Abstract
The magnetic proximity effect (MPE) attracts much attention as a promising way for introducing ferromagnetism into a nonmagnetic electrontransport channel. Although the range of MPE is generally limited to the interface, it is extended to several tens of nm in highquality semiconductor bilayers consisting of a nonmagnetic quantum well (QW) and an underlying ferromagnetic semiconductor (FMS) layer. To elucidate the mechanism of this longrange MPE, it is essential to observe the magnetically proximitized electronic structure of the nonmagnetic semiconductor. Here, by investigating the Shubnikov  de Haas oscillations in nonmagnetic ntype InAs QW / FMS (Ga,Fe)Sb bilayers, we successfully observe the spinpolarized Fermi surface of the InAs QW. The spontaneous spinsplitting energy in the conduction band of the InAs QW reaches 18 meV when applying a negative gate voltage. This large and gatetunable spinpolarized Fermi surface of a magnetically proximitized InAs QW provides an ideal platform for novel spintronic and topological devices.
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Introduction
Introducing magnetism into highmobility semiconductors while preserving their intrinsic electronic properties has long been required. In the field of spinbased electronics (spintronics), this provides pathways to realizing highspeed and lowpower electronic devices that are compatible with the present semiconductor technology. Also, in a newly emerging field of topologybased electronics, narrowgap semiconductors such as InAs and InSb, when interfaced with swave superconductors, are likely to host topologically protected excitation states such as Majorana fermions, which are highly promising for faulttolerant quantum computing^{1,2}. In these systems, an important ingredient is spinsplitting in the electronic structure; however, it is always necessary to apply a strong magnetic field to induce spin splitting in previous works. Unfortunately, the strong magnetic field suppresses or destroy the superconductivity in the heterostructures and hampers observation of the Majorana zero mode in the superconducting gap. The use of an external magnetic field also makes it difficult to individually control each Majorana fermion, and this severely limits the scalability of the quantum computing system. A spontaneous and gatevoltagetunable spinsplit energy band in the semiconductor channel is thus highly demanded. In this context, ferromagnetic semiconductors^{3,4,5}, which are grown by doping a large amount (a few ~20%) of magnetic elements (such as Mn and Fe) into conventional semiconductors, are promising material candidates. However, the magnetic elements in ferromagnetic semiconductors are not only randomly distributed and work as scattering centers, but also alter the original semiconductor band structure by forming impurity bands; these features usually hinder highspeed device operation.
To solve these problems, utilizing a magnetic proximity effect (MPE) is one of the most promising pathways^{6,7,8,9,10,11,12}. In a semiconductorbased nonmagnetic (NM)/ferromagnetic (FM) bilayer system, the coupling between the carriers in the NM channel with the magnetization of the FM layer occurs via small penetration of carrier’s wavefunction at the NM/FM interface, as illustrated in Fig. 1a, leading to a spinsplit energy band in the NM channel. When the NM semiconductor is a quantum well (QW), high coherency of the twodimensional (2D) carriers largely extends the magnetic coupling range to the whole QW thickness (~40 nm)^{13}. In addition, the MPE can be enhanced/suppressed by applying a negative/positive gate voltage to shift the 2D electron wavefunction towards or away from the FM layer. Therefore, the spin splitting in the NM semiconductor channel, if it is successfully induced by the MPE from the neighboring FM layer, in principle can be largely modulated by external electrical means (see Supplementary Note 1).
In our previous work, we observed a giant proximity magnetoresistance (MR) in ntype InAs (15–40 nm)/(Ga,Fe)Sb (15 nm, Fe 20%) bilayers, which is induced by a strong MPE between the two layers^{13}. In this system, (Ga,Fe)Sb is a ptype ferromagnetic semiconductor (insulating at low temperature <20 K) with high Curie temperature over 300 K^{14,15}, while the ntype InAs layer works as a 2D electron channel with high mobility. In our recent study of the magnetotransport properties of InAs/(Ga,Fe)Sb bilayers, we found that the SdH oscillations observed under a magnetic field applied perpendicular to the film plane disappear when the magnetic field is applied in the film plane. This result indicates that the InAs layer is a 2D system. Also, the conductivity of (Ga,Fe)Sb is about 100 times larger than that of InAs, particularly at low temperature. Therefore, more than 99% of the electron carriers flow in the InAs channel in the InAs/(Ga,Fe)Sb bilayer heterostructure, and the contribution from the (Ga,Fe)Sb layer is negligible. The lattice mismatch between InAs and (Ga,Fe)Sb is only of the order of 0.1%, which allows epitaxial growth of highquality singlecrystalline heterostructures. Furthermore, the InAs/(Ga,Fe)Sb interface is a typeIII heterostructure, meaning that the bottom of the conduction band of InAs is lower than the top of the valence band of (Ga,Fe)Sb at the NM/FM interface. This enables large penetration of the electron wavefunction of the InAs QW into the (Ga,Fe)Sb layer, and consequently very strong MPE is induced in the InAs channel. Note that the Fermi level in (Ga,Fe)Sb is pinned at the Fe impurity band located slightly above the valence band top in the band gap. Therefore, (Ga,Fe)Sb works as a potential barrier for the electron carriers in the InAs layer. Since InAs is a material candidate for many important applications such as highspeed transistors, THz emitters and detectors, and Majorana physics, the centered interest here is then how the electronic structure, in particular the spindependent Fermi surface, of the InAs QW is affected by the MPE.
In this Letter, we investigate the electronic states in InAs QW/(Ga,Fe)Sb by studying Shubnikov  de Haas (SdH) oscillations controlled by applying a gate voltage V_{g}. SdH oscillations originate from Landau quantization of the electronic structure of a material under a strong external magnetic field (Fig. 1b). Analyses of the SdH oscillations provide essential information of the Fermi surface, such as the surface area and cyclotron mass, and as we will show later, the existence of the spinsplit energy states^{16,17,18}. The observation of SdH oscillations requires high electron mobility of the NM channel, which was not available in the previous metallic NM/FM systems^{19,20,21}. In the semiconductorbased InAs QW/(Ga,Fe)Sb system, the high carrier coherency of the InAs QW allows us to measure the SdH oscillation and characterize the detail of the Fermi surface. We successfully quantify the spontaneous spin splitting energy of the InAs QW and its dependence on V_{g}. The spinsplit energy by MPE reaches 18 meV with a topgate voltage V_{g} = −19 V. This value of spin splitting energy is the largest ever reported among those induced by MPE to the best of our knowledge. (see Supplementary Note 2)
Results
We grow two heterostructures (sample A and sample B) consisting of ntype InAs (15, 12 nm) QW/(Ga,Fe)Sb (15 nm, Fe 20%)/AlSb (100 nm) /AlAs (10 nm)/GaAs (100 nm) on semiinsulating GaAs (001) substrates, respectively, by molecular beam epitaxy (Supplementary Fig. S1a). The growth temperature is 550 °C for the GaAs and AlAs layers, 470 °C for the AlSb layer, 250 °C for the (Ga,Fe)Sb and InAs layers. Bright and streaky in situ reflection highenergy electron diffraction (RHEED) patterns of InAs (Supplementary Fig. S1b) and (Ga,Fe)Sb (Supplementary Fig. S1c) indicate good crystal quality and smooth surface during the growth.
We first pattern the samples into 100 × 400 μm^{2} Hall bars using standard photolithography and Ar ion milling, and then form several electrodes (source S, drain D, and electrodes for transport measurements) using a sputtering deposition and liftoff process of an Au (50 nm)/Cr (5 nm) film. An insulating Al_{2}O_{3} layer (~100 nm) is deposited at 170 °C on the Hall bars using atomic layer deposition (ALD). Finally, we form a topgate electrode, again by a sputtering deposition and liftoff of an Au (50 nm)/Cr (5 nm) film (Supplementary Fig. S1d). We apply V_{g} between the gate electrode and the source electrode, and measure the longitudinal resistance R_{xx} by a standard fourterminal method and Hall resistance R_{xy} simultaneously. Note that most of the current flows in the ntype InAs QW at low temperature (≤20 K). The 2D conductivity is obtained as \({\sigma }_{{xx}}=\left({R}_{{xx}}w/l\right)/ \left({\left({R}_{{xx}}w/l\right)}^{2}+{R}_{{xy}}^{2}\right)\), where w and l are the Hall bar width and length, respectively. In this study, an external magnetic field B is always applied perpendicular to the film plane. For the analysis of SdH oscillations, pretreatment of the data before Fourier transform is described in Methods.
We estimate the electron density n from R_{xy} of sample B (mobility μ = 3500 cm^{2}/Vs) measured at 3 T and 2 K. The temperature dependence of the electron carrier density and mobility are shown in Supplementary Fig. S2. The Hall coefficient is always negative, which indicates that electron conduction is dominant. In our recent study of the magnetotransport properties of InAs/(Ga,Fe)Sb bilayers, we have found the oddparity MR, which behaves as an odd function against B and the resistance change reaches almost 13.5% of the total resistance at 10 T^{22}. Although the oddparity MR is also observed in this study, this phenomenon is out of scope of this paper and not considered. Therefore, all the MR and Hall resistance data in this work are obtained by extracting only the even and odd components against B from the raw data, respectively.
Figure 2a shows the B dependence of R_{xx}, which is normalized to the zerofield resistance value (=[R_{xx}(B)−R_{xx}(0 T)]/R_{xx}(0 T)), when applying various gate voltages V_{g} (from −12 V to 5 V) at 2 K. Clear SdH oscillations at all the V_{g} values indicate the high crystal quality of the InAs QW despite the low growth temperature (250 °C). Fast Fourier transformed (FFT) spectra of the SdH oscillation are shown in Fig. 2b, where the longitudinal axis is the normalized amplitude of the FFT spectra, and the horizontal axis is the FFT frequency. The shift of the main peaks with applying V_{g} represents the change of Fermi surface area. The relation between the 2D carrier density n and the frequency F of the SdH oscillation is simply described as
where h and e are the Planck constant and the electron charge, respectively^{23}. The change of n with V_{g} estimated by Eq. (1) is shown in the inset of Fig. 2b, which indicates that our FET device works properly. We note that, however, the resolution of these B dependence of R_{xx} does not allow an observation of the spindependent Fermi surface of the InAs QW (see Supplementary Note 3).
To probe the spindependent Fermi surface of the InAs QW under MPE, we measure the V_{g} and n dependences of the SdH oscillations at a constant B. Note that, as shown in Fig. 3a, both n and V_{g} can always be uniquely related. Experimental results of V_{g} dependence of the conductivity σ_{xx} – V_{g} and n dependence of oscillating components Δσ_{xx} – n are shown in Fig. 3b and c, respectively. The oscillatory behavior of σ_{xx} is diminished with decreasing B, which is typical of SdH oscillations. The FFT spectra of those in Fig. 3c are shown in Fig. 3d. These spectra have two peaks, pointed by blue and red triangles (named F_{−} and F_{+}, respectively) in Fig. 3d, at all the magnetic fields B from 8 to 14 T. Our selfconsistent calculations show that only the first quantization level of the InAs QW is occupied, given the thickness of 12–15 nm and the range of the carrier densities shown in Fig. 3a^{13}. This clearly excludes the multiple quantized levels in the InAs QW as the origin of this twopeak feature. As discussed in the next paragraph, we find that these twopeak feature in the ndependent SdH oscillations reflects the spinsplit Fermi surface of the InAs QW, induced by MPE from the FM (Ga,Fe)Sb layer.
According to the Lifshitz–Kosevich (LK) theory, SdH oscillations in a Fermi surface without considering the spin degree of freedom can be described as^{23,24}
where Δσ_{xx} is the oscillatory component of the longitudinal conductivity, A is the normalization factor which depends on neither temperature nor magnetic field. X = 2π^{2}k_{B}T/ħω_{c}, where k_{B} is the Boltzmann constant, ħ is the reduced Planck constant, ω_{c} is the cyclotron frequency defined by eB/m^{*}, T_{D} is the Dingle temperature, and 2πβ is the phase shift caused by the Berry phase. Using Eqs. (1) and (2), the dependence of SdH oscillation on n under a constant B can be described as:
where \({A}^{{\prime} }=AX/ \, {\sinh} X \, [\exp \left(2{\pi }^{2}{k}_{{{{{{\rm{B}}}}}}}{T}_{{{{{{\rm{D}}}}}}}/\hslash {\omega }_{{{{{{\rm{c}}}}}}}\right)]\). When considering the spin splitting in these Landau levels, a phase shift corresponding to the spinsplitting energy should be introduced into the oscillatory term in Eq. (3). Because shifting the Fermi level between two continuous Landau levels (energy gap = ℏω_{c}) adds a 2π phase shift in a quantum oscillation, a spin splitting energy ΔE yields a phase shift of \(\pm {2\pi} [(\Delta E/2)/(\hslash {\omega }_{c})]\) for the spinup/down state, respectively^{16}. In the InAs/(Ga,Fe)Sb bilayers, the spin splitting in the InAs QW includes a Zeeman splitting due to B and a spontaneous splitting energy induced by MPE (=Δ_{MPE}), respectively. To include the MPE into ΔE, we assume
Where g is Lande’s effective gfactor (=8)^{25}, μ_{B} is Bohr’s magneton, Δ_{0} (>0) and ξ (>0) are fitting parameters. Here, the second term Δ_{MPE} = \({\varDelta }_{0}\xi n\) is assumed to be a linear function of n, which is based on the results estimated from our analysis of the MR curves shown in Fig. 2a (See Supplementary Note 4). Also, because ΔE is much smaller than the Fermi energy, it is reasonable to assume equal carrier densities in the spinup and spindown states, which are equivalent to n/2. Then from Eq. (4) we can rewrite Eq. (3) as
where
and \({A}_{\pm }\) is the amplitude of each spin component without the reduction factor. R(n) is the phenomenological reduction factor of SdH oscillation (See Methods). For simplicity, m^{*} is assumed to be 0.07m_{0}, which is independent of n. This is reasonable considering the weak ndependence of m* (See Supplementary Note 5). The Berry phaserelated term β is taken to be zero because the InAs conduction band should be topologically trivial^{26,27}. Equation (5) clearly indicates that there are two oscillation frequencies, F_{−} and F_{+} in the ndependent SdH oscillations, which is consistent with the twopeak feature observed experimentally. Furthermore, from Eq. (7) both F_{−} and F_{+} are inversely proportional to B, which is consistent with the experimental observation shown in the inset of the Fig. 3d. We note that when the ndependent MPE is absent, i.e. ξ = 0, the oscillation contains only one frequency.
From the fitting parameters c_{+} and c_{−} in the inset of the Fig. 3d, we obtained ξ = 1.0 ×10^{−11} [meV.cm^{2}] from Eq. (7). Also, we can determine \({\phi }_{\pm }\) by fitting Eq. (5) to the Δσ_{xx} data measured at B = 14, 13, 12 and 11 T, which are shown in Fig. 3e and Supplementary Fig. S11, respectively. In this fitting, we set F_{+}, and F_{−} as 8.1 ×10^{−13} cm^{2} and 1.0 ×10^{−12} cm^{2}, respectively, at 14 T from the spectra in Fig. 3d. We note that Δ_{0} cannot be uniquely determined from Eq. (8) because \({\phi }_{\pm }\) have uncertainty of 2πN (N ∈ Z). However, a reasonable solution can be achieved by choosing the minimum Δ_{0} so that Δ_{MPE} become positive; Δ_{0} and λ are estimated to be 39 meV and 8.4 ×10^{12} cm^{−2}, respectively. From these values of ξ and Δ_{0}, the spontaneous spinsplitting energy Δ_{MPE} as a function of V_{g} is obtained as shown in Fig. 3f (Note that we only show the region where Δ_{MPE} > 0). The increase of Δ_{MPE} as negative V_{g} is applied is due to the enhancement of the MPE. The maximum value of Δ_{MPE} (\({\it{\varDelta}}_{{{{{{\rm{MPE}}}}}}}^{\max }=\) 18 meV at V_{g} = −19 V) is 4 times larger than the previous one (=3.8 meV) estimated from fitting the MR curves using a phenomenological Khosla–Fischer model in a similar InAs/(Ga,Fe)Sb heterostructure^{13}. Here, we argue that in our InAs/(Ga,Fe)Sb heterostructures, the Rashba or Dresselhaus effect cannot cause the spinsplitting feature as observed in this study. First, to elucidate the role of the Rashba effect, we performed selfconsistent calculations of the electronic structure in the InAs/(Ga,Fe)Sb bilayer heterostructures under various gate voltages (electron carrier densities). The calculations are conducted in the range of n = 2 ~ 3.5 × 10^{12} cm^{−2}, corresponding to V_{g} = –20 ~ –5 V in Fig. 3f. Note that the Fermi level in the (Ga,Fe)Sb layer is pinned at the Fe impurity band, which is slightly above the top of the valence band in the band gap^{28}. We show the calculation results in Supplementary Fig. S12. As shown in Supplementary Fig. S12a, the potential energy of the InAs quantum well (the conduction band bottom of InAs) is strongly asymmetric at V_{g} = –5 V, inducing an electric field pushing the electron carriers in the InAs quantum well towards the Al_{2}O_{3} interface. However, the InAs quantum well potential becomes more symmetric as negative V_{g} up to –20 V is applied (Supplementary Fig. S12b, c, d), which causes the electron carrier density to decrease. Because the Rashba effect is induced by the internal electric field due to the inversion symmetry breaking in the growth direction of InAs, the selfconsistent calculation results indicate that the Rashba effect in the InAs/(Ga,Fe)Sb bilayer heterostructure should decrease when V_{g} is changed from –5 V to –20 V. This trend is opposite to our experimental observation of the gate voltage dependence of the spinsplitting energy, thus we exclude the Rashba effect as a possible origin. Meanwhile, because the Dresselhaus effect is always much weaker than the Rashba effect in the InAs/GaSb QW, it should have a negligible influence on the spin splitting in InAs/(Ga,Fe)Sb^{29}. From these analyses, we concluded that the spinsplitting energy in the InAs quantum well is induced by the MPE from the ferromagnetic (Ga,Fe)Sb layer. Comparing with the previous MR curve fitting procedure, the SdH oscillations directly probe the electronic states and Fermi surface of the InAs channel, and thus give a more reliable characterization of the spinsplit Fermi surface. No other NM/FM bilayers of any materials can achieve such a large and highly gatetunable spontaneous spinsplitting energy via MPE thus far. The large Δ_{MPE} induced by a gate voltage in the NM InAs QW has great advantages for realizing magnetic fieldfree topological quantum devices. In junctions of swave superconductor Al/InAs^{30}, which is one of the typical structures for realizing Majorana bound states, the theoretically required spinsplitting energy is only 0.1 meV. The MPE in the present InAs/(Ga,Fe)Sb can induce this large spin splitting energy without any external magnetic field. In addition, since Δ_{MPE} is controllable by the gate voltage, appearance/disappearance of Majorana bound states can be modulated locally by electrical means. This feature is essential for operations such as braiding Majorana bound states, which is required for the topological quantum computation^{31,32}.
In conclusion, we investigate the spindependent electronic structure of InAs/(Ga,Fe)Sb heterostructures by studying the gatevoltage dependence of SdH oscillations, and clarify a spontaneous large spinsplitting energy in the Fermi surface due to MPE. The estimated spin splitting reaches 18 meV, which is the largest ever reported among those induced by MPE to the best of our knowledge. This feature is increased by applying a negative gate voltage and enhancing the penetration of the electron wavefunction into the FM (Ga,Fe)Sb layer. These findings confirm that the semiconductorbased NM InAs/FM (Ga,Fe)Sb bilayer system is promising and pave the pathway to establish novel spintronic and topologybased devices utilizing MPE.
Methods
Data pretreatment for SdH oscillation
Pretreatment of the SdH oscillation data is necessary for deconvoluting the quantum oscillation spectra, since the SdH oscillations are usually masked by nonoscillating component. Here, we subtracted the background using a polynomial function up to the ninth order and extracted the oscillatory component. Then, we carried out the wellestablished pretreatment procedure for Fourier transforming of the SdH oscillations. First, we perform zeropadding to the data. Generally, the number of data points of the SdH oscillations is not enough to obtain the sufficient resolution in frequency. Zeropadding is a useful procedure to recover the resolution. Before the FFT, arrays of “0” data with finite length (in this paper, we set 2^{15} as the length.) are added to the start and the end of the data to increase the number of data points. Next, we multiply the Hann window as the window function to recover the periodicity and it avoids the artificial peak near zero frequency in the FFT spectrum. The discrete window function consists of the same number N of data points as the data set, and is given by
where n is an integer. Finally, we performed fast Fourier transform on the treated dataset.
Phenomenological reduction factor in the analysis of SdH oscillation
Since the n dependence of A’ in Eq. (3) is not clearly acknowledged, we introduce the phenomenological reduction factor R(n), which originates from the Dingle reduction factor (λ is a fitting parameter). When increasing n, the mobility decreases with n due to the enhancement of electronelectron scattering, which leads to the reduction of the SdH oscillation amplitude.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
A part of this work was conducted at Advanced Characterization Nanotechnology Platform of the University of Tokyo, supported by “Nanotechnology Platform” of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. This work was partly supported by GrantsinAid for Scientific Research (19K21961, 20H05650, 23K17324), CREST program (JPMJCR1777) and PRESTO Program (JPMJPR19LB) of Japan Science and Technology Agency, the UTECUTokyo FSI research granting program, the Murata Science Foundation and Spintronics Research Network of Japan (SpinRNJ).
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L.D.A. and M.T. planned the study. H.S., K.T. and L.D.A. designed the experiments. H.S. and K.T. grew the samples, performed sample characterizations and transport properties, fabricated the FET devices. H.S., K.T., L.D.A. discussed on the mechanism and performed theoretical calculations. All authors discussed intensively and wrote the manuscript.
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Shiratani, H., Takiguchi, K., Anh, L.D. et al. Observation of large spinpolarized Fermi surface of a magnetically proximitized semiconductor quantum well. Commun Phys 7, 6 (2024). https://doi.org/10.1038/s42005023014856
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DOI: https://doi.org/10.1038/s42005023014856
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