Observation of spontaneous spin-splitting in the band structure of an n-type zinc-blende ferromagnetic semiconductor

Large spin-splitting in the conduction band and valence band of ferromagnetic semiconductors, predicted by the influential mean-field Zener model and assumed in many spintronic device proposals, has never been observed in the mainstream p-type Mn-doped ferromagnetic semiconductors. Here, using tunnelling spectroscopy in Esaki-diode structures, we report the observation of such a large spontaneous spin-splitting energy (31.7–50 meV) in the conduction band bottom of n-type ferromagnetic semiconductor (In,Fe)As, which is surprising considering the very weak s-d exchange interaction reported in several zinc-blende type semiconductors. The mean-field Zener model also fails to explain consistently the ferromagnetism and the spin-splitting energy of (In,Fe)As, because we found that the Curie temperature values calculated using the observed spin-splitting energies are much lower than the experimental ones by a factor of 400. These results urge the need for a more sophisticated theory of ferromagnetic semiconductors.

12) Page 6, lines 124-125, the authors state that the spin splitting of device B is larger than A because of the higher Fe concentration. However, they don't mention what theory or model they base that prediction on. The authors say that mean-field Zener doesn't apply, yet they still want to say that the Tc scales with Fe doping. They should explain what theory supports their statement. 13) Page 6, lines 135-141, the authors explain the magnetic field dependence of the spin splitting device A at 50K, and they claim that the data indicate a g-factor of 621, which is a giant g-factor induced in InFeAs which is larger than observed in II-VI DMSs. Here I actually strongly disagree with the authors. Their M vs T data are too difficult to fit the Tc accurately, therefore they don't really know if all ferromagnetism in sample A is gone by 50K. In fact, there is plenty of spontaneous magnetization at 50K in their data. Thus the anomalously large g-factor is likely just because 50K isn't high enough to make that sample exhibit pure paramagnetism. Only if the authors did the field dependence at higher temperature (where there is no M_spont) and fitted with Brillouin function could this be believable. I am not convinced that InFeAs shows anomalously large s-d exchange based on the data shown. 14) Page 6, lines 145-147, authors state that there is still room to increase Tc in InFeAs, but this rather vague. The authors should be more specific and say how much more room there is to increase Fe doping or increase the n-type doping. Specifically, what are the defect limits? Solubility limits? 15) Page 8, lines 183-187, the authors conclude that the 4-fold symmetry in the tunneling region is much smaller than observed in the diffusion region because SOI is weaker than in the VB. But this conclusion is contradictory to the authors' main conclusion, which is that s-d exchange is larger than p-d exchange, however here they are stating the opposite. To be more specific in this section, the authors are stating that the VB spin splitting is larger than the CB splitting, which necessarily requires that p-d exchange is greater than s-d. However, their main conclusion is that p-d exchange = 0, and s-d is anomalously large. This contradiction needs to be resolved. 16) Page 9, lines 201-202, the authors state that s-d exchange is generally considered to be very weak. The wording should be changed, the s-d exchange is well measured in a wide range of DMSs, so it is not considered to be any value, it is just well measured. Rather than using such unqualified comparisons, the authors should directly state, we measure XXX s-d exchange which is different than the range of s-d exchange XX splitting observed in all other DMSs. 17) Page 9, lines 204-205, the authors should note that the N0_alpha values they observed are basically identical to the values observed in all II-VI DMSs. This means that the s-d exchange they measure is not surprisingly large, but actually exactly what would be expected based on measurements in II-VI DMSs carried out a few decades ago. 18) Page 9, lines 213-214, the authors state that the measured s-d exchange splitting is not large enough to account for the observed Tc based on the mean-field Zener model. I agree with this statement. However, I would suggest that rather than stating that the Zener model is a failure (when in fact it explains magnetism in a number of DMSs), they can state that InFeAs in fact have a different physical mechanism for the magnetism, which the Zener model does not capture.
19) Page 10, lines 230-232, the authors are explaining how Be doping is used as both a donor and acceptor. This explanation needs to be a bit more detailed because of the general Nature audience. They need to explain that Be is amphoteric and can sit on either In or As site depending on the substrate temperature and therefore lead to either n or p type doping. 20) Fig2a-d, why are the vertical axes arbitrary units? In Fig. 1 the authors have dI/dV in units of mA/V, therefore d^2I/dV^2 should be units of mA/V^2. 21) Fig2e, the Delta_E values need to have error bars to reflect the uncertainty in the fitting procedure. Fig. 3a-d, the vertical axis now has units of A/V^2, however the axis is not labeled, so we can't actually know what size of the signal. They should include axes labels. Also, the plotted fitting curves don't appear to be offset Lorentzians, but more complex functions. It is appropriate for the authors to include the fit function in the manuscript. Fig. 4c, if authors carried out fitting, the data points should have error bars to show the uncertainty of the fit parameters.

23)
Reviewer #3 (Remarks to the Author): There has been a long standing issue of what is the mechanism for stabilization of ferromagnetism in ferromagnetic semiconductors (FMS). Dietl and Ohno proposed that a mean field (MF) Zener model explains the ferromagnetic behavior of FMS (Ref. 1 and Ref. 2). An equation modeling the behavior was proposed that described the magnetic properties in GaMnAs quite well. However predictions for other FMS were less than satisfactory. Curie temperatures of GaMnN, InMnAs and InMnSb differed from experimental observation as much as several hundred degrees. While Dietl and Ohno continue to stand by the theory, the experimental community does not. Ahn et al investigate the magnetotransport characteristics of InFeAs Esaki diodes using tunneling spectroscopy. From spectroscopic analysis spin-splitting energies of 40-50 meV were measured. This is quite large and the authors indicate that this is the reason why large Tc is observed in InFeAs. The observed Tc can't be explained by the MF Zener model. Thus the authors conclude a different model is needed but do not offer an alternative. The work is of interest to the FMS and wider magnetics communities. The reviewer concludes that either the theory is wrong or the measurements are wrong or possibly both. The use of tunneling spectroscopy data is fraught with experimental issues. First the band structures of both the conduction band and valence band are needed. The authors assume that the valence and conduction bands are parabolic. They ignore the contributions to the density of states from the heavy hole band and light hole band in the valence band. Note k.p calculations of the valence band of InAs and InMnAs have been published see M. A. Meeker et al PRB 2015. Some discussion is needed regarding tunneling and band structure. Error analysis of tunneling spectroscopy data is needed. Also iron in InAs may have several charge states. The authors should discuss this since it may modify the tunneling spectra. As to alternative theories, Huang and Wessels noted that Fe in InAs is resonant with the conduction band see reference K. Huang J. Appl. Phys 64 6770 1988. From this they concluded that a vacuum referred binding energy (VRBE) model is relevant for transition metal doping of InAs and other III-V semiconductors. A model was subsequently proposed that transition metals with d-levels resonant with the semiconductor conduction or valence band should be a good FMS with high Curie temperatures (B. Wessels, New Journal of Physics 2008). Semiconductors with transition metals with d levels well within the band gap will not be good FMS as in the case of GaMnN. The InFeAs alloy studied here seems to support this VRBE model since the authors claim that Fe level is resonant with the conduction band. The breakdown of the Zener model was discussed by Wessels, New Journal of Physics 2008. It is somewhat puzzling that the authors ignore the large body of literature on the InMnAs system in their introduction which has shown high Tc behavior. Also there is prior literature of Fe levels in InAs that should be discussed in light of their work. The main conclusion is that there is major disagreement between the MF Zener theory of Dietl and Ohno and tunneling spectroscopy results presented in this work. Other comments: There is always confusion with possible magnetic precipitates in the Fe-As system. Are there any? Note Be is an acceptor in III-V semiconductors see typo on line 73.
First of all, we would like to thank all the reviewers for their valuable and constructive comments, which helped us improve the quality of our paper. In the following, we address and answer all the comments and questions, point by point. We also show revised parts in the revised main manuscript and Supplementary Information. (In the revised main manuscript, the revised parts are colored.)

Re-evaluation of the spin split energy E of device A
The two spin-Esaki diode devices (A and B) studied in this work differ in Fe concentration (6% and 8%, respectively) and electron density (due to co-doping of Be double donors in the (In,Fe)As layer in device B). Both of the two diode devices show two-valley structures in the d 2 I/dV 2 -V curves  3a-d in the main manuscript), corresponding to the splitting of the majority spin conduction band (CB) and minority spin CB of (In,Fe)As. For the two-valley structures, we fitted the sum of two Lorentzian curves to determine the valley center positions of the majority and minority spin CBs (Vmajor and Vminor, respectively) [see eq.(R5) in page R20]. The spin split energy E of the (In,Fe)As layers was estimated by the difference between Vmajor and Vminor. We found that this estimation is appropriate in device B, but needs to be corrected in device A as explained below. Note that, however, the main conclusions remain unchanged. Figures R1a and b show the band profiles of the p-n junctions in the two devices A and B, respectively, at low temperature (ferromagnetic state) and bias voltage V = 0, Vmajor, and Vminor. In these figures, we set the Fermi energy at the zero point, and denote the energy levels of the majority and minority spin CB bottom edges of (In,Fe)As and the valence band (VB) top of p + -InAs as Emajor, Eminor, and Ep, respectively. The spin split energy E of (In,Fe)As is therefore given by Eminor -Emajor. In device A, because Vmajor is positive whereas Vminor is negative (data shown in Figs. 2a,b of the main manuscript), the Fermi level EF lies above the band edge of the majority spin CB and below that of the minority spin CB ("half-metallic" band structure, i.e. Emajor < 0, Eminor > 0), as illustrated in Fig. R1a. At Vmajor, the band edge of majority spin CB of (In,Fe)As is aligned with the top of the VB of p + -InAs, thus eVmajor = -Emajor + Ep. However, at Vminor, which is negative, the band edge of the minority spin CB of (In,Fe)As is aligned with the quasi-Fermi level of the p + -InAs, thus eVminor = -Eminor. Therefore, we have the following relation: This means that the difference between Vmajor and Vminor overestimates the spin split energy E of (In,Fe)As CB in device A by Ep.
To obtain the accurate value of Ep, we calibrated the Be flux of our MBE system by growing one control sample composed of, from the surface, 500 nm-thick GaAs:Be/50 R3 nm-thick GaAs grown on semi-insulating GaAs (001) substrate. The hole density of the GaAs:Be in the control sample was measured to be 1.02×10 18 cm -3 , thus in our p + -InAs with the same Be concentration, Ep = 8.3 meV [we used the effective masses of heavy hole and light hole to be 0.41m0 and 0.026m0, respectively, reported in W. Nakwaski et al. Physica B 210, 1-25 (1995)]. The new E values of device A were obtained by subtracting 8.3 meV from the e(Vmajor -Vminor) values of device A and are shown in Fig.   R2 (=Revised Fig. 2e in the revised manuscript). Note that this correction does not affect any main conclusions of our paper.
On the other hand, in device B, because both Vmajor and Vminor are positive (data shown in Figs. 2c,d of the main manuscript), the Fermi level EF in (In,Fe)As lies above both the majority and minority spin CB bottom edges as illustrated in Fig. R1b (thus, Emajor < 0 and Eminor < 0). At Vmajor (Vminor), the band edge of the majority (minority) spin CB of (In,Fe)As is aligned with the top of the VB of p + -InAs. Therefore we have the following relations: Thus the difference between Vmajor and Vminor corresponds exactly to the spin split energy E of (In,Fe)As CB in device B. No correction for device B is needed.

Corresponding revised parts:
 We corrected the E data of device A in revised Figs. 2e and 3e in the revised manuscript. The effective g-factor of (In,Fe)As estimated at 50 K is therefore corrected to 478 (page 8, line 178).

Responses to reviewer #1:
1) There are a number of typographical errors e.g. line 215: "calls for reconsiderations on the chemical trend" should be "calls for reconsideration of the chemical trend" line 229: "donors in (In,Fe)As layer", should probably be "donors in the (In,Fe)As layer", I would suggest a careful proof-reading of the manuscript and the supplementary material. On another note, the features in highlighted by the black arrows Figure 1, c and d, are quite subtle, "kink" might be a more fitting description than "step-like".
again, x appears without being declared, however ℏ and m * are declared without being mentioned. It can be assumed m * and ℏ are related to 3 once the relevant reference have been read, but these need to be made explicit. Such errors do not inspire confidence for such crucial calculations.
Our response: Corresponding revised parts: We declared the variable x as the concentration of the magnetic atoms in the in-line equation in page 2, line 38 of the revised manuscript.  In page 12, line 275 of the revised manuscript, we added the equation (2) next to R5 the equation (1) as follows: Here, S is the spin angular momentum of each magnetic atom, kB is the Boltzmann constant, N0 is the cation density, AF is the Fermi liquid constant, x is the Fe  is the density of states (DOS) at the Fermi level EF, ℏ is the reduced Planck constant, m * is the electron effective mass. In the revised manuscript, these are explicitly described.
3) In Fig. 2e, the temperature dependence of E, shows roughly E of 48 meV and 40 meV for devices B and A, respectively, with corresponding TC of 65 K and 45 K. This is actually consistent with the prediction of mean-field Zener model: TC ~ (N0) 2 ~ E 2 (if the differences in x are ignored). A more careful comparison between the observation of E vs T and theoretical prediction from mean-field Zener model would be more convincing.
Our response: The reviewer pointed out that the change of the Curie temperature (TC) and spin split energy (E) in the two devices A and B seem to be understood as a result of the increase of N0since TC ~ (N0) 2 ~ E 2 in the mean-field Zener model, if we neglect the difference in the Fe concentration x. However, as we explained in the Discussion section of the manuscript, it is the large discrepancy between the N0values estimated from TC and E of the same sample that clearly indicates the failure of the mean-field Zener model for FMSs, at least in the case of (In,Fe)As. For example in device A, the N0value estimated fromE (= 31.7 meV) is 0.21 eV, while that estimated from TC (= 45K) is 4.5 eV, which differ by 20 times. In Fig. R3a, we show the experimental E -T data and the theoretical predictions from the mean-field Zener (MFZ) model of device A (the experimental data are red circles, and the theoretical curves are green and blue curves, respectively). The two theoretical curves are calculated assuming the total angular momentum J = 5/2 for Fe 3+ state. The green curve was calculated with the experimental value TC of 42 K, which yields N04.5 eV and E = 675 meV. The blue curve was calculated with the experimentalvalue E of 32 meV, corresponding to R6 N0= 0.21 eV and TC = 0.093 K. Both curves largely deviate from the experimental E -T data. Therefore, the mean-field Zener model either underestimates TC or overestimates E by one or two orders of magnitude. Our results clearly indicate that the absolute values of TC and E cannot be described consistently by the mean-field Zener model, at least in the case of (In,Fe)As. On the other hand, if we treat the E and TC values as separated parameters, not related to each other by the descriptions of the mean-field Zener model, we found that the Brillouin-function fittings are in considerably good agreement with the E -T data of devices A and B. In Fig. R3b, we show the E -T data of devices A and B and two Brillouin-function fitting curves (dotted red and blue curves). These two Brillouin-function curves are generated by TC = 42K and E = 32 meV (device A, dotted red curve) and TC = 65K and E = 50 meV (device B, dotted blue curve). The two dotted curves explain quite well the experimental data in devices A and B. These results indicate that although the mean-field Zener model proposed by T. Dielt et al. [Phys. Rev. B 63, 195205 (2001)] fails, the magnetic properties of (In,Fe)As can be described quite satisfactorily by other mean-field approaches rather than the Zener model. Corresponding revised parts: R7  In page 7, line 153 in the revised manuscript, we added the comparison between the E -T data of devices A and B with the Brillouin-function fitting curves: "We also show in Fig The plots making the comparison between the different field strengths is interesting, could the data from -1T to +1T be made available in the supplementary material.
Our response: In the following, we explain possible reasons for this complicated behavior.
First, we show in Fig. R5a the dI/dV -V curves of device B at 3.5 K and under 0 T (black) and 1 T (red). One can see that the shallowing of the majority spin valley in the d 2 I/dV 2 -V curve is caused by the increase of the dI/dV after the end of the direct tunnelling region (V ~ Vmajor) of the Esaki diode (indicated by the red arrow in Fig. R5a).
As illustrated in Fig. R5b, at the end of the tunnelling region (V ~ Vmajor), the (In,Fe)As conduction band (CB) bottom (majority spin CB bottom) is lifted to the same energy as the p + -InAs valence band (VB) top, and direct tunnelling from CB to VB is suppressed.
Therefore, the increase of the dI/dV after the end of the tunnelling region reflects the tunnelling conductance due to other indirect tunnelling processes, such as magnon-assisted tunnelling, phonon-assisted tunnelling, gap-state assisted tunneling, or their combinations.
In Fig. R5c, we plot d 2 I/dV 2 -V curves of device B, measured at 3.5 K under various magnetic fields H from -1 T to 1 T applied in the film plane (the data at 0 T and 1 T are the same as those plotted in Fig. R4c). To show the dependence of the asymmetry between the majority and minority spin's valleys on the magnetic field H, we plot in Fig. R5d the difference in the d 2 I/dV 2 values at the majority and minority spin's valley center, d 2 I/dV 2 = d 2 I/dV 2 (Vmajor) -d 2 I/dV 2 (Vminor), as a function of the magnetic field H. We see that d 2 I/dV 2 -H shows the same nonlinear behavior under positive and negative H. This result indicates that the indirect tunnelling process in device B is magnetic-field dependent.
Here, we propose a scenario of gap-state assisted tunnelling through paramagnetic Fe-induced gap states at the interface of the p-n junction: At the interface or in the depletion region of the p-n junction, some Fe gap states can exist due to diffusion of Fe atoms from the (In,Fe)As electrode. The energy levels of these paramagnetic Fe-induced states are close to the CB bottom of (In,Fe)As [see K. Huang et al., J. Appl. Phys 64, 6770 (1988)]. Thus, electrons at the CB bottom of (In,Fe)As can indirectly tunnel to the VB top of p + -InAs through these paramagnetic Fe gap states after the end of the direct R9 tunneling region. Using these diagrams, we will explain the behavior of the two spin valleys in the d 2 I/dV 2 -V curves in Figs. R4a, c, and d, as follows. At 3.5 K and 0 T (Fig. R6a), the Fe gap-state assisted tunnelling occurs mainly from the majority spin CB, whose energy is close to that of the Fe gap states, to the p +-InAs VB through the paramagnetic Fe gap states that have the same spin magnetic moment.
At 3.5 K and 1 T (Fig. R6b), however, there are more Fe gap states whose magnetic moments aligned with the majority spins in the CB of (In,Fe)As. Thus, indirect tunnelling from the majority spin CB is enhanced. This explains the increase of the dI/dV after the end of the direct tunneling region (Fig. R5a) and the shallowing of the majority spin valley of the d 2 I/dV 2 -V curves (Fig. R4c) when H was applied.

R11
 Behavior of the d 2 I/dV 2 -V curves in Fig. R4d (

device B, 50 K)
In the upper panel of Fig. R4d, the majority spin valley is shallower than the minority spin valley at 0 T, and becomes slightly deeper when applying H, although it is still shallower than the minority spin valley. We can explain this behavior as follows. At 50 K, electrons from both the majority spin (blue arrow) and minority spin (red arrow) CBs can tunnel through the paramagnetic Fe gap states and contribute to the indirect tunnelling current, as shown in Fig. R6c. This is possible because of the smaller spin split energy of (In,Fe)As CB and phonon-assisted processes existing at 50 K. This explains why the majority spin valley at 50 K (Fig. R4d) is shallower than that at 3.5 K ( Fig. R4c) at 0 T. However, when H was applied, more magnetic moments of the paramagnetic Fe states are aligned with the majority spin in the CB of (In,Fe)As, and the indirect tunnelling from the minority spin CB is partly suppressed, as shown in Fig.   R6d. Therefore the total indirect tunnelling current decreases, which explains why the majority spin valley of the d 2 I/dV 2 -V curve at 50 K in devices B becomes less shallow (slightly deeper) with applying H as seen in Fig. R4d.
 Behavior of the d 2 I/dV 2 -V curves in Fig. R4a (device A, 3

.5 K)
In the upper panel of Fig. R4a (Fig. 3a in the main manuscript), the majority spin valley is shallower than the minority spin valley even at 0 T, and the two spin valleys change very little with H. Figure R7 shows the schematic energy diagrams of the gap-state assisted tunnelling through paramagnetic Fe states (green lines) in device A at bias voltages V = Vmajor and V = Vminor. In device A, because the Fermi level of (In,Fe)As lies above the bottom of the majority spin CB but below that of the minority spin CB, the tunneling direction of electrons at V = Vminor is opposite to that at V = Vmajor, as illustrated in Fig. R7 (please also refer to Fig. R1 of this Response Letter). At 0 T and V = Vmajor (Fig. R7a) the Fe gap-state assisted tunnelling current is contributed only by electrons in the majority spin CB (blue arrow) of (In,Fe)As, because the minority spin CB (red arrow) is empty. Meanwhile, at V = Vminor (Fig. R7b,d) the minority spin electrons in the VB of p + -InAs, however, cannot tunnel into the minority spin CB of (In,Fe)As through the Fe gap states because the energy levels of the Fe gap states are lower than the minority CB bottom edge. Therefore the Fe gap-state assisted tunnelling current at V = Vminor is zero. This difference of the Fe gap-state assisted tunnelling currents in the cases of V = Vmajor and V = Vminor explains why the majority spin valley is shallower than the minority spin valley at 0 T. We also note that because the electron density n of (In,Fe)As is lower (~1×10 18 cm -3 ) than device B, the depletion layer of the p-n junction extends more into the (In,Fe)As side. Due to the lack of carriers inside the R12 depletion region of (In,Fe)As, more Fe atoms act as paramagnetic Fe states, which increases the number of the Fe gap states. This situation further enhances the Fe gap-state assisted tunnelling current at V = Vmajor in device A in comparison with that of device B.
When applying H = 1 T (Fig. R7c), the number of majority spin Fe gap states increases.
However, due to the small electron density n in the CB of (In,Fe)As layer, an increase in the number of majority spin Fe gap states (which is already quite large at 0 T) does not yield any large effect. This is why the gap-state assisted tunnelling current shows almost no change. Besides, other magnetic-field-independent mechanism (ex. phonon-assisted tunnelling) may be dominant in device A. However due to small n of the (In,Fe)As layer, the indirect tunnelling current from the majority spin CB is almost unchanged.

R13
Corresponding revised parts:  We added Fig R5, R6, and R7 (as Supplementary Figure S3, S4, and S5), and our explanation of the behaviors of the two spin valleys in the d 2 I/dV 2 -V curves of the two devices A and B in Supplementary Information as Supplementary Note 3. Fig. 3e the magnetic field dependence of E for device A is in a paramagnetic state at 50 K, why is the linear behavior of the Zeeman effect is not observed?

5) In
Our response: The Curie temperature (TC) of the (In,Fe)As layer in device A is 42 ~ 45 K, estimated from the temperature dependence of the spin split energy E (Fig. R2  Corresponding revised parts:  In page 8 line 178, we changed "giant g-factor" to "giant effective g-factor".
 We added a short explanation of the behavior of the E -H curve of device A at 50 K in the revised manuscript, page 8, line 179: "We note that the measurement temperature of 50 K is very close to TC of device A. …effectively enhance the g-factor." 6) The section dealing with the analysis of the "Magnetic anisotropy of the band structure of (In,Fe)As" need a bit more care. Could the difference in the curves of +1T and -1T fields be due to sample alignments?
Our response: The difference in the dI/dV -V curves at 1T and -1T is caused by an odd function contribution of the magnetic field H, due to the Hall effect in the p + -InAs substrate, as R14 explained below.
The devices were placed meticulously in the center position of the space between the two poles of our electromagnet (misalignment, if any, should be in millimeter order). In principle, with the center position as the origin, the distribution of magnetic field in this sample space is an even function of the position. Figure R8a shows the position dependence of H in the sample space of our electromagnet measured by a Gaussmeter, which confirmed the even-function symmetry of the magnetic field. Therefore misalignment of the sample from the center position, if any, cannot generate a response that is an odd function of H. In Fig. R8b, we show a general situation when the sample is placed in the X-Y plane under a magnetic field H applied in the X direction (HX). The currents (red lines) flow R15 through the mesa diode in the Z direction. In the thick p + -InAs substrate, however, the currents can flow in the X-Y plane (yellow areas in Fig. R8b). Because the top mesa is not located exactly at the center of the substrate, different path lengths are expected for currents flowing in different directions in the X-Y plane (see the top-view in Fig. R8b).
In these yellow areas, the magnetic field HX and the currents in the Y-direction induce ( Fig. 4b) The reason for fitting the experimental data is rather poorly introduced and seems to be more justified after consideration of facts.
Our response: The TAMR results in Fig. 4a  The results in the tunnelling region show a four-fold symmetry and another higher-order (eight-fold) symmetry. The four-fold symmetry reflects the cubic symmetry of the zinc-blende crystal structure of (In,Fe)As. Meanwhile, the eight-fold symmetry has been observed in the crystalline anisotropy magnetoresistance (AMR) of (In,Fe)As [see P. N. Hai et al., Appl. Phys. Lett. 100, 262409 (2012)]. Therefore we fitted these data by a four-fold term and an eight-fold term.
On the other hand, the data in the diffusion region are dominated by two-fold terms.
The symmetry axis of the two-fold symmetry in the diffusion region rotates by 45 degrees (from [010] to [110]) as the bias voltage V increases from 0.42 to 0.49 V. This indicates that there are at least two two-fold terms with different symmetry axes in this region. Therefore, we fitted the data in this region by two two-fold terms (with the symmetry axes along [110] and [010]) and a four-fold term which is typical of zinc blende structure. Corresponding revised parts:  In page 10, line 227 of the revised manuscript, we added the explanation of the R16 fitting process: "One can see the TAMR results in …symmetry axes in this region." Curiously Fig 4(b) 1 which shows the cross sectional from Fig.4(a) data at 50mV, in the tunneling region of sample A, the raw data does not appear to be as symmetric as the curve fit. Is there a plausible explanation for the asymmetric behavior of the peaks? Our response: As described above and in Fig. 4b-1, in the tunnelling region the (dI/dV) signal shows a four-fold symmetry and another higher-order (eight-fold) symmetry. The asymmetric behavior of the peaks in Fig. 4b-1 is quite weak and changes randomly at different bias voltages in the tunnelling region. We think that this asymmetry is due to the measurement noise and does not represent any meaningful physics.

Responses to reviewer #2:
1) Is the (In,Fe)As n ++ or n-type? I think this distinction is important.
Our response: Although the electron density n of the (In,Fe)As layers in the two samples cannot be . Therefore, (In,Fe)As thin films in this work are n + type. Corresponding revised parts:  In page 4, line 81 of the revised manuscript, we described the rough estimation of the electron density in our two samples: "Although the electron density of the (In,Fe)As layers in … are in the order of 1×10 19 and 1×10 18 cm -3 , respectively 19,22,23 ."  We revised the notation n-(In,Fe)As to n + -(In,Fe)As and p-InAs to p + -InAs in several places in the revised manuscript.

2)
Abstract: authors mention that weak nature of s-d exchange is a "common belief". I have to ask them to reword this statement, because s-d exchange is well measured in II-VI and III-V semiconductors over the last few decades, so it is not a belief, but a result of many measurements. Our response: Following the reviewer's comments and suggestions, we have revised several parts in the explanation of the experimental principles and methods, as follows: Fig. R9 (= Fig. 1c in the revised manuscript).

Corresponding revised parts:
We explained the reason why the spin-split DOS can be probed using I-V measurement R19 results in the tunnelling region.  In page 4, line 96, we added: "Because the tunnelling conductance is proportional to the …and temperature dependence of the "kink" structure."  We added schematic viewgraphs that explain the change in DOS of the (In,Fe)As CB and the corresponding change in dI/dV -V curves at T > TC and T < TC, as shown in Fig. R9 (the same as Fig. 1c of the revised main manuscript).

7)
Page 4, lines 88-89, the authors mention that there should be no current in the band gap region, however, they are not mentioning the possibility of thermionic current from electrons and holes thermally hopping and diffusing over the barrier. Their explanation is only valid at 0K.

Our response:
Although there is contribution of the thermionic current, we think that the contribution of the paramagnetic Fe gap-state assisted tunneling is more important, because we can explain the experimental magnetic-field and temperature dependence of the majority and minority spin valleys, as discussed in our responses to the comment 4) of the reviewer 1. We added comments on the origins of the tunneling current in the band-gap region of our devices.

10)
Page 5, lines 117-118, the authors conclude that the two-valley structure corresponds to the spin-splitting of the CB. This is the central premise of the study, and I fully agree with this conclusion.
Our response: We thank the reviewer for his encouraging comment.

11)
Page 5, lines 121-123, the authors discuss that the two valleys are fitted to extract the spin splitting Delta_E, however they don't provide the fit function. They should at a minimum provide the fit function, explain its physical justification, how many fit parameters are used, and what is the uncertainty of the extracted parameters, i.e. error bars. This is especially important for the data near TC, where clearly the broadening of the peaks means that the fit uncertainty must necessarily diverge.
Our response: To analyze the two-valley features of the d 2 I/dV 2 -V curves, we used the following fitting function, which is the sum of two Lorentzian curves and a linear offset:

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Corresponding revised parts:  In page 6, line 138 of the revised manuscript, we added the explanation of the fitting process as: "To analyse these two-valley structures, we use …which is smaller than 1 meV in almost all the data points.".  We revised Fig. 2e, Fig. 3e, and Fig. 3f, and added error bars to the E data points.

12)
Page 6, lines 124-125, the authors state that the spin splitting of device B is larger than A because of the higher Fe concentration. However, they don't mention what theory or model they base that prediction on. The authors say that mean-field Zener doesn't apply, yet they still want to say that the Tc scales with Fe doping. They should explain what theory supports their statement.
Our response: While the mean-field Zener model fails to explain the high TC value in our case by a factor of 100, the E -T experimental data follow the Brillouin-function fitting, indicating that the magnetic properties of (In,Fe)As can be described quite satisfactorily by other mean-field approaches rather than the Zener model (Please also see Fig. R3 and our responses to the comment 3) of reviewer 1). Within a mean-field theory framework, as long as there are s-d exchange interactions in the system, we think it is quite natural to expect that the spin split energy E increases with increasing the Fe concentration x, irrespective of the Zener model. This is because the more Fe doping into the lattice leads to the higher probability that free electrons can interact with Fe spins.
On the other hand, from the experimental results, at 3.5 K the ratio E(device A) : E(device B) = 50 : 32 = 1.6, which is larger than the ratio x (device A) : x (device B) = 8 : 6 = 1.33. This indicates that the experimental values of E(at 0 K) do not simply increase proportionally with x in both devices, which deviates from the theoretical prediction of the mean-field Zener model. Corresponding revised parts:  In page 7, line 153 in the revised manuscript we added the discussion on the comparison of the experimental E -T data of devices A and B with the mean-field theories: "We also show in Fig. 2e the two Brillouin-function fitting curves … which will be discussed later in the Discussion section."

13)
Page 6, lines 135-141, the authors explain the magnetic field dependence of the spin splitting device A at 50K, and they claim that the data indicate a g-factor of 621, which is a giant g-factor induced in InFeAs which is larger than observed in II-VI

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DMSs. Here I actually strongly disagree with the authors. Their M vs T data are too difficult to fit the Tc accurately, therefore they don't really know if all ferromagnetism in sample A is gone by 50K. In fact, there is plenty of spontaneous magnetization at 50K in their data. Thus the anomalously large g-factor is likely just because 50K isn't high enough to make that sample exhibit pure paramagnetism. Only if the authors did the field dependence at higher temperature (where there is no M_spont) and fitted with Brillouin function could this be believable. I am not convinced that InFeAs shows anomalously large s-d exchange based on the data shown.
Our response: We agree with the reviewer that the temperature of 50 K is not high enough to make the (In,Fe)As thin film in device A exhibit pure paramagnetism. However, measurements at temperatures much higher than TC, which are required to accurately estimate the g-factor of paramagnetic (In,Fe)As, are very difficult because of the broadening of the tunnelling spectroscopy at high temperatures. Thus, the g-factor at 50 K is considered as an "effective" g-factor. We made a comment on this and deleted the comparison of the g-factor value in (In,Fe)As with the values in II-VI DMSs in the main manuscript.
Corresponding revised parts:  In page 8 line 178, we changed the phrase "giant g-factor" to "giant effective g-factor".  In page 8, line 179 of the revised manuscript, we added a comment: "We note that the measurement temperature of 50 K is very close to TC …is difficult because of the broadening of the tunnelling spectroscopy at high temperatures"  We deleted the following sentence in the previous manuscript: "which is larger than that observed in II-VI diluted magnetic semiconductors (DMSs) 21 ."

14)
Page 6, lines 145-147, authors state that there is still room to increase Tc in InFeAs, but this rather vague. The authors should be more specific and say how much more room there is to increase Fe doping or increase the n-type doping. Specifically, what are the defect limits? Solubility limits?
Our response: We expect that the Curie temperature (TC) of (In,Fe)As can be largely increased by increasing either the Fe concentration x or the electron density n, as is commonly observed in carrier-induced ferromagnetic semiconductors. Since the Fe concentration x (6 and 8%) and electron density n (~1×10 19 cm -3 ) in the present (In,Fe)As samples are still far below the maximum values achieved in Mn-doped III-V FMSs (the maximum

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Mn doping concentration is ~20% and the maximum hole density is ~10 21 cm -3 ), there is still much room for increasing either x or n, and consequently TC in (In,Fe)As. The highest x that has been reported so far for (In,Fe)As is 9% [see Hai, P. N. et al., Appl. Phys. Lett. 101, 182403 (2012)], but this can be increased by optimizing the growth conditions or using special techniques such as delta doping. On the other hand, the control of n by chemical doping so far has been limited only to the use of Be or Si.
Although Be or Si atoms are doped in (In,Fe)As, n is limited to at most 1×10 19 cm -3 due to their amphoteric behavior and low activation rates in InAs, especially at low growth temperature. Searching for good donors, possibly by using group VI elements such as Te or increasing n by electrical gating, are intriguing methods that may increase n to the order of 10 20 cm -3 . Corresponding revised parts:  In page 8, line 191 of the revised manuscript, we revised our statement on the possibility of increasing TC in (In,Fe)As: "The temperature range where large spin-split energy in …that may increase n to the order of 10 20 cm -3 ."

15)
Page 8, lines 183-187, the authors conclude that the 4-fold symmetry in the tunneling region is much smaller than observed in the diffusion region because SOI is weaker than in the VB. But this conclusion is contradictory to the authors' main conclusion, which is that s-d exchange is larger than p-d exchange, however here they are stating the opposite. To be more specific in this section, the authors are stating that the VB spin splitting is larger than the CB splitting, which necessarily requires that p-d exchange is greater than s-d. However, their main conclusion is that p-d exchange = 0, and s-d is anomalously large. This contradiction needs to be resolved.
Our response: In this paper we discussed about the s-d exchange interaction energy and its relation to TC, but made no comment on the p-d exchange interaction. Actually, from the experimental data (the I-V curves) of our Esaki diodes, we were able to obtain neither the information about the spin-dependent structure of the valence band of (In,Fe)As nor the p-d exchange interaction energy in this material. Therefore we would like to leave the discussion on the p-d exchange interaction of (In,Fe)As open for future studies. We think that it is possible (and natural) that the p-d exchange interaction is even larger than the s-d exchange interaction in (In,Fe)As. Therefore there is no contradiction in our conclusions. Corresponding revised parts:  In page 4, line 90 of the revised main manuscript, we added a comment: "(we do R24 not take into account spin splitting of the VB of (In,Fe)As, because it is away from the Fermi level thus irrelevant to the present study)".

16)
Page 9 Our response: As described in page R5-R6, the N0 value that we estimated using the mean-field Zener model largely varies depending on which equation (Eq. (1) or Eq. (2)) is usded: For example in device A, the N0value estimated fromE (= 32 meV) is 0.21 eV, while that estimated from TC (= 45K) is 4.5 eV, which differ by one order of magnitude. This is because the mean-field Zener model is not applicable to (In,Fe)As. Thus, we now consider that any quantitative comparison of the N0 values between (In,Fe)As and II-VI DMSs is inappropriate and would like to make no comparison. This revision does not affect the main conclusions of our paper.
We deleted our statement in the main manuscript in p.12: ", which is highly surprising because the s-d exchange interaction is generally considered to be very weak in ZB-type semiconductors".

Corresponding revised parts:
 We deleted the following statement in page 12 in the main manuscript: " , which is highly surprising because the s-d exchange interaction is generally considered to be very weak in ZB-type semiconductors"

18)
Page 9, lines 213-214, the authors state that the measured s-d exchange splitting is not large enough to account for the observed Tc based on the mean-field Zener model. I agree with this statement. However, I would suggest that rather than stating that the Zener model is a failure (when in fact it explains magnetism in a number of DMSs), they can state that InFeAs in fact have a different physical mechanism for the R25 magnetism, which the Zener model does not capture.
Our response: As the reviewer suggested, we have revised the statement about the failure of the mean-field Zener model. One of our conclusions is that the mean-field Zener model cannot explain the magnetic properties of (In,Fe)As and some other FMSs, such as high-TC narrow-gap FMS (Ga,Fe)Sb reported recently [see N. T. Tu et al., Appl. Phys. Lett. 108, 192401 (2016)]. This indicates that the mean-field Zener model is not a universal model for FMSs, and searching for other appropriate unified model for FMSs is strongly required.

Corresponding revised parts:
 We revised the Discussion section, from page 13, line 285 in the main manuscript: "Besides the case of (In,Fe)As, the breakdown of the mean-field Zener model …thus remains an unsolved theoretical challenge.", and cited works of (In,Mn)As,  Fig. 2e. In most of the data points, the error bars are smaller than the data point's size. Fig. 3a-d, the vertical axis now has units of A/V^2, however the axis is not labeled, so we can't actually know what size of the signal. They should include axes labels. Also, the plotted fitting curves don't appear to be offset Lorentzians, but more complex functions. It is appropriate for the authors to include the fit function in the manuscript.

22)
Our response: Corresponding revised parts:  We have revised Fig. 3; we added labels in the vertical axes of Fig. 3a-d, and error bars in Fig. 3e,f. In most of the data points, the error bars are smaller than the point's size.  In page 6, line 138 of the revised manuscript, we added the explanation of the fitting process as follows: "We fitted these two-valley features by two Lorentzian curves …which is smaller than 1 meV in almost all the data points." Fig. 4c, if authors carried out fitting, the data points should have error bars to

23)
show the uncertainty of the fit parameters.
Our response: We conducted the fitting of the ( V  Fig. 4c are smaller than the size of the data points and cannot be seen. Corresponding revised parts:  We have revised Fig. 4, added error bars in Fig. 4c. In most of the data points, the error bars are even smaller than the point's size.

Responses to reviewer#3:
1) The reviewer concludes that either the theory is wrong or the measurements are wrong or possibly both. The electron-filled CB bottom and the possible Fe impurity band (IB) in the n + -(In,Fe)As side, and the empty VB top (the heavy hole (HH) and light hole (LH) bands) in the p + -InAs side. Thus the dI/dV -V curves in this tunnelling region reflect the DOSs of these bands at various temperatures and magnetic fields.
For the p + -InAs VB, we have to consider the following two things: -Because the p + -InAs is non-magnetic, the magnetic field dependence of the HH and LH bands should be very weak (the Zeeman energy of InAs at 1 T is smaller than 1 meV).

R28
-Although the HH and LH bands can split with strain at low temperature, the strain-induced splitting is also very small because the p + -InAs layer was grown on the lattice-matched InAs substrate with very little or no strain. Furthermore, the strain-induced splitting, if any, would not show Brillouin-function-like temperature dependence or magnetic field dependence, contrasting to the spin splitting observed in the d 2 I/dV 2 -V curves in Fig. 2a-d and Fig. 3a-f in the main manuscript.
Therefore, from the magnetic field and temperature dependence of the d 2 I/dV 2 -V curves in Fig. 2 and Fig.3, we conclude that the change in the DOS of the VB of p + -InAs is not the origin of the observed experimental results.
Furthermore, in the n + -(In,Fe)As side, we have to consider two band components; the CB and the Fe-related IB.
-The Fe-related IB may show temperature and magnetic field dependence, but the spin split energy (~ 2 eV) of the d-band should be much larger than the observed value (32 ~ 50 meV).
-From the TAMR results shown in Fig. 4 of the main manuscript, the tunnelling from the Fe-related IB to the VB of p + -InAs seems to be prohibited by the difference in orbital symmetry. The dI/dV -V data in the tunnelling region show only very weak four-fold symmetry and eight-fold symmetry of the CB of (In,Fe)As ( Fig. 4a and Therefore, by investigating the temperature and magnetic field dependence, we concluded that the splitting behavior in the d 2 I/dV 2 -V curves presented in this work is caused by the spin splitting in the CB of (In,Fe)As.
Next we discuss the charge states of Fe in (In,Fe)As. Although we did not conduct any direct measurement of the charge states of Fe, most of the Fe atoms are found to replace In sites in the neutral Fe 3+ state. There are two reasons for this assignment; we have observed in (In,Fe)As 1) weak dependence of the electron density on the Fe concentration, and 2) weak temperature dependence of the electron mobility which indicates that the scattering by neutral impurities is dominant [See our previous study: Hai, P. N., et al., Appl. Phys. Lett. 101, 182403 (2012)]. As mentioned above, the tunnelling of electrons from the Fe-related IB into the VB of p + -InAs seems to be prohibited by orbital symmetry. Therefore the position of the Fe-related IB (which depends on the charge state of Fe) is not relevant to the analysis of the tunnelling spectroscopy in the tunnelling region of our diode devices. Note that in the diffusion region, where the transport of electrons from the Fe-related IB is no longer prohibited,

R29
the TAMR results (Fig. 4 in the main manuscript) indicate that the Fe-related IB is close to the CB bottom or/and the VB top of (In,Fe)As, as described in the main manuscript (p.11-12). These results are consistent with the positions of the Fe 3+ and Fe 2+ charge states in InAs reported by Huang and Wessels [J. Appl. Phys 64, 6770 (1988)].

Corresponding revised parts:
 In page 4, line 96 in the revised manuscript, we explained that the contribution from the VB of p + -InAs, if any, can be distinguished by measuring the temperature and magnetic field dependence of the d 2 I/dV 2 -V curves: "Because the tunnelling conductance is proportional to … by investigating the magnetic field and temperature dependence of the "kink" structure."  In page 8, line 186, we excluded the VB of p + -InAs from the origin of the two-valley structure in the d 2 I/dV 2 -V curves: "It is obvious that the VB of p + -InAs cannot generate this large spin splitting …correspond to the majority spin and minority spin CB of (In,Fe)As."  In page 12, line 264 in the revised manuscript, we added a comment: "This indicates that the Fe-related IB is irrelevant to the spin splitting observed in the tunnelling region of the two devices A and B.".

2)
As to alternative theories, Huang and Wessels noted that Fe in InAs is resonant with the conduction band see reference K. Huang J. The main conclusion is that there is major disagreement between the MF Zener theory of Dietl and Ohno and tunneling spectroscopy results presented in this work. Our response:

R30
We thank the reviewer for informing us of related literatures that we were unaware of, which helps us to improve the integrity and quality of our paper. We also agree with the suggestion of the reviewer and have revised the Discussion section to comply with it.
To estimate the energy position of the Fe levels in InAs, we have measured the band structure of (In,Fe)As (Fe 6%) using angle-resolved photoemission spectroscopy (ARPES), as shown in Fig. R11. By measuring at the resonant energy of Fe core-levels, we were able to observe the Fe-related IB close to the CB bottom of ( Here we have significantly revised the Discussion section in the main manuscript. We introduce more experimental reports on high TC narrow-gap FMSs such as (In,Mn)As,

3) Other comments:
There is always confusion with possible magnetic precipitates in the Fe-As system. Are there any?
Note Be is an acceptor in III-V semiconductors see typo on line 73.
Our response: We have conducted careful and systematic studies on the structural properties of