Chapter 2: Chiral Phonons in Materials

Chiral phonons were theoretically predicted in 2015 and first widely reported experimentally in 2018 for monolayer WSe₂, revolutionizing our understanding of phonon physics. This chapter explores the materials that host chiral phonons, from valley-coupled phonons in WSe₂ and MoS₂ monolayers to intrinsic chirality in 3D crystals like α-quartz and tellurium. We examine the symmetry requirements, experimental signatures, and temperature-dependent behavior that make these materials platforms for exploring phonon angular momentum.

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2.1 Chiral Phonons in 2D Transition Metal Dichalcogenides

Crystal Structure and Symmetry

Monolayer transition metal dichalcogenides (TMDs) with the formula MX₂ (M = Mo, W; X = S, Se, Te) exhibit a unique trigonal prismatic structure belonging to the D_3h point group.

Key TMD Materials

E' and E'' Phonon Modes at Γ Point

The E' and E'' modes in TMD monolayers are doubly degenerate at the Brillouin zone center (Γ point). Although these 2D irreps can be expressed in a circular polarization basis, the degeneracy at Γ implies no intrinsic energy splitting between opposite circular components.

Physical characteristics (material dependent):

• E' mode (~250-260 cm⁻¹ for WSe₂): In-plane optical mode where M and X atoms vibrate in opposite directions
• E'' mode (frequency varies by material): Often involves out-of-plane motion and remains an optical mode in typical TMDs
• Circular basis is a convenient representation; net PAM and selection rules are most prominent at K/K'

Valley-Phonon Coupling at K and K' Points

The most remarkable feature of TMD monolayers is the coupling between valley degree of freedom and chiral phonons.

Selection rules for valley-phonon coupling:

$$\text{K valley} \xrightarrow{\sigma^+} |\text{E}^+\rangle \quad (\text{right-handed})$$

$$\text{K' valley} \xrightarrow{\sigma^-} |\text{E}^-\rangle \quad (\text{left-handed})$$

This coupling arises from:

1. Conservation of angular momentum: Photon helicity + valley angular momentum = phonon angular momentum
2. Symmetry constraints: D_3h symmetry enforces opposite helicity coupling at K and K'
3. Electron-phonon matrix element: Valley-dependent deformation potential

Valley-Selective Optical Excitation

The valley degree of freedom can be selectively addressed using circularly polarized light:

Experimental manifestations:

• Circularly polarized Raman spectroscopy: Different intensities for E' modes under σ⁺ and σ^- excitation
• Valley photoluminescence: Preferential emission from one valley under polarized excitation
• Intervalley scattering: Chiral phonons mediate K ↔ K' valley transitions

Intervalley Scattering via Chiral Phonons

Chiral phonons provide a unique mechanism for intervalley scattering:

$$|K, n\rangle + |\text{E}^-\rangle \longrightarrow |K', n-1\rangle$$

$$|K', n\rangle + |\text{E}^+\rangle \longrightarrow |K, n-1\rangle$$

where $n$ is the electronic state index. This process:

• Conserves total angular momentum (valley + phonon)
• Requires momentum transfer ~K - K' (large momentum transfer)
• Is temperature-dependent (population of chiral phonons)
• Determines valley lifetime and valley Hall effect

2.2 3D Chiral Crystals

Beyond 2D materials, chiral phonons also exist in three-dimensional crystals with specific symmetry properties.

α-Quartz: First 3D Observation

In 2018, Zhu et al. reported the first observation of chiral phonons in a 3D crystal: α-quartz (SiO₂).

Key experimental findings:

• Raman circular dichroism: Intensity difference $I_{R} - I_{L} \neq 0$ for certain phonon modes
• Temperature dependence: Chiral phonon signature persists up to α-β phase transition (~573°C)
• Mode-specific chirality: Not all phonons are chiral; depends on symmetry representation

Screw Axis Symmetry

The key symmetry element enabling 3D chiral phonons is the screw axis:

Tellurium and Selenium

Tellurium (Te) and selenium (Se) are elemental crystals with helical chain structures exhibiting intrinsic chirality:

Physical manifestations:

• Circular photogalvanic effect: Photocurrent direction depends on light helicity
• Phonon Hall effect: Transverse phonon thermal current under temperature gradient
• Valley-like behavior in 3D: Chiral phonons couple to pseudospin in Weyl semimetal phases

Chiral Organic Crystals

Organic molecular crystals with chiral molecules also host chiral phonons:

2.3 Phonon Dispersion with Chirality

Chirality-Induced Band Splitting

In materials with broken time-reversal or mirror symmetry, phonon chirality can induce band splitting along certain directions:

$$\omega_+(\mathbf{q}) \neq \omega_-(\mathbf{q})$$

where $\omega_{\pm}$ are frequencies of phonons with opposite chirality at wavevector $\mathbf{q}$.

Phonon Dispersion in WSe₂

For WSe₂ monolayer, representative features often discussed include:

1. Γ point: E' and E'' modes are doubly degenerate (circular components are basis choices)
2. K and K' points: Valley-resolved chiral character emerges; opposite valleys host opposite handedness
3. General q-points: Small wavevector-dependent effects can appear; details are material- and calculation-dependent

Acoustic vs Optical Chiral Phonons

2.4 Temperature Dependence

Thermal Population of Chiral Phonons

The population of chiral phonons follows the Bose-Einstein distribution:

$$n(\omega, T) = \frac{1}{e^{\hbar\omega / k_B T} - 1}$$

At finite temperature:

• Low T (k_BT << ℏω): Exponentially suppressed phonon population
• High T (k_BT >> ℏω): Classical limit $n \approx k_B T / \hbar\omega$

Implications for valley dynamics:

• Intervalley scattering rate $\Gamma_{KK'} \propto n(\omega_E)$
• Valley lifetime $\tau_v \propto 1/\Gamma_{KK'}$
• At room temperature (300 K), E' modes (~250 cm⁻¹ ≈ 360 K) are partially populated

Anharmonic Effects

Phonon-phonon interactions modify chiral phonon properties at elevated temperatures:

1. Frequency shift: $$\omega(T) = \omega_0 + \Delta\omega_{\text{anharmonic}}(T)$$ Typically $\Delta\omega < 0$ (softening) and $|\Delta\omega| \sim$ few cm⁻¹
2. Linewidth broadening: $$\Gamma(T) = \Gamma_0 + \Gamma_{\text{anharmonic}}(T)$$ Determines phonon lifetime $\tau = \hbar/\Gamma$
3. Chirality degradation: Phonon angular momentum mixing via three-phonon processes

Phase Transitions

Some materials undergo structural phase transitions that affect chiral phonons:

2.5 Computational Modeling: Valley-Phonon Coupling

We now implement a simplified model of valley-phonon coupling in TMD monolayers using Python.

import numpy as np
import matplotlib.pyplot as plt
from scipy.linalg import eigh

# Parameters for WSe2 monolayer
class TMDParameters:
    def __init__(self):
        # Lattice constant (angstrom)
        self.a = 3.32
        # Phonon frequencies (cm^-1)
        self.omega_E_prime = 250.0  # E' mode
        self.omega_E_double_prime = 310.0  # E'' mode
        # Valley coupling strength (meV)
        self.g_valley = 5.0
        # Temperature (K)
        self.T = 300
        # Boltzmann constant (meV/K)
        self.k_B = 0.0861733

def bose_einstein(omega_cm, T, k_B=0.0861733):
    """
    Bose-Einstein distribution
    omega_cm: phonon frequency in cm^-1
    T: temperature in K
    k_B: Boltzmann constant in meV/K
    """
    # Convert cm^-1 to meV: 1 cm^-1 ≈ 0.124 meV
    omega_meV = omega_cm * 0.124
    if T == 0:
        return 0
    x = omega_meV / (k_B * T)
    return 1.0 / (np.exp(x) - 1)

def valley_phonon_hamiltonian(k_vec, params):
    """
    Construct valley-phonon coupled Hamiltonian
    k_vec: momentum vector [kx, ky] in units of 1/a
    params: TMDParameters object

    Returns 4x4 Hamiltonian in basis |K,+>, |K,->, |K',+>, |K',->
    where +/- denote E+ and E- chiral phonon states
    """
    kx, ky = k_vec
    k_norm = np.sqrt(kx**2 + ky**2)

    # Phonon energies (in meV)
    E_phonon = params.omega_E_prime * 0.124

    # Valley-phonon coupling (momentum-dependent)
    g = params.g_valley * k_norm

    # Hamiltonian matrix
    H = np.array([
        [E_phonon, 0, g, 0],           # |K,+>
        [0, E_phonon, 0, g],           # |K,->
        [g, 0, E_phonon, 0],           # |K',+>
        [0, g, 0, E_phonon]            # |K',->
    ])

    return H

def plot_valley_phonon_dispersion():
    """
    Plot phonon dispersion with valley coupling
    """
    params = TMDParameters()

    # Define k-path: Γ -> K
    n_k = 100
    k_path = np.linspace(0, 1.0, n_k)  # in units of K point distance

    eigenvalues = np.zeros((n_k, 4))

    for i, k in enumerate(k_path):
        # K point is at (4π/3a, 0)
        k_vec = np.array([k * 0.5, k * 0.866])
        H = valley_phonon_hamiltonian(k_vec, params)
        eigvals, eigvecs = eigh(H)
        eigenvalues[i, :] = eigvals

    # Convert to cm^-1
    eigenvalues_cm = eigenvalues / 0.124

    plt.figure(figsize=(10, 6))
    for band in range(4):
        plt.plot(k_path, eigenvalues_cm[:, band], linewidth=2)

    plt.xlabel('k along Γ→K (units of K)', fontsize=12)
    plt.ylabel('Phonon Energy (cm⁻¹)', fontsize=12)
    plt.title('Valley-Phonon Coupled Dispersion in WSe₂', fontsize=14)
    plt.axhline(y=params.omega_E_prime, color='gray', linestyle='--',
                alpha=0.5, label='Uncoupled E\' mode')
    plt.legend(['Band 1', 'Band 2', 'Band 3', 'Band 4', 'Uncoupled'],
               fontsize=10)
    plt.grid(True, alpha=0.3)
    plt.tight_layout()
    plt.savefig('valley_phonon_dispersion.png', dpi=300)
    plt.show()

def plot_temperature_dependent_scattering():
    """
    Plot temperature-dependent intervalley scattering rate
    """
    params = TMDParameters()
    temperatures = np.linspace(10, 500, 100)

    # Scattering rate proportional to phonon population
    scattering_rates = np.zeros_like(temperatures)

    for i, T in enumerate(temperatures):
        n_phonon = bose_einstein(params.omega_E_prime, T, params.k_B)
        # Simplified scattering rate (arbitrary units)
        scattering_rates[i] = (n_phonon + 1)  # emission + absorption

    plt.figure(figsize=(10, 6))
    plt.plot(temperatures, scattering_rates, linewidth=2, color='darkblue')
    plt.xlabel('Temperature (K)', fontsize=12)
    plt.ylabel('Intervalley Scattering Rate (arb. units)', fontsize=12)
    plt.title('Temperature Dependence of K↔K\' Scattering via E\' Phonons',
              fontsize=14)
    plt.axvline(x=300, color='red', linestyle='--', alpha=0.5,
                label='Room Temperature')
    plt.grid(True, alpha=0.3)
    plt.legend(fontsize=10)
    plt.tight_layout()
    plt.savefig('temperature_scattering.png', dpi=300)
    plt.show()

# Execute visualizations
if __name__ == "__main__":
    print("=== TMD Valley-Phonon Coupling Simulation ===\n")

    params = TMDParameters()
    print(f"Material: WSe₂ monolayer")
    print(f"E' phonon frequency: {params.omega_E_prime} cm⁻¹")
    print(f"Valley coupling: {params.g_valley} meV")
    print(f"Temperature: {params.T} K\n")

    # Calculate phonon population at room temperature
    n_300K = bose_einstein(params.omega_E_prime, 300)
    print(f"Phonon population at 300K: {n_300K:.3f}")
    print(f"This means ~{n_300K:.1f} phonons per mode on average\n")

    print("Generating plots...")
    plot_valley_phonon_dispersion()
    plot_temperature_dependent_scattering()
    print("Done!")

Code Explanation

1. TMDParameters class: Stores material-specific parameters for WSe₂
2. bose_einstein function: Calculates thermal phonon population
3. valley_phonon_hamiltonian: Constructs 4×4 Hamiltonian coupling K and K' valley phonon states
4. plot_valley_phonon_dispersion: Shows how valley coupling modifies phonon bands
5. plot_temperature_dependent_scattering: Demonstrates temperature effect on intervalley transitions

2.6 Phonon Chirality in Different Material Classes

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd

class ChiralPhononMaterial:
    """Base class for materials hosting chiral phonons"""
    def __init__(self, name, dimension, symmetry):
        self.name = name
        self.dimension = dimension  # '2D' or '3D'
        self.symmetry = symmetry
        self.chiral_modes = []

    def add_chiral_mode(self, frequency, angular_momentum, symmetry_label):
        """Add a chiral phonon mode"""
        self.chiral_modes.append({
            'frequency': frequency,  # cm^-1
            'L_z': angular_momentum,  # in units of hbar
            'label': symmetry_label
        })

    def get_summary(self):
        """Return summary of material properties"""
        return {
            'Name': self.name,
            'Dimension': self.dimension,
            'Symmetry': self.symmetry,
            'Num_Modes': len(self.chiral_modes)
        }

# Define materials database
def create_materials_database():
    materials = []

    # WSe2 monolayer
    wse2 = ChiralPhononMaterial('WSe₂', '2D', 'D₃ₕ')
    wse2.add_chiral_mode(250, +1, 'E\'+')
    wse2.add_chiral_mode(250, -1, 'E\'-')
    wse2.add_chiral_mode(310, +1, 'E\'\'+')
    wse2.add_chiral_mode(310, -1, 'E\'\'-')
    materials.append(wse2)

    # MoS2 monolayer
    mos2 = ChiralPhononMaterial('MoS₂', '2D', 'D₃ₕ')
    mos2.add_chiral_mode(383, +1, 'E\'+')
    mos2.add_chiral_mode(383, -1, 'E\'-')
    materials.append(mos2)

    # α-Quartz
    quartz = ChiralPhononMaterial('α-SiO₂', '3D', 'P3₁21')
    quartz.add_chiral_mode(207, +1, 'A+')
    quartz.add_chiral_mode(207, -1, 'A-')
    quartz.add_chiral_mode(465, +1, 'E+')
    quartz.add_chiral_mode(465, -1, 'E-')
    materials.append(quartz)

    # Tellurium
    te = ChiralPhononMaterial('Te', '3D', 'P3₁21')
    te.add_chiral_mode(92, +1, 'A₁+')
    te.add_chiral_mode(92, -1, 'A₁-')
    te.add_chiral_mode(121, +1, 'E+')
    te.add_chiral_mode(121, -1, 'E-')
    materials.append(te)

    return materials

def plot_chiral_mode_comparison(materials):
    """Plot comparison of chiral phonon frequencies across materials"""
    fig, ax = plt.subplots(figsize=(12, 8))

    colors_2d = '#e74c3c'
    colors_3d = '#3498db'

    y_positions = []
    y_labels = []

    for i, mat in enumerate(materials):
        y_pos = i * 2
        y_positions.append(y_pos)
        y_labels.append(f"{mat.name}\n({mat.dimension})")

        # Get frequencies of chiral modes
        frequencies = [mode['frequency'] for mode in mat.chiral_modes]
        unique_freqs = sorted(list(set(frequencies)))

        for freq in unique_freqs:
            color = colors_2d if mat.dimension == '2D' else colors_3d
            ax.scatter(freq, y_pos, s=150, c=color, alpha=0.7,
                      edgecolors='black', linewidth=1.5, zorder=3)

    ax.set_yticks(y_positions)
    ax.set_yticklabels(y_labels, fontsize=11)
    ax.set_xlabel('Phonon Frequency (cm⁻¹)', fontsize=13)
    ax.set_title('Chiral Phonon Frequencies Across Materials', fontsize=15,
                 fontweight='bold')
    ax.grid(True, alpha=0.3, axis='x')

    # Legend
    ax.scatter([], [], c=colors_2d, s=150, label='2D Materials')
    ax.scatter([], [], c=colors_3d, s=150, label='3D Materials')
    ax.legend(fontsize=11, loc='upper right')

    plt.tight_layout()
    plt.savefig('chiral_phonon_comparison.png', dpi=300)
    plt.show()

def create_materials_table(materials):
    """Create summary table of materials"""
    data = []
    for mat in materials:
        freqs = [mode['frequency'] for mode in mat.chiral_modes]
        data.append({
            'Material': mat.name,
            'Dimension': mat.dimension,
            'Symmetry': mat.symmetry,
            'Chiral Modes': len(mat.chiral_modes),
            'Freq Range (cm⁻¹)': f"{min(freqs):.0f}-{max(freqs):.0f}"
        })

    df = pd.DataFrame(data)
    print("\n=== Chiral Phonon Materials Database ===\n")
    print(df.to_string(index=False))
    print("\n")

    return df

# Execute
if __name__ == "__main__":
    materials = create_materials_database()
    df = create_materials_table(materials)
    plot_chiral_mode_comparison(materials)

Exercises

Summary

In this chapter, we explored the materials landscape hosting chiral phonons:

• 2D TMDs (WSe₂, MoS₂, MoSe₂): D_3h symmetry enables E' and E'' chiral modes with valley-phonon coupling
• Valley degree of freedom: K and K' valleys couple to opposite chirality phonons via optical selection rules
• 3D chiral crystals: α-quartz, Te, Se, and organic crystals with screw axis symmetry (3₁ or 3₂)
• Phonon dispersion: Chirality induces band splitting and momentum-dependent coupling
• Temperature effects: Bose-Einstein statistics govern phonon population and scattering rates

The next chapter will focus on experimental techniques for detecting and characterizing chiral phonons, including circularly polarized Raman spectroscopy and ultrafast methods.