Nature of the PerpendiculartoParallel Spin Reorientation in a Mndoped GaAs Quantum Well: Canting or Phase Separation?
Abstract
It is well known that the magnetic anisotropy in a compressivelystrained Mndoped GaAs film changes from perpendicular to parallel with increasing hole concentration . We study this reorientation transition at in a quantum well with Mn impurities confined to a single plane. With increasing , the angle that minimizes the energy increases continuously from 0 (perpendicular anisotropy) to (parallel anisotropy) within some range of . The shape of suggests that the quantum well becomes phase separated with regions containing low hole concentrations and perpendicular moments interspersed with other regions containing high hole concentrations and parallel moments. However, due to the Coulomb energy cost associated with phase separation, the true magnetic state in the transition region is canted with .
During the past decade, the ferromagnetic transition temperature of Mndoped GaAs films has been quickly approaching room temperature jung05 . Theoretical studies of Mndoped GaAs jung06 based on the KohnLuttinger (KL) model have been quite successful at modeling and predicting much of the behavior found experimentally. In agreement with theory abol01 ; dietl01 , experiments show that at low temperatures, the magnetic anisotropy of films under compressive strain is perpendicular or out of the plane when the hole concentration is small, transforming to parallel or in the plane as increases saw04 . Although Mndoped GaAs quantum wells have also been studied theoretically for several years brey00 ; mac00 , little is known about the nature of the spin reorientation in a quantum well. We show that the spin reorientation in a quantum well happens in three stages: for small hole concentrations, the angle of the magnetization with respect to the film normal is 0 so that the magnetization lies perpendicular to the plane; for high hole concentrations, so that the magnetization lies in the plane of the quantum well. In between, the moments are either canted with or phaseseparated with regions containing low hole concentrations and perpendicular moments interspersed with regions containing high hole concentrations and parallel moments.
The magnetic anisotropy of a quantum well sensitively depends on the spinorbit coupling. In pure GaAs, the spinorbit coupling plays two roles. First, it lowers the energy of the band compared to the band at the point by about 320 meV. Since the lower band is rarely occupied by any holes, it is commonly ignored. Second, spinorbit coupling changes the energies of the holes so that heavy () and light () holes carry angular momentum and , respectively, along their momentum direction . In the absence of elastic strain, the energies of the light and heavy hole bands in bulk GaAs are degenerate at the point.
In a quantum well bounded by , the square of the component of the momentum is quantized. For the two lowest wavefunctions and of the quantum well, . Due to the difference between the light and heavy hole masses, the confinement of the holes in a quantum well breaks the degeneracy of the bands at the point with . To simplify the following discussion, we shall discuss hole rather than electron bands so that hole energies increase quadratically like for small . Including the effects of lattice strain, the energy gap between the and subbands of at the point is
(1) 
where and eV is the deformation potential dietl01 . Hence, compressive strain () plays qualitatively the same role as carrier confinement within the quantum well. As the quantum well becomes narrower, the dominant contribution to this splitting comes from the confinement of the holes and not from strain, which is typically less than 0.5%. For a quantum well with 20 layers or fewer, the contribution of strain to can be safely neglected. On the other hand, GaAs films with larger than about 50 cannot be treated as quantum wells because too many wavefunctions would be required. In such films, the main contribution to the band splitting comes from strain abol01 rather than the confinement of the holes. Regardless of their origin, the energy splittings produce a gap in the spinwave spectrum that allows a twodimensional layer of Mn spins to order ferromagnetically melk07 .
The carriers in the lower, and upper, subbands have bandmasses and , respectively, where and dietl01 ; melk07 . The coupling of the holes with the Mn spins is included by treating the Mn spins classically and by assuming that Mn impurities with concentration are restricted to the plane. We shall denote as the number of holes for each of the cations in the central plane. For small , the holes occupy a small portion of the Brillouin zone centered around . Since the holes then interact with many different Mn moments, the precise locations and structure of the Mn impurities are not important and their interactions with the holes may be treated within a meanfield approximation. The exchange coupling of the Mn spins with the holes is then given by , where are the hole spins and the sum is over all Mn sites. Comparing with the potential used in Refs.abol01 and dietl01 , we obtain the exchange coupling , where eV is estimated from photoemission measurements oka98 and is the number of layers in the quantum well. Here, is the Ga lattice constant in the plane. It follows that eV.
While our results are supported by numerical calculations that include both wavefunctions and , a qualitative understanding of the magnetic anisoptropy can be obtained from a simplified model that considers only . Due to the typically small Mn concentrations, the demagnetization fields can be neglected compared to the anisotropy introduced by the electronic band structure. If the Mn moments are tilted an angle away from the axis, then to linear order in , the two lower subbands are split by () or 0 (), as shown in Fig.1 for . By contrast, the two upper bands are split by () or (). When only the lowest subband is populated by holes, the energy difference is of order , so that the anisotropy is perpendicular. When the two lower subbands are both occupied by holes, the energy difference is of order because more holes occupy the lowest subband and perpendicular anisotropy still dominates. But when holes begin to occupy the lower of the two upper subbands, the energy difference becomes positive and of order due to the larger splitting of the upper subbands when . Assuming that , the reorientation transition occurs close to the filling where the chemical potential crosses . Very roughly, this implies that the transition from perpendicular to parallel anisotropy occurs when , independent of the Mn concentration.
The scenario of magnetic reorientation would be reversed when tensile strain overcomes the effects of carrier confinement so that . Because perpendicular or parallel anisotropy dominates for compressive or tensile strain only at very low hole concentrations, the behavior of the anisotropy at higher hole concentrations has led to the oftenheard statement shen97 ; liu03 ; thev06 that compressive or tensile strain is associated with parallel or perpendicular anisotropy.
We now examine more closely the details of the spin reorientation. With both wavefunctions and included, there are 8 hole bands rather than 4 for every point. Since , the Mn spins only couple to the holes in , with projection . The wavefunctions and are coupled by the offdiagonal terms in the KL Hamiltonian with matrix elements proportional to , which vanishes for but is given by for and . The energy of the KL plus exchange (KLE) model is obtained by first diagonalizing the Hamiltonian written as an 8 by 8 matrix in and space. The resulting eigenvalues are integrated over up to the Fermi level melk07 .
Fixing and , the energy is then minimized with respect to . For , is plotted versus in Fig.2. As shown, the angle corresponding to the minimum energy smoothly increases from 0 at to at . Of course, the chemical potential satisfies the condition . We have plotted versus in the inset to Fig.2. This “S” shaped curve is typical of phase separation, where regions of different coexist. The phase with high or low hole density is metastable so long as or , while the phase is unstable when . Performing a Maxwell construction for the data in Fig.2, we find that phase separation occurs between regions with hole concentration and and regions with and . The phaseseparated region is bracketed by the solid circles in Fig.2. As the hole concentration increases from to , the total area of the regions with parallel or perpendicular anisotropy increases or decreases, respectively. A phaseseparated phase is stable only if the KLE Hamiltonian is expanded in more than one .
However, phase separation of the quantum well into regions with low and high hole concentrations costs Coulomb energy, which was not taken into account by the KLE model. The size of the phaseseparated regions will be determined by a balance between the cost in Coulomb energy and the energy gained by phase separation. The Coulomb and phaseseparation energies are estimated by supposing that a chargedensity wave in the Mn plane oscillates between fillings and within a rectangular region in the plane of length and width . As described in the inset to Fig.3, a region of length and filling and a region of length and filling are separated by an interface of length . The total length of the rectangular region is , which is measured in units of the inplane lattice constant . For overall filling , charge conservation requires that .
To calculate the energy gained by creating a phaseseparated mixture, we subtract the energy of a uniform phase with , yielding the blue dotdashed curve in Fig.3 unphys . The dielectric constant for GaAs is used to evaluate the Coulomb energy cost, given by the red dashed curve in Fig.3. The Fermi wavelength is evaluated for this uniform filling with both Mn orientations. While the perpendicular is a bit smaller than the parallel , the calculated total energies are both positive. We conclude that phase separation is prevented by the cost in Coulomb energy for and that the Mn moments will be canted for fillings between and . This is also the case for smaller Mn dopings which are slightly more unfavorable to phase separation. In the homogeneous phase with filling , the canting angle is not affected by the Coulomb energy and can be taken directly from our results.
Since the longrange electric field contribution to the Coulomb energy may be suppressed in a double quantumwell, where the excess electronic charge on one quantum well is offset by the deficit on the other, a double quantumwell may exhibit phase separation rather than canting. By tuning the distance between the individual quantum wells, it may be possible to control the transformation from canting to phase separation. The dependence of the hole density on the magnetization angle will induce a coupling between spin and chargedensity excitations analogous to the ones observed in electronic systems giu04 . The emergence of phase separation will soften the transverse chargedensity mode in a double quantum well.
In Fig.4, we plot the phase diagram of the magnetic orientation in the quantum well, leaving open the possibility of phase separation. The canted region is just a bit narrower than the phaseseparated region shown in the figure. The lower bound to the canted region is given fairly accurately by for all . By contrast, the upper bound increases from 0.0135 at to 0.018 at . Hence, both the phaseseparated and canted regions grow as the Mn concentration increases.
Due to the large size of the gap energy , we do not expect our results to change very much with increasing temperature so long as eV. When the Mn concentration is distributed along the width of the quantum well, our results with effective Mn concentration will be unchanged provided that . We have also examined the consequences of moving the Mn plane from the center of the quantum well to . Since , the Mn moments in this plane couple equally to the holes of both wavefunctions. For , the canted region between and is shifted substantially upwards by the displacement of the Mn plane.
Although the spin reorientation transition has not been studied in a quantum well, several experiments have been performed on thin films. By measuring the remanent magnetization along different field directions, Sawicki et al. saw04 estimated that the change from perpendicular to parallel magnetic anisotropy in 400 nm films with or 0.05 occurs when cm or 0.0045 holes per cation. Using angledependent xray magnetic circular dichroism to study 50 nm films with or 0.08, Edmonds et al. edm06 observed this reorientation transition at a hole concentration of about 5 times higher or 0.0225 holes per cation.
The hole concentration in a quantum well may be controlled using a fieldeffect transistor, such as the one built by Ohno et al. ohno00 . Of course, the canting or phase separation of the Mn moments would be easier to observe if the spinreorientation transition happened at higher values of . As discussed above, the easiest way to enhance the hole filling of the spin reorientation in a narrow quantum well is to displace the Mn plane by 1/6 of the quantumwell width. The relevant ingredients of the spinreorientation transition are the singleparticle gap , the spinorbit coupling in the valence band, and the magnetic coupling of the Mn ions with holes. Accordingly, we believe that our predictions extend well beyond the simplest confinement potential discussed here to a broad class of quantumwell potentials.
This research was sponsored by the U.S. Department of Energy Division of Materials Science and Engineering under contract DEAC0500OR22725 with Oak Ridge National Laboratory, managed by UTBattelle, LLC.. This research was also supported by NSF awards DMR0453518, DMR0548011, EPS0132289 and EPS0447679. Research carried out in part at the University of North Dakota Computational Research Center, supported by NSF EPS0132289 and EPS0447679.
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(13)
The energy gained due to phase separation is
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