Abstract
The current state of information technology accentuates the dichotomy between processing and storage of information, with logical operations carried out by charge-based devices and non-volatile memory based on magnetic materials. The main obstacle for a wider use of magnetic materials for information processing is the lack of efficient control of magnetization. Reorientation of magnetic domains is conventionally carried out by non-local external magnetic fields or by externally polarized currents1,2,3. The efficiency of the latter approach is enhanced in materials where ferromagnetism is carrier-mediated4, because in such materials the control of carrier polarization provides an alternative means for manipulating the orientation of magnetic domains. In some crystalline conductors, the charge current couples to the spins by means of intrinsic spin–orbit interactions, thus generating non-equilibrium electron spin polarization5,6,7,8,9,10,11 tunable by local electric fields. Here, we show that magnetization can be reversibly manipulated by the spin–orbit-induced polarization of carrier spins generated by the injection of unpolarized currents. Specifically, we demonstrate domain rotation and hysteretic switching of magnetization between two orthogonal easy axes in a model ferromagnetic semiconductor.
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Main
In crystalline materials with inversion asymmetry, intrinsic spin–orbit interactions couple the electron spin with its momentum ℏk. The coupling is given by the Hamiltonian , where ℏ is the reduced Planck constant and is the electron spin operator (for holes should be replaced by the total angular momentum J). Electron states with different spin projection signs on are split in energy, analogous to the Zeeman splitting in an external magnetic field. In zinc-blende crystals such as GaAs there is a cubic Dresselhaus term12 , whereas strain introduces a term that is linear in k, where Δɛ is the difference between strain in the and directions13. In wurtzite crystals or in multilayered materials with structural inversion asymmetry, there also exists the Rashba term14 , which has a different symmetry with respect to the direction of k, , where is along the axis of reduced symmetry. In the presence of an electric field, the electrons acquire an average momentum ℏΔk(E), which leads to the generation of an electric current in the conductor, where is the resistivity tensor. This current defines the preferential axis for spin precession 〈Ω(j)〉. As a result, a non-equilibrium current-induced spin polarization is generated, the magnitude of which 〈JE〉 depends on the strength of various mechanisms of momentum scattering and spin relaxation5,15. This spin polarization has been measured in non-magnetic semiconductors using optical7,8,9,11,16 and electron spin resonance17 techniques. It is convenient to parameterize 〈JE〉 in terms of an effective magnetic field Hso. Different contributions to Hso have different current dependencies (∝j or j3), as well as different symmetries with respect to the direction of j, as schematically shown in Fig. 1c,d, enabling one to distinguish between spin polarizations in different fields.
To investigate interactions between the spin–orbit-generated magnetic field and magnetic domains, we have chosen (Ga,Mn)As, a p-type ferromagnetic semiconductor18,19 with zinc-blende crystalline structure similar to GaAs. Ferromagnetic interactions in this material are carrier-mediated20,21. The total angular momentum of the holes J couples to the magnetic moment F of Mn ions by means of antiferromagnetic exchange . This interaction leads to the ferromagnetic alignment of magnetic moments of Mn ions and equilibrium polarization of hole spins. If further, non-equilibrium spin polarization of the holes 〈JE〉 is induced, the interaction of the hole spins with magnetic moments of Mn ions enables one to control ferromagnetism by manipulating J. Magnetic properties of (Ga,Mn)As are thus tightly related to the electronic properties of GaAs. For example, strain-induced spin anisotropy of the hole energy dispersion is largely responsible for the magnetic anisotropy in this material. (Ga,Mn)As, epitaxially grown on the (001) surface of GaAs, is compressively strained, which results in magnetization M lying in the plane of the layer perpendicular to the growth direction, with two easy axes along the [100] and [010] crystallographic directions22,23. Recently, control of magnetization by means of strain modulation has been demonstrated24. In this letter, we use spin–orbit-generated polarization 〈JE〉 to manipulate ferromagnetism.
We report measurements on two samples fabricated from (Ga,Mn)As wafers with different Mn concentrations. The devices were patterned into circular islands with eight non-magnetic ohmic contacts, as shown in Fig. 1a and discussed in the Methods section. In the presence of a strong external magnetic field H, the magnetization of the ferromagnetic island is aligned with the field. For weak fields, however, the direction of magnetization is primarily determined by magnetic anisotropy. As a small field (5<H<20 mT) is rotated in the plane of the sample, the magnetization is re-aligned along the easy axis closest to the field direction. Such rotation of magnetization by an external field is demonstrated in Fig. 2. For the current , the measured Rx y is positive for M||[100] and negative for M||[010]. Note that Rx y, and thus also the magnetization, switches direction when the direction of H is close to the hard axes [110] and , confirming the cubic magnetic anisotropy of our samples. The switching angles where Rx y changes sign are denoted as ϕH(i) on the plot.
In the presence of both external and spin–orbit fields, we expect to see a combined effect of Hso+H on the direction of magnetization. For small currents (a few microamperes) Hso≈0, and Rx y does not depend on the sign or the direction of the current. At large d.c. currents, the value of ϕH(i) becomes current dependent and we define ΔϕH(i)(I)=ϕH(i)(I)−ϕH(i)(−I). Specifically, for , the switching of magnetization occurs for I=+0.7 mA at smaller ϕH(1) than for I=−0.7 mA, ΔϕH(1)<0. For the magnetization switching, the I dependence of the switching angle is reversed, ΔϕH(3)>0. There is no measurable difference in switching angle for the and [100]→[010] transitions (ΔϕH(2,4)≈0). When the current is rotated by 90∘ , we observe ΔϕH(2)>0, ΔϕH(4)<0 and ΔϕH(1,3)≈0. Figure 2c shows that ΔϕH(2)(I) decreases as current decreases and drops below experimental resolution of 0.5∘ at I<50 μA. Similar data are obtained for sample B (see Supplementary Fig. S4).
The data can be qualitatively understood if we consider an extra current-induced effective magnetic field Heff, as shown schematically in Fig. 1b. When an external field H aligns the magnetization along one of the hard axes, a small perpendicular field can initiate magnetization switching. For I||[110], the effective field aids the [100]→[010] magnetization switching, whereas it hinders the switching. For ϕH(1)≈90∘ and ϕH(3)≈270∘, where and magnetization transitions occur, Heff||H does not affect the transition angle, ΔϕH(2,4)=0. For , the direction of the field Heff||[110] is reversed relative to the direction of the current, compared with the I||[110] case. The symmetry of the measured Heff with respect to I coincides with the unique symmetry of the strain-related spin–orbit field (Fig. 1c).
The dependence of ΔϕH(i) on various magnetic fields and current orientations is summarized in Fig. 3a,b. Assuming that the angle of magnetization switching depends only on the total field Heff+H, we can extract the magnitude Heff and angle from the measured ΔϕH(i), thus reconstructing the whole vector Heff. Following a geometrical construction shown in Fig. 3d and taking into account that ΔϕH(i) is small, we find that
and θ can be found from the comparison of switching at two angles. We find that θ≈90∘, or for and . To further test our procedure, we carried out similar experiments with small current I=10 μA but constant extra magnetic field having the role of Heff. The measured δ H(ΔϕH) coincides with the applied δ H within the precision of our measurements. (See Supplementary Fig. S5.)
In Fig. 3c, Heff is plotted as a function of the average current density 〈j〉 for both samples. There is a small difference in the Heff versus 〈j〉 dependence for and . The difference can be explained by considering the current-induced Oersted field HOe∝I in the metal contacts. The Oersted field is localized under the pads, which constitutes only 7% (2.5%) of the total area for sample A (B). The Oersted field has the symmetry of the field shown in Fig. 1d, and is added to or subtracted from the spin–orbit field, depending on the current direction. Thus, Heff=Hso+HOe for and Heff=Hso−HOe for . We estimate the fields to be as high as 0.6 mT under the contacts at I=1 mA, which corresponds to HOe≈0.04 mT (0.015 mT) averaged over the sample area for sample A (B). These estimates are reasonably consistent with the measured values of 0.07 mT (0.03 mT). Finally, we determine Hso as an average of Heff between the two current directions. The spin–orbit field depends linearly on j, as expected for strain-related spin–orbit interactions: dHso/dj=0.53×10−9 and 0.23×10−9 T cm2 A−1 for samples A and B respectively.
We now compare the experimentally measured Hso with theoretically calculated effective spin–orbit field. In (Ga,Mn)As, the only term allowed by symmetry that generates Hso linear in the electric current is the Ωɛ term, which results in the directional dependence of Hso on j precisely as observed in experiment. As for the magnitude of Hso, for three-dimensional J=3/2 holes we obtain
where E is the electric field, g* is the Luttinger Landé factor for holes, μB is the Bohr magneton and nh,l and τh,l are densities and lifetimes for the heavy (h) and light (l) holes. Detailed derivation of Hso is given in the Supplementary Information. Using this result, we estimate dHso/dj=0.6×10−9 T cm2 A−1 assuming nh=n≫nl and τh=mh/(e2ρ n), where ρ is the resistivity measured experimentally, and using Δɛ=10−3, n=2×1020 cm−3. The agreement between theory and experiment is excellent. It is important to note, however, that we used GaAs band parameters25 mh=0.4 m0, where m0 is the free electron mass, g*=1.2 and C=2.1 eV Å. Although the corresponding parameters for (Ga,Mn)As are not known, the use of GaAs parameters seems reasonable. We note, for example, that GaAs parameters adequately described tunnelling anisotropic magnetoresistance in recent experiments26.
Finally, we demonstrate that the current-induced effective spin–orbit field Hso is sufficient to reversibly manipulate the direction of magnetization. Figure 4a shows the ϕH dependence of Rx y for sample A, showing the magnetization switching. If we fix H=6 mT at ϕH=72∘, Rx y forms a hysteresis loop as current is swept between ±1 mA. Rx y is changing between ±5 Ω, indicating that M is switching between the [010] and directions. Short (100 ms) 1 mA current pulses of alternating polarity are sufficient to permanently rotate the direction of magnetization. The device thus performs as a non-volatile memory cell, with two states encoded in the magnetization direction, the direction being controlled by the unpolarized current passing through the device. The device can be potentially operated as a four-state memory cell if both the [110] and directions can be used to inject current. We find that we can reversibly switch the magnetization with currents as low as 0.5 mA (current densities 7×105 A cm−2), an order of magnitude smaller than by polarized current injection in ferromagnetic metals1,2,3, and just a few times larger than by externally polarized current injection in ferromagnetic semiconductors4.
Methods
The (Ga,Mn)As wafers were grown by molecular beam epitaxy at 265 ∘C and subsequently annealed at 280 ∘C for 1 h in nitrogen atmosphere. Sample A was fabricated from a 15-nm-thick epilayer with 6% Mn, and sample B from a 10-nm-thick epilayer with 7% Mn. Both wafers have a Curie temperature Tc≈80 K. The devices were patterned into 6- and 10-μm-diameter circular islands to decrease domain pinning. Cr/Zn/Au (5 nm/10 nm/300 nm) ohmic contacts were thermally evaporated. All measurements were carried out in a variable-temperature cryostat at T=40 K for sample A and at 25 K for sample B, well below the temperature of (Ga,Mn)As-specific cubic-to-uniaxial magnetic anisotropy transitions27, which has been measured to be 60 and 50 K for the two wafers. The temperature rise for the largest currents used in the reported experiments was measured to be <3 K.
Transverse anisotropic magnetoresistance Rx y=Vy/Ix is measured using the four-probe technique, which ensures that possible interfacial resistances, for example, those related to the antiferromagnetic ordering in the Cr wetting layer28, do not contribute to the measured Rx y. The d.c. current Ix was applied either along the [110] (contacts 4–8 in Fig. 1a) or along the (contacts 2–6) direction. Transverse voltage was measured in the Hall configuration, for example, between contacts 2–6 for . To ensure uniform magnetization of the island, magnetic field was ramped to 0.5 T after adjusting the current at the beginning of each field rotation scan. We monitor Vx between different contact sets (for example, 1–7, 4–6 and 3–5) to confirm the uniformity of magnetization within the island.
To determine the direction of magnetization M, we use the dependence of Rx y on magnetization29:
where , are the resistivities for magnetization oriented parallel and perpendicular to the current, and is an angle between magnetization and current. In a circular sample, the current distribution is non-uniform and the angle between the magnetization and the local current density varies throughout the sample. However, the resulting transverse anisotropic magnetoresistance depends only on ϕM. For the current-to-current-density conversion, we model our sample as a perfect disc with two point contacts across the diameter. The average current density in the direction of current injection is 〈j〉=2I/(π a d), where a is the disc radius and d is the (Ga,Mn)As layer thickness. In a real sample, the length of contact overlap with (Ga,Mn)As ensures that j changes by less than a factor of 3 throughout the sample. A detailed discussion of the current distribution and of measurements of joule heating can be found in the Supplementary Information.
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Acknowledgements
This work was supported in part by NSF under the grants ECS-0348289 (L.P.R.) and DMR-0603752 (J.K.F.).
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A.C. and M.O. fabricated samples and carried out measurements, X.L. and J.K.F. designed and grew (Ga,Mn)As wafers specifically for these experiments, Y.L.-G. developed the theory and L.P.R. designed and supervised the experiments.
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Chernyshov, A., Overby, M., Liu, X. et al. Evidence for reversible control of magnetization in a ferromagnetic material by means of spin–orbit magnetic field. Nature Phys 5, 656–659 (2009). https://doi.org/10.1038/nphys1362
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DOI: https://doi.org/10.1038/nphys1362
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