# Ty
# --- DRUDE MODEL ---
# Métal : ions + free e- | n = N/V = p * Na * z / M
# Hyp : e- free/indpt (perfect gas), instant events (relaxat° time tau), continuous scattering.
# Prob scat durat° dt : dS = dt / tau
# Prob no scat btw 0 -> t : P(t) = e^(-t/tau)
# <t> = tau

# --- ELECTRICAL TRANSPORT ---
# J = nq <v> = sigma * E = n(q^2 * tau / m) * E = nq * mu * E
# mu = |q| * tau / m  (mobility)
# sigma = n * q^2 * tau / m = n * |q| * mu (conductivity)
# vd = -mu * E (drift velocity)
# p = 1/sigma (resistivity) | Matthiessen Rule : 1/tau = 1/tau_th + 1/tau_lattice

# --- FREE PATH ---
# l = <v> * tau = vth * tau | d btw 2 scatt.
# Collisions break periodicity crystal.
# dp(t)/dt = F - p/tau (variat° momentum) | damping frictional term
# p = m * v
# dv/dt = 0 -> steady regime
# Lorentz Force : F = q(E + v ^ B)

# --- HALL EFFECT ---
# courant plongé dans champs mag perpendiculaire —> 
# création champs élec transverse 
# Accumulat° charge e- 1 face -> autre (+) (mvt electrons)
# Deviat° charges -> E_Hall compense F_Lorentz
# E_Hall = Rh * J * B  -> steady rate.
# Rh = -1 / (n * q)

# --- THERMAL PROPERTIES ---
# E = d * grad(T) | (Seebeck) T != -> création E
# J = nq <v> = sigma * E = 0 -> <v> = 0
# vdeltaT = 1/2 (v(x-l) - v(x+l)) = -ldv/dx = -ldv/dT x dt/dx
# <v> = - (tau/6) * (dv^2/dT) * grad(T) + (q * tau / m) * E = 0
# E = (tau/6 * dv^2/dT) / (q * tau / m) * grad(T)
# = 1/3q x d/dt(1/2(mv^2)grad T
# Cv = n * dEa/dT = (1/V)x dE/dT
# Jq = -K * grad(T) (Fourier Law)
# K / (sigma * T) = L (Wiedemann Law)

# --- SOMMERFELD MODEL (QUANTIC E-) T=0 ---
# 1D : k = n * pi / L 
# H_hat * psi = E * psi 
# (Fermi) En = h_bar^2 * k^2 / (2m)
# v = 10vth = hk/m
# 3D : kx = 2 * nx * pi / Lx (BVK) + Bloch th.
# kf = (3 * pi^2 * n)^(1/3)
# Ef = h_bar^2 * (3 * pi^2 * n)^(2/3) / (2m)

# --- DENSITY OF STATES ---
# g(k) = 2L^3 / (2pi)^3 = 2V / (2pi)^3 (nb at/deltak)
# N = deltakzb x g(k) (souvent 2N)
# g(e) = C * sqrt(E) car dN = g(e) * dE

# --- LATTICE & BLOCH ---
# Reciprocal lattice : T . T* = 2pi
# a* = (2pi/V) * (b ^ c) | V*= (2pi)^3 / V
# Bragg : 2d sin(theta) = n * lambda

# --- BLOCH THEOREM ---
# H_hat * psi = E * psi | V(r + T) = V(r)
# psi_k(r) = u_k(r) * e^(i * k.r)
# u_k(r + T) = u_k(r)
# En,k = En,-k (Invariance)

# --- BVK : INFINITE CRYSTAL ---
# psi(x + Lx, y, z) = psi(x, y, z)
# Structure bands : E(k + G) = E(k)

# --- NATURE SOLIDES ---
# NB = total nb e- / nb available states in BZ = Nq/2N = q/2
# q = p * Za (nb at żonę elem* nb e- at)
# q odd : metals (unfilled band)
# q even : insulator, SC -> bandgap Eg (1-3 eV)

# --- CALCULATION METHOD (LCAO / TIGHT BINDING) ---
# 1. Base atomique : H_hat * phi_a(x) = Ea * phi_a(x) 
# 2. Fonction d'onde cristalline (Bloch) : 
# psi(x) = sum_n ( c_n * phi_a(x - na) ) avec c_n = e^(ikna) 
# 3. Énergie E(k) = <psi|H_hat|psi> / <psi|psi> 
# 4. Intégrales de saut (Near neighbour approximation) : 
# - <phi_n | H_hat | phi_n> = alpha 
# - <phi_n’ | H_hat | phi_n> = -beta (Dirac, n, n+/-1)
# 5. Développement : 
# E(k) = (1/N) * sum_n * sum_n' [ e^(ik(n'-n)a) * <phi_n|H_hat|phi_n'> ] 
# E(k) = alpha + (-beta)*e^(ika) + (-beta)*e^(-ika) 
# E(k) = alpha - 2*beta*cos(ka)

# --- KP METHOD (k.p) ---
# 1. Équation pour u_nk (partie périodique) : 
# [ (P_hat + h_bar*k*Id)^2 / 2me + V(r) ] * u_nk = Enk * u_nk 
# 2. Développement de l'opérateur : 
# [ P_hat^2/2me + (h_bar*k)^2/2me + h_bar(k.P_hat)/me + V(r) ] * u_nk = Enk * u_nk 
# 3. État non-perturbé (k=0) : 
# [ P_hat^2/2me + V(r) ] * u_n0 = En0 * u_n0 
 4. Opérateur de perturbation W_hat : 
# W_hat = (h_bar*k)^2 / 2me * Id + h_bar(k.P_hat) / me 
# 5. Correction d'énergie (Perturbation ordre 2) : 
# En(k) = En(0) + <un0|W_hat|un0> + Sum_n' [ |<un0|W_hat|un'0>|^2 / (En0 - En'0) ]
# W = (h_bar*k)^2 / 2me + h_bar/me * (k.p_hat)

# --- EFFECTIVE MASS ---
# 1/m* = (1/h_bar^2) * (d^2E / dk^2)
# m* > 0 (convex U) | m* < 0 (concave )
# <v> = 1/h_bar * dEk/dk
# a = F_ext / m*
# h_bar * dk/dt = F

# --- HOLE CONCEPT ---
# qh = -e = +e
# Holes in unoccupied states (m* < 0)

# --- PHONONS ---
# Quantized normal mode of vibration of crystal lattice.
# Acoustic (in phase) vs Optical (out of phase—> dipole oscillant qui interagit Lum)
# Nb modes : 3N | Acoustic -> 3 | Optical -> 3N - 3

# --- SEMICONDUCTORS (SC) ---
# Direct gap : GaAs (optoelectronic)
# Indirect gap : Si (need phonon)(microelectronics, solar cell)
# Optics : SC bcs Bandgap allows photon emission.
# nb free e- = metals>SC (cond can be turned by doping —> PN)pe your text here