Coexisting electronic smectic liquid crystal and superconductivity in a Si square-net semimetal

This study utilizes scanning tunneling microscopy and numerical calculations to reveal the coexistence of short-ranged charge stripe (smectic) order and superconductivity in the Si square-net semimetal NaAlSi, attributing the intertwined phases to kinetic energy suppression on specific p-orbital hole pockets of the Fermi surface.

The Gist

Imagine a bustling city where the citizens are electrons. Usually, in a metal, these electrons zip around chaotically like a crowd at a music festival—moving in all directions, filling every space. But in certain special materials, these electrons decide to organize themselves into patterns, much like people forming lines or clusters.

This paper is about a specific material called NaAlSi (a crystal made of Sodium, Aluminum, and Silicon) where the electrons do something very strange and beautiful: they form a "liquid crystal" pattern while the whole material is also becoming a superconductor.

Here is the story of what they found, explained simply:

1. The Two Superpowers: Superconductivity and "Liquid Crystals"

First, let's define the two main characters:

• Superconductivity: This is when electricity flows with zero resistance. It's like a highway where cars (electrons) can drive at infinite speed without ever hitting a bump or using any fuel.
• Electronic Liquid Crystal: In our everyday world, liquid crystals are used in screens (like your phone). They are a state of matter that is somewhere between a liquid (flowing) and a solid (ordered).
  • Nematic: Imagine a crowd of people all facing the same direction (like soldiers), but standing randomly. They broke the "rotation" symmetry (they aren't a circle anymore).
  • Smectic: Now imagine those same people forming neat, parallel rows or stripes. They broke both rotation symmetry and translation symmetry (they aren't just facing one way; they are lined up in specific spots).

In this paper, the electrons in NaAlSi form stripes (smectic order). They line up in parallel rows, creating a "traffic jam" pattern, but they do it while the material is also superconducting.

2. The Discovery: A Dance of Stripes

The scientists used a super-powerful microscope called a Scanning Tunneling Microscope (STM). Think of this microscope as a tiny, sensitive finger that can feel the "energy" of individual electrons on the surface of the crystal.

What they saw was amazing:

• The Stripes: The electrons weren't spread out evenly. They formed short, wavy stripes of high and low density.
• The Instability: These stripes were fragile. If you looked at them for a while, they would shift and rearrange themselves. It's like watching a sand dune shift in the wind; the pattern is there, but it's not locked in stone. This suggests the stripes are purely made of electron behavior, not a physical crack in the rock.
• The Twist: The direction of these stripes changes depending on the energy level. Above a certain energy, the stripes run North-South. Below that energy, they run East-West. It's as if the electrons are playing a game of "switcheroo" based on how much energy they have.

3. The Connection: The "Cooper Pair" Dance

Superconductivity happens when electrons pair up (called Cooper pairs) and dance together. Usually, you expect this dance to be uniform everywhere.

However, in this material, the scientists found that the strength of the superconducting dance changes depending on where you are in the electron stripes.

• Where the electron density is high (the stripe), the superconducting gap is one size.
• Where the density is low (between stripes), the gap is a different size.

It's as if the superconducting dance floor has a wavy texture. The electrons are dancing harder in some spots and softer in others, perfectly synchronized with the stripe pattern. The authors call this a "Cooper pair liquid crystal." The superconductivity itself has become striped!

4. Why Does This Happen? (The Physics Explanation)

Why do the electrons do this? The scientists used computer simulations to figure it out.

Imagine the electrons are sitting on a flat, round table (the Fermi surface).

1. The Flat Top: In this material, there are two large, flat "pockets" of electrons (like two flat-topped hills).
2. The Energy Saving: Nature loves to save energy. The electrons realized that if they broke the symmetry and formed stripes, they could lower their energy significantly. It's like a group of people realizing that if they all sit in a specific pattern, they can all fit more comfortably.
3. The Mechanism: The electrons interact with the "flat tops" of their energy pockets. By forming stripes, they create a new pattern that allows them to settle into a lower energy state. It's a bit like a Jahn-Teller effect (a fancy term for when a molecule distorts to become more stable), but happening with the whole electron sea.

5. Why Is This a Big Deal?

Usually, we only see these complex "liquid crystal" electron patterns in materials with d-orbitals (like high-temperature superconductors made of copper). These are materials known for being "strongly correlated" (the electrons are very picky about each other).

NaAlSi, however, is made of p-orbitals (Sodium, Aluminum, Silicon). These are usually considered "simple" materials where electrons don't interact much.

• The Surprise: Finding this complex, stripe-forming behavior in a "simple" p-orbital material is a huge surprise. It's like finding a complex, synchronized dance routine in a group of people who usually just walk randomly.
• The Implication: This suggests that the link between superconductivity and electronic liquid crystals is much more universal than we thought. It might happen in many more materials than we previously believed.

Summary

In short, this paper describes a material where electrons spontaneously organize into shifting, fragile stripes (a smectic liquid crystal) while the material is also superconducting. The superconductivity itself gets "striped" to match the electrons. This happens because the electrons find a way to save energy by organizing themselves around flat spots in their energy landscape. It's a rare and beautiful example of how simple atoms can create complex, liquid-crystal-like behaviors in the quantum world.

Technical Summary

1. Problem and Motivation

Electronic liquid crystals (nematic and smectic phases) are spontaneous symmetry-breaking states often observed in unconventional superconductors, particularly in $d$-orbital systems like cuprates and iron-based superconductors. However, their occurrence in $s$- or $p$-orbital systems, which typically lack strong electron correlations, is unexpected and poorly understood.

The specific material of interest is NaAlSi, a nodal-line semimetal with a bulk superconducting transition temperature ($T_c$) of $\approx 7.2$ K. While its superconductivity has been speculated to be unconventional due to a low density of states at the Fermi level ($E_F$), this remains controversial. The central problem addressed is whether NaAlSi hosts an electronic liquid crystal phase (specifically a smectic charge order) and how this order interacts with its superconducting state.

2. Methodology

The authors employed Scanning Tunneling Microscopy (STM) and Scanning Tunneling Spectroscopy (STS) combined with Density Functional Theory (DFT) calculations.

• Sample Preparation: High-quality NaAlSi crystals were synthesized, cleaved in situ under ultra-high vacuum at 77 K to expose flat Na-terminated surfaces with $C_{4v}$ symmetry.
• STM/STS Measurements:
  • Measurements were performed at low temperatures ($T \approx 1.5$ K, effective electron temperature $T_{eff} \approx 350$ mK).
  • Tunneling conductance ($dI/dV$) was measured to map the local density of states (LDOS).
  • To minimize tip-height artifacts, the authors utilized normalized conductance $L(r, V) = [dI/dV(r, V)] / [I(r, V)/V]$.
  • Fast Fourier Transforms (FFT) of the conductance maps were analyzed to identify charge ordering wavevectors.
  • Spatial mapping of the superconducting gap ($\Delta$) was performed by fitting coherence peaks in $dI/dV$ spectra.
• Theoretical Calculations:
  • DFT calculations (using VASP) were performed to determine the surface electronic band structure.
  • Wannier functions were constructed to model surface states and simulate the spectral function.

3. Key Contributions and Results

A. Observation of Short-Range Smectic Order

The study reveals a short-ranged charge stripe (smectic) order in NaAlSi, a rare finding in a $p$-orbital system.

• Symmetry Breaking: The charge order breaks both rotational ($C_4 \to C_2$) and translational symmetries of the lattice, defining it as a smectic phase rather than a nematic one.
• Incommensurability: The stripe periodicity is incommensurate with the atomic lattice, with a period of approximately $4.25a_0$ (where $a_0$ is the lattice constant).
• Fluctuations: The order is fragile and exhibits strong spatial fluctuations. The stripe configuration can reconfigure abruptly under mild perturbations, suggesting a purely electronic origin rather than a structural lattice distortion.
• Energy Dependence: The orientation of the stripes is energy-dependent. Stripes oriented along one axis dominate above $E_F$, while a $90^\circ$ rotated pattern appears below $E_F$. This indicates that the two orthogonal symmetry-breaking configurations reside at different energies.

B. Intertwined Superconductivity and "Pair Liquid Crystal"

The authors resolved a spatial modulation of the superconducting gap amplitude ($\Delta$) that is in phase with the charge stripe order.

• Gap Modulation: The superconducting coherence peaks show a periodic variation in energy spacing ($2\Delta$) that matches the period of the charge stripes.
• Secondary Order: Unlike standard Charge Density Wave (CDW) systems where the CDW is the primary order and induces a secondary pair density wave (PDW), the authors argue that here the primary order is a smectic liquid crystal. Consequently, the modulated superconducting state is termed a secondary "pair liquid crystal."
• Domain Walls: The study observed domain walls (DWs) separating $90^\circ$ rotated stripe domains. The superconducting gap size remained uniform across these DWs, suggesting no direct competition or strong coupling between the gap magnitude and the stripe orientation at the DW, but rather a global intertwining of the orders.

C. Theoretical Mechanism

DFT calculations provide a driving mechanism for the observed smectic order:

1. Band Structure: The Fermi surface consists of a small electron pocket (Al $s$-orbital) and two large, flat-topped hole pockets (Si $p$-orbitals) oriented orthogonally.
2. Nematic Instability: A lifting of degeneracy between the $p_x$ and $p_y$ hole pockets (via a band Jahn-Teller effect) lowers the energy by sinking the flat top of one pocket below $E_F$. This breaks rotational symmetry (nematic state).
3. Smectic Instability: A subsequent nesting-driven instability occurs where the remaining Fermi contour interacts with the electron band. This reconstruction opens a gap at the new zone boundary defined by the wavevector $q_{str}$, further lowering the energy and breaking translational symmetry (smectic state).
4. Explanation of Energy Asymmetry: The model explains why different stripe orientations appear above and below $E_F$: only one hole band interacts with the electron band above $E_F$, while both do so below $E_F$.

4. Significance

• Extension to $p$-orbital Systems: This work demonstrates that electronic liquid crystal behavior is not exclusive to strongly correlated $d$-orbital systems but can emerge in $p$-orbital semimetals with specific band structures (flat bands near $E_F$).
• Novel Phase Identification: It identifies a rare smectic electronic phase (breaking both rotational and translational symmetry) that is distinct from conventional CDWs due to its strong fluctuations and energy-dependent orientation.
• Superconductivity Mechanism: The discovery of a "pair liquid crystal" suggests that unconventional superconductivity in NaAlSi may be intimately linked to these electronic symmetry-breaking fluctuations, offering a new perspective on the interplay between charge order and superconductivity in nodal-line semimetals.
• Theoretical Insight: The proposed mechanism involving flat-topped hole pockets and band Jahn-Teller effects provides a roadmap for understanding similar phenomena in other square-net semimetals (e.g., Sb-based systems).

In summary, the paper establishes NaAlSi as a unique platform where a short-range smectic charge order coexists and intertwines with superconductivity, driven by the specific topology of its $p$-orbital derived Fermi surface.