Alfven waves consider a solid with an equal concentration n of electrons of mass m, and holes of mass m_h. This situation may arise in a semimetal or in a compensated semiconductor. Place the solid in a uniform magnetic field B = B_z. Introduce the coordinate ζ = x + iy appropriate for circularly polarized motion, with ζ having time dependence e^-iwt. Let w_ℓ = eB/m_ℓC and w_h = eB/m_hc.

(a) In CGS units, show that ζ_e = eE⁺/mℓw(w + wℓ); ζ_h = – eE⁺/rn_hw(w – w_h) are the displacements of the electrons and holes in the electric field E^– e^–iwt = (E_x + iE_y) e^{- iwt}.

(b) Show that the dielectric polarization P+ = ne(ζh – ζe) in the regime w << wℓ, wh may be written as P⁺ = nc²(m_h + m_e)E⁺/B², and the dielectric function ε(w) = ε₁ + 4πP⁺/E⁺ = ε₁ + 4πc²p/B², where ε₁, is the dielectric constant of the host lattice and p = n(m_ℓ + m_h) is the mass density of the carriers. If ε₁ may be neglected, the dispersion relation w²ε (w) = c²K² becomes, fur electromagnetic waves propagating in the z direction, w² = (B²/4πp)K². Such waves are known m Alfven waves; they propagate with the constant velocity B/ (4πp)^1/2. If B = 10kG; n = 10¹⁸ cm^-3; m = 10^-27g, the velocity is ~10⁸ cm s^-1. Alfven waves have been observed in semi-metals and in electron-hole drops in germanium.