Phonon Angular Momentum Hall Effect

• Phonon Angular Momentum Hall Effect is a transverse transport phenomenon where phonon angular momentum flows perpendicular to a thermal gradient driven by intrinsic and extrinsic Berry curvature effects.
• Advanced magnetometry and first-principles simulations corroborate measurements of edge magnetization and quantify Hall conductivity across varied crystalline solids.
• Material engineering through strain, doping, and optical pumping offers a tunable strategy to optimize PAMHE for innovative phononic and spintronic device applications.

The phonon angular momentum Hall effect (PAMHE) is a transverse transport phenomenon in which the angular momentum carried by phonons flows perpendicularly to an applied thermal gradient, analogous to the electronic spin Hall and magnon Nernst effects. This effect relies on the existence of nontrivial Berry curvature in phonon bands, the presence of both transverse and longitudinal acoustic phonons, and can manifest both intrinsic (adiabatic, Berry curvature–driven) and extrinsic (nonadiabatic, disorder-mediated) mechanisms. The resulting transverse phonon angular momentum flux leads to edge accumulation and, in ionic crystals with nonzero Born effective charge, to edge magnetization detectable by advanced magnetometry. The phenomenon is supported and quantified in crystalline solids ranging from simple acoustic lattices to complex 2D materials with chiral phonon band structures.

1. Fundamental Formalism and Mechanisms

The PAMHE arises from the interplay of intrinsic and extrinsic contributions to the phonon magnetic moment, denoted $m_{qs}$ for a phonon mode with wavevector $q$ and branch $s$. The general expression is

$$m_{qs} = m_{qs}^{\rm int} + m_{qs}^{\rm SS} + m_{qs}^{\rm SJ}$$

where $m_{qs}^{\rm int}$ is the adiabatic (“intrinsic”) term present under the Born–Oppenheimer approximation, $m_{qs}^{\rm SS}$ is the skew‐scattering term due to asymmetric electron–phonon transition rates, and $m_{qs}^{\rm SJ}$ is the side‐jump term arising from coordinate shifts during scattering. The intrinsic term is directly associated with the Berry curvature $\Omega^{\rm ph}(q)$ of the phonon bands:

$$m_{qs}^{\rm int} = -\,(q\times u_{qs})\cdot\Omega^{\rm ph}(q)$$

where $u_{qs}$ is the phonon polarization vector. The extrinsic terms, $m_{qs}^{\rm SS}$ and $m_{qs}^{\rm SJ}$, emerge when the phonon relaxation rate $\eta_p$ and electronic scattering rate $\eta_e$ become comparable, as in highly disordered systems or those with strong phonon–electron coupling. They are governed by nonlocal (interband) Berry curvature elements, scattered electron magnetic moments, and band-dependent rates (Xue et al., 6 Jan 2025).

2. Semiclassical Transport and Hall Conductivity

Under a longitudinal temperature gradient $\nabla T$, the nonequilibrium phonon distribution acquires a correction $\delta n_{qs}$, driving the transverse (Hall) angular momentum current:

$$J^L_x = \sum_{q,s} \ell_{qs}\,v_{qs,x}\,\delta n_{qs} = \sigma^L_{xy}(-\nabla_y T)$$

with

$$\ell_{qs} = m_{qs}\,\frac{\partial f_0}{\partial\varepsilon_{qs}}, \quad \sigma^L_{xy} = \sum_{qs} \frac{m_{qs}\,v_{qs,x}\,v_{qs,y}\,\tau_{qs}}{T}\left(-\frac{\partial f_0}{\partial\varepsilon_{qs}}\right)$$

where $v_{qs}$ is the phonon group velocity, $\tau_{qs}$ the relaxation time, and $f_0$ the Bose–Einstein distribution. Both intrinsic and extrinsic $m_{qs}$ contribute directly to the Hall conductivity and can be of comparable magnitude, especially when $\tau_p\approx\tau_e$ (Xue et al., 6 Jan 2025).

In the linear response/Kubo formalism for bosonic Bogoliubov–de Gennes phonon systems, the intrinsic Hall conductivity is written as:

$$\sigma_{\rm PAM}^{xy} = -\frac{k_B}{\hbar}\sum_{s} \int_{\rm BZ} \frac{d^3k}{(2\pi)^3}\, \Omega_{s}^{xy}(k)\, f(\omega_s(k),T)$$

where $\Omega_{s}^{xy}(k)$ is the Berry curvature and $f(\omega,T)$ is a mode-dependent thermal occupation function (Park et al., 2020).

3. Chiral Phonons, Pseudo-Angular Momentum, and Berry Curvature

Chiral phonons in systems with $C_3$ symmetry (such as MoSi$_2$As$_4$ or MoS$_2$) possess well-defined pseudo-angular momentum (PAM), $l_{\rm ph}(q) = l_{\rm ph}^o(q) + l_{\rm ph}^s(q)$, the sum of orbital and self-rotation components. Under circularly polarized light or a temperature gradient, these chiral modes can be selectively populated, resulting in a valley-contrasting or handedness-dependent Hall response in both electronic and phononic degrees of freedom (Zhang et al., 2022).

The general mechanism for the phonon-PAM Hall effect requires:

• Phonon bands with nonzero Berry curvature
• Each band and momentum state labeled by well-defined $l_{\rm ph}^{\lambda}(q)$

The transverse PAM current is then:

$$J_l^\mu = \sum_\lambda \int_{BZ} \frac{d^2q}{(2\pi)^2} l_{ph}^\lambda(q)\, v^{an}_{\lambda,\mu}(q)\, n_\lambda(q)$$

where the anomalous velocity is $v^{an}_\lambda(q) = -F\times\Omega_\lambda(q)$ for driving force $F$, and $n_\lambda(q)$ is the phonon occupation. The resulting conductivity is proportional to the sum over product of $l_{ph}$ and Berry curvature (Zhang et al., 2022).

In materials such as MoSi$_2$As$_4$, the number of chiral-phonon modes and enhanced spin-orbit coupling result in a larger phonon-PAM Hall response (up to 80% larger than in monolayer MoS$_2$), with additional tunability through optical pumping and photon helicity (Zhang et al., 2022).

4. Interplay of Intrinsic and Extrinsic Effects

The relative strengths of intrinsic and extrinsic contributions to PAMHE depend sensitively on material parameters:

• Impurity concentration, phonon–electron coupling, and disorder modify relaxation rates, shifting the balance between Berry-curvature-dominated ("adiabatic") and nonadiabatic skew/side-jump mechanisms.
• Lattice strain directly alters the phonon Berry curvature, modifying both $m^{\rm int}$ and the extrinsic amplitudes.
• Doping modifies which electronic bands participate in skew scattering, permitting sign control and enhancement of $m^{\rm SS}$.
• In two-band Dirac systems (e.g. gapped graphene), the extrinsic skew-scattering contribution can be strongly amplified by small gaps and long phonon lifetimes:

$$m_{K}^{SS} \propto \frac{\tau_p}{\tau_e}\, m^e_0\,\Omega^{mol}_0\,\frac{1}{-\omega+2\varepsilon_0}$$

(Xue et al., 6 Jan 2025).

5. Edge Accumulation and Magnetization

The transverse PAM current due to PAMHE leads to edge accumulation of phonon angular momentum in finite samples:

$L^z_{\rm edge}(x) \simeq \tau_L\, j_x^{L^z}(x)$

where $\tau_L$ is the relaxation time for angular momentum. For ionic lattices with nonzero Born effective charge $Z^*$, this angular momentum accumulation yields a measurable edge magnetization

$$M_{\rm edge}(x) = \sum_\alpha \mu_\alpha\, \Delta L_\alpha^z(x)$$

where $\mu_\alpha = (eZ_\alpha^*)/(2M_\alpha)$. In crystals with two sublattices of opposite $Z^*$, this edge magnetization is nonzero and directly related to the underlying PAM accumulation (Park et al., 2020).

Typical edge magnetizations are on the order of $10^{-5}$–$10^{-4}$ A/m, with per-cell moments $\Delta\mu \sim 10^{-13}$–$10^{-12} \mu_B$ at $T \sim 100$ K in simple cubic insulators (Park et al., 2020).

6. Experimental and Computational Perspectives

Detection of PAMHE and the associated edge magnetization is experimentally challenging due to the small scale of the effect, but accessible with advanced techniques:

• Magneto-optical Kerr effect (MOKE) and polar Kerr microscopy for surface magnetization
• Scanning NV-center magnetometry with nanotesla sensitivity and sub-50-nm resolution
• Scanning SQUID or Hall-probe magnetometry for coarse but sensitive measurements

PAMHE is maximized in crystals hosting both transverse and longitudinal acoustic modes, chiral phonon bands with large Berry curvature, and broken inversion or time-reversal symmetry. Optically pumped phonon populations, especially in 2D materials with strong SOC and valley contrast, provide additional pathways to probe and manipulate PAMHE (Zhang et al., 2022).

First-principles approaches combining density-functional perturbation theory, phonon–CEF coupling, and Berry curvature evaluation across the Brillouin zone support predictive materials exploration, as exemplified in studies of NaYbSe$_2$ and MoSi$_2$As$_4$ (Pai et al., 2022, Zhang et al., 2022).

7. Material Engineering and Future Directions

Routes to engineer and enhance PAMHE include:

• Selection of materials with symmetries (e.g., $C_{3x}$, noncentrosymmetric) permitting robust chiral-phonon PAM labels
• Engineering band crossings and Berry curvature hotspots by applying strain, magnetic fields, or designing vibronic–phonon resonance
• Doping and disorder engineering to optimize the phonon–electron coupling regimes and suppress undesired relaxation

The ability to control both magnitude and sign of PAMHE via external fields, strain, and optical pumping suggests the potential for functional phononic and spintronic devices leveraging angular momentum currents—distinct from charge or pure spin Hall effects and accessible even in insulators (Xue et al., 6 Jan 2025, Park et al., 2020, Zhang et al., 2022, Pai et al., 2022).