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A cuprate superconductor is a copper-oxide material exhibiting superconductivity, with layered copper-oxygen units central to its crystal structure. Chemical composition and oxygen content must accompany a quoted superconducting transition temperature; the nominal recipe of a mixed-phase pellet is not the composition of its superconducting constituent. The formal hole count in a cuprate illustrates the distinction between composition and electronic interpretation.

For a chosen homogeneous composition ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_{4-\delta }$, assign formal oxidation states $+3$ to lanthanum, $+2$ to barium and $-2$ to oxygen. If $v$ is the mean formal copper oxidation state, charge neutrality requiresThe excess formal positive charge per copper relative to $+2$ is therefore $p_{\mathrm{formal}}=x-2\delta$. For stipulated $x=0.12$ and $\delta =0.01$, it is $0.10$; at $\delta =0.06$, it is zero. These are bookkeeping examples, not measurements of mobile carriers. The symbol $x$ here belongs to this declared formula, not the original mixed-phase recipe. A whole-pellet elemental ratio cannot simply be assigned to each constituent phase.

Charge neutrality alone does not locate holes. The lecture reviews oxygen-related carrier evidence and charge-$2e$ pairing from the Josephson effect and fluxoid and flux quantization in a superconductor. Pair charge does not establish phonon-mediated BCS theory for cuprate superconductors.^[ref]

High-temperature superconductivity is a historical name for superconductivity above earlier metallic benchmarks. The original Bednorz-Müller experiment operated at tens of kelvins, requiring deep cooling; cuprate superconductors opened a new materials family.^[ref]

A higher superconducting transition temperature enlarges the available operating range, but useful transport also needs a sufficient superconducting critical current at the operating temperature and magnetic field. In a ceramic, connections between grains can limit the whole specimen even when individual grains support superconductivity. This connects the materials problem to superconducting weak links and the Josephson effect. The parallel-path model of a superconducting ceramic below isolates why connectivity matters.

The clearest examples are the cuprate superconductors. Their unusual properties should be compared with the clean, weak-coupling, fully gapped s-wave electron–phonon limit of BCS theory; that limit is not a definition of all possible BCS-like paired states.

Cuprates are layered copper oxides with strongly anisotropic transport. Their undoped parents are antiferromagnetic Mott insulators or, more specifically for many compounds, charge-transfer insulators, despite partially filled bands in a simple band picture. Doping produces a superconducting dome rather than simply improving an ordinary weakly interacting metal. This points to strong electronic correlations. Near optimal doping, an extended temperature range often shows approximately linear electrical resistivity, termed strange-metal behaviour, instead of the low-temperature $T^2$ law of a conventional Fermi liquid. Neither law applies at every temperature and doping.

The dominant Cooper pair symmetry is spin-singlet $d_{x^2-y^2}$, with a representative gapIt changes sign under a quarter turn and has nodes on the zone diagonals. Phase-sensitive experiments detect that sign change; it is not established by a large transition temperature alone. ^[ref] Clean nodal quasiparticles give power-law low-temperature responses, for example electronic heat capacity $C\propto T^2$ and penetration-depth change $\lambda (T)-\lambda (0)\propto T$, in contrast with activated responses of a fully gapped s-wave state. Disorder, dimensional crossover and competing phases can alter these powers.

The cuprate pseudogap is a partial, momentum-dependent loss of low-energy electronic spectral weight extending above the superconducting transition in underdoped materials. It need not be identical to the superconducting gap, and observing it alone does not identify its microscopic origin or prove preformed pairs. ^[ref]

Short coherence lengths, anisotropy and low phase stiffness in underdoped systems make thermal phase and vortex fluctuations important. Many cuprates are strong type-II superconductors, with large London penetration depths relative to their superconducting coherence lengths. Gap-to-transition-temperature ratios and isotope effects vary with doping and need not obey the simplest weak-coupling benchmarks $2\Delta (0)/(k_{\mathrm{B}}{T}_c)=3.53$ and $T_c\propto M_{\mathrm{ion}}^{-1/2}$. A weak or variable isotope effect by itself does not identify the pairing mechanism.

These features motivate strongly correlated pairing descriptions, often involving spin fluctuations, but do not uniquely establish a microscopic pairing glue. Other high-temperature families need not share every cuprate property: iron-based superconductors are multiband and may be nodeless. The robust distinction is the observed combination of correlated normal states, unusual gap symmetry and important collective fluctuations, rather than high $T_c$ alone.

The t-J model approximates the low-energy sector of a repulsive Hubbard model with $U\gg t$ near one electron per site. Large on-site repulsion suppresses double occupancy. Virtual hopping out of and back into that low-energy sector produces antiferromagnetic superexchange with $J=4t^2/U$. One conventional effective Hamiltonian iswith projected electron operator ${\tilde{c}}_{i\sigma }=c_{i\sigma }(1-n_{i\bar{\sigma }})$. At half filling, nearest-neighbor hopping cannot move an electron without entering a forbidden doubly occupied state, leaving the leading spin-exchange model. Doped holes can move and couple the charge dynamics to these spin correlations. Additional terms, including three-site processes at order $t^2/U$, are omitted in this simplified Hamiltonian.

The exchange term energetically favors a nearest-neighbor spin singlet. With $b_{ij}^{†}=(c_{i\uparrow }^{†}{c}_{j\downarrow }^{†}-c_{i\downarrow }^{†}{c}_{j\uparrow }^{†})/\sqrt{2}$ in the no-double-occupancy subspace, ${\mathbf{S}}_i\cdot {\mathbf{S}}_j-n_i{n}_j/4=-b_{ij}^{†}{b}_{ij}$. A square-lattice pairing ansatz can therefore use bond singlets with opposite signs along the two axes. This d-wave bond order hasIt changes sign under a quarter-turn and vanishes along the zone diagonals. It is an even-parity, spin-singlet pairing pattern, not a statement of triplet pairing or necessarily broken time reversal. Phase-sensitive interference experiments distinguish this sign structure from a merely anisotropic nonnegative gap in cuprate superconductors.^[ref]

The model captures strong local repulsion, antiferromagnetism and mobile doped carriers. Its interacting phase diagram is not solved by writing down the Hamiltonian: a mean-field d-wave ansatz is a proposed approximation, not a proof that every parameter choice has that exact ground state or a complete microscopic explanation of every high-temperature superconductor.