions to magnetically ordered moments in Sr2IrO4”  In _Physical Review B_ 101, 2020, pp. 155110
  * [51] H. Watanabe, T. Shirakawa and S. Yunoki  “Microscopic study of a spin-orbit-induced Mott insulator in Ir oxides”  In _Physical Review Letters_ 105, 2010, pp. 216410
  * [52] S. Tagliati, V.. Krasnov and A. Rydh  “Differential membrane-based nanocalorimeter for high-resolution measurements of low-temperature specific heat”  In _Review of Scientific Instruments_ 83, 2012, pp. 055107
  * [53] N.. Fortune et al.  “Wide range thin-film ceramic metal-alloy thermometers with low magnetoresistance”  In _arXiv preprints_ , pp. arxiv.org/abs/2306.06950