CONFINAMIENTO NANOSC´OPICO EN ESTRUCTURAS ... - It works!

180 Publicaciones224 J. Planelles, J. I. Climente and J. L. MovillaThere is by now a vast literature both experimental and theoretical on QRs,including simple 1D models which can grasp basic behaviors of these multiplyconnectedsystems [15]. In the present paper we revisit at an elementary level themost significant quantum effects produced by a magnetic field on a quantum ring.2. Aharonov-Bohm effectThe Hamiltonian of a charged particle in a magnetic field reads,Ĥ =(ˆp − eA)22m e+ V , (1)where ˆp is the canonical moment, A the potential vector, e the particle chargeand V the spatial confining potential. If the magnetic field is axial and constant,⃗B = B 0⃗ k, we may choose the potential vector ⃗ A = (−12 y B 0, 1 2 xB 0,0) so that theHamiltonian eq. 1 turns into:Ĥ = − 2∇ 2 − eB ˆLz + e2 B 2ρ 2 + V = ˆp2 z+2m e 2m e 8m e 2m Ĥ2D HO − eB ˆLz + V , (2)e 2m ewhere Ĥ2D HOis the 2D harmonic oscillator Hamiltonian.If the vertical confinement is severe so that we can approximately separatevariables and only consider the vertical ground state, and additionally, the in-planeconfinement is zero or parabolic, the eigenvalues (Landau levels) grow linearly 1with the magnetic field and never intersect.Now, if the particle is spatially confined in a hollow cylinder and we apply anaxial magnetic field inside the inner radius a only, i.e., B = B 0 if 0 < ρ < a andB = 0 otherwise, we may choose the following potential vector:{ 12 ⃗A = A φ ⃗u φ =B ρ⃗u φ 0 < ρ < aBa 22ρ ⃗u φ a < ρ < ∞ , (3)where ⃗u φ is the unitary vector tangential to the cylindric surface, perpendicularto the half plane φ = constant and pointing in the direction of increasing azimuthangle φ. The potential vector ⃗ A defined in Eq. 3 is continuous at ρ = a, where Bhas a step-like discontinuity 2 . The selected potential vector fulfills the Coulombgauge, ∇A = 0. Then, ˆp and A commute and the Hamiltonian eq. 1 describingour system (which is located in the interval a < ρ < ∞) becomes:Ĥ = − 22m e∇ 2 + iem eBa 22ρ 2 ∂∂φ + e2 B 2 a 48m e ρ 2 + V . (4)1 In actual 3D confinements it may grow quadratically [14].2 Note that although we may choose another potential vector yielding the same magnetic field,no gauge will allow us to select A = 0 in all the region where the system is located because thegauge-invariant flux Φ = ∫ BdS = ∮ Adl is not zero.