Chemical doping: composition, carriers and disorder

Scope

Chemical doping changes a material's composition; it does not supply a universal carrier counter. Distinguish the starting mixture, measured site occupancy, formal ionic charge and mobile carriers. Substitution can also change disorder, bond geometry and competing order. Oxygen removal or chain ordering adds another control variable. A superconducting dome is therefore a relationship among specified samples and criteria, not a recipe transferable between families. R2 R3 R4 R5

Notation matters even before measurement. Takagi's 1989 abstract uses $(\mathrm{La}_{1-x}\mathrm{Sr}_x)_2\mathrm{CuO}_4$ through $x=0.175$. Expanding the formula gives $\mathrm{La}_{2-2x}\mathrm{Sr}_{2x}\mathrm{CuO}_4$: in $\mathrm{La}_{2-y}\mathrm{Sr}_y\mathrm{CuO}_4$, therefore, $y=2x$ and the same endpoint is $y=0.35$. For example, site fraction $x=0.075$ corresponds to $y=0.15$, or $\mathrm{La}_{1.85}\mathrm{Sr}_{0.15}\mathrm{CuO}_4$. This is a composition conversion, not a measured hole density or an experimentally selected optimum. The publisher title and abstract were checked; full methods remain outside this review. R1

Mathematical assumptions

Charge neutrality is useful bookkeeping under explicit ionic assumptions. Taking La/Nd as $3+$, Sr as $2+$, Ce as $4+$ and O as $2-$ gives formal average Cu valence $2+x+2\delta$ in $\mathrm{La}_{2-x}\mathrm{Sr}_x\mathrm{CuO}_{4+\delta}$, and $2-x-2y$ in $\mathrm{Nd}_{2-x}\mathrm{Ce}_x\mathrm{CuO}_{4-y}$. Stoichiometric infinite-layer $\mathrm{Nd}_{1-x}\mathrm{Sr}_x\mathrm{NiO}_2$ gives formal Ni valence $1+x$. These balances do not locate holes among orbitals or establish an itinerant carrier density. Oxygen deviations and compensating defects change the balance. R1 R5

Even the elementary weak-field two-band Drude model yields $R_H=(n_h\mu_h^2-n_e\mu_e^2)/[e(n_h\mu_h+n_e\mu_e)^2]$, with $e>0$. Mobilities weight the response. For compensation, $n_e=n_h=n$, this becomes $(\mu_h-\mu_e)/[en(\mu_h+\mu_e)]$; its sign can reverse at fixed density. This illustrative identity is not a fit to correlated transport. Kasahara explicitly discusses why that simple model fails for the magnitude of the observed Hall response. R4

Liang's empirical YBCO calibration uses $p=11.491u+5.17\times10^9u^6$, $u=1-c/c_0$, with $c_0=1.18447\,\mathrm{nm}$ and $c$ measured at 22°C. It depends on chain-order assumptions and reference compositions. The relation $T_c/T_{c,\max}=1-82.6(p-0.16)^2$ has two algebraic branches and fails near the material's suppressed one-eighth region. Neither expression is a universal oxygen-to-hole conversion; a fitted lattice parameter and a directly measured carrier density are different evidence. R2

Experimental criteria

Measure composition on the specimens whose transitions are compared. Ni's Co-substituted Ba122 series used wavelength-dispersive spectroscopy (WDS): actual Co fraction was about 0.74 times the nominal fraction for that growth series. A nominal 0.10 batch averaged approximately 0.074. Its phase diagram distinguishes resistance, magnetization and heat capacity. Resistive offset means maximum-slope extrapolation to zero, rather than an independently established instrumental zero. The reported maximum near 23 K spans measured fractions approximately 0.058–0.074. R3

For YBCO, oxygen content alone omits chain configuration. Liang determined oxygen content by weight analysis with a stated precision of $\pm0.002$ in $x$; that is not an uncertainty in planar hole concentration $p$. The chain-ordered ortho-II, ortho-VIII and ortho-III preparations use $x=0.50$, $0.67$ and $0.75$. The superconducting $T_c$ is the midpoint of field-cooled magnetization in 1 Oe along the c axis; the lattice parameter is measured separately at 22°C.

The fully oxygenated reference has $T_c=85.3$ K. With the paper's $T_{c,\max}=94.3$ K, algebraic inversion of the empirical parabola gives $p\approx0.126$ or $0.194$. The paper uses the latter because it treats that reference as slightly overdoped. Temperature alone does not choose a branch; the lower algebraic root also lies near the one-eighth region where the uncorrected calibration is unreliable. The quoted $p$ is a model-dependent estimate, not a directly measured carrier count. Record oxygen/order history, calibration assumptions and uncertainty with the result. R2

In the P-substituted Ba122 author manuscript, composition was measured by energy-dispersive X-ray analysis. Figure 2 separates onset and zero-resistivity temperatures. Version 3 reports zero resistance from $x\geq0.20$ and a maximum transition around 31 K, while observing a narrow transition at $x=0.33$. Do not turn “isovalent” into proof of unchanged Fermi surfaces, scattering or carrier populations. Nor does a similar phase diagram make chemical substitution physically identical to applied pressure. R4

Li's nickelate study uses 8–10 nm films on SrTiO3, capped and reduced from perovskite precursors. The reported preparation couples approximately 25 nm caps to 280°C for 4–6 h and approximately 2 nm caps to 260°C for 1–3 h. The source reports no notable differences between those cap groups at the same composition; these are its preparation conditions, not a universal synthesis prescription. The in-plane lattice constants are locked to the substrate at 3.91 Å in the checked measurements.

The sampled superconducting compositions are $x=0.15$, $0.175$, $0.20$ and $0.225$, inside the reported $0.125<x<0.25$ dome. Figure 2 defines $T_{c,10\%R}$ and $T_{c,90\%R}$ using 10% and 90% of each sample's resistance at 20 K. Symbols average multiple samples at the same $x$; error bars span the observed extrema, not a stated standard error or confidence interval. The comparison cuprate curve uses a 50% criterion. These temperatures cannot be pooled as one uniform definition of $T_c$.

For the sampled $x=0$, $0.10$, $0.125$ and $0.25$ films, the caption reports no superconducting sign down to below 50 mK. That is a finite measurement bound, not an exact zero transition temperature or a precisely located continuous phase boundary. The 20 K resistance reference is also not a transition. No numerical nickelate $T_c$ is digitized here, and the small Hall response remains interpreted through multiple bands. R5

Limits and disputes

Sidis's author manuscript compares reported 3% Ni-substituted YBCO ($T_c=80$ K) with 1% Zn-substituted YBCO ($T_c=78$ K). The samples were chosen for similar transition temperatures, not equal impurity fractions. Their reported impurity ratios were inferred from the suppression of $T_c$ relative to the pure system; Ni content and homogeneity were additionally checked by microprobe. A corresponding independent Zn assay is not stated in the selected passage. These two samples alone do not determine an independent dose-response slope or a universal impurity ranking.

In the checked 20–50 meV neutron window, the Ni sample's normal-state response has a finite detection upper limit; an undetected signal is not exact zero. The low-temperature-minus-above-$T_c$ spectra peak near 35 meV for Ni and 38 meV for Zn. Those are neutron spectral energies, not transition temperatures or direct gap measurements. The manuscript's Figure 2 caption reverses the sample order relative to its panel labels and surrounding text: the panels label (a) 3% Ni and (b) 1% Zn. This account follows the panel labels and text together and flags the disagreement; final-version comparison and raw background subtraction remain unchecked. The proposed impurity scenarios do not uniquely establish a pairing mechanism. R6

Keep author versions and specimen labels attached to every value. Ni's v1 specific-heat paragraph and Figure 8 label $x=0.078$, while its WDS example averages $x=0.074$ and its phase-summary maximum spans $0.058\leq x\leq0.074$. Their numerical proximity does not establish one specimen identity; this review does not silently replace 0.078 with 0.074.

Kasahara's author versions report different sampled ranges and zero-resistivity thresholds:

These are version-specific reports, not a statistical uncertainty interval or a continuously measured phase boundary. Version 3 Figure 2 labels the onset curve above the zero-resistivity curve, but its caption assigns their colors in the opposite order. Read the criteria and labels together; no precise onset/zero value is digitized here. Both versions caution that superconductivity in the low-composition coexistence region may lack bulk character. Neither a zero-resistance path nor simultaneous anomalies alone prove homogeneous microscopic coexistence. Full journal-version comparison and raw-data reproduction remain open. R3 R4

A dome does not establish a unique pairing mechanism, homogeneous microscopic coexistence or a universal optimal carrier concentration. The nickelate manuscript notes sample-to-sample resistivity variations associated with microstructure and does not establish the origin of its small dip within the dome. Preserve that uncertainty; numerical resemblance to a cuprate anomaly is not an ordering measurement. R5

Discovery implications

Compare the quantities before comparing the domes:

This table is an editorial comparison of the checked experiments. Keep nominal composition, measured composition, formal valence and transport-model parameters separate; link each value to a specimen, preparation, probe and source version. Reduction, annealing or a static measurement alone is not this review's evidence of thermodynamic equilibrium. For the YBCO reference and nickelate records, equilibrium remains unknown at the checked source scope; this does not assert nonequilibrium. The calculations verify only branch arithmetic; they do not optimize a synthesis or validate a carrier-density calibration. R2 R5

Related topics

Compare cuprates, iron-based superconductors, nickelates, pressure tuning, zero resistance and specific heat.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.