condensed-matter-physics

# Condensed Matter Physics ## What I Do I provide comprehensive condensed matter physics tools including crystal structure analysis, electronic band theory, superconductivity, magnetism, and semiconductor physics for materials science applications. ## When to Use Me - Crystal structure analysis - Electronic band calculations - Superconducting material properties - Magnetic material behavior - Semiconductor device design - Phonon dispersion calculations ## Core Concepts - **Crystal Lattices**: Bravais lattices, Miller indices - **Band Theory**: Bloch theorem, band gaps, DOS - **Superconductivity**: BCS theory, Ginzburg-Landau - **Magnetism**: Ferromagnetism, antiferromagnetism - **Phonons**: Lattice vibrations, thermal conductivity - **Semiconductors**: Doping, heterostructures - **Topological Phases**: Berry phase, Chern numbers - **Strong Correlation**: Hubbard model, Mott insulators ## Code Examples ### Crystal Structure ```python import numpy as np def reciprocal_lattice(a1, a2, a3): V = np.dot(a1, np.cross(a2, a3)) b1 = 2 * np.pi * np.cross(a2, a3) / V b2 = 2 * np.pi * np.cross(a3, a1) / V b3 = 2 * np.pi * np.cross(a1, a2) / V return b1, b2, b3 def miller_indices(h, k, l, a): d = a / np.sqrt(h**2 + k**2 + l**2) return d def bragg_angle(d, lambda_): return np.arcsin(lambda_ / (2 * d)) a = 4.0 # Angstroms h, k, l = 1, 1, 1 d = miller_indices(h, k, l, a) print(f"d-spacing for (111): {d:.4f} Angstroms") a1 = np.array([a, 0, 0]) a2 = np.array([0, a, 0]) a3 = np.array([0, 0, a]) b1, b2, b3 = reciprocal_lattice(a1, a2, a3) print(f"Reciprocal lattice vectors: {b1}") ``` ### Band Theory ```python def free_electron_energy(k, m): hbar = 1.055e-34 return hbar**2 * np.dot(k, k) / (2 * m) def nearly_free_electron(k, G, V_G, a): return hbar**2 * k**2 / (2*m) + V_G * np.cos(np.dot(G, k) * a) def tight_binding_hopping(t, k, a): return -2 * t * np.cos(np.dot(k, a)) def density_of_states(E, E_F, sigma): return np.exp(-(E - E_F)**2 / (2*sigma**2)) / (sigma * np.sqrt(2*np.pi)) def berry_phase(A_k): return np.sum(A_k) % (2*np.pi) def zak_phase(A_k): return np.imag(np.sum(np.log(A_k))) def chern_number(berry_curvature): return int(np.sum(berry_curvature) / (2*np.pi)) ``` ### Superconductivity ```python def bc_gap_delta(T, Tc, delta0=1.76): if T >= Tc: return 0 return delta0 * np.sqrt(1 - (T/Tc)**3) def london_penetration_depth(lambda_L, n_s, m): mu0 = 4e-7 * np.pi return np.sqrt(m / (mu0 * n_s * e**2)) def critical_field_hc(T, Tc, hc0): return hc0 * (1 - (T/Tc)**2) def superconducting_energy_gap(delta, V=1): return 2 * delta def ginzburg_landau_coherence_length(xi, Tc, m, hbar): return xi def abrikosov_vortex_lattice(nu, H): return nu * H T = 4.0 Tc = 9.2 delta = bc_gap_delta(T, Tc) print(f"Gap at T=4K: {delta:.4f} meV") ``` ### Magnetism ```python def exchange_energy(J, S1, S2): return -2 * J * np.dot(S1, S2) def zeeman_energy(g, mu_B, B, S): return -g * mu_B * np.dot(B, S) def curie_temperature(J, z, S): return 2 * z * J * S * (S + 1) / (3 * k_B) def susceptibility_chi(C, T): return C / T def langevin_function(x): return coth(x) - 1/x def brillouin_function(J, x): return (2*J + 1)/(2*J) * coth((2*J+1)*x/2) - 1/(2*J) * coth(x/2) def neutron_scattering_factor(F_hkl): return np.sum(b_j * np.exp(2*np.pi*i * (h*x_j + k*y_j + l*z_j))) J = 0.5 z = 8 S = 0.5 k_B = 8.617e-5 # eV/K Tc = 2 * z * J * S * (S + 1) / (3 * k_B) print(f"Curie temperature: {Tc:.1f} K") ``` ### Semiconductor Physics ```python def effective_mass(m*, m_e): return m* / m_e def carrier_concentration(ni, T): return ni def intrinsic_carrier_concentration(Eg, T, Nc, Nv): k_B = 8.617e-5 return np.sqrt(Nc * Nv) * np.exp(-Eg / (2 * k_B * T)) def fermi_level(Eg, ni, N_d, N_a): return Eg/2 + k_B * T * np.log(N_d/N_a) def mobility_mu(E, T, mu0): return mu0 * (T/T0)**alpha def hall_coefficient(RH, p, n, mu_p, mu_n): return 1 / (p * e) * (1 - (n * mu_n**2) / (p * mu_p**2 + n * mu_n**2)) def bandgap_narrowing(delta_Eg, N): return delta_Eg * (N / N0)**(1/2) Eg = 1.12 # eV for Si T = 300 ni = intrinsic_carrier_concentration(Eg, T, 2.8e19, 1.04e19) print(f"Intrinsic carrier concentration: {ni:.2e} cm^-3") ``` ## Best Practices 1. **Periodic Boundary Conditions**: Use for infinite crystals 2. **K-point Sampling**: Ensure convergence 3. **Temperature Effects**: Include thermal expansion 4. **Spin-Orbit Coupling**: Important for heavy elements 5. ** Hubbard U**: Use DFT+U for correlated systems ## Common Patterns ```python # Wannier functions def maximally_localized_wannier(): pass # DFTB calculations def density_functional_tight_binding(): pass ``` ## Core Competencies 1. Crystal structure and symmetry 2. Electronic band structure 3. Superconductivity theory 4. Magnetic properties 5. Semiconductor physics