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A superconductor whose pairing order parameter has the angular or crystal-symmetry structure of d-wave superconductivity. The sign of the gap depends on momentum direction and nodes can permit low-energy fermionic Bogoliubov quasiparticles.

In weak-coupling conventional superconductivity, a useful guide is $k_B{T}_c\sim \hslash {\omega }_{\mathrm{ph}}\exp [-1/{\lambda }_{\mathrm{eff}}]$: an appreciable phonon scale and an effective attraction after Coulomb effects are favorable. A large density of states at the Fermi level can strengthen pairing, while light constituent masses can raise relevant phonon frequencies. Increasing electron–phonon coupling without limit is not a general strategy: lattice instabilities, competing order and changes to the phonon spectrum constrain it, and stronger coupling requires treatment beyond the simple exponential expression.

For unconventional high-temperature superconductivity, strong electronic interactions and magnetic fluctuations can support sign-changing pairing, as in d-wave superconductivity. Layered structures and proximity to magnetic order may enhance useful correlations, but can also promote competing insulating or density-wave states. Sufficient mobile-carrier phase stiffness is needed to turn pairing into coherent superconductivity; strong pairing alone does not guarantee a large transition temperature. Disorder that is relatively harmless to an isotropic conventional pair state can be pair-breaking for sign-changing gaps. There is no universal recipe established by these qualitative conditions: the pairing interaction, symmetry, carrier density, competing phases and phase coherence must all be considered.

In three dimensions an ideal Bose gas can saturate its excited-state population at chemical potential zero, with the remaining particles forming a Bose-Einstein condensate. The transition follows from the Bose-Einstein distribution integrated against its density of states; below the ideal-gas transition, $n_0/n=1-(T/T_c{)}^{3/2}$. Condensation alone does not supply the finite Landau critical velocity of an interacting superfluid: the ideal quadratic spectrum has ${\inf }_{p}E(p)/p=0$.

By contrast, a weak attractive interaction near a Fermi surface produces the Cooper instability and the BCS gap equation. Its logarithmic enhancement permits pairing at arbitrarily weak attraction in the ideal model. The low-temperature excitations have $E=\sqrt{{\xi }^2+{\Delta }^2}$, and in weak coupling $k_B{T}_c=(e^{{\gamma }_E}/\pi )\Delta (0)\simeq 0.567\Delta (0)$. This transition is controlled by pair formation when phase fluctuations are small; deeply bound pairs instead motivate a Bose-condensation description with their own density and effective mass.

For the given ellipsoidal dispersion, set $q_x=k_x$, $q_y=k_y$ and $q_z=k_z/\sqrt{\gamma }$. Then $\epsilon ={\hslash }^2{q}^2/(2m)$ while the integration measure is $d^{3}k=\sqrt{\gamma }{d}^{3}q$. Thus the anisotropic ideal-gas density of states per unit volume and per internal state isThe full continuum momentum range is assumed; this is an anisotropic three-dimensional gas, not yet a discrete stack of two-dimensional bands.

At the Bose transition, substitute $x=\epsilon /(k_{B}T)$:The integral is $\Gamma (3/2)\zeta (3/2)\simeq 2.315$. Solving for the Bose-Einstein condensation temperature for an anisotropic effective mass givesThe heavier $z$-direction effective mass increases the available excited-state count and lowers the condensation temperature at fixed number density.

For a balanced two-spin Fermi gas, let $n_F$ denote total fermion number density and take $N_s$ to be the single-spin density of states used in the conventional gap equation. Counting the filled ellipsoid givesIf a spin-summed density of states is used instead, the coupling convention in the supplied integral must change accordingly; inserting a factor two into only one side would change the exponent incorrectly.

For ${\epsilon }_c\ll {\epsilon }_F$, replace the density of states across the interaction window by $N_s({\epsilon }_F,\gamma )$. Writing ${\lambda }_0=|g|N_{s0}$, the zero-temperature integral becomesand thereforeThe second expression additionally requires ${\lambda }_0{\gamma }^{1/3}\ll 1$. The explicit density dependence is ${\lambda }_0=|g|m(3{\pi }^2{n}_F{)}^{1/3}/(2{\pi }^2{\hslash }^2)$. This mass-anisotropy enhancement of a weak-coupling BCS gap initially raises the pairing scale, while the ideal Bose condensation scale decreases as ${\gamma }^{-1/3}$.

Large anisotropy weakens transport and coherence in the heavy direction. The separation of pair formation and phase ordering then matters, so a large mean-field gap need not imply a large actual $T_c$. A qualitative sketch has $T_c$ initially rising with the pairing scale and eventually turning down when coherence is the smaller scale. If preformed fermion pairs behave as bosons, use $n_B=n_F/2$ and pair mass about $2m$; in this continuum model this gives $k_B{T}^{\mathrm{BEC}}\simeq 0.218{\epsilon }_{F0}{\gamma }^{-1/3}$.

The turnover cannot be located reliably by taking the weak-coupling formulas to $\gamma \to \infty$: ${\epsilon }_F$ falls, ${\lambda }_0{\gamma }^{1/3}$ grows, the narrow-window assumption fails and the pair chemical potential and mass require a self-consistent treatment. The smooth curve in the figure is explicitly an illustrative interpolation of two extrapolated scales, not such a calculation. If actual decoupled two-dimensional layers are approached, ordinary ideal-gas finite-temperature condensation is absent and interacting phase order requires a different analysis, including the Berezinskii–Kosterlitz–Thouless transition when appropriate.

The cuprates make the distinction between pairing and phase coherence relevant: they are layered and have anisotropic motion, but their strong correlations, lattice band structure and predominantly d-wave superconductivity are outside this attractive continuum $s$-wave model. Anisotropy by itself is therefore not an explanation or quantitative fit of cuprate transition temperatures.

The superconducting order is a coherent pair amplitude, schematically $\Delta \propto ⟨c_{\downarrow }{c}_{\uparrow }⟩=|\Delta |e^{i\varphi }$. Under a global fermion phase change $c\mapsto e^{i\alpha }c$, it acquires phase $2\alpha$. A symmetry-broken thermodynamic description selects a pair phase. This must be distinguished from electromagnetic gauge freedom: a local gauge transformation changes the phase and potentials together and is a redundancy of the description, not an observable symmetry that the material literally violates.

A conventional centrosymmetric s-wave superconductor usually has an even-parity orbital pair state and a spin singlet state, consistent with the antisymmetry of two fermions. Its ordinary zero-field equilibrium state preserves time reversal. The electromagnetic coupling makes phase dynamics and screening part of the Anderson-Higgs mechanism in a superconductor; the Meissner effect is a measurable consequence. The condensate phase description does not imply that electric charge is no longer conserved.

In an unconventional superconductor, the gap can transform nontrivially under the crystal point group or have other nonstandard pairing structure. A common example is d-wave superconductivity, with sign changes and possible nodes in the gap. This does not automatically imply an additional observable fourfold-rotation breaking: a rotation that reverses the gap's sign can be accompanied by a gauge phase change. A nematic superconducting state instead breaks an actual crystal rotation symmetry in its gauge-invariant responses.

A chiral combination such as $p_x+i{p}_y$ can break time reversal, while a spin-triplet order parameter can also distinguish spin directions when the applicable spin-rotation and spin-orbit constraints permit it. Helium-3 superfluidity illustrates paired-fermion order with richer spin and orbital structure. These possibilities must be tested rather than equated: unconventional pairing does not by itself mean spin-triplet pairing, time-reversal breaking, or nematicity. Gap angular dependence, phase-sensitive interference, spin response and spontaneous magnetic signatures probe different symmetry questions. The phase-sensitive test of d-wave pairing is more specific than inferring a gap sign merely from its magnitude.