Can monolayer ZrX2 electrides host both electron- and hole-doped superconductivity?

A bipolar superconductor candidate built from flat bands

The paper asks whether a single two-dimensional electride family can support superconductivity on both sides of the carrier-doping axis, and what controls the difference between the two sides. Monolayer ZrX2 (X = Cl, Br, I) is the test case: a semiconducting electride in which two excess electrons per formula unit sit at hexagonal interstitial sites rather than on atoms, written as [ZrX2]2+·2e−.

The motivation is a contrast the authors draw with monolayer InSe, where electron doping gives a Tc of 3.4 K at 0.2 e/cell while hole doping has been reported at 55–75 K. The authors ask whether other electron-rich 2D materials behave similarly, and what decides whether hole doping produces high-Tc superconductivity or magnetism. Their answer is framed around where the excess electrons sit and which orbital character the flat bands carry.

Doping is simulated, not chemically applied

All results come from density functional theory calculations with Quantum ESPRESSO, using PAW pseudopotentials and the PBE functional. Carrier doping is introduced by changing the total electron count against a compensating jellium background, so the crystal structure is not altered by dopants or disorder. This is a clean but idealized way to dope: it isolates the electronic effect of added or removed charge.

Phonons and electron-phonon coupling were computed with density functional perturbation theory on 8×8×1 q-point meshes and dense k-point meshes up to 96×96×1. Tc was evaluated with the McMillan–Allen–Dynes formula using a screened Coulomb repulsion of μ* = 0.1. Because that formula assumes weak coupling, the authors also checked Tc with the Allen–Dynes modification; the corrected values are slightly higher but keep the same doping dependence.

The monolayers themselves are predicted to be stable: phonon dispersions show no imaginary frequencies, ab initio molecular dynamics at 300 K shows thermodynamic stability, and cleavage energies of 0.300–0.318 J/m2 are below graphene's 0.36 J/m2, which the authors read as evidence that monolayers can be exfoliated. These are calculated stability indicators, not measured samples.

Electron doping reaches high Tc at tiny carrier counts

Electron doping produces a dome-shaped Tc versus concentration curve for all three compounds. Superconductivity turns on at 0.01 e/f.u. for ZrBr2 and ZrI2 and at 0.02 e/f.u. for ZrCl2. The maxima are 30.0 K for ZrBr2 at 0.01 e/f.u., 22.5 K for ZrCl2 at 0.05 e/f.u., and 18.1 K for ZrI2 at 0.01 e/f.u. Beyond the optimum, Tc falls monotonically, and at 0.35 e/f.u. in ZrCl2 imaginary phonon frequencies appear, marking a lattice instability.

The mechanism the authors give is a flat conduction-band minimum. Near the conduction-band minimum, the band forms six Mexican-hat-like features around Γ, creating a weakly dispersive ring and a sharp density-of-states peak. For ZrCl2 at 0.05 e/f.u., the Fermi-level density of states is 18.10 states/eV/spin/f.u., compared with 7.70 for electron-doped InSe. The coupling constant λ is 2.29, with over 90% coming from acoustic phonons in the 50–150 cm−1 range, versus λ = 0.65 for electron-doped InSe. High density of states plus strong low-frequency acoustic coupling is the paper's explanation for the high Tc.

Hole doping needs far more carriers for a small Tc

Hole doping behaves differently. Superconductivity does not appear at low hole concentrations; a threshold is required, at 0.2 h/f.u. for ZrCl2 and 0.25 h/f.u. for ZrBr2 and ZrI2. Even then, Tc is only about 2 K, an order of magnitude below the electron-doped maxima.

The threshold has a band-structure origin. In ZrCl2 at 0.15 h/f.u., the Fermi level lies above the Γ-point band and only a K-point hole pocket exists. At 0.2 h/f.u., the Fermi level reaches the Γ-point band, adding a pocket where the flat dz2 band resides and producing a sharp jump in the density of states. The coupling is also weaker: at 0.2 h/f.u. in ZrCl2, λ = 0.48, dominated by acoustic modes near K in the 105–125 cm−1 range, and the Fermi-level density of states is 10.10 states/eV/spin/f.u., both well below the electron-doped case.

Higher hole doping pushes the system toward magnetism

The hole-doped superconducting window closes because the system becomes magnetic. All three compounds stay nonmagnetic up to a critical hole concentration, then develop a finite magnetic moment: about 0.65 h/f.u. for ZrCl2 and ZrBr2 and 0.6 h/f.u. for ZrI2. For ZrI2 the ferromagnetic state is stable across the relevant range, while ZrCl2 and ZrBr2 move from a weak stripe antiferromagnetic order to ferromagnetic order as doping increases.

The authors trace this to the progressive emptying of the interstitial band. As holes increase, the Fermi level moves toward a density-of-states peak, and the Stoner criterion N(εF)I > 1 is satisfied. In ZrI2, the density of states drops from 2.86×2 states/eV in the nonmagnetic case at 0.6 h/f.u. to 2.84 states/eV in the ferromagnetic case. Superconductivity is suppressed and gives way to the magnetic state. The comparison with LaBr2, where one excess electron per formula unit half-fills an interstitial band and produces intrinsic ferromagnetism, supports this reading.

Where the excess electrons sit sets the asymmetry

The paper's central comparison is with InSe and LaBr2. In InSe, the flat band at the valence-band maximum comes from In-In σ bonds, so hole doping metallizes that flat band directly and produces a high density of states and strong coupling. In ZrX2, the flat band relevant to electron doping is the conduction-band minimum, dominated by Zr dz2 orbitals, while the interstitial electrons fill the topmost valence band.

The authors conclude that the spatial location and orbital character of the excess electrons determine where flat bands sit relative to the band edges, and that this in turn governs both the strength of superconductivity and the nature of the competing order. That is an interpretation built from the band structures and charge densities, not a directly measured quantity.

Predicted Tc values remain computational

The Tc values are outputs of the McMillan–Allen–Dynes formula applied to computed electron-phonon coupling, not measured transition temperatures. Doping is modeled by a jellium background rather than by real dopants, gating, or intercalation, so the calculations do not address how such carrier densities would be achieved experimentally or what disorder real doping would introduce. The magnetic states are likewise calculated spin configurations.

Within those limits, the paper's specific priority is the asymmetry itself: the electron-doped side reaches 30.0 K at 0.01 e/f.u. in ZrBr2, while the hole-doped side needs 0.2–0.25 h/f.u. to reach about 2 K and then loses to magnetism near 0.6–0.65 h/f.u. For readers comparing 2D electrides as bipolar superconductivity platforms, the useful discriminator is not whether both doping signs superconduct, but how far the flat band sits from the Fermi level and which orbital carries it.

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