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Transition between 1D and 0D spin transport studied by Hanle precession

The precession of electron spins in a perpendicular magnetic field, the so called Hanle effect, provides an unique insight into spin properties of a non-magnetic material. In practice, the spin signal is fitted to the analytic solution of the spin Bloch equation, which accounts for diffusion, relaxation and precession effects on spin. The analytic formula, however, is derived for an infinite length of the 1D spin channel. This is usually not satisfied in the real devices. The finite size of the channel length $l_{\text{dev}}$ leads to confinement of spins and increase of spin accumulation. Moreover, reflection of spins from the channel ends leads to spin interference, altering the characteristic precession lineshape. In this work we study the influence of finite $l_{\text{dev}}$ on the Hanle lineshape and show when it can lead to a two-fold discrepancy in the extracted spin coefficients. We propose the extension of the Hanle analytic formula to include the geometrical aspects of the real device and get an excellent agreement with a finite-element model of spin precession, where this geometry is explicitly set. We also demonstrate that in the limit of a channel length shorter than the spin relaxation length $λ_{s}$, the spin diffusion is negligible and a 0D spin transport description, with Lorentzian precession dependence applies. We provide a universal criterion for which transport description, 0D or 1D, to apply depending on the ratio $l_{\text{dev}}/λ_{s}$ and the corresponding accuracy of such a choice.