Unconventional Superconductors: Pairing Symmetry and Experimental Probes

Hello! Welcome back to our course.

In our last lesson on Andreev reflection, we operated under the assumption that the superconducting gap, $\Delta$, was a constant value, independent of direction. This isotropic gap is a hallmark of conventional superconductors. Today, we will challenge this assumption and delve into the rich and complex world of unconventional superconductivity.

Our focus will be on contrasting the simple, isotropic s-wave pairing of conventional superconductors with the anisotropic d-wave pairing found in many high-temperature superconductors like the cuprates. By the end of this lesson, you will understand the fundamental symmetry principles that dictate the nature of the Cooper pair wavefunction and how powerful experimental techniques, particularly Angle-Resolved Photoemission Spectroscopy (ARPES), can directly visualize the structure of the superconducting gap in momentum space.

1. The Symmetry of the Cooper Pair: Why 's' and 'd'?

The classification of pairing states (s-wave, p-wave, d-wave, etc.) is not arbitrary; it's a direct consequence of the fundamental quantum statistics of electrons. A Cooper pair is a bound state of two electrons, which are identical fermions. Therefore, the total wavefunction of the pair, $\Psi ({\mathbf{r}}_1,s_1;{\mathbf{r}}_2,s_2)$, must be antisymmetric upon exchange of the two electrons.

$\Psi ({\mathbf{r}}_2,s_2;{\mathbf{r}}_1,s_1)=-\Psi ({\mathbf{r}}_1,s_1;{\mathbf{r}}_2,s_2)$

We can separate the wavefunction into a spatial part, $\varphi ({\mathbf{r}}_1-{\mathbf{r}}_2)$, which depends on the relative coordinates of the electrons, and a spin part, $\chi (s_1,s_2)$. The overall antisymmetry requirement imposes a strict constraint on the symmetry of these two parts.

To understand this more formally, please read the following introductory section.

As the text explains, the logic is as follows:

• Spatial Part $\varphi (\mathbf{r})$: The symmetry of the spatial part under inversion ($\mathbf{r}\to -\mathbf{r}$) is given by its parity, $(-1{)}^l$, where $l$ is the relative orbital angular momentum quantum number of the pair.
  • Even Parity: $l=0,2,4,\dots$ (s-wave, d-wave, etc.). The spatial wavefunction is symmetric: $\varphi (-\mathbf{r})=\varphi (\mathbf{r})$.
  • Odd Parity: $l=1,3,5,\dots$ (p-wave, f-wave, etc.). The spatial wavefunction is antisymmetric: $\varphi (-\mathbf{r})=-\varphi (\mathbf{r})$.
• Spin Part $\chi (s_1,s_2)$: The two electron spins can combine into two configurations:
  • Spin Singlet (S=0): Antisymmetric spin wavefunction, $\frac{1}{\sqrt{2}}(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)$.
  • Spin Triplet (S=1): Symmetric spin wavefunctions, $|\uparrow \uparrow ⟩$, $|\downarrow \downarrow ⟩$, and $\frac{1}{\sqrt{2}}(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)$.

For the total wavefunction to be antisymmetric, we must combine a symmetric spatial part with an antisymmetric spin part, or vice-versa. This leads to the fundamental classification:

• Conventional (s-wave) & d-wave: Even $l$ ($l=0$ and $l=2$, respectively). These states must be spin-singlets.
• p-wave: Odd $l$ ($l=1$). This state must be a spin-triplet.

2. The Anisotropic Gap Function and its Nodes

The spatial symmetry of the Cooper pair wavefunction $\varphi (\mathbf{r})$ directly translates to the momentum-space dependence of the superconducting gap, $\Delta (\mathbf{k})$.

• s-wave ($l=0$): The wavefunction is spherically symmetric. The gap $\Delta (\mathbf{k})={\Delta }_0$ is isotropic—it has the same constant magnitude everywhere on the Fermi surface. This is the simple case we have considered so far.

• d-wave ($l=2$): The wavefunction has a more complex, anisotropic structure. The most prominent example, relevant for the cuprate high-temperature superconductors, is the $d_{x^2-y^2}$ state. In this case, the gap function has a strong momentum dependence, which on a simple 2D square lattice can be written as:
  $\Delta (\mathbf{k})={\Delta }_0(\cos (k_{x}a)-\cos (k_{y}a))$
  where $a$ is the lattice constant.

The most profound consequence of this anisotropy is the existence of nodes: points or lines on the Fermi surface where the gap vanishes. For the $d_{x^2-y^2}$ gap, $\Delta (\mathbf{k})=0$ whenever $|k_x|=|k_y|$. These correspond to the diagonals of the 2D Brillouin zone.

Crucially, the gap function not only goes to zero at the nodes, but it also changes sign across a nodal line.

The presence of nodes has a dramatic effect on the low-energy properties of the superconductor. While an s-wave superconductor has a hard gap in its density of states (DOS), an unconventional superconductor with nodes will have states available at arbitrarily low energies.

To see the formalism behind these different gap structures, the following reading expands on the generalized BCS theory.

3. Experimental Probes: How to "See" the Gap Anisotropy

Theoretical predictions are one thing, but how do we experimentally confirm the $d_{x^2-y^2}$ nature of the gap in materials like the cuprates? We need a probe that is sensitive to momentum $\mathbf{k}$.

3.1 Angle-Resolved Photoemission Spectroscopy (ARPES)

ARPES is the most direct technique for mapping the electronic band structure and, consequently, the superconducting gap. As you know from your background in photoluminescence, spectroscopy involves exciting particles and analyzing what comes out. In ARPES, high-energy photons strike a material, causing electrons (photoelectrons) to be ejected. By measuring the kinetic energy and the emission angle of these electrons, one can reconstruct their original energy and momentum inside the solid.

Under the sudden approximation, ARPES directly measures the single-particle spectral function $A(\mathbf{k},\omega )$. A peak in $A(\mathbf{k},\omega )$ corresponds to a well-defined electronic state with energy $\hslash \omega$ and momentum $\mathbf{k}$.

The opening of a superconducting gap means that the electronic states at the Fermi level ($\omega =0$) are pushed to higher binding energies. The magnitude of this energy shift at a given $\mathbf{k}$ is precisely the superconducting gap $\Delta (\mathbf{k})$. By performing this measurement for many different momenta $\mathbf{k}$ across the Fermi surface, we can literally map out the gap's anisotropy.

The following resource provides a beautiful demonstration of this.

As the paper shows, the ARPES experiment provides a stunning confirmation of the $d_{x^2-y^2}$ picture:

• Along the Cu-O bond direction (e.g., $\mathbf{k}=(\pi ,0)$, the "anti-node"), the spectra show a clear shift below $T_c$, indicating a maximal gap.
• Along the zone diagonal (e.g., $\mathbf{k}\approx (\pi /2,\pi /2)$, the "node"), the spectra are essentially unchanged across $T_c$, indicating a zero gap.

By collecting data across the entire Brillouin zone, a full momentum-space map of $|\Delta (\mathbf{k})|$ can be constructed, revealing the four-leaf clover pattern characteristic of the $d_{x^2-y^2}$ state.

3.2 Phase-Sensitive Probes

ARPES is incredibly powerful for measuring the magnitude of the gap, $|\Delta (\mathbf{k})|$. However, it is not directly sensitive to the phase of the order parameter, i.e., the sign change across the nodal lines. Proving this sign change requires a different class of experiments based on quantum interference.

A classic example involves a Superconducting Quantum Interference Device (SQUID) that incorporates a d-wave superconductor. By fabricating a loop that forces Cooper pairs to tunnel along paths with different signs of the order parameter (e.g., along the a-axis and b-axis of a cuprate), an intrinsic $\pi$ phase shift is introduced into the interference condition. This leads to a measurable shift in the SQUID's magnetic flux response by half a flux quantum (${\Phi }_0/2$), a "smoking gun" signature for the d-wave state. The "Introduction to Unconventional Superconductivity" text you looked at earlier discusses these phase-sensitive tests in detail in the sections titled "Phase sensitive tests - interference and frustration" and "Other phase sensitive test based on scattering states" if you're interested in exploring further.

Conclusion

Today, we've moved beyond the simple isotropic model of superconductivity and into the far richer landscape of unconventional pairing. We saw that fundamental quantum statistics provide a deep reason for the existence of different pairing symmetries.

Key Takeaways:

• Pairing Symmetry: The antisymmetry of the two-electron wavefunction links the orbital angular momentum ($l$) to the spin state (singlet/triplet). Conventional s-wave ($l=0$) and unconventional d-wave ($l=2$) pairing are both spin-singlet states.
• Gap Anisotropy: While the s-wave gap is isotropic, the d-wave gap is highly anisotropic, featuring nodes (zeros) and a characteristic sign change across these nodes. The $d_{x^2-y^2}$ state is the established symmetry for cuprate superconductors.
• Experimental Probes:
  • ARPES directly measures the momentum dependence of the gap magnitude $|\Delta (\mathbf{k})|$, providing a map of the nodes and anti-nodes.
  • Phase-sensitive SQUID experiments confirm the crucial sign change of the d-wave order parameter by detecting interference effects caused by an intrinsic $\pi$ phase shift.

Preview of the Next Lesson

The phase of the wavefunction, which was so critical for understanding d-wave superconductivity today, is a concept of profound importance in modern condensed matter physics. In our next lesson, we will begin our exploration of topological phases of matter by generalizing this idea. We will introduce the Berry phase, a geometric phase acquired by a quantum system as its parameters are cyclically varied, which forms the mathematical foundation for understanding phenomena like the quantum Hall effect.