  Large spin splitting in the conduction band (CB) and valence band (VB) of
ferromagnetic semiconductors (FMSs), predicted by the influential mean-field
Zener model[1,2] and assumed in many spintronic device proposals[3-8], has
never been observed in the mainstream p-type Mn-doped FMSs[9-15]. Here using
tunnelling spectroscopy in Esaki-diode structures, we report the observation of
such a large spontaneous spin-splitting energy ({\Delta}E = 31.7 - 50 meV) in
the CB bottom of n-type FMS (In,Fe)As, which is surprising considering the very
weak s-d exchange interaction reported in several zinc-blende (ZB) type
semiconductors[16,17]. The mean-field Zener model also fails to explain
consistently the ferromagnetism and the spin splitting energy {\Delta}E of
(In,Fe)As, because we found that the Curie temperature (TC) values calculated
using the observed {\Delta}E are much lower than the experimental TC by a
factor of 400. These results urge the need for a more sophisticated theory of
FMSs. Furthermore, bias-dependent tunnelling anisotropic magnetoresistance
(TAMR) reveals the magnetic anisotropy of each component of the (In,Fe)As band
structure [CB, VB, and impurity band (IB)]. The results suggest that the energy
range of IB overlaps with the CB bottom or VB top, which may be important to
understand the strong s-d exchange interaction in (In,Fe)As[18,19].
