# Shallow core levels, or how to determine the doping and $T_{c}$ of
Bi2Sr2CaCu2O8+δ and Bi2Sr2CuO6+δ without cooling

Tonica Valla<EMAIL_ADDRESS>Donostia International Physics Center,
20018 Donostia-San Sebastian, Spain
Condensed Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, New York 11973, USA
Asish K. Kundu  Condensed Matter Physics and Materials Science Department,
Brookhaven National Laboratory, Upton, New York 11973, USA
National Synchrotron Light Source II, Brookhaven National Laboratory, Upton,
New York 11973, USA  Petar Pervan  Institut za fiziku, Bijenička 46, HR-10000
Zagreb, Croatia
Ivo Pletikosić  Condensed Matter Physics and Materials Science Department,
Brookhaven National Laboratory, Upton, New York 11973, USA
Department of Physics, Princeton University, Princeton, NJ 08544, USA  Ilya K.
Drozdov  Zebin Wu  Genda D. Gu  Condensed Matter Physics and Materials Science
Department, Brookhaven National Laboratory, Upton, New York 11973, USA

###### Abstract

Determining the doping level in high-temperature cuprate superconductors is
crucial for understanding the origin of superconductivity in these materials
and for unlocking their full potential. However, accurately determining the
doping level remains a significant challenge due to a complex interplay of
factors and limitations in various measurement techniques. In particular, in
Bi2Sr2CuO6+δ and Bi2Sr2CaCu2O8+δ, where the mobile carriers are introduced by
non-stoichiometric oxygen $\delta$, the determination has been extremely
problematic. Here, we study the doping dependence of the electronic structure
of these materials in angle-resolved photoemission and find that both the
doping level, $p$, and the superconducting transition temeprature, $T_{c}$ can
be precisely determined from the binding energy of the Bi $5d$ core-levels.
The measurements can be performed at room temperature, enabling the
determination of $p$ and $T_{c}$ without cooling the samples. This should be
very helpful for further studies of these materials.

###### pacs:

74.25.Kc, 71.18.+y, 74.10.+v, 74.72.Hs

##  I Introduction

Cuprate superconductivity arises from the intricate interactions between the
electrons within the copper-oxygen planes. Doping, or the introduction of
foreign atoms that shift the effective hole (or electron) concentration away
from the half-filling is fundamental to turn the antiferromagnetic parent Mott
insulator into a high temperature superconductor [1, 2, 3, 4, 5, 6, 7].
However, 38 years after the discovery, the mechanism of high-$T_{c}$
superconductivity is still unresolved. One of the reasons is related to
uncertainties in the doping level of studied materials. The doping level in
cuprates has been notoriously difficult to measure directly and accurately [8,
9]. No single technique provides a definitive picture of the doping level. In
addition, the doping level $p$, defined as the concentration of holes (or
electrons) away from the half filling, may differ from the concentration of
mobile carriers contributing to transport, $p_{tr}$ [10, 11, 12]. Hall effect
and thermoelectric power offer indirect estimates of mobile carriers $p_{tr}$,
but the Hall coefficient and thermopower in cuprates are strongly dependent on
temperature [9, 8, 10, 11, 12]. Each method has uncertainties, and the
respective values vary substantially. In practice, $p$ is usually calculated
backward from the measured $T_{c}$ by assuming the putative parabolic
$T_{c}-p$ dependence that is considered universal for all the cuprates [9, 8].
However, only in a very limited number of materials such as La2-xSrxCuO4 and
La2-xBaxCuO4, can the doping be approximately determined from chemical
composition as $p\approx x$. These two materials however, have very different
$T_{c}-p$ dependencies, illustrating the invalidity of the ”universal”
parabolic $T_{c}-p$ dome [13]. In particular, in Bi2Sr2CuO6+δ (Bi2201) and
Bi2Sr2CaCu2O8+δ (Bi2212), in which the doping is induced by introduction of
non-stoichiometric oxygen $\delta$, the determination has been very difficult
[9, 8]. Bi2212 and Bi2201 have been prototypical cuprate high-Tc
superconductors (HTSC), with their phase diagrams heavily studied by many
different techniques. In particular, these materials have been a perfect
subject of angle resolved photoemission spectroscopy (ARPES) and scanning
tunneling microscopy (STM) studies due to their ease of cleaving. ARPES and
STM have been crucial for understanding cuprate superconductivity as these two
techniques have provided the invaluable information about different phenomena
and their developments with doping, mostly in Bi2212. The $d$-wave symmetry of
the superconducting gap [14, 15], the normal state gap (pseudogap) [16, 17,
18], the quasiparticle (QP) self-energy [19, 15, 20, 21] and, more recently,
the reconstruction of the Fermi surface (FS) [22, 23] are few notable
examples.

Until recently, the most studied cuprate, Bi2212, could only be doped within a
limited range on the overdoped side, leaving a crucially important region of
the phase diagram, where $T_{\mathrm{c}}\rightarrow{0}$, out of reach. Only
very recently, has it become possible to extend the overdoped range beyond the
point at which superconductivity vanishes by annealing the in-situ cleaved
samples in ozone [24, 21]. This has made it possible to monitor the
development of electronic excitations as superconductivity weakens and finally
completely disappears, allowing a better understanding of its origins [21].

By modifying the doping level of the as-grown crystals, through annealing of
the in-situ cleaved samples either in vacuum, for underdoping, or in ozone,
for overdoping, we were able to span the wide region of the phase diagram,
ranging from strongly underdoped, to strongly overdoped, where the
superconductivity was completely suppressed. By measuring the volume of the
underlying Fermi surface, we were able to precisely determine the doping level
directly from ARPES. That allowed us to follow the development of spectral
features with doping with unprecedented clarity and construct the phase
diagram that correctly maps different phases and properties of both
Bi2Sr2CaCu2O8+δ and Bi1.8Pb0.4Sr2CuO6+δ. As an interesting result, we have
discovered that the binding energy of the Bi 5d core levels may serve as a
precise indicator of doping levels and $T_{c}$ in both materials. As the
measurements of Bi 5d core levels can be performed very quickly and at room
temperature, they can provide extremely efficient and precise determination of
both quantities, without the need for cooling the samples.

##  II Materials and Methods

The experiments were done in an experimental facility that integrates oxide-
MBE with ARPES and SI-STM capabilities within the common vacuum system [24,
21, 25]. The starting samples were synthesized by the traveling floating zone
method and the superconducting transition temperature ($T_{c}$) was measured
by spin susceptibility down to 1.8 K. The as-grown Bi2201 samples were not
superconducting and they required a vacuum annealing to achieve
superconductivity with the maximal $T_{c}\approx 20-25$ K. The as-grown Bi2212
samples were slightly overdoped ($T_{c}=91$ K) single-crystals. The samples
were clamped to the sample holder and cleaved with the Kapton tape in the
ARPES preparation chamber (base pressure of $3\times 10^{-8}$ Pa). Thus, the
use of Ag-epoxy was completely eliminated, resulting in perfectly flat cleaved
surfaces and unaltered doping level (no need for epoxy curing at elevated
temperatures). The cleaved samples were then annealed in-situ in the ARPES
preparation chamber to different temperatures ranging from 450 to 950 K for
several hours, resulting in the loss of oxygen and underdoping. For
overdoping, the cleaved as-grown samples were transfered to the MBE chamber
(base pressure of $8\times 10^{-8}$ Pa) where they were annealed in $3\times
10^{-3}$ Pa of distilled O3 at $650-750$ K for 1 hour. After the annealing,
samples were cooled to room temperature in the ozone atmosphere and transfered
to the ARPES chamber (base pressure of $8\times 10^{-9}$ Pa). Annealing of as
grown crystals in O3 resulted in increased doping in the near-surface region,
as evidenced by the increased hole Fermi surface, reduced spectral gap and its
closing temperature. The most of the crystal’s volume remained near the
optimal doping upon ozone annealing. The thickness of the overdoped surface
layer was in the sub-micron range, as only the thinnest, semi-transparent re-
cleaved flakes showed the significant reduction in $T_{c}$ in susceptibility
measurements. Therefore, the only measure of $T_{c}$ in the overdoped regime
was spectroscopic: the temperature induced changes in the quasiparticle peak
intensity, as well as the leading edge position clearly indicate $T_{c}$ [15,
26, 21]. In-situ annealing in vacuum resulted in homogeneous samples and the
surface $T_{c}$ measured by ARPES showed no variation with repeated re-
cleaving of the annealed crystal and was in a good agreement with the bulk
susceptibility measurements. Our estimate of $T_{c}$ of the overdoped surfaces
was within $\pm 5$ K, except for the samples falling outside of the
superconducting dome, for which the estimate was limited by the base
temperature that could be reached with our cryostat (10 K).

The ARPES experiments were carried out on a Scienta SES-R4000 electron
spectrometer with the monochromatized HeI (21.22 eV) radiation (VUV-5k). The
total instrumental energy resolution was $\sim$ 4 meV. Angular resolution was
better than $\sim 0.15^{\circ}$ and $0.4^{\circ}$ along and perpendicular to
the slit of the analyzer, respectively. The Bi 5$d$ core level spectra were
obtained by using the He II (40.8 eV) radiation at the total instrumental
resolution of 20 meV.

##  III Results

Figure 1: Development of the Fermi surface of Bi2Sr2CaCu2O8+δ (Bi2212) and
Bi1.8Pb0.4Sr2CuO6+δ (Bi2201) with doping. (a-c) Fermi surface of Bi2212 at
various stages of doping, as indicated. (d-f) Fermi surface of Bi2201 for
vacuum annealed (d), as grown (e), and as-grown, after annealing in ozone (f).
Spectral intensity, represented by the gray-scale contours, is integrated
within $\pm 3$ meV around the Fermi level. Solid and dashed curves represent
the Fermi surfaces obtained from tight-binding (TB) approximation that best
fit the experimental data. The area enclosed by the TB lines that best
represent the experimental data is calculated and used for determination of
the doping parameter $p$. The maps were recorded at 12 K, for Bi2212, and at
22 K (normal state) for Bi2201, respectively.

Figure 1 shows the photoemission intensity from the narrow energy window
around the Fermi level ($\pm$3 meV), representing the Fermi surfaces of the
Bi2212 (a-c) and Bi2201 (d-f) samples, at different doping levels. The tight
binding (TB) contours, representing the best fits to the experimental Fermi
surfaces are also shown [27]. The bare in-plane band structure is approximated
by the tight-binding formula:

$E_{\pm}(k)=\mu-2t(\cos k_{x}+\cos k_{y})+4t^{\prime}\cos k_{x}\cos
k_{y}-2t^{\prime\prime}(\cos 2k_{x}+\cos 2k_{y})\pm t_{\perp}(\cos k_{x}-\cos
k_{y})^{2}/4$

For Bi2212, $t_{\perp}\neq 0$ and $\pm$ is for anti-bonding (bonding) state.
$\mu$ is chemical potential. The hopping parameters that best describe the
Fermi surfaces of selected measured samples are given in Table 1.

The Bi2201 has a single Fermi sheet, while the Bi2212 has two, the bonding
(marked by the solid TB contour) and the antibonding one (dashed TB contour).
The “shadow” FS, shifted by $(\pi,\pi)$ relative to the intrinsic one is also
visible. In addition, in the Bi2212 and in the ozone annealed Bi2201, the
supermodulation replicas are also visible. Both structures originate from the
structural supermodulations and are shown to minimally affect the electrons in
the Cu-O planes [28, 29, 30, 23].

Figure 2: Determination of the Fermi wave-vector for gaped states. (a)
Dispersion at $k_{x}=\pi/a$ along the Y-M-X line corresponding to the Bi2212
sample with $T_{c}=50$ K $(p=0.25)$. (b) The same for the sample with
$T_{c}=30$ K $(p=0.09)$. The red lines represent the peak position in the
momentum distribution curves (MDC) fitted by Lorentzians. The arrows indicate
positions were the MDC derived dispersion starts becoming ”vertical”. (c) The
energy distribution curves (EDC) for the two samples from (a) and (b),
corresponding to the wave vector at which dispersion saturates vertically. The
arrows mark the corresponding gap magnitudes, $\Delta_{0}$, from Drozdov et al
[24].  Figure 3: Phase diagram of Bi2212 and Bi2201 . Superconducting
transition temperature, $T_{c}$, for Bi2212 (black circles) and Bi2201 (red
triangles) plotted versus experimentally determined doping, $p$. The maximal
gap magnitude, $\Delta_{0}$, for Bi2212 (blue circles) is plotted on the right
axis. The experimentally determined Bi2212 superconducting dome [24] is
represented by black dashed line. The superconducting dome for Bi2201 is
represented by the red dashed line [31].

For each sample, the number of carriers is obtained directly from the
Luttinger count of the area enclosed by the Fermi contour, $p=2A_{FS}-1$. For
the Bi2212, both the bonding and the antibonding states are counted,
$A_{FS}=(A_{-}+A_{+})/2$, originating from two Cu-O planes per unit cell.

The Fermi wave-vectors, $k_{F}$, for Bi2212 were obtained at the base
temperature, $T\sim 10-15$ K, in the superconducting state for most of the
samples. In the samples that displayed the sharp Bogoliubov’s quasiparticles
(BQP), and those were all the superconducting samples except for the most
underdoped one, $p=0.09$, the Fermi points were extracted from energy
distirbution curves (EDC), at the position where the BQP had a maximal energy
(Fig. 2(a,c)). In the heavily underdoped sample ($p=0.09$), the BQPs were not
observed and the EDCs were monotonic functions of energy, lacking any peak-
like structure at low energies (red curve in Fig. 2(c)). We note that both in
the case when the BQPs were present and when they were absent, the peaks in
momentum distribution curves (MDC) at “sub-gap” energies stopped dispersing,
forming a vertical dispersion of the residual intensity inside the gap (the
peak was at constant $k$ as $E$ was varied inside the gap), as shown in Fig.
2(a, b), illustrating two different doping cases. In the samples with BQPs
present, that $k$ position was found to be equal to the position of back-
folding point in Bogoliubov’s dispersion, or to the true $k_{F}$ point from
the normal state. In the strongly underdoped samples where BQP did not exist,
the $k$ point where the vertical dispersion is observed is taken to be the
$k_{F}$.

Regarding the precision to which the doping level can be experimentally
determined from the Luttinger count, we note that the states in the antinodal
region of the Brillouin zone are much broader in the strongly underdoped
samples than those in the overdoped ones, as illustrated in Fig.2. The
broadening might be due to the inhomogeneity in doping and/or the increased
scattering (self-energy) in strongly underdoped samples. Additional factor is
the bilayer splitting, which was not clearly resolved in the Bi2212 at
$p=0.09$. Regardless of the broadening mechanism, for the broader MDC peaks,
the peak positions could be determined with less certainty, necessarily
leading to larger uncertainties in the doping level. This was already
reflected in the larger $x-$axis error bars in our previous publications [24,
32, 33], where we estimated that $\Delta p/p\approx 2\Delta k_{F}/k_{F}$.
Here, $\Delta p$ and $\Delta k_{F}$ could be treated as uncertainties
originating from standard deviations in the peak positions in fitting of MDC
curves. However, if the MDC peaks are broad due to an intrinsic
inhomogeneities (in doping, for example) in these heavily underdoped samples,
then $\Delta k_{F}$ should be regarded as originating from a distribution of
such inhomogeneities. Consequently, the width of the MDC peaks, and not only
uncertainty in their positions should be taken into account when determining
an uncertainty in the doping level. Although we believe that the
inhomogeneities do not play the dominant role, as will be discussed later, we
acknowledge that the uncertainty in the MDC peak position is not the only
limiting factor in determining the doping level of a strongly underdoped
sample. Therefore, our $x-$axis error bars in Fig. 2 and Fig. 5 now reflect
the momentum width of the MDC peaks averaged over the Fermi surface and not
uncertainties in the peak positions.

Table 1: Tight-binding parameters for Bi2201 and Bi2212 samples from Fig.1 Sample |  $\mu$ (eV) |  $t$ (eV) |  $t^{\prime}$ (eV) |  $t^{\prime\prime}$ (eV) |  $t_{\perp}$ (eV)
---|---|---|---|---|---
Bi2212(a) | 0.34 | 0.39 | 0.12 | 0.045 | 0.108
Bi2212(b) | 0.445 | 0.36 | 0.108 | 0.036 | 0.108
Bi2212(c) | 0.467 | 0.36 | 0.108 | 0.036 | 0.108
Bi2201(d) | 0.42 | 0.36 | 0.1 | 0.036 | 0
Bi2201(e) | 0.49 | 0.36 | 0.1 | 0.036 | 0
Bi2201(f) | 0.545 | 0.37 | 0.1 | 0.036 | 0
Figure 4: Doping dependence of Bi 5d core levels in Bi2212 and Bi2201. The Bi
5d levels spectra for several Bi2212 (middle) and Bi2201 (bottom) samples at
different doping levels, as indicated. The top spectrum corresponds to a 1.5 Å
thick film of Bi on as grown Bi2212. The presented spectra were obtained by
using the He II radiation at $12-20$ K. Spectra were also recorded at 300 K,
with no changes observed with temperature. The thick lines represent the fits
of the Bi 5$d_{5/2}$ core level with Lorentzian peak on a cubic background.
The spectra for Bi2212 and Bi/Bi2212 are offset in $y$-direction for clarity.

By having a direct measure of the doping level, we can now plot various other
quantities versus independently determined $p$ and construct our own doping
phase diagrams. For example, we can plot $T_{c}$ measured by ARPES (as
described in Drozdov et al [24]), or by the bulk susceptibility measurements,
versus doping, as shown in Fig. 3 for both Bi2212 and Bi2201. We show the
doping phase diagram with the superconducting phases for both Bi2212 from
Drozdov et al [24] and the Bi2201 from Valla et al [31]. We also show the
maximal gap magnitude, $\Delta_{0}$, as a function of doping for Bi2212. These
phase diagrams generally agree with the published ones for these compounds [9,
8, 15, 6, 34]. The only adjustment needed for Bi2212 was the slight shift of
the SC dome, now centered at $p=0.18$. We also note that we have now revised
the $\Delta_{0}$ value for the $p=0.09$ sample. As can be seen in Fig. 2(c),
the EDC from that sample lacks any discernible peaks and the gap cannot be
determined in a conventional way, by associating it with either the peak or
the leading edge energy [35]. In that case, we think that a much better
estimate for the gap magnitude is the energy at which the MDC-derived
dispersion starts turning vertical, as illustrated in Fig. 2. In cases where
the BQPs do exist, that point coincides with the maximum in the Bogoliubov
dispersion and the associated peak energy of the corresponding EDC. In the
$p=0.09$ sample, there is no peak in EDC, but the upturn in the MDC-derived
dispersion exists, indicating the near-vertical dispersion of in-gap
intensity. We identify this point as a much better estimate for the gap
magnitude in the cuprate superconductors without BQPs. In fact, we believe
that this method might be appropriate for any gaped system with no sharp
quasiparticle peaks.

Also shown in Fig. 3 is the phase diagram for Bi2201. However, as Bi2201 is
typically co-doped with Pb or La, its phase diagram is not unique and
comparisons should be done only for the same compounds.

Now we want to point to another quantity of Bi2212 and Bi2201 spectra,
seemingly unrelated to superconductivity, but which displays the doping
dependence that can be used to directly and quickly determine the doping level
and $T_{c}$ in these materials, without even performing any $T$-dependent
measurements. This quantity is the binding energy of the Bi $5d$ core-level
states inside the Bi-O planes.

Figure 4(a) shows the Bi 5$d$ region of the photoemission spectra
corresponding to several Bi2212 and Bi2201 samples at different doping levels.
We see that with the increase in doping, the states shift to lower binding
energies, both in Bi2212 and Bi2201. The Bi 5$d$ and Sr 4$p$ levels both shift
when the doping is varied, albeit by different amounts. They both move to
lower binding energies with doping. This indicates that both Bi and Sr are
always completely oxidized, with a stable valence, in their corresponding
planes and that they cannot be more oxidized. Then, with more oxygen content
in the sample upon ozone annealing, the Bi and Sr core levels shift to lower
binding energies due to a better screening of the photohole in an increasingly
metallic environment of the progressively more doped Cu$-$O planes. This is
consistent with conventional description of core-hole screening in metals
[36].

It is interesting to note that the width of the Bi 5$d$ states is nearly the
same for all the samples. If the doping was very inhomogeneous in the heavily
underdoped samples, one would expect that the Bi 5$d$ peaks should also become
broader. Therefore, we do not think that the doping inhomogeneity is as severe
as some scanning tunneling microscopy studies suggest [37]. Consequently, the
inhomogeneity cannot be entirely responsible for the observed broadening of
the Fermi surface in the antinodal region in strongly underdoped samples.

We note that the observed relationship between the doping and the binding
energy of Bi $5d$ states works regardless of the doping mechanism. In
particular, we have discovered that any metallic species adsorbed on the
surface of Bi2212 will donate electrons to it and consequently will reduce the
hole doping at the interface [38, 33]. The hole doping can be reduced so
severely that the films on Bi2212 often make the interface non-
superconducting, rendering the proximity effect at the metal - HTSC interfaces
impossible [39, 33, 38].

For an illustration, we also show a spectrum of the thin Bi film deposited on
Bi2212 that was near the optimal doping ($p=0.2$, $T_{c}\approx 91$ K). The Bi
$5d_{5/2}$ level corresponding to the Bi$-$O plane was shifted to $-25.86$ eV,
while the state corresponding to the film was at $-23.87$ eV. The Bi film
itself was not connected and the bands were not formed yet. Its main effect on
the Bi2212 samples was donating electrons to the Cu$-$O planes.

The shifts in the Bi $5d_{5/2}$ level are summarized in Fig. 5. The first two
points for Bi2212 in Fig. 5(a) ($p\approx 0$ and $p\approx 0.045$) correspond
to the doping levels after depositing Bi films on an optimally doped Bi2212
sample, clearly indicating severe underdoping, outside the superconducting
phase. The dependence on the doping level for both Bi2212 (blue symbols) and
Bi2201 (green symbols) indicate that in both materials the Bi $5d$ levels
shift closer to the Fermi level nearly linearly with increasing doping (the
parameters for some of the Bi2212 samples used in this study are given in
Table 2). One significant distinction between Bi2201 and Bi2212 is that there
is a $\sim 0.5$ eV difference in the Bi 5$d_{5/2}$ level energy at similar
doping levels. This points to a higher metallicity and a better screening of
charges in Bi2212 than in Bi2201 - not unexpected, as the two Cu$-$O similarly
doped planes should certainly screen better than a single one.

Figure 5: The phase diagrams of Bi2212 and Bi2201. (a) The binding energy of
Bi $5d_{5/2}$ level in Bi2212 (blue circles) and Bi2201 (green triangles)
samples as a function of doping on the right-hand scale. $T_{c}$ is also shown
on the left-hand scale. (b) The doping level, $p$ as a function of binding
energy of Bi $5d_{5/2}$ core level in Bi2212 and Bi2201. Dashed lines
represent parabolic fits of the data.

Figure 5 (b) shows the inverse dependence, $p$ vs $E_{B}(5d_{5/2})$, the main
result of this study that allows the direct determination of the doping level
of Bi2212 and Bi2201 samples by measuring solely the spectra of Bi $5d$ core
levels. In the studied doping regime, the parabolic fit can be used to
determine the doping level from the binding energy $E_{B}$ of the Bi
$5d_{5/2}$ level:

$p=a+bE_{B}+c(E_{B})^{2}$ (1)

where $a=-77.468$, $b=6.4799$ eV-1 and $c=-0.13474$ eV-2 for Bi2212 and
$a=-177.06$, $b=14.261$ eV-1 and $c=-0.28641$ eV-2 for Bi2201.

Table 2: The parameters of several Bi2212 samples from Fig.5 $p$ |  $T_{c}$ (K) | treatment | Bi $5d_{5/2}$ (eV))
---|---|---|---
0 | NA | Bi film on as grown | 25.86
0.09 | 30 | vacuum ann. (950 K) | 25.70
0.14 | 85 | vacuum ann. (800 K) | 25.55
0.16 | 92 | vacuum ann. (450 K) | 25.49
0.20 | 91 | as grown | 25.40
0.25 | 50 | O3 annealed | 25.21
0.29 | 0 | O3 annealed | 25.05
0.31 | 0 | O3 annealed | 25.02

By having the doping level determined, one can then use the ”universal”
formula to determine $T_{c}$ of an unidentified sample [9]:

$T_{c}=T_{c}^{max}[1-82.6(p-p^{max})^{2}]$ (2)

where $T_{c}^{max}=94$ K, and $p^{max}=0.18$ for our modified Bi2212 dome. The
parameters for Bi2201 depend on Pb or La co-doping and were not precisely
determined for our Bi1.8Pb0.4Sr2CuO6+δ sample.

##  IV Discussion

As demonstrated in Fig. 5, our study enables an easy determination of $p$, a
quantity previously notoriously difficult to determine, and $T_{c}$, without
even cooling the sample. The determination can be performed in the in-lab
environment, using an UV-lamp and He II excitation, as in our study. The
measurements can be performed very quickly, due to a large cross section for
photoemission of Bi $5d$ levels at He II excitation. However, the other
excitation sources (synchrotron or in-lab x-ray sources) and different Bi (or
Sr) core levels could be used for the determination. For different core
levels, the slope of the doping dependence might be slightly different, but
the overall shape (nearly linear) should be preserved. In that case, re-
calibration of the binding energy vs $p$ relationship could be performed with
just two different samples with known $T_{c}$. We also expect that other core
levels with stable oxidation states could serve as indicators of the doping
level of Cu-O planes in other cuprates. Their shift to lower binding energies
would indicate better screening due to higher metallicity of the neighbouring
Cu-O planes as the hole doping increases. We hope that this new method will be
very helpful in constructing precise phase diagrams in future studies of
cuprate superconductors.

Finally, we note that some earlier studies where the doping was varied not by
oxygen non-stoichiometry, but by substituting Ca by Y, display somewhat
different behavior [40]. Our present study cannot be directly compared, as we
have not studied how the hole doping and core levels depend on Ca$-$Y, Bi$-$Pb
and other non-oxygen non-stoichiometries. Unfortunately, such studies have
never been done systematically. We think that the main problem with Tjernberg
et al [40] was a lack of an access to the actual doping level of Cu$-$O
planes. The shifts were presented vs. yttrium content that does not equate the
doping $p$.

In some other studies, the doping levels were deduced from the Luttinger
count, as required, but the observed core levels shifts were interpreted as
rigid shifts due to variation of chemical potential $\mu$ during doping [41].
Even though there were obvious inconsistencies with that scenario, as the
different core levels shifted differently, that fact was ignored for a long
time. Our present study clearly demonstrates that the shift in chemical
potential is not the main reason of the observed core levels shifts. Our study
provides an independent and consistent estimate of the chemical potential
$\mu$ and of its dependence on the measured doping level from the tight
binding approximation for the valence band, as shown in Table 1 for selected
samples. If we take this as a realistic estimate, it is obvious that $\mu$
shifts roughly 4-5 times less than the studied Bi $5d$ core levels over the
same doping range and cannot be responsible for the core level shifts observed
here. On the other hand, if the increased screening due to higher metallicity
of the doped Cu$-$O planes is the main driver of the observed shifts, then we
would expect that as the material turns from $p-$type into $n-$type (by
depositing a metallic film on an underdoped sample, for example), the Bi and
Sr core levels should eventually start shifting back, near $p=0$. Previous
studies on different families of $n-$type cuprates were inconclusive and
contradicting [42, 43], indicating that further studies are needed. Again, the
precise determination of $p$ (via the Luttinger count) and the binding
energies of the core levels should be performed simultaneously. We note that
our least doped point $(p\approx 0)$, where the underdoping was achieved by Bi
deposition, suggests that it should be possible to turn the surface layers of
Bi2212 into $n-$type and resolve this issue in future studies.

Irrespectively of the origin of the observed shifts, we have discovered a
simple and efficient method of determining the doping level of the Cu-O planes
and $T_{c}$ of Bi-based cuprate superconductors from purely spectroscopic
measurements that does not require cooling.

##  V acknowledgments

This work was supported by the US Department of Energy, Office of Basic Energy
Sciences, contract no. DE-SC0012704. I.P. was supported by ARO MURI program,
grant W911NF-12-1-0461. T.V. acknowledges the support from the Red guipuzcoana
de Ciencia, Tecnología e Innovación – Gipuzkoa NEXT 2023 from the Gipuzkoa
Provincial Council.

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