---
title: Bilayer Triple-Q Magnetic State
url: https://www.emergentmind.com/topics/bilayer-triple-q-state
type: topic
---

# Bilayer Triple-Q Magnetic State

The bilayer triple-$Q$ state is a noncollinear, noncoplanar magnetic ground state stabilized in certain ultrathin transition-metal bilayers and multilayers, notably in Mn-based films on heavy-element substrates. This state is characterized by a superposition of three symmetry-equivalent spin spirals within each atomic layer, with interlayer coupling that enforces tetrahedral angles both within and between layers. The resulting configuration features zero net spin moment but can support a finite “topological” orbital magnetization due to nontrivial spin chirality. The stabilization and orientation of the bilayer triple-$Q$ state depend sensitively on higher-order magnetic interactions and anisotropic exchange couplings.

## 1. Spin Structure and Tetrahedral Geometry

In a single Mn atomic layer on a hexagonal lattice, the triple-$Q$ (“3$Q$”) state is formed by superposing three spin spirals with propagation vectors $\mathbf{Q}_1$, $\mathbf{Q}_2$, and $\mathbf{Q}_3$ pointing along the three symmetry-equivalent $\overline{\Gamma}{-}\text{M}$ directions of the Brillouin zone. This generates a four-sublattice magnetic unit cell, with the spin directions on the sublattices given by the corners of a tetrahedron in spin space:
\[
\begin{align*}
\hat{\mathbf n}_1 &= \frac{(+1,+1,+1)}{\sqrt{3}} \\
\hat{\mathbf n}_2 &= \frac{(+1,-1,-1)}{\sqrt{3}} \\
\hat{\mathbf n}_3 &= \frac{(-1,+1,-1)}{\sqrt{3}} \\
\hat{\mathbf n}_4 &= \frac{(-1,-1,+1)}{\sqrt{3}}
\end{align*}
\]
The nearest-neighbor spins thus subtend tetrahedral angles $\arccos(-1/3) \approx 109.5^\circ$. The spin at site $i$ is assigned as $\mathbf{S}_i = S\,\hat{\mathbf n}_{p(i)}$, $p(i)\in\{1,2,3,4\}$.

In the bilayer extension, each layer forms its own 3$Q$ texture. The ideal “bilayer triple-$Q$” state ($3Q_{\rightleftarrows}$, *Editor's term*) features a relative lateral ($2\pi/3$) phase shift between the two layers, ensuring that vertical nearest-neighbor spins across the two layers also subtend the tetrahedral angle. Both intra-layer and interlayer neighbor pairs have $109.5^\circ$ spin separation. The vector structure of the bilayer 3$Q$ state can be written as:
\[
\begin{aligned}
\mathbf{S}^T(\mathbf{r}^T) &= \frac{1}{\sqrt{3}} \sum_{n=1}^3 \hat{\mathbf{e}}_n\,\cos(\mathbf{Q}_n \cdot \mathbf{r}^T) \\
\mathbf{S}^B(\mathbf{r}^B) &= \frac{1}{\sqrt{3}} \sum_{n=1}^3 \hat{\mathbf{e}}_n\,\cos\bigl(\mathbf{Q}_n \cdot \mathbf{r}^B + \tfrac{2\pi}{3}\bigr)
\end{aligned}
\]
with $\hat{\mathbf{e}}_n$ being mutually coplanar or tetrahedral unit vectors.

## 2. Microscopic Energetics and Exchange Hamiltonians

The stability of the bilayer triple-$Q$ state depends on a balance of magnetic interactions:
- **Heisenberg exchange:** Dominant nearest-neighbor intralayer couplings, e.g., $J_1^\parallel = -36.78$ meV for hcp-Mn/Ir(111), favor antiferromagnetism.
- **Dzyaloshinskii–Moriya interaction (DMI):** In multilayer systems, the DMI may be zero or nonzero depending on the substrate and stacking. In Mn/Ir(111), $D_1^\parallel \approx 2.82$ meV.
- **Anisotropic symmetric exchange (ASE):** For example, in Pd/Mn/Re(0001), $J_\mathrm{ASE}(1\textrm{NN}) = -0.30$ meV is the leading spin–orbit term coupling the triple-$Q$ state to the lattice [2307.09764].
- **Higher-order exchange interactions (HOI):** Both intralayer and interlayer, including biquadratic, three-site, and four-site terms. In Mn/Ir(111), interlayer “odd” HOI (e.g., $Y_1^{\perp,\textrm{odd}}\approx -5.0$ meV) are decisive for the energetic preference for the triple-$Q$ state [2506.05091].
  
The general atomistic Hamiltonian incorporates these terms:
\[
\begin{aligned}
H = &-\sum_{i\neq j} J_{ij} (\mathbf{S}_i \cdot \mathbf{S}_j)
-\sum_{i\neq j} \mathbf{D}_{ij} \cdot (\mathbf{S}_i \times \mathbf{S}_j) \\
&\phantom{{}={}} - \text{HOI terms} - K_u \sum_i (S_i^z)^2
\end{aligned}
\]
  
The inclusion of interlayer higher-order terms is crucial: pairwise interlayer exchange tends to favor collinear RW-AFM ($1Q$) states. Only when interlayer “odd” higher-order terms are included does the system stabilize the ideal bilayer 3$Q$ ground state.

## 3. Lattice Symmetry, State Selection, and Spin-Orbit Effects

Without anisotropic exchange, the 3$Q$ tetrahedron is free to rotate in spin space. ASE uniquely selects the 3$Q^1$ orientation, with one spin out-of-plane and three in-plane $120^\circ$ apart, as the minimum-energy state. For $J_\mathrm{ASE}<0$ (e.g., Pd/Mn/Re(0001)), this selects the $3Q^1_{\textrm{M}}$ domain, ensuring a single magnetic domain with threefold symmetry and a robust pattern in spin-polarized STM. Spin–orbit interactions such as magnetocrystalline anisotropy (MAE) and DMI cancel due to symmetry for the ideal tetrahedral 3$Q$ state, leaving ASE as the only relevant spin–orbit term [2307.09764].

Modifications from ideality, such as increased sixth-order HOI or lattice stacking variations (e.g., hcp-stacking), can distort the 3$Q$ angles, reducing the symmetry and introducing rotational domains. For example, in hcp-stacked Mn/Re(0001), the tetrahedral angle can increase to $\alpha\approx55^\circ$ [2307.09764], yielding a distorted 3$Q$.

## 4. Energetic Hierarchy and DFT Results

Density functional theory and atomistic modeling yield the following energetic landscape for representative systems [2506.05091]:

| State                            | $E$ (meV/Mn) | $\Delta E$ (meV, reference) |
|-----------------------------------|--------------|-----------------------------|
| Bilayer ideal $3Q_{\rightleftarrows}$ | $-23.3$      | $-23.3$                     |
| Bilayer non-ideal $3Q_{\rightrightarrows}$ | $+7.8$       |  $+7.8$                     |
| Bilayer $1Q$ RW-AFM$_{\rightleftarrows}$ | $0$          | $0$                         |

The energy gain of the ideal bilayer $3Q_{\rightleftarrows}$ over the collinear $1Q$ state ($\approx23$ meV/Mn) arises only when interlayer HOI are included. In Pd/Mn and Rh/Mn on Re(0001), analogous stabilization energies $\Delta E_{3Q-1Q}\approx 20$–$40$ meV/Mn are observed [2307.09764].

## 5. Topological Spin Textures and Orbital Magnetization

The noncoplanar spin texture of the triple-$Q$ state gives rise to a finite scalar spin chirality:
\[
\chi_{ijk} = \mathbf{S}_i \cdot (\mathbf{S}_j \times \mathbf{S}_k)
\]
This chirality generates a “topological” orbital moment per Mn via the real-space Berry-phase mechanism, even in the absence of net spin moment:
- In Pd/Mn bilayers (almost ideal 3$Q$), $|m_\mathrm{orb}|\approx 0.21\,\mu_B$ per Mn.
- In Rh/Mn bilayers (closer to perfect tetrahedral order), $|m_\mathrm{orb}|\approx 0.36\,\mu_B$ [2307.09764].
- In the model Mn/Ir(111) bilayer 3$Q_{\rightleftarrows}$, $M_\mathrm{orb}^z\approx 0.02\,\mu_B$ per Mn layer ($0.04\,\mu_B$ per two Mn), with vanishing net spin magnetization [2506.05091].

Distortions from the ideal 3$Q$ angles reduce the orbital moment, as quantified numerically.

## 6. Variants, Competing Phases, and Robustness in Magnetic Bilayers

Alternative triple-$Q$ states and related multi-$Q$ textures arise in models with additional interactions or external fields. For example, in bilayer systems with staggered DMI, multiple triple-$Q$ phases emerge depending on interlayer coupling and field strength. Low-field triple-$Q$ states can display uniform nonquantized scalar chirality of sign opposite to skyrmion crystals, while high-field triple-$Q$ states approach the symmetric limit with nearly equal amplitude modulations in both layers [2112.07072]. In all cases, whether the state achieves nontrivial integer topological charge depends on the specific amplitude and phase relations of the $Q$-vector superpositions.

## 7. Experimental Signatures and Implications

Experimental realization and imaging of the bilayer triple-$Q$ state have been achieved using spin-polarized STM in Pd/Mn and Rh/Mn bilayers on Re(0001), directly confirming the predicted orientation, lack of rotational domains, and expected magnetic supercell contrast amplitudes ($\delta z\approx\pm10$ pm in Pd/Mn) [2307.09764]. DFT-based STM simulations reproduce the hexagonal and stripe contrasts observed under tip-magnetization rotation.

The theoretical and computational evidence supports that the bilayer triple-$Q$ state is stabilized by a precise interplay of lattice symmetry, inter- and intralayer higher-order exchange, and, where relevant, anisotropic exchange couplings. Its zero net spin moment but large topological orbital magnetization make it a promising prototype for antiferromagnetic spintronic and orbitronic applications in ultrathin multilayers, leveraging emergent phenomena tied to the topology of noncoplanar spin textures [2506.05091].

Source: https://www.emergentmind.com/topics/bilayer-triple-q-state