Valleytronics Transition Metal Dichalcogenides Berry Curvature Optical Orientation Valley Hall
# TMDC Valleytronics: Optical Circular Dichroism, Berry Curvature Anomalous Velocity, and Valley Hall Transport
## 1. Introduction to Valleytronics in Two-Dimensional TMDCs
Valleytronics—the control and manipulation of charge carriers in different valley (momentum-space regions) of energy bands—represents a new frontier in quantum transport and optoelectronics. Transition metal dichalcogenides (TMDCs) such as MoS₂, WSe₂, and MoSe₂ possess two inequivalent valleys at the K and K' points of the hexagonal Brillouin zone, separated by the entire zone boundary. Due to time-reversal and valley-exchange symmetry breaking, these valleys exhibit opposite Berry curvature, enabling valley-specific optical selection via circular dichroism and generating valley Hall currents (Hall transport in valley rather than charge-spin space). The valleytronic degree of freedom is as robust as charge and spin, yet offers unique advantages: valleys are distinguishable by optical selection rules (σ⁺ and σ⁻ circular polarization excite K and K' respectively), manipulable by magnetic fields, and coupled to orbital angular momentum. This section develops the microscopic TMDC Hamiltonian, derives Berry curvature and anomalous velocity, and explores valley-dependent optical and transport phenomena.
## 2. Crystal Structure and Symmetry of Monolayer TMDCs
Monolayer TMDCs crystallize in a honeycomb lattice where transition metal atoms (Mo, W) form a hexagonal sublattice, sandwiched between two layers of chalcogen atoms (S, Se, Te). The point group symmetry is D₃h (broken inversion symmetry). The electronic band structure near the Fermi level is dominated by d-orbital states from the transition metal, which hybridize with p-orbitals from chalcogens, creating a complex band structure with multiple valleys.
Two equivalent valleys at K and K' (at the edges of the Brillouin zone, related by time-reversal symmetry) contain electrons with opposite valley indices tau = +1 (K valley) and tau = -1 (K' valley). The energy-momentum dispersion is approximately two-fold degenerate at each valley, reflecting the pseudo-spin (orbital angular momentum of d-electrons).
## 3. Tight-Binding Hamiltonian and K/K' Point Valleys
Near each K (or K') point, the electronic structure can be described by a 2×2 effective Hamiltonian acting on the pseudo-spin (↑, ↓) basis (representing d_{z²} and d_{x²-y²} orbital states):
where:
- v_F approx 5 times 10^5 m/s is the Fermi velocity
- Delta approx 1.5-2 eV is the band gap (direct gap at K, K')
- λ is the spin-orbit coupling strength (~0.05-0.1 eV)
- tau = pm 1 labels the valley
- sigma are Pauli matrices acting on pseudo-spin
- s_z acts on real spin (↑/↓)
The first term (Dirac-like kinetic term) dominates at low energies. The gap term Delta sigma_z opens the gap at the K point. The spin-orbit term couples pseudo-spin and real spin, creating a spin-valley-locked structure: electrons with spin ↑ at K valley have orbital momentum ↓, and vice versa at K'.
## 4. Band Structure and Carrier Properties
Diagonalizing the Hamiltonian yields the valence and conduction band dispersions:
At the K point (mathbf{k} = 0), the band gap is E_c - E_v = Delta. The bands are non-parabolic and show Dirac-like features (especially in the conduction band near K). Due to the spin-orbit coupling, each band is split into two spin-polarized subbands, with opposite spin-valley locking at K and K' valleys.
Typical TMDC parameters (MoS₂):
- Direct band gap at K: E_g approx 1.8-1.9 eV (room temperature)
- Effective masses in conduction band: m_e approx 0.5 m_e (electron mass)
- Effective masses in valence band: m_h approx 0.6 m_e
## 5. Berry Curvature and Geometric Phase
The Berry curvature, a geometric property of the Bloch wave function, is defined as:
For the TMDC conduction band near the K point, the Berry curvature has a characteristic form:
where a is the lattice constant and t approx hbar v_F a / 2 is the hopping integral. At the band gap (k = 0), the Berry curvature is singular (diverges), reflecting the special geometry. At large k, it falls off as 1/k^3.
Crucial feature: The Berry curvature at K is opposite in sign to that at K' (due to time-reversal symmetry breaking by broken inversion). This asymmetry is the foundation of valley-contrasting phenomena.
## 6. Optical Selection Rules and Circular Dichroism
The optical transition matrix element between valence and conduction bands depends on photon helicity (circular polarization). For circularly polarized light:
where mathbf{e}_sigma is the circular polarization vector (sigma = +1 for σ⁺ right-circularly-polarized, sigma = -1 for σ⁻ left-circularly-polarized).
The selection rule arises from orbital angular momentum conservation: the photon carries orbital angular momentum hbar (along z), and the electron transition must conserve total angular momentum. Due to the pseudo-spin structure and spin-orbit coupling, the transition matrix elements are valley-selective:
This means σ⁺ light (right-handed) predominantly excites electrons in the K valley, while σ⁻ (left-handed) excites K' valley electrons. The resulting valley-contrasting optical absorption is circular dichroism:
## 7. Valley Hall Effect and Anomalous Velocity
When an electric field E is applied to a TMDC, carriers experience not only the Lorentz force but also a Berry curvature-induced anomalous velocity:
This term is independent of band occupancy and arises from the geometric phase accumulated during Bloch oscillations. In a TMDC with opposite Berry curvatures at K and K' valleys, the anomalous velocities are:
The two valleys deflect in opposite directions (anomalous Hall effect), giving rise to the valley Hall effect: a charge-current-driven valley current or, conversely, a valley-current-driven charge current.
## 8. Valley Hall Conductivity and Transport
The valley Hall conductivity tensor is defined as:
where f_n is the Fermi-Dirac distribution, Omega_{n,z}^{tau} is the z-component of Berry curvature for band n at valley τ, and tau = pm 1.
For a single TMDC layer with both spin-split bands, the valley Hall conductivity is:
where the valley Hall conductivity coefficient C_{VH} is a topological integer (related to the Chern number of a given valley). For MoS₂, C_{VH} approx 0.5-1, depending on the Fermi level position and temperature.
This is remarkable: the valley Hall conductivity is quantized in units of e^2/h—a topological property independent of scattering, sample disorder, or microscopic details!
## 9. Exciton Valley Physics and Dark Excitons
When an electron and hole are bound by Coulomb attraction (exciton formation), the resulting composite pseudo-particle inherits valley properties. Exciton valley polarization (preferential formation in K vs K' valley) can be generated via circularly polarized light and measured via photoluminescence dichroism.
Importantly, dark excitons (optically inactive states due to spin and/or valley selection rules) dominate the low-temperature exciton population. These dark states have lifetimes orders of magnitude longer (microseconds to milliseconds) than bright excitons, making them excellent candidates for optical storage and quantum information.
## 10. Measurement Techniques and Experimental Signatures
Valley physics in TMDCs is probed by:
1. Circular Dichroism: Measure differential absorption Delta alpha(ω) = alpha_{σ+} - alpha_{σ-}. Peak Delta alpha indicates valley polarization; width reveals valley scattering timescale.
2. Valley Hall Transport: Apply transverse electric field and measure longitudinal valley current (via spin Hall effect in adjacent ferromagnet or spin-to-charge conversion layer).
3. ARPES with Circular Dichroism: Angle-resolved photoemission spectroscopy using σ⁺/σ⁻ photons resolves valley-specific band structure and Berry curvature.
4. Second Harmonic Generation (SHG): Nonlinear optical response is valley-sensitive; SHG intensity reverses between K and K' valleys.
Experimental achievements:
- Valley Hall conductivity measured at ~0.5 × (e²/h) in bilayer MoS₂
- Valley lifetime extracted from time-resolved Kerr rotation: τ_valley ~ 100 fs to 1 ps (disorder-limited)
- Dark exciton valley coherence times: 100 ps to 1 ns (phonon-limited)
## 11. Python Numerical Solver: Berry Curvature and Valley Transport
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad
from scipy.interpolate import interp2d
# Physical constants
HBAR = 1.0546e-34 # J·s
E_CHARGE = 1.6022e-19 # C
PLANCKS_H = 6.6261e-34 # J·s
def tmdc_hamiltonian(kx, ky, valley=1, Delta=1.5, lambda_so=0.05):
"""
TMDC tight-binding Hamiltonian near K/K' point.
H_τ(k) = ℏ v_F (τ k_x σ_x + k_y σ_y) + (Δ/2) σ_z + λ τ (1-σ_z)/2 s_z
In 2x2 form (pseudo-spin basis): ↑ = d_z², ↓ = d_x²-y²
Args:
kx, ky: wave vector components (1/nm)
valley: +1 (K) or -1 (K')
Delta: band gap (eV)
lambda_so: spin-orbit coupling (eV)
Returns:
H: 2x2 Hamiltonian matrix (eV)
"""
v_F = 5e5 # Fermi velocity (m/s) = 5e-4 (nm/fs)
v_F_nm_fs = v_F * 1e-15 * 1e9 # Convert to nm/(fs) × ℏ units
# Kinetic energy (Dirac-like)
KE_matrix = HBAR * v_F_nm_fs * np.array([
[0, valley * kx - 1j * ky],
[valley * kx + 1j * ky, 0]
], dtype=complex)
# Band gap
Gap_matrix = (Delta / 2) * np.array([
[1, 0],
[0, -1]
], dtype=complex)
# Spin-orbit term (pseudo-spin / real-spin coupling)
SO_matrix = (lambda_so / 2) * valley * np.array([
[-1, 0],
[0, 1]
], dtype=complex)
H = KE_matrix + Gap_matrix + SO_matrix
return H
def berry_curvature_k_space(kx, ky, band_index, delta_k=1e-5):
"""
Compute Berry curvature at (kx, ky) via finite difference.
Ω_z(k) = -Im[⟨∂_kx u | ∂_ky u⟩ - ⟨∂_ky u | ∂_kx u⟩]
Approximated numerically using finite differences.
Returns:
Omega_z: Berry curvature component (nm²)
"""
# Compute wave functions at nearby points
H = tmdc_hamiltonian(kx, ky, valley=1)
evals, evecs = np.linalg.eigh(H)
psi = evecs[:, band_index]
# ∂_kx
H_kx = tmdc_hamiltonian(kx + delta_k, ky, valley=1)
evals_kx, evecs_kx = np.linalg.eigh(H_kx)
psi_kx = evecs_kx[:, band_index]
# ∂_ky
H_ky = tmdc_hamiltonian(kx, ky + delta_k, valley=1)
evals_ky, evecs_ky = np.linalg.eigh(H_ky)
psi_ky = evecs_ky[:, band_index]
# Berry curvature connection
dpsi_kx = (psi_kx - psi) / delta_k
dpsi_ky = (psi_ky - psi) / delta_k
# Ω_z ~ -Im[⟨ψ | ∂_kx ψ⟩ ⟨∂_ky ψ | ψ⟩ - (x ↔ y)]
berry_conn_x = -1j * np.vdot(psi, dpsi_kx)
berry_conn_y = -1j * np.vdot(psi, dpsi_ky)
Omega_z = np.imag(berry_conn_x * np.conj(berry_conn_y) -
berry_conn_y * np.conj(berry_conn_x))
return Omega_z
def valley_hall_conductivity(kx_range, ky_range, E_fermi=0.5):
"""
Compute valley Hall conductivity by integrating Berry curvature.
σ_VH = (e²/h) ∫ (d²k/(2π)²) f(E_k) Ω_z(k)
Returns:
sigma_VH: valley Hall conductivity (e²/h units)
"""
dk = kx_range[1] - kx_range[0]
sigma_VH = 0
for kx in kx_range:
for ky in ky_range:
H = tmdc_hamiltonian(kx, ky, valley=1)
evals, evecs = np.linalg.eigh(H)
# Conduction band (band_index = 1)
E_c = evals[1]
f_c = 1.0 / (1 + np.exp((E_c - E_fermi) / 0.026)) # Fermi-Dirac at 300K
Omega_z = berry_curvature_k_space(kx, ky, band_index=1)
sigma_VH += f_c * Omega_z * (dk**2 / (2 * np.pi)**2)
return sigma_VH
def optical_selection_rules(kx, ky, polarization=1):
"""
Compute optical absorption for σ+ (polarization=1) and σ- (polarization=-1).
Selection rule: σ+ excites K, σ- excites K' valley.
Matrix element ∝ ⟨c | e·p | v⟩
"""
H = tmdc_hamiltonian(kx, ky, valley=1)
evals, evecs = np.linalg.eigh(H)
# Conduction and valence band states
psi_c = evecs[:, 1] # Conduction band
psi_v = evecs[:, 0] # Valence band
# Dipole moment along σ± directions
# Simplified: |⟨c | σ_x ± i σ_y | v⟩|²
sigma_x = np.array([[0, 1], [1, 0]], dtype=complex)
sigma_y = np.array([[0, -1j], [1j, 0]], dtype=complex)
if polarization == 1: # σ+
dipole = np.vdot(psi_c, (sigma_x + 1j * sigma_y) @ psi_v)
else: # σ-
dipole = np.vdot(psi_c, (sigma_x - 1j * sigma_y) @ psi_v)
absorption = np.abs(dipole)**2
return absorption
def circular_dichroism(kx_range, ky_range):
"""
Compute circular dichroism α_+ - α_-.
"""
dichroism = np.zeros((len(ky_range), len(kx_range)))
for i, ky in enumerate(ky_range):
for j, kx in enumerate(kx_range):
alpha_plus = optical_selection_rules(kx, ky, polarization=1)
alpha_minus = optical_selection_rules(kx, ky, polarization=-1)
dichroism[i, j] = alpha_plus - alpha_minus
return dichroism
# Main execution
if __name__ == "__main__":
print("="*70)
print("TMDC VALLEYTRONICS: BERRY CURVATURE AND VALLEY HALL TRANSPORT")
print("="*70)
# Momentum space grid
k_range = np.linspace(-0.2, 0.2, 41) # 1/nm
print(f"\nTMDC Parameters (MoS₂-like):")
print(f" Band gap Δ = 1.5 eV")
print(f" Spin-orbit coupling λ = 0.05 eV")
print(f" Fermi velocity v_F = 5×10⁵ m/s")
# Compute Berry curvature field
print(f"\nComputing Berry Curvature Field...")
berry_field = np.zeros((len(k_range), len(k_range)))
for i, ky in enumerate(k_range):
for j, kx in enumerate(k_range):
berry_field[i, j] = berry_curvature_k_space(kx, ky, band_index=1)
print(f" Max Berry curvature: {np.max(np.abs(berry_field)):.3e} nm²")
print(f" Min Berry curvature: {np.min(np.abs(berry_field)):.3e} nm²")
# Circular dichroism
print(f"\nComputing Circular Dichroism...")
dichroism = circular_dichroism(k_range, k_range)
print(f" Max dichroism: {np.max(dichroism):.3e}")
print(f" Valley polarization detectable via σ+/σ- ratio")
# Plotting
fig, axes = plt.subplots(2, 2, figsize=(13, 11))
# Plot 1: Berry Curvature in k-space
ax = axes[0, 0]
im1 = ax.contourf(k_range, k_range, berry_field, levels=30, cmap='RdBu_r')
ax.plot(0, 0, 'y+', markersize=15, markeredgewidth=2, label='Γ point')
ax.set_xlabel('$k_x$ (nm$^{-1}$)', fontsize=11)
ax.set_ylabel('$k_y$ (nm$^{-1}$)', fontsize=11)
ax.set_title('Berry Curvature Ω_z(k) - Conduction Band (K valley)', fontsize=12, fontweight='bold')
ax.set_aspect('equal')
plt.colorbar(im1, ax=ax, label='Ω$_z$ (nm$^2$)')
# Plot 2: Band structure along K-Γ-K'
ax = axes[0, 1]
k_path = np.linspace(-0.2, 0.2, 200)
E_c_vals = []
E_v_vals = []
for k in k_path:
H = tmdc_hamiltonian(k, 0, valley=1)
evals = np.linalg.eigvalsh(H)
E_v_vals.append(evals[0])
E_c_vals.append(evals[1])
ax.plot(k_path, E_c_vals, 'b-', linewidth=2.5, label='Conduction')
ax.plot(k_path, E_v_vals, 'r-', linewidth=2.5, label='Valence')
ax.fill_between(k_path, E_v_vals, E_c_vals, alpha=0.2, color='gray')
ax.axhline(y=0, color='k', linestyle='-', alpha=0.3)
ax.axvline(x=0, color='k', linestyle='--', alpha=0.5)
ax.set_xlabel('$k_x$ (nm$^{-1}$) along K-Γ-K\'', fontsize=11)
ax.set_ylabel('Energy (eV)', fontsize=11)
ax.set_title('TMDC Band Structure', fontsize=12, fontweight='bold')
ax.legend()
ax.grid(True, alpha=0.3)
# Plot 3: Circular Dichroism
ax = axes[1, 0]
im2 = ax.contourf(k_range, k_range, dichroism, levels=20, cmap='RdBu_r')
ax.set_xlabel('$k_x$ (nm$^{-1}$)', fontsize=11)
ax.set_ylabel('$k_y$ (nm$^{-1}$)', fontsize=11)
ax.set_title('Optical Circular Dichroism: α$_{\\sigma^+}$ - α$_{\\sigma^-}$', fontsize=12, fontweight='bold')
ax.set_aspect('equal')
plt.colorbar(im2, ax=ax, label='|⟨c|σ$_+$|v⟩|² - |⟨c|σ$_-$|v⟩|²')
# Plot 4: Valley-dependent anomalous velocity
ax = axes[1, 1]
E_fermi_sweep = np.linspace(-0.5, 2, 50)
sigma_VH_vs_EF = []
for E_f in E_fermi_sweep:
sigma_VH = valley_hall_conductivity(k_range, k_range, E_fermi=E_f)
sigma_VH_vs_EF.append(sigma_VH)
ax.plot(E_fermi_sweep, sigma_VH_vs_EF, 'purple', linewidth=2.5)
ax.fill_between(E_fermi_sweep, 0, sigma_VH_vs_EF, alpha=0.3, color='purple')
ax.axvline(x=0.75, color='r', linestyle='--', alpha=0.7, label='Dirac point')
ax.set_xlabel('Fermi Level $E_F$ (eV)', fontsize=11)
ax.set_ylabel('Valley Hall Conductivity (e²/h)', fontsize=11)
ax.set_title('Valley Hall Conductivity vs Fermi Level', fontsize=12, fontweight='bold')
ax.legend()
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('tmdc_valleytronics.png', dpi=150, bbox_inches='tight')
print(f"
[Plot saved as 'tmdc_valleytronics.png']")
print("
" + "="*70)
print("TMDC VALLEYTRONICS ANALYSIS COMPLETE")
print("="*70)## References
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2. Cao, T., et al. (2012). "Valley-selective Circular Dichroism of Monolayer Molybdenum Disulphide." *Nature Communications*, 3(1), 887.
3. Schaibley, J. R., et al. (2016). "Valleytronics in 2D Materials." *Nature Reviews Materials*, 1(11), 16055.
4. Yuan, H., et al. (2013). "Generation and Detection of Giant Transient Valley Currents in MoS₂." *Nature Physics*, 9(9), 563-569.
5. Yao, W., et al. (2008). "Valley Contrasting Physics in Graphene: Magnetic Moment and Topological Transport." *Physical Review B*, 77(23), 235406.