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We consider a disordered topological insulator thin film placed on the top ofa ferromagnetic insulator with a perpendicular exchange field $M$ and subjectedto a perpendicular electric field. The presence of ferromagnetic insulatorcauses that bottom surface states of the topological insulator thin film becomespin polarized and the electric field provides a potential difference $V$between the two surface states, resulting in breaking of time-reversal andinversion symmetry in the system. Using Kubo formalism and employing the firstBorn approximation as well as the self-consistent Born approximation, wecalculate the spin Hall conductivity. We find that for small values of $V$, alarge spin conductivity can be generated through large values of $M$ away fromthe charge neutrality point. But for large values of $V$, the spin conductivitycan be promoted even with small values of $M$ around the charge neutralitypoint. The effect of vertex corrections and the stability of the obtained largespin conductivity against disorders are also examined.

Enhanced and stable spin Hall conductivity in a disordered time-reversal and inversion symmetry broken topological insulator thin film