We investigate the effects of magnetic and nonmagnetic impurities on the two-dimensional surface states of three-dimensional topological insulators (TIs). Modeling weak and strong TIs using a generic four-band Hamiltonian, which allows for a breaking of inversion and time-reversal symmetries and takes into account random local potentials as well as the Zeeman and orbital effects of external magnetic fields, we compute the local density of states, the single-particle spectral function, and the conductance for a (contacted) slab geometry by numerically exact techniques based on kernel polynomial expansion and Greens function approaches. We show that bulk disorder refills the suface-state Dirac gap induced by a homogeneous magnetic field with states, whereas orbital (Peierls-phase) disorder perserves the gap feature. The former effect is more pronounced in weak TIs than in strong TIs. At moderate randomness, disorder-induced conducting channels appear in the surface layer, promoting diffusive metallicity. Random Zeeman fields rapidly destroy any conducting surface states. Imprinting quantum dots on a TIs surface, we demonstrate that carrier transport can be easily tuned by varying the gate voltage, even to the point where quasi-bound dot states may appear.
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