In a study of the ferromagnetic phase of a multilayer digital ferromagnetic
semiconductor in the mean-field and effective-mass approximations, we find the
exchange interaction to have the dominant energy scale of the problem,
effectively controlling the spatial distribution of the carrier spins in the
digital ferromagnetic heterostructures. In the ferromagnetic phase, the
majority and minority carriers tend to be in different regions of the space
(spin separation). Hence, the charge distribution of carriers also changes
noticeably from the ferromagnetic to the paramagnetic phase. An example of a
design to exploit these phenomena is given.Comment: 4 pages, 3 figures. Submitted to Phys. Rev. 

A. Ghazali

A. Van Esch

B. Lee

G. Kioseoglou

H. Ohno

I. Vurgaftman

J. Fernández-Rossier

L. Brey

L. J. Sham

L. Loureiro da Silva

M.L. Reed

R.K. Kawakami

S. Sanvito

S.A. Wolf

T. Dietl

T. Jungwirth

X. Chen

English

arXiv

In a study of the ferromagnetic phase of a multilayer digital ferromagnetic semiconductor in the mean-field and effective-mass approximations, we find the exchange interaction to have the dominant energy scale of the problem, effectively controlling the spatial distribution of the carrier spins in the digital ferromagnetic heterostructures. In the ferromagnetic phase, the majority-spin and minority-spin carriers tend to be in different regions of the space (spin separation). Hence, the charge distribution of carriers also changes noticeably from the ferromagnetic to the paramagnetic phase. An example of a design to exploit these phenomena is given here.We acknowledge the Spanish Ministry of Education for financial support and support by Grant Nos. DARPA/ONR N0014-99-1-109 and NSF DMR 0099572

Fernández-Rossier, Joaquín

Sham, L.J.

Spin separation in digital ferromagnetic heterostructures

RUa Reposity University of Alicante

PHYSICAL REVIEW B 66, 073312 ~2002!Spin separation in digital ferromagnetic heterostructuresJ. Ferna´ndez-Rossier* and L. J. ShamDepartment of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093~Received 18 January 2002; revised manuscript received 16 April 2002; published 30 August 2002!In a study of the ferromagnetic phase of a multilayer digital ferromagnetic semiconductor in the mean-fieldand effective-mass approximations, we find the exchange interaction to have the dominant energy scale of theproblem, effectively controlling the spatial distribution of the carrier spins in the digital ferromagnetic hetero-structures. In the ferromagnetic phase, the majority-spin and minority-spin carriers tend to be in differentregions of the space ~spin separation!. Hence, the charge distribution of carriers also changes noticeably fromthe ferromagnetic to the paramagnetic phase. An example of a design to exploit these phenomena is given here.DOI: 10.1103/PhysRevB.66.073312 PACS number~s!: 75.70.Cn, 75.50.Pp, 75.10.2bThe research in proactive use of the spins of the carriersto add a new dimension to electronics starts a new areaknown as spintronics.1 The recent discoveries2,3 of ferromag-netism with high Curie temperatures in a number of conven-tional semiconductors doped with magnetic impurities holdpromise for the implementation of spintronics in semicon-ductors.Inhomogeneously doped semiconductors, such as the p-njunction, play a crucial role in conventional electronic de-vices. Their properties depend on the distribution of the itin-erant carriers, which is governed by the Coulomb interactionof the carriers with the impurities and with other carriers. Inthis paper we study the charge and spin distributions of theitinerant carriers in semiconductors d doped with magneticimpurities, such as GaMnAs.4,5 Our theory is within themean-field, effective-mass, and virtual-crystal approxima-tions. In addition to the electrostatic forces, the itinerant car-riers have an exchange interaction with the magnetic impu-rities. Our calculation shows that the magnetization of thehigh Mn concentration in the d layers in the ferromagneticphase gives rise to a spin-dependent potential experienced bythe carriers comparable in order of magnitude to the chargepotential of the d layer. The effect of the spin-dependentpotential on the inhomogeneous spin distribution of the car-riers was noted by Loureiro da Silva et al.6 in multilayeredGaMnAs with 5% magnetic impurities. By contrast, the ddoping with a nominal concentration per atomic plane of25–50% Mn atoms gives rise to a qualitatively different phe-nomenon of spin separation. This creates the possibility ofthe magnetic control of the itinerant carriers by manipulatingthe magnetization of the Mn ions. In particular, the spin-dependent potential may be used to influence the spin depen-dence of the distribution of itinerant carriers. The majoritycarriers7 accumulate in the region of the Mn layers whereasminority carriers are repelled from the Mn region. In theparamagnetic phase, the spin potential averages to zero andthe magnetic influence disappears. To show the potential ofthe spin separation for device applications, we give an ex-ample of how heterostructures may be designed to engenderspin separation.This paper is organized as follows. A brief review of therelevant theoretical formalism precedes the application to es-tablish the general principle of magnetic control and to showhow it works in two specific systems. The first is the experi-0163-1829/2002/66~7!/073312~4!/$20.00 66 0733mental system of multiple digital layers of Refs. 4 and 5. Wefind the phenomenon of spin separation in the ferromagneticphase. Specifically, for appropriate carrier densities and in-terlayer distances, the minority carriers are located mainly inthe interlayer spacer, with a very small overlap with the mag-netic atoms whereas the majority carriers are mainly locatedin the layers of magnetic atoms. Second, we design a doubleGaAs quantum well with AlGaAs barriers and a d layer ofMn in the middle of one well and a d layer of an acceptor~Be! in the middle of the other. When the GaMnAs well ischanged from the paramagnetic phase to the ferromagneticphase, a net transfer of charge and spin from the Be well tothe Mn well is produced, because of the appearance of thedominant magnetic interaction. As a result, there is a poten-tial drop across the structure as well as a spin polarization inthe Be well.Our calculations are based on the mean-field model,8–12,6with two types of spins: localized d electrons with magneticmoments of 5/2 Bohr magnetons and itinerant carriers ~holesbecause of the Mn and Be doping!. There is a Heisenbergspin exchange between the itinerant carrier and the Mn delectron, which is responsible for the ferromagnetism. Thehole energy bands are based on the effective-mass approxi-mation with the kinetic energy given by the bands of the hostGaAs. The effective potential induced by the dopants is theonly change for the carrier subbands, as in the virtual crystalapproximation. The Coulomb interaction between the holesis taken into account in the Hartree approximation. The in-plane energy dispersion of the holes subbands is describedby a parabolic model that neglects the spin-orbit interaction.The omission of the spin-orbit effect, which has also beenused in Refs. 6 and 8–10, greatly simplifies the numericalcomputation. On the other hand, it leads to an overestimateof the polarization of the Mn spins induced by the interactionwith the carriers. We correct for this by using a smaller valueof the coupling constant J of the Heisenberg exchange,which reproduces the Curie temperature obtained in our pre-vious calculation including the spin-orbit interaction.12 Be-cause the charge and spin distributions are averaged proper-ties of the system, we argue that the qualitative aspects of theeffects discussed in this paper would remain if the spin-orbitinteraction is included.Both model systems studied here are taken to be transla-tionally invariant in the x-y plane normal to the growth axis.©2002 The American Physical Society12-1BRIEF REPORTS PHYSICAL REVIEW B 66, 073312 ~2002!Thus, the Mn concentration cM(z) varies only along the zaxis. We assume that, in addition to the magnetic impuritiesthat act as acceptors in GaMnAs, there is a distribution ofdonors, cc(z), which partially compensate the Mn acceptors,so that the total density of holes is smaller than the density ofMn, as is observed.2 For simplicity, we assume that the spa-tial distributions of the donor and acceptor impurities are thesame except for a multiplicative constant. From the chargeneutrality condition, P1Cc5CM , where P, Cc , and CM arethe average densities of holes, of the compensating impuri-ties, and of the Mn impurities, respectively.In the mean-field approximation, the effect of the mag-netic impurities on the itinerant carriers is given by a spin-dependent potential:Vs5Js2 cM~z !^M ~z !& , ~1!where s561 denotes the spin directions, J is the exchange-coupling constant between the itinerant carrier and the Mnspin, and ^M (z)& stands for the local average magnetizationof Mn. The Mn polarization is produced by the molecularfield created by the spin polarization of the itinerant carriersas well as the external magnetic field. At temperature T andzero external magnetic field, the Mn magnetization is givenby the usual Brillouin function:M ~z !5SBSS JkBT p1~z !2p2~z !2 D , ~2!where ps(z) is the spin-dependent density of the itinerantcarriers.The effective Schro¨dinger equation for the holes has aself-consistent potential Ve(z)1Vs(z), where Ve(z) is theelectrostatic potential experienced by the hole and is givenby,d2dz2Ve~z !54pe2e@cc~z !2cM~z !1p1~z !1p2~z !# . ~3!The hole eigenenergy is the sum of the quantized energyEn ,s from the motion in the z direction and the plane-waveenergy ekW i5\2k i2/2m i from the in-plane motion. The holewave function is the product of the bound-state envelopefunction in z times the plane wave normal to the z axis. Thespin-dependent hole density is given by:ps~z !5(nE d2kW i~2p!2ufn ,s~z !u2e (En ,s1ekW i2m)/kBT11, ~4!where m is the chemical potential.It is illuminating to estimate the strength of the spin-dependent potential ~1! for a single magnetic digital layer.Throughout the paper we take the hole effective-mass tensorto be m i50.11 and mz50.37 times the free-electron massand model the distribution of the Mn in a digital layer asGaussian:07331cM~z !5CMApDexp@2~z/D!2# . ~5!In a digital layer of Ga0.5Mn0.5As the total concentration ofMn is CM53.1331014 cm2253.13 nm22. Transmissionelectron microscopy experiments4 reveal that the Mn isspread over two or three atomic planes. This correspondsroughly to the Gaussian half-width of D50.5 nm. The ex-change integral of J5150 meV nm3 used in our previouswork12 to calculate the Curie temperature is reduced to J5100 meV nm3 here in order to compensate the absence ofspin orbit in the present calculation. The maximum of thespin-dependent potential in Eq. ~1! for saturated magnetiza-tion is 441.5 meV in comparison with the charge well depthof about 177 meV. The peak to valley splitting is twice themaximum value. In a diluted magnetic semiconductor quan-tum well, the magnetic potential is one order of magnitudesmaller than in a digital layer. Hence, the magnetic control ismore practical in digital layers than in quantum wells.The magnetic potential is attractive for the majority car-riers and repulsive for the minority carriers. As the tempera-ture is increased, the magnetization decreases and so does themagnetic potential, vanishing above the Curie temperature.Accordingly, the itinerant carrier density profiles change sig-nificantly as the temperature varies. Both the potentials andthe density profiles are shown in Fig. 1 for a system of N54 digital layers, with an interlayer separation D540 ML~GaAs monolayers!, or 11.3 nm, and a density of holes perlayer p51.331013 cm22. In order to improve numericalconvergence, we locate 100 meV step potential barriers atz5650 nm. Similar results are obtained for systems be-tween 1 and 10 layers. In the left panels of Fig. 1 we presentthe potential profiles at T540 K ~thick dashed line!, whenthe system is paramagnetic. In the same panels we show theFIG. 1. Left upper panel: spin-dependent potentials for majoritycarriers ~solid line! and minority ~thin dot-dashed line! at T55 K~the ferromagnetic phase! and total potential at T540 K ~the para-magnetic phase! ~thick dashed line!. Left bottom panel: magnifica-tion of the upper panel. Right panel: total carrier density profiles forT55 K ~solid line, the ferromagnetic case! for T540 K ~long-dashed line, the paramagnetic case!. Notice the logarithmic scale inthe vertical axis.2-2BRIEF REPORTS PHYSICAL REVIEW B 66, 073312 ~2002!total potential for majority ~attractive! and minority ~repul-sive! carriers for spin-dependent potentials at T55 K whenthe system is ferromagnetic. The lower panel is a magnifica-tion of the center of the upper panel. There it can be betterappreciated the potential in the paramagnetic phase, whoseonly contribution is electrostatic. The large magnitude of thespin-dependent part of the potential is apparent in the upperleft panel. The effect on the density profile of the itinerantcarriers is shown in the right panel. In the ferromagnetic case~thin solid line! the carriers tend to pile up in the magneticlayers. In the paramagnetic case ~long-dashed line! the car-rier distribution is more spread out, with more carriers in thespacer layers relative to the ferromagnetic case.Figure 2 shows the resultant spin-dependent density pro-files for two temperatures. In the ferromagnetic case, the dis-tributions of majority and minority carriers are distinct. Themajority carriers are localized in the magnetic layers whereasthe minority carriers are almost totally expelled from it ~no-tice the logarithmic scale in the vertical axis!. This ‘‘spinseparation’’ phenomenon is, of course, absent in the para-magnetic case. We have checked that as the interlayer dis-tance decreases or the hole density increases the spin sepa-ration diminishes. Spin fluctuations of Mn beyond the mean-field approximation are a major source of spin-flip scattering.Hence, we expect the spin-flip scattering to be reduced whenthe spin separation is larger.Results of Fig. 1 show how the exchange interaction canovercome the electrostatic interactions in the case of a digitallayer. We propose here a heterostructure in which this effectis made more apparent and can be measured. It is a doublequantum well of GaAs with GaAsAl barriers, d doped withone layer of Mn in the middle of the left well and one layerof the acceptor Be in the middle of the right well. For illus-tration but not necessity, we choose the density profile of theBe to be identical to that of the Mn ions plus its compensat-ing charges. We take the density of holes per digital layer ineither well of 1.331013 cm22. The digital layer of Mn has atotal density of 3.1331013 cm22 and a Gaussian distribu-tion as in Eq. ~5! with D50.5 nm.Above the Curie temperature the system is totally sym-metric ~by the construction of the model! and so is the chargeFIG. 2. Spin-dependent hole density profiles for T55 K major-ity carriers ~solid!, minority carriers ~long dashed!, and for T540 K ~long dashed!.07331distribution ~shown in the middle column of Fig. 3!. At lowtemperatures the majority carriers with spin antiparallel tothe Mn are attracted by the magnetic layer. As a result, thereis a spin-dependent charge transfer from the nonmagnetic tothe magnetic well ~the left column of Fig. 3!, which gener-ates an internal electric field and a potential drop. In therightmost panel we plot both the potential drop and the Mnaverage magnetization as a function of the temperature. Thepotential drops follow the average magnetization and theirmaximum is close to 20 meV at zero temperature. Largervalues can be obtained for smaller carrier densities, but thisdecreases the Curie temperature.12 The charge imbalance is1.231012 cm22 or 4.6% of the total hole density. Thecharge-transfer effect also takes place for other values of thecarrier density, as well as for structures with a different den-sity of Be and Mn. As a result of the rearrangment of thecarriers in the heterostructure, as the system goes from theparamagnetic to the ferromagnetic phase, large changes inthe in-plane and vertical transport properties can be expectedas well.Tight-binding calculations for a superlattice of digitalGaAs/GaAsMn ~Ref. 13! give very similar results to thoseobtained using an envelope function approach,12 validatingthe use of the effective-mass approximation in d doped sys-tems. A density functional-calculation for a superlattice ofdigital GaAs/MnAs predicts a half-metallic behavior.14 Ourresults should be compared with a similar calculation includ-ing compensating impurities, 50% of Mn and larger inter-layer distances.In conclusion, we have studied the spatial distribution ofthe carrier spins in digital ferromagnetic heterostructures.Our main results are: ~i! The distribution of carriers in digitalferromagnetic heterostructures of GaAsMn is largely con-trolled by the exchange interaction that overcomes the elec-trostatic interaction. ~ii! Large changes in the spatial distri-FIG. 3. Left upper/lower panels: spin-dependent potentials/density profiles for majority holes ~solid line! and minority holes~thin dot-dashed line! at T55 K ~the ferromagnetic phase!. Themiddle panels: spin-dependent potentials and density distributionsat T540 K ~the paramagnetic phase!—same notation as above.Right panel: potential drop ~solid line! and magnetization ~dashedline! versus temperature.2-3BRIEF REPORTS PHYSICAL REVIEW B 66, 073312 ~2002!bution of carriers take place when ferromagnetism isswitched off ~for instance, by increasing the temperatureabove the Curie temperature!. ~iii! In some situations theminority carriers are totally separated from the Mn layersand the majority carriers. This might have consequences intransport properties and the anomalous Hall effect.07331We wish to thank D. D. Awschalom, R. Kawakami,A. Gossard, E. Gwinn, L. Brey, and B. H. Lee for stimulatingdiscussions. We acknowledge the Spanish Ministry ofEducation for financial support and support by GrantNos. DARPA/ONR N0014-99-1-109 and NSF DMR0099572.*Present Address: Department of Physics, University of Texas atAustin, Austin, TX 78712.1 S.A. Wolf and D. Treger, IEEE Trans. Magn. 36, 2748 ~2000!.2 H. Ohno et al., Appl. Phys. Lett. 69, 363 ~1996!; A. Van Eschet al., Phys. Rev. B 56, 13103 ~1997!.3 S. Sonoda et al., cond-mat/0108159; M.L. Reed et al., Appl.Phys. Lett. 79, 3473 ~2001!.4 R.K. Kawakami et al., Appl. Phys. Lett. 77, 2379 ~2000!.5 X. Chen et al., Bull. Am. Phys. Soc. 46, 508 ~2001!;cond-mat/0203361 ~unpublished!; G. Kioseoglou et al., Appl.Phys. Lett. 80, 1150 ~2002!.6 L. Loureiro da Silva et al., Appl. Phys. Lett. 79, 3305 ~2001!.7 Throughout the paper the words majority and minority carriersrefer to the spin degree of freedom.The word carrier refers al-ways to the holes, which are the relevant particles in the system2of interest ~III-V-Mn semiconductors!.8 T. Jungwirth, W.A. Atkinson, B.H. Lee, and A.H. MacDonald,Phys. Rev. B 59, 9818 ~1999!.9 B. Lee, T. Jungwirth, and A.H. MacDonald, Phys. Rev. B 61,15606 ~2000!; L. Brey and F. Guinea, Phys. Rev. Lett. 85, 2384~2000!.10 A. Ghazali, I.C. da Cunha Lima, and M.A. Boselli, Phys. Rev. B63, 153305 ~2001!.11 T. Dietl et al., Science 287, 1019 ~2000!; T. Dietl, H. Ohno, andF. Matsukura, Phys. Rev. B 63, 195205 ~2001!.12 J. Ferna´ndez-Rossier and L.J. Sham, Phys. Rev. B 64, 235323~2001!.13 I. Vurgaftman and J.R. Meyer, Phys. Rev. B 64, 245207~2001!.14 S. Sanvito and N.A. Hill, Phys. Rev. Lett. 87, 267202 ~2001!.-4

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Spin separation in digital ferromagnetic heterostructures

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