Flat band magnetism in LaBr

Chengcheng Xiao, Nicholas Bristowe, Arash Mostofi

@iconxicon

Chengcheng-Xiao

E-mail: cx219@ic.ac.uk

Main Takeaways

LaBr monolayer is a magnetic 2D material.

There is an electronic flat band in the system and can be described using a simple tight binding model - The Dice model.

The flat band and magnetism can be controlled by external strain.

Methods

First-principles calculations were performed under the density functional theory (DFT) formalism, as implemented in the pseudopotential VASP, QuantumEspresso, and all-electron ELK code.
Exchange-correlation treatment:
- GGA-PBE
- Hybrid functional (HSE06)
- PBE+U (with Hubbard corrections)
Self-consistent Hubbard calculation performed using in-house developed code based on pythTB.

structure

Bulk experimentally synthesized*

*Ref: Monohalogenide der Lanthanoide, Hj. Mattausch, et.al, ZAAC, 1980

E_{\text{bind}} =\frac{(E_{\text{full layer}} - 2 \times E_{\text{half layer}})}{A_{\text{contacting}}}

AFM

Spacegroup:
Binding energy:

0.11 \mathrm{eV/ Å^{2}}

\mathrm{P\bar{3}m1}

bulk Structure

Bulk structure [White]
- Spacegroup: R-3m
Monolayer [Yellow]
- Spacegroup: P-3m1
Exfoliation energy:

Monolayer could be easily exfoliated from bulk.

Stability

Phonon dispersion - Dynamical stability
Formation energy - Chemical stability

Chemically metastable, but experiementally synthesized*

*Ref: Monohalogenide der Lanthanoide, Hj. Mattausch, et.al, ZAAC, 1980

Flat band

We turn our focus to the bands near the Fermi level.
Middle band is fairly flat comparing to other bands and it induces magnetism due to Stoner instability.

AFM

Ground state

Controlling Flat band

Strain can be used to control the shape of the flat band, and magnetic ground states.

2% biaxial tensile strain:

AFM to FM phase transition

Band width of the middle band change with strain.

Controlling Flat band

Strain can be used to control the shape of the flat band, and magnetic states.

At around 2% biaxial tensile strain, system undergoes phase transition from AFM to FM.

AFM

MagnetiC site

Spin density ( )

\rho_{\uparrow} - \rho_{\downarrow}

Off atom electrons contributes the most!

AFM

Monolayer LaBr is an 2D magnetic material with a magnetic moment of

1.7 \mu_B .

Interstitial sites

Spin density shows two interstitial sites. ELF gives us another doubly occupied site.

AFM

ELECTRIDES!

Electride materials:

ionic compounds in which electrons are localized at interstitial sites.

Some 3D Electrides:
- Na2He [under high pressure]
- C12A7:e- (12CaO·7Al2O3)
- LaCoSi
- Na [under high pressure]

Applications:
- Electron emitters
- Superconductors
- Battery anodes
- Catalysts

Electron in an octahedral interstitial site

Bader analysis

Bader analysis: the overall space is partitioned by the so-called zero flux plane where the charge density is at a minimum with zero flux.

Monolayer LaBr system is indeed an electride system with three interstitial sites.

Dice Model

The Dice model:
- Sites arranged in a triangle lattice.
- One possible hopping quenched (here, \( t_{23} \)).
- The remaining hoppings cause destructive interference.

Wavefunction plot for the middle band @ Gamma

Flat band!

Extended Hubbard Model

To understand the underlying mechanism of magnetism relating to the flat band, we employed extended Hubbard model.
The electron-electron interaction is taken into consideration in a mean-field self-consistent fashion.

\begin{aligned} \mathcal{H}=&-t \sum_{\langle ij\rangle} \sum_{\sigma}\left(c_{i \sigma}^{\dagger} c_{j \sigma}+c_{j \sigma}^{\dagger} c_{i \sigma}\right) \\ \end{aligned}

\begin{aligned} &+U \sum_{i} n_{i \uparrow} n_{i \downarrow}+X \sum_{\langle ij \rangle} \sum_{\sigma}\left(c_{i \sigma}^{\dagger} c_{j \sigma}+c_{j \sigma}^{\dagger} c_{i \sigma}\right)\left(n_{i,-\sigma}+n_{j,-\sigma}\right) \\ &+J \sum_{\langle ij \rangle} \left(\frac{1}{2}\sum_{\alpha,\beta} c_{i\alpha}^{\dagger} \sigma_{\alpha\beta} c_{i\beta}\right) \left(\frac{1}{2}\sum_{\alpha,\beta} c_{j\alpha}^{\dagger} \sigma_{\alpha\beta} c_{j\beta}\right) \end{aligned}

results

Model parameters were obtained using hopping parameters from Wannier transforming the DFT Hamiltonian for both the AFM and FM case.
Linear interpolating the strained parameters: AFM to FM phase transition indeed happens at around 2% tensile strain.