Spin dynamics and quantum transport in magnetic semiconductor quantum structures

Spin dynamics and quantum transport in magnetic semiconductor quantum structures

Journal of Magnetism and Magnetic Materials 200 (1999) 130}147 Spin dynamics and quantum transport in magnetic semiconductor quantum structures D.D. ...

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Journal of Magnetism and Magnetic Materials 200 (1999) 130}147

Spin dynamics and quantum transport in magnetic semiconductor quantum structures D.D. Awschalom *, N. Samarth Department of Physics, University of California, Santa Barbara, CA 93106, USA Department of Physics, Pennsylvania State University, University Park, PA 16802, USA Received 23 January 1999; received in revised form 22 April 1999

Abstract Quantum structures derived from magnetic semiconductors serve as a powerful arena within which to study the interplay between quantum electronics and thin "lm magnetism. In particular, the semiconductor aspects of these #exible systems allow direct access to the electronic spin degrees of freedom using both magneto-optical as well as magnetotransport probes. Here we provide an overview of recent developments in the experimental study of II}VI magnetic semiconductor quantum structures, with particular emphasis on the dynamical behavior of "eld-tunable electronic spin states and spin-dependent quantum transport.  1999 Elsevier Science B.V. All rights reserved. Keywords: Spin dynamics; Quantum transport; Semiconductors

1. Introduction Semiconductor quantum structures have played an important role in the development of contemporary devices used in the read-out, transfer and processing of information. The most prominent examples of such applications are quantum well diode lasers in CDROMs, high electron mobility transistors (HEMTs) in wireless communications and resonant tunneling diodes in direct broadcast receivers/transmitters [1]. At the same time, the technology responsible for the engineering of these materials has been fruitfully exploited in studies of fundamentally new physics such as the integer/fractional quantum Hall e!ects and mesoscopic physics

* Corresponding author. E-mail address: [email protected] (D.D. Awschalom)

[2]. The pursuit of tailored magnetic (metallic) heterostructures has independently taken a parallel path along which basic discoveries such as giant magneto-resistance [3] are directly linked with applications in memory storage, resulting in the recent invention of ultrahigh density hard disks [4]. Unlike semiconductor quantum devices that depend on the manipulation of electronic charge, these metal-based quantum devices explicitly depend on the exploitation of electronic spin. It is natural to envisage a merging of these disparate areas, with a view to developing a semiconductorbased framework for &magneto-electronics' [5]. For instance, there have already been studies that couple quantized energy levels in semiconductor nanostructures either with local magnetic "elds created by integrated submicron ferromagnetic structures [6,7] or with the exchange "elds of magnetic ions incorporated into the semiconductor lattice

0304-8853/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 4 2 4 - 2

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itself [8,9]. In addition, spin-based semiconductor device schemes have been proposed [10] and their practical realization is being examined in ongoing experiments [11]. Further examples of potential quantum spin devices include single electron spin transistors that rely on spin-dependent tunneling into a magnetic quantum dot and magnetic "eld e!ect transistors that employ carrier injection into spin-polarized transport channels. For instance, Fig. 1 illustrates a scheme in which spin transport exploits the enhanced spin splitting within a mesoscopic magnetic semiconductor structure. In a magnetic two-dimensional electron gas, the transport takes place through spin-polarized states that are strongly exchange coupled with local spins as a result of quantum con"nement. One may then seek to understand the length scales over which spin coherence is maintained in such systems and question the relationship between this distance and other relevant scales such as the phase coherence length and elastic mean free path. It is important within this context to develop model systems and techniques that allow one to directly address the physics underlying spin transport and scattering in magnetically active quantum semiconductor structures. Quantum structures derived from the II}VI magnetic semiconductors (MS) in particular provide a valuable framework for probing the dynamical interplay between carrier transport, electronic spin scattering, quantum con"nement and magnetic dimensionality. Such information emerges from spatio-temporally resolved magneto-optical spectroscopy [12,13], as well as quantum and mesoscopic magneto-transport [14]. Another exciting aspect of spin dynamical studies of quantum structures is the possibility of establishing, storing and manipulating the coherence of

Fig. 1. Schematic of a magnetically gated &spinFET' in which the transport in a spin-polarized channel is controlled using fringe "elds from an epitaxial ferromagnet.

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electronic and magnetic spins in solid-state systems [15,16] for testing concepts in quantum computation [17,18]. We note in particular that recent optical experiments probing spin relaxation and coherent spin transport in doped (non-magnetic) semiconductors reveal surprisingly long relaxation times [19}21], indicating that spin transport may be more &robust' than charge transport. In this paper, we provide an overview of the physics of II}VI MS quantum structures, beginning with an introduction to their basic properties and then proceeding to discuss aspects of spin dynamics and transport in di!erent &spin-engineered' systems. The quantum structures of interest here are derived from II}VI MS alloys that incorporate transition metals (principally, Mn>) on the group II site of a zinc-blende lattice (Fig. 2). The exchange coupling between the con"ned electronic states and the local moments results in striking magnetic-"eld-induced e!ects at low temperatures such as an exciton spin splitting that can be as large as &100 meV in magnetic "elds of a few Tesla, a giant &quantum con"ned' Faraday e!ect characterized by a Verdet constant of &10 degrees/cm T [22], spin-polarized

Fig. 2. Summary of the energy gap (4.2 K) and lattice parameter of some II}VI MS alloys. The lines representing ternary alloys are merely guides for the eye; band-gap bowing e!ects are often present and in some cases (e.g. (Zn,Mn)Se) result in large departures from the linear interpolations shown. Note that all these materials are stabilized in the zinc-blende form when grown by MBE on GaAs and have a direct band gap.

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quantum transport [23,24] and a magneto-resistance that can be as large as &700% in a "eld as small as 1 kG [25]. We note that these spectacular magneto-electronic e!ects are restricted to cryogenic temperatures; however, the detailed level of understanding of the II}VI MS alloys [26,27] and the substantial progress in the control over the epitaxial growth of quantum structures based on these materials provides a viable basis for proofof-concept magneto-electronic devices. The physical insights gained from low-temperature studies of these materials will be useful in developing more practical devices from newer materials such as the III}V MS heterostructures where large magnetoelectronic e!ects are possible above liquid-nitrogen temperatures [9].

2. Exchange interactions in II}VI MS quantum structures The II}VI MS quantum structures are primarily fabricated using the molecular beam epitaxy (MBE) of semiconductor alloys such as (Zn,Mn)Se, wherein a transition-metal ion (Mn>) randomly substitutes the group II cation in a II}VI semiconductor lattice. Since the addition of Mn into a II}VI semiconductor typically increases the band gap and also changes the lattice parameter, the simplest MS heterostructures consist of a strained, binary, nonmagnetic quantum well (QW) #anked by MS barriers (Fig. 3a). The most common * and earliest studied * examples of such QWs are ZnSe/ (Zn,Mn)Se and CdTe/(Cd,Mn)Te. Typically, such QWs are fabricated on (1 0 0)GaAs substrates, after the deposition of an appropriate bu!er layer that accommodates the lattice mismatch with the substrate. Although early work in this "eld was limited to these two material systems [28], subsequent progress in MBE growth has expanded this family of quantum structures to include a variety of band alignments, strain con"gurations and MS alloys. Some examples of these are illustrated in Fig. 3 and will be described in more detail in the next section. As with any semiconductor quantum structure, &band-gap engineering' techniques allow one to create materials with arti"cially tailored electronic potentials by manipulating physical parameters such

Fig. 3. Examples of di!erent &spin-engineered' MS heterostructures: (a) a magnetic barrier QW structure in which the magnetic ions are located in the barrier; (b) a magnetically coupled double quantum well; (c) a &digital magnetic heterostructure' in which the magnetic ions are nominally incorporated into the QW region in discrete, quasi-2D layers; (d) a magnetic 2DEG in which modulation doping is employed to create a 2D Fermi sea that is in contact with magnetic ions; (e) a type-II magnetic QW structure in which only conduction band states are con"ned to the magnetic region.

as layer composition, strain and layer thickness. For instance, ZnSe/(Zn,Mn)Se QWs can be embedded in varying strain con"gurations that enable the adjustment of the relative energies of the light- and heavy-hole bands, whilst the choice of di!erent materials in a heterostructure determines whether electrons and holes are con"ned to the same spatial regions (as in &type-I' ZnSe/(Zn,Cd,Mn)Se QWs) or are spatially isolated from each other (as in &type-II' ZnTe/(Cd,Mn)Se QWs). In addition to band-gap engineering, however, the presence of local moments allows the &spin engineering' of new phenomena through the control of two classes of exchange interactions: E The d}d super exchange between the d-electrons of the magnetic ions; this is a short-ranged

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antiferromagnetic interaction and is characterized by a nearest-neighbor exchange integral of &10 K. E The sp}d exchange between the d-electrons and the band electrons/holes; this interaction is ferromagnetic (potential exchange) for conduction band states and is predominantly antiferromagnetic (kinetic exchange) for valence band states. The latter interaction enhances the spin splitting of the band states in an external magnetic "eld. In an MS quantum structure, this results in a spin-dependent con"ning potential that is magnetically tuned, and hence leads to striking magneto-optical and magneto-transport responses. The enhanced electronic spin splitting of band states is essentially proportional to the sample magnetization and also to the overlap between con"ned electronic states and magnetic moments. For instance, the spin splitting of conduction band states in the presence of an external magnetic "eld B applied along the z-axis is given by *E"gk B#f (t)(N a)1S 2 (1)  X where g is the intrinsic electronic g-factor, a is the s}d exchange integral, 1S 2 is the statistically averX aged projection of the Mn> moment along z, N is  the number density of Mn> ions and f (t) is the wave-function overlap between the con"ned state and local moments. To account for antiferromagnetic spin}spin correlations between Mn> ions, 1S 2 is described by an empirical modi"cation of X the standard Brillouin function B (x) for S", by   using the parameters S (which is the saturation  value for the spin of an individual Mn> ion and is smaller than ) and the modi"ed temperature ¹ "   ¹#¹ :  1S 2"N S B (5k B/k ¹ ). (2) X      We note that for a given distribution of magnetic spins on an MS lattice, there is a well-de"ned statistical population of magnetic ions, with isolated spins, pairs of spins, triplets, etc. Due to the antiferromagnetic coupling between nearest-neighbor Mn> spins, the magnetization is dominated by the paramagnetic response of isolated single spins.

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The e!ect of a magnetic "eld on the con"ned electronic states in an MS heterostructure is readily probed using steady-state magneto-optical spectroscopic techniques such as magneto-photoluminescence (magneto-PL), magneto-absorption and the Faraday/Kerr e!ect. Such spectra are typically dominated by electric dipole allowed excitonic transitions that follow well-known selection rules. For instance, in the Faraday geometry (magnetic "eld parallel to the direction of light propagation), the opposite spin states of the e1}hh1 exciton (S "$1) are populated by opposite circuX lar polarizations of light (p> and p\). The large spin splitting of excitonic states is hence easily observed in either polarization-resolved PL or absorption spectra, yielding a simple and direct manifestation of the physics represented by Eq. (1). Since the intrinsic g-factor for electrons in wide band gap II}VI semiconductors is small (e.g. g&1.1 in (Zn,Cd)Se [13,16]), the spin splitting is dominated by the second term in Eq. (1). For convenience, this may be viewed in terms of a "eld- and temperature-dependent &e!ective' g-factor that can reach values as large as &100. The spin splitting of other band states may also be analyzed in a similar way; we note that the exchange integral (and hence the spin splitting) for the heavy-hole states is typically &5 times larger than that for conduction band and light-hole states. Hence, in most optical experiments that probe heavy-hole excitons, the spin splitting is dominated by that of the valence band states.

3. Spin engineering of II}VI MS quantum structures The &spin engineering' of magneto-optical and magneto-transport phenomena in MS quantum structures is essentially achieved by tailoring the wave-function overlap f (t) in Eq. (1) and also by controlling the magnetization via the composition and dimensionality of the MS layers. This allows one to de"ne di!erent classes of spin-engineered quantum structures, each of which is designed to address qualitatively distinct physics. We now provide a brief overview of some of the important categories of MS quantum structures.

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3.1. Magnetic barrier QWs This con"guration is generically described by the band alignment shown in Fig. 3a. Here, the quantum-con"ned electronic states interact with the magnetic ions through the exponential tails of the wave function. In the presence of an external magnetic "eld, the con"ning barrier for spin-up electrons is raised and that for spin-down states is lowered, so that the con"ned states show a spindependent Zeeman shift. This Zeeman tunability of the quantum con"nement is a potentially powerful tool for the characterization of MS quantum structures and was initially exploited for the determination of band o!sets [28,29]. In these experiments, the magnetic "eld essentially allows the quantum con"nement to be continuously varied within a single sample, so that the energy of con"ned states can be mapped out as a function of the con"nement potential. Detailed band structure calculations may then be employed to calculate the band o!sets that best explain the experimentally observed Zeeman shifts. The reliability of the band o!set values determined from such experiments is of course limited by the various approximations involved in the relevant band structure calculations (e.g. single band versus multiband k.p, self-consistent inclusion of electron}hole interactions, etc.). A more important limitation of the data analysis in early work stems from the assumption of an ideal interface and a bulk-like magnetization for the MS barrier regions, neither of which are completely realistic scenarios. More recent analyses of (Cd,Mn)Te/CdTe QWs have tried to account for the fact that &surface' Mn spins at an ideal interface experience an enhanced paramagnetism in comparison to the bulk due to the reduction in the number of nearest neighbors; however, such surface e!ects are found to greatly underestimate the observed paramagnetic response of the Mn spins [30]. This has led to detailed studies that have attempted to model the interface in CdTe/(Cd,Mn)Te QWs, inferring that the smearing of the interface can range from 1 to 5 monolayers (ML), depending on the details of the growth [31,32]. The experimental indications of interface broadening here are quite unequivocal: for instance, the Zeeman splitting of a QW constructed

using a (Cd,Mg)Te/CdTe/(Cd,Mn)Te heterostructure shows a larger spin splitting when the MS barrier precedes the non-magnetic (Cd,Mg)Te barrier in the growth sequence. It should be cautioned, however, that the detailed interface pro"les deduced from such analysis are predicated on the ability to make a credible physical model of the situation, ranging from accurate calculations of the band structure to the microscopic formulation of the magnetization of an inhomogeneously diluted MS lattice. 3.2. Magnetically coupled double quantum wells The magnetically coupled double quantum well (MCDQW) structure provides an interesting variation on a classic con"guration that has been extensively studied in more mainstream semiconductor quantum structures [33]. As shown in Fig. 3b, this type of structure relies on the interaction of quantum-con"ned states with magnetic ions that reside in a thin barrier between two QWs, so that the coupling between the QWs can be Zeeman tuned using a spin-dependent barrier. A powerful aspect of such a structure is that one may independently vary the electronic degrees of freedom (quantum con"nement) and the magnetic degrees of freedom (composition and dimensionality of the MS barrier). A variety of type-I ZnSe/Zn Cd Se double quantum wells coupled \V V by a thin Zn Mn Se barrier have been fabricated \V V and probed using steady-state magneto-PL and magneto-absorption [32,34]. These experiments allow a detailed understanding of the magnetically induced changes in the band structure as the spindependent coupling between the QWs is tuned by a magnetic "eld. It is interesting to also note that the magnetization of the thin coupling layer can also be probed using magneto-optics, providing a sensitive energy-resolved magnetometry of paramagnetic monolayers whose magnetism is inaccessible using conventional SQUID techniques. 3.3. Spin superlattices In the examples provided earlier, the magnetically induced changes in con"nement are not large enough to completely overcome the primary

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con"ning potential created by the self-consistent interplay between the intrinsic band o!sets and the electron-hole Coulomb force. The "rst realization of dramatic magnetic-"eld-induced changes in quantum con"nement were obtained using ZnSe/ (Zn,Fe)Se QWs where both the strain and material composition were chosen so that a very small valence band o!set could be overcome by the spin splitting of the heavy-hole states, hence producing magnetically induced transformations between type-I and type-II band alignments [35]. This concept was then extended to the idea of a &spin superlattice' in which the components of the heterostructure are chosen so that both the conduction band and valence band o!sets are small. A magnetic "eld hence induces a complete spatial separation of exciton spin states with spin-up excitons in the barriers and spin-down excitons in the wells (illustrated in Fig. 4). This unusual spin-dependent con"nement has been demonstrated in ZnSe/(Zn,Mn)Se [36], ZnSe/(Zn,Fe)Se [37] and (Cd,Mg)Te/(Cd,Mn)Te [38] superlattices. The

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application of a small magnetic "eld initially induces a type-I spin-dependent con"ning potential where both spin-up and spin-down excitons are con"ned within the magnetic layers. With increasing "eld, however, the large spin splitting of the heavy-hole states in the magnetic layers creates a type-I band alignment for spin-down excitons and a type-II alignment for spin-up excitons. Eventually, both the conduction band and heavyhole splitting overcome the respective band o!sets to produce a complete spatial separation of the opposite exciton spin states * or a &spin superlattice', with spin-up excitons con"ned to the nonmagnetic regions and spin-down excitons con"ned to the magnetic regions. The changes in band alignment from type-I to type-II and back to type-I are directly witnessed in steady-state magneto-optical spectroscopy, manifesting themselves as systematic changes in the absorption coe$cients and the Zeeman shifts for the respective polarization-resolved transitions. 3.4. Digital magnetic heterostructures

Fig. 4. Illustration of the formation of a spin superlattice in ZnSe/(Zn,Mn)Se. At B"0, for Mn concentrations below &10%, band-gap bowing e!ects lead to band o!sets that create a weak type-I band alignment with both electron and hole spin states localized in the (Zn,Mn)Se layers. On application of a magnetic "eld, the band alignment "rst undergoes a transformation to type-II, followed by the formation of a spin superlattice.

A qualitatively di!erent approach to the spin engineering of MS quantum structures was recently introduced, wherein the magnetic species are nominally restricted to discrete spatial regions [22]. In one example of such &digital magnetic heterostructures' (DMHs), fractional monolayers of MnSe are incorporated into the con"ning region of a type-I ZnSe/(Zn,Cd)Se QW in a periodic manner, hence allowing the fabrication of type-I QW in which the con"ned electronic states interact strongly with the magnetic ions (Fig. 3c). The use of a shortperiod superlattice in the QW region has distinct advantages over a random quarternary alloy. First, the inhomogeneous broadening of the con"ned electronic levels is minimized: for instance, exciton absorption spectra in these samples have a fullwidth at half-maximum of &6 meV, despite the complex composition of the QW region. Further, the quasi-2D arrangement leads to an enhanced paramagnetic response of even large local concentrations of Mn>, with the lower dimensionality leading to a suppression of spin glass [39] and antiferromagnetic order [40]. For instance, neutron di!raction studies have shown that the NeH el

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temperature of MnSe layers decreases with the layer thickness down to three monolayers, below which no evidence of an antiferromagnetic phase transition has yet been observed [40]. Finally, from a pragmatic viewpoint, the overall composition of a digital QW region is far more reproducible and easily characterized in comparison to that of a random quartenary alloy. Given the "ndings of interfacial mixing and growth front segregation in CdTe/(Cd,Mn)Te, it is relevant to question the extent to which the nominally 2D regions of magnetic ions are spatially localized. In the absence of detailed magneto-optical studies of interfaces in (Zn,Mn)Se-based systems (similar to those probing CdTe/(Cd,Mn)Te interfaces), this is a di$cult question to answer in a quanti"able manner. However, there are several circumstantial pieces of evidence (X-ray di!raction, RHEED oscillations, studies of n-doping) that suggest that * although the 2D picture of a DMH system is obviously idealized * the intermixing pro"le in the (Zn,Mn)Se-related systems probably has a relatively narrow FWHM (&1 ML). A systematic study of the steady-state magnetooptical response of DMH structures has been carried out with the aim of elucidating the e!ects of rearranging a "xed number of magnetic ions within the span of a con"ned excitonic wave function. The samples used in these experiments consist of 120 As Zn Cd Se/ZnSe single QWs containing a     "xed total amount of MnSe (three monolayers) incorporated into the well in a series of equispaced planes, with the number of planes and in-plane Mn density varying from sample to sample as follows: a single three-monolayer barrier of MnSe (1; 3 ML), three equispaced single monolayers of MnSe (3;1 ML), six one-half monolayers (6; ML),  12 one-quarter monolayers (12; ML), 24 one eighth monolayers (24; ML), and a non-mag netic control (NM). The local quasi-2D Mn> concentrations are hence 100, 50, 25 and 12.5%, respectively, with the average Mn composition in each sample maintained at &8%. Fig. 5 illustrates a simple model for the envelope functions of con"ned conduction electron states calculated from a single-electron solution to the SchroK dinger equation for three geometries using known band structure parameters. This demonstrates how this

Fig. 5. Envelope wave functions for con"ned states in 3 DMH structures calculated using a simple one-band approach. For heuristic purposes, the randomly diluted regions of (Zn,Cd,Mn)Se are approximated as MnSe &barriers' of fractional monolayer width.

scheme allows control of the in-plane magnetic spin density, the shape of the band electron wave functions and the spatial overlap of these wave functions with the localized Mn spins. We emphasize the heuristic intent of this model: a realistic calculation of the con"ned wave functions in these digital QWs is beyond the scope of our present knowledge of microscopic details, such as the relevant pseudo-potentials, the exact nature of the magnetic regions, etc. Low-temperature static PL and absorption studies reveal sharp heavy-hole exciton peaks with small Stokes shifts (&3 meV) whose FWHM (&6 meV) are attributed to inhomogeneous broadening caused by nonmagnetic alloy and well width #uctuations. As discussed earlier, an applied magnetic "eld splits these excitonic features into lower (spin-down, S "#1) and higher energy X (spin-up, S "!1) states due to the sp}d exchange X interaction (Fig. 6a). Note that the spin splittings are signi"cantly larger than the inhomogeneous absorption linewidths even in modest magnetic "elds (1 T), providing a model two-level spin system. The variation of the spin splitting with the composition of the DMH QW is shown in Fig. 6b, indicating that successive dilution of the magnetic layers markedly increases the g-factor. This order of magnitude change cannot be explained solely by the larger electronic overlap with the Mn spins and is due to an increase in the number of uncompensated paramagnetic spins as the quasi-2D Mn concentration f is decreased, leading to a superlinear dependence on f. The earlier assertions that intermixing and segregation in this material system are not as severe as in the CdTe/(Cd,Mn)Te system

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3.5. Magnetic two-dimensional electron gases

Fig. 6. (a) Optical density (o.d) of the p> (spin down) and p\ (spin up) components of the e1}hh1 excitonic absorption in a DMH sample containing (24;) monolayer regions of MnSe  embedded in a ZnSe/(Zn,Cd)Se QW. The data is measured in a "eld of 1 T at 4.2 K. (b) Zeeman shifts of the p> and p\ components of the e1}hh1 exciton absorption peaks in di!erent DMH samples.

appear to be consistent with the clear increase in spin splitting between the 12; and 24;   samples, where the spacing between the magnetic regions is 3 and 1.5 monolayers, respectively. A recent calculation of the spin splitting in these DMH samples (assuming a statistical distribution of Mn spins in idealized 2D layers) is able to reproduce the Zeeman splitting quite well, again adding to the suspicion that the Mn ions are well localized during the epitaxial growth process [41,42].

While earlier studies of MS quantum structures mainly focused on optical studies of undoped samples, recent progress in modulation doping has enabled the study of the in#uence of the magnetic moments on quantum transport in both macroscopic [23,24] and mesoscopic [43}45] samples. The two-dimensional electron gas (2DEG) formed in modulation-doped semiconductor heterostructures is a well-established model 2D system for fundamental studies of electron}electron interactions both in the presence and absence of disorder. By extending this concept to include spin}spin interactions between a 2DEG and local moments [46], magnetic 2DEGs have been realized in which spin-dependent e!ects are maximized while simultaneously maintaining the sample mobility in a regime that allows clean studies of quantum transport [23,24]. The generic structure employed is described by in Fig. 3d; in such structures, carriers are transferred from an n-doped ZnSe layer to a DMH quantum well created by periodically inserting fractional monolayers of MnSe into (Zn,Cd)Se. Since the introduction of the magnetic atoms into such a lattice degrades this mobility, it is important to design optimal architectures that simultaneously maximize spin e!ects and the 2DEG mobility. The best mobility obtained so far for the non-magnetic &host' material (ZnSe/(Zn,Cd)Se) is &40 000 cm/V s, while for magnetic 2DEGS it drops to &20 000 cm/V s [47]. More recently, high-mobility magnetic 2DEGs have also been realized in modulation-doped (Cd,Mg)Te/ (Cd,Mn)Te QWs [48]; here, the mobility of 2DEGs created in the non-magnetic host lattice (Cd,Mg)Te/ CdTe is substantially higher than in the ZnSe/ (Zn,Cd)Se system, with a record mobility of &120 000 cm/V s recently reported [49]. As in the case of the (Zn,Cd)Se-based systems, the introduction of Mn rapidly degrades this mobility, but magnetic 2DEGs have been obtained with mobility in the range 20 000}60 000 cm/V s, depending on the Mn composition. We note for completeness that there were at least two earlier attempts to exploit the spin degree of freedom in magnetic 2DEGs but these were frustrated either by contributions of opposite sign

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from spin}orbit e!ects [50] or by low-mobility samples [51]. 3.6. Magnetic semiconductor quantum dots and wires All the MS quantum structures described so far are two dimensional in nature. The dramatic changes in the electronic density of states with further reductions in dimensionality makes it extremely attractive to study the properties of MS quantum wires and dots. Two approaches are underway for fabricating such structures: self-assembled growth and lateral patterning. The former is motivated by the successful fabrication of CdSe quantum dots by strained layer epitaxy and the observation of zero-dimensional (0D) states in these nanostructures [52,53]. The strained layer epitaxy of (Cd,Mn)Se quantum dots is a natural extension of this work (see for instance Fig. 7). Other approaches include the coupling of 0D states

Fig. 7. AFM (phase contrast) image of self-assembled quantum dots formed during the strained layer epitaxy of (Cd,Mn)Se (&5% Mn) on a (1 0 0)ZnSe surface. The spatial variation of Mn composition during the formation of such dots is at present unknown. Our studies reveal that such CdSe-based quantum dots are typically between 1.5 and 3 nm in height and have lateral base diameters in the range 10}20 nm. The area of the scan shown is 600 nm;600 nm.

in strained layer CdSe quantum dots to a vicinal MS layer [54] or the creation of 0D states within a magnetic QW via the strain "eld of nearby CdSe quantum dots [53]. Finally, MS nanostructures may also be fabricated using lateral patterning either via electron beam lithography [43}45] or using focused ion beams [55]. Thus far, the reported attempts at laterally patterned MS nanostructures have been restricted to the mesoscopic regime (with dimensions in the range 200}500 nm). Ongoing work, however, shows that electron beam lithography in conjunction with wet chemical etching is capable of producing high-quality MS nanostructures with lateral dimensions as small as 20 nm [56].

4. Exciton spin scattering in MS quantum wells and superlattices As mentioned in the introduction, we believe that it is vital to develop an understanding of dynamical spin phenomena in MS quantum structures. One of the most thoroughly studied aspects of electronic spin dynamics in these structures is the spin-#ip scattering of excitons, measured using both PL and Faraday rotation in the time domain. In these techniques, the magnetic "eld is oriented in the Faraday geometry (i.e. parallel to the direction of light propagation) and a pump pulse of circularly polarized light preferentially populates a speci"c excitonic spin state (e.g. S "#1). In time-resolved X PL, the time-dependent decay of excitons from both spin-up and spin-down states is monitored using an upconversion technique in which a timedelayed probe pulse interrogates a time slice of the emitted PL [57]. The time-resolved PL intensity I is separated into its circularly polarized components I(p>) and I(p\). The time-resolved PL polarization de"ned as (I(p>)!I(p\))/(I(p>)#I(p\)) then represents the time-dependent exciton spin population, hence providing a real-time view of the spin-#ip scattering process. In time-resolved Faraday rotation, the time-dependent decay of a population of spin-polarized excitons generates changes in the magnetization of the system, measured through the Faraday rotation (FR) imparted to a weaker time-delayed linearly polarized probe

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pulse [58]. We now discuss the insights gained into the exciton spin-#ip process during studies of three classes of quantum structures: spin superlattices, magnetically coupled double quantum wells and digital magnetic heterostructures. 4.1. Exciton spin scattering in spin superlattices As described earlier, the spin superlattice is an unusual system in which the magnetic "eld leads to a spatial separation of electronic spin states. The spin-dependent dynamics of spin-polarized excitons in a Zn Mn Se/ZnSe spin superlattice \V V have been directly observed by time-resolved (150 fs) upconversion PL spectroscopy [59]. The excitonic recombination lifetimes and spin relaxation rates are seen to be strongly dependent on both the energy and spatial location of spin states in the superlattice, displaying dynamical behavior which is markedly di!erent from that seen in ordinary quantum structures, including those composed of traditional MS heterostructures (Fig. 8). At low "elds, where both spin-up and spin-down exciton states are localized in the MS layer, the time- and polarization-resolved PL data reveal a spin scattering time of order tens of picoseconds; this scattering time is also observed to increase with the magnetic ion concentration. These results are consistent with predictions of spin scattering due to the sp}d exchange interactions [60], and stand in strong contrast with earlier expectations that spin scattering in type-I magnetic semiconductor QWs would be dominated by non-magnetic electron}hole exchange considerations [57]. In addition, a marked change in the dynamics is observed as the heterostructure goes through magnetic-"eld-induced changes in band alignment and subsequent spin superlattice formation. In contrast to the dynamics at lower "elds, the time-resolved polarization after SSL formation is essentially instantaneous, indicating an extremely rapid spin-#ip scattering of carriers from the non-magnetic layers into the magnetic QWs on a time scale much shorter than the exciton recombination lifetime. Since the ZnSe layers in these samples are expected to be relatively unstrained, the mixing of light- and heavy-hole states in the ZnSe regions may be responsible for the observed behavior.

Fig. 8. (a) Magnetic "eld dependence of PL lifetimes of the p> and p\ e1}hh1 exciton transitions in a ZnSe/(Zn,Mn)Se spin superlattice. A dramatic decrease in the PL lifetime for the p\ state upon formation of the SSL at B &0.5 T. (b) The initial 4 ps of the polarization-resolved PL with linearly polarized excitation before ( T) and after (3 T) SSL formation. 

4.2. Exciton spin scattering in magnetically coupled DQWs The MCDQW provides an ideal template for using femtosecond optical spectroscopies to probe spin-#ip scattering under conditions of strong electronic con"nement and reduced magnetic dimensionality [61}63]. Studies of the time-resolved PL in MCDQW samples indicate that the spin-#ip scattering of excitons is characterized by a time scale of 10}20 ps, in agreement with expectations based on the role of sp}d exchange interactions [60]. This is illustrated by the data in Fig. 9, where we show the time dependence of the PL polarization from an MCDQW sample of ZnSe/(Zn,Cd)Se containing a 3 monolayer barrier of (Zn,Mn)Se. Fig. 9 shows that } in the absence of a magnetic "eld } the spin-#ip scattering process for the degenerate states is &symmetric' and is characterized by a spin-#ip scattering time in the range 10}20 ps. (We note that the exciton recombination lifetime in

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spin-down states increases with the initial energy of the oriented excitons, implying the surprising observation of concurrent spin and energy relaxation, a process for which there is currently no satisfactory theoretical explanation. 4.3. Exciton spin scattering in DMH samples As discussed earlier, the combination of large exciton spin splitting and narrow inhomogeneous linewidths in DMH samples creates a convenient realization of a two-level spin system. As might be expected for a two-level system in thermal equilibrium, the polarization of the DMH samples scales with the Zeeman energy splitting *E and is given by P(*E)"tanh(*E/2k ¹). The e!ective temperature ¹"35 K&3 meV/k , implying that * at ¹"5 K * the spin-polarized exciton populations do not reach thermodynamic equilibrium. Fig. 10 shows the time-resolved polarization of the emitted PL from di!erent DMH samples when spin-down (spin-up) exciton populations are created with 120 fs pulses of p>(p\) circularly polarized light. The data for the samples is taken at di!erent magnetic Fig. 9. Time-resolved PL from carriers excited in an MCDQW in which 4 nm QWs of (Zn,Cd)Se are coupled by a 4 monolayer barrier of (Zn,Mn)Se (20% Mn). The data is shown in increasing magnetic "elds for carriers injected (a) spin down, and (b) spin up. Applying a magnetic "eld removes the degeneracy between exciton spin states, hence a!ecting the spin relaxation. Studies as a function of Mn concentration also show that the electronic spin dynamics are also dependent on the statistical clustering of Mn ions in MS.

this sample is somewhat longer * decreasing from &60 ps at B"0 to &35 ps at high "elds.) The application of a magnetic "eld breaks the spin degeneracy and hence introduces an asymmetry in this dynamical response: the spin-#ip process from spin-down to spin-up states is almost completely suppressed, while the spin-#ip rate for spin-up states is relatively "eld independent. As has been argued recently, the variation of the spin scattering rate with the Zeeman splitting can be attributed to the e!ects of phase-space "lling [41,42]. These time-resolved PL experiments have also revealed that the rate of exciton spin decay from spin-up to

Fig. 10. Time-resolved PL polarization at ¹"4.6 K from different DMH samples excited 25 meV above the zero-"eld PL peak from excitons initially oriented spin down and spin up. The data is measured at a "xed value of the exciton splitting (*E"3 meV for all samples except for the 1;3 ML sample where it is 5 meV).

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"elds but at the same value of the spin splitting, showing that the time evolution of the two spin populations in the 2D DMH structures is determined solely by the Zeeman splitting, where the spin relaxation for each DMH sample is measured at *E"3 meV. Static measurements of the FR clearly show both n"1 and 2 hh-exciton resonances (Fig. 11a). Further, the combination of narrow absorption lines and large spin splitting in low "elds results in enormous optical rotations * a single 120 As DMH in B(50 mT elicits a rotation of h "22 deg/T, cor$ responding to a Verdet constant of &1.8;10 deg/cm T. Fig. 11b shows the time-resolved FR in the regime where contributions sensitive to the relative populations of the two Zeeman-split exciton states dominate. The initial decay of h matches the rapid spin scattering and carrier $ recombination seen in the time-resolved PL data, and is only weakly dependent on temperature. Note that when pumping the spin-up state, a majority of these carriers have spin-#ipped by &6 ps, resulting in the negative value of h . $ A characteristic spin-scattering time q (*E) is  obtained from the initial slope of the time-resolved PL and FR data. The scattering time corresponding to the spin #ip of spin-up excitons to spin-down excitons is shown as a function of the exciton spin splitting in Fig. 12. The results indicate that q for  spin reversal from spin-up to spin-down is a monotonically decreasing function of the spin splitting (*E) in all the 2D DMH structures, while q in the  1;3 ml system is only weakly dependent on (*E).

Fig. 11. (a) Static di!erential FR in a (24;) DMH sample at  4.2 K, showing clearly resolved n"1 and 2 heavy-hole exciton transitions. Note that this &quantum con"ned Faraday e!ect' is characterized by huge peak value of the Verdet constant. (b) Time-resolved FR in the (24;) DMH sample showing a spin #ip scattering time of a few picoseconds.

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Fig. 12. The spin-#ip scattering times at 4.6 K in various DMH samples plotted as a function of exciton spin splitting. Open (closed) symbols are derived from the time-resolved polarization (Faraday rotation).

Quite surprisingly, q does not intrinsically vary  with the 2D spin distribution. A recent calculation of spin-#ip scattering rates in these DMH samples clearly shows that the magnetic "eld dependence of the scattering rates originates in the variation of the Fermi factors * or phase space "lling * that accompany the large spin splitting of band edges [41,42].

5. Optically probed spin coherence in MS quantum structures While the experiments described in the earlier section probe spin-#ip scattering, a di!erent perspective on dynamical spin behavior is obtained when time-resolved Faraday/Kerr rotation is used to probe the evolution of coherent superpositions of spin states. Here, a femtosecond optical pump pulse (&50 pJ) tuned to a zero-"eld excitonic resonance coherently photoexcites oppositely spin-polarized excitons and generates changes in the magnetization of the system, measured through the FR imparted to a weaker time-delayed linearly polarized probe pulse [58]. With the magnetic "eld oriented in the Faraday geometry, the pump pulse is linearly

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polarized, whilst in the Voigt geometry, it is circularly polarized. In contrast to time-resolved PL measurements, the time-resolved FR reveals not only the electronic dynamics of spin-polarized excitons in the quantum well, but also the evolution of perturbations to the Mn sublattice after the excitons have recombined. A similar measurement can be performed in the re#ection geometry, leading to the measurement of the time-resolved Kerr e!ect and the evolution of free carrier dynamics in the structure [19]. In the "rst experiments employing this technique, a linearly polarized pump pulse propagating in the Faraday geometry was used to create a coherent superposition of spin-split exciton states in an MCDQW [62,63]. The pump-induced FR then exhibits an oscillatory dependence on time, with the period of the oscillations corresponding to the spin splitting (Fig. 13a). The observations may be loosely viewed as arising from the &quantum beating' of the coherently excited states, with the decay

of the oscillations over a few picoseconds providing a measure of spin dephasing. More rigorous theoretical analysis of the problem demonstrates that the observation of these exciton spin beats arises from exciton}exciton correlations [64], where interband coherence times are quite short. A dramatic di!erence is observed in the Voigt geometry (i.e. magnetic "eld orthogonal to the direction of light propagation) (Fig. 13b) [15,16] The frequency of the quantum beats observed now corresponds to the spin splitting of only the conduction band electronic states and the intraband decoherence times are in tens of picoseconds. This decoupling between the spin precession of electrons and heavy holes occurs because the heavy holes are constrained to a point along the growth axis because of strain and quantum con"nement. Consequently, the heavy holes spin scatter very rapidly in a few picoseconds, leaving the conduction band electrons to precess independently with a longer decoherence time. Finally, after the conduction band electrons have completely spin relaxed, GHz oscillations are observed in the Voigt geometry after the THz electronic oscillations have subsided (Figs. 14a and b). This arises from the coherent oscillation of the Mn ions induced by the transient hole-exchange "eld (Fig. 14c) and corresponds to a time-domain electron spin resonance experiment. The decay of these Mn oscillations yields a direct measure of their "eld-dependent transverse relaxation time ¹ in  monolayer geometries, and provides an ultrasensitive means of measuring conduction electron spin resonance in quantum structures.

6. Quantum transport in magnetic two-dimensional electron gases

Fig. 13. Time-resolved FR in a ZnSe/(Zn,Cd)Se DQW structure in which the wells are coupled by a 4 monolayer (Zn,Mn)Se barrier (10% Mn). In (a), we show data measured in the Faraday geometry, with linearly polarized excitation and in (b), the experiment is carried out in the Voigt geometry, with a circularly polarized excitation. Note the dramatic di!erences in both the frequency and decay time of the oscillations.

The classic signature of a 2DEG with a modest degree of disorder is the integer quantum Hall e!ect (IQHE) in which the application of a magnetic "eld perpendicular to the 2DEG plane results in a vanishing longitudinal sheet resistance (o ) and VV a quantized Hall resistance (o "(h/le)) when an VW integer number (l) of Landau levels are "lled. In magnetic 2DEGs, detailed studies of the IQHE and of the quantum oscillations in o reveal that * as VV

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Fig. 14. (a) Time-resolved Faraday rotation in the Voigt geometry at long time delays, showing the "nal oscillations of the electrons superimposed on an induced precession of the Mn spins in the MCDQW sample described in Fig. 4. (b) Evolution of the Mn precession, showing long-lived free-induction decay characterized by a spin relaxation time ¹ of the order of  nanoseconds. (c) Schematic illustrating the torque exerted on an Mn moment by the hole exchange "eld.

a result of the exchange-enhanced spin splitting * the Landau levels involved in quantum transport are completely spin resolved even at relatively high temperatures (Fig. 15). Further, a direct measurement of this spin splitting using magnetooptical spectroscopy [65] and the subsequent construction of a Landau level diagram clearly show a substantial spin polarization of the 2DEG in modest magnetic "elds at low temperatures. Fig. 16 shows that in the quantum regime * de"ned by u q'1 where u is the cyclotron frequency and ! ! q is the quantum lifetime * all the IQHE states of interest are purely separated by a cyclotron gap; the character of the IQHE states in such 2DEGs is hence fundamentally di!erent from that of traditional 2DEGs where pairs of IQHE states such as l"1, 2 or 3, 4 may be spin resolved but are separated by a spin gap. Ironically, the introduction of large spin e!ects in magnetic 2DEGs renders the energy level structure of these 2DEGs closer to that

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Fig. 15. IQHE in a magnetic 2DEG sample containing a modulation-doped DMH quantum well of (Zn,Cd)Se with equispaced,  monolayers of MnSe. The single particle scattering time de duced from a standard analysis of the quantum oscillations is roughly 0.35 ps. Note that * due to the large exchange-enhanced spin splitting * all the integer states are clearly spin resolved right from the onset of quantum oscillations at &2 T (i.e. from l"8 onwards), despite the presence of disorder and at temperatures as high as 4.2 K. The anomalous shapes of the IQHE plateaus at low temperatures are probably the result of sample disorder. The measurements shown here are made on a mesa-etched Hall bar using DC techniques.

envisaged in theories of the quantum Hall e!ect that ignore the presence of spin [66]. In addition to providing a model system for studying spin-polarized quantum transport, magnetic 2DEGs also constitute a new testing ground for the interplay between electron}electron interactions, spin polarization and disorder. This has been examined in both the weakly localized and strongly localized regimes. In both cases, two generic features are present in the magneto-transport: a striking background MR is observed that is positive at low "elds and negative at high "elds; these characteristics are present even when the magnetic "eld is parallel to the plane of the 2DEG, indicating that

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Fig. 16. Landau level fan diagram for the magnetic 2DEG sample described in Fig. 15. Solid (dashed) lines correspond to spin-down (spin-up) states. The dark solid line shows the variation of the Fermi energy with magnetic "eld. Parameters used in this calculation are: E "7 meV at B"0, mH"0.14m and $   ¹"360 mK. The spin-splitting parameters used are obtained by "tting the magneto-optical data in Fig. 3: ¹ "2.1 K and  a saturation conduction band spin splitting of 12.9 meV.

there are important contributions stemming from the spin splitting of electronic states and/or the magnetization of the sample. While there is no detailed model for the negative MR at present, the behavior is qualitatively consistent with the suppression of spin-disorder scattering as the paramagnetic landscape is smoothened by a magnetic "eld. The positive MR in weakly localized samples (k l '1, where k is the Fermi $ $ wave vector and l is the elastic scattering length)  has been attributed to the e!ects of the spin splitting on the disorder-modi"ed electron}electron interactions, extending to 2D a perturbative "eld theory [67] that had earlier explained similar behavior in 3D MS alloys [68]. A deeper examination of the MR in gated magnetic 2DEGs as a function of carrier density and temperature, however, shows that this interpretation is at best incomplete [69], particularly because the perturbative constraint imposed by the theory (k l <1) excludes signi"cant  regimes of experimental interest. As shown in Fig. 17, in strongly localized magnetic 2DEGs, the positive MR can be extremely dramatic at low temperatures (&700% in "elds as low as 0.1 T). This striking MR has been shown to follow a

Fig. 17. The longitudinal MR of a low-density (N "1.33;10 cm\) magnetic 2DEG at three di!erent tem1 peratures. The giant positive MR at low "elds is attributed to the suppression of spin-dependent hopping paths by the spin splitting. The critical "elds B and B demarcate quantum phase ! ! transitions between insulating and quantum Hall liquid states. The inset shows the magnetic "eld variation of the Hall resistivity o at ¹"0.9 K. VW

universal scaling law that arises as a consequence of the suppression of spin-dependent hopping paths when localized states with an on-site correlation energy undergo a large spin splitting. Finally, Fig. 17 also shows that at high "elds, these strongly localized 2DEGs undergo a phase transition to a quantum Hall liquid. In the magnetic 2DEGs described above, the magnetization of the paramagnetic lattice is largely una!ected by the presence of a 2DEG. This is consistent with the well-established view that the Mn}Mn exchange in II}VI magnetic semiconductors is dominated by short-ranged antiferromagnetic superexchange; contributions from carrier-mediated mechanisms such as the Bloembergen}Rowland and RKKY interactions are unimportant [26,27]. A recent mean-"eld calculation [70] has re-examined the carrier-mediated exchange in heavily doped II}VI MS in the presence of delocalized or weakly localized carriers. The principal prediction is that a dominant ferromagnetic RKKY Mn}Mn interaction can be produced

D.D. Awschalom, N. Samarth / Journal of Magnetism and Magnetic Materials 200 (1999) 130}147

by manipulating factors such as the carrier concentration, the dimensionality of the sample, quantum con"nement and * perhaps * disorder. In a simpli"ed picture that ignores the in#uence of disorder and electron}electron interactions, the Curie} Weiss temperature H associated with the carriermediated RKKY exchange is given by (apart from some constants) H&o(E )If (t), $

(3)

where o(E ) is the density of states of the carriers at $ the Fermi energy E , I is the s}d (p}d) exchange $ integral for electrons (holes), and f (t) represents the wave-function overlap of the carriers with the magnetic moments in the lattice. We note that * in contrast to metals * the period of the oscillatory RKKY interaction here is much longer than the nearest-neighbor magnetic ion distance. Consequently, the interaction is purely ferromagnetic and a ferromagnetic phase will result if the RKKY exchange overwhelms the antiferromagnetic superexchange (i.e. H'¹ ). Since H is enhanced by  increasing the density of states and the carrier-ion exchange, an attractive system for observing a ferromagnetic transition is a magnetic 2D hole gas (2DHG) in a II}VI MS quantum well where both the e!ective mass and the p}d exchange are much larger than for electrons. To verify this prediction, Haury et al. [71] carried out a magneto-optical study of modulation p-doped (Cd,Mn)Te quantum wells containing a 2DHG with a sheet concentration in the range (1.6}3.2);10 holes cm\, as deduced from the Moss}Burstein shift between photoluminescence excitation and PL spectra. The key observation is that the PL spectra reveal a doublet structure at zero magnetic "eld below a critical temperature ¹ "1.8 K, suggesting the onset of a ferromagnetic ! phase; such behavior is not observed in an undoped control sample. From the temperature- and "elddependence of the PL spectra, the Curie}Weiss temperature H"¹ #¹ is determined. This ex!  periment is quite suggestive of the ferromagnetic phase anticipated by the mean "eld model; more de"nitive evidence, however, will have to await additional measurements such as magneto-transport in these magnetic 2D hole gases.

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7. Conclusions and summary This review has focused on experiments that probe spin dynamics and transport because the fundamental understanding of these problems is likely to play an important role in the development of future magnetolectronic device concepts that incorporate these materials. These experiments have measured the characteristic time scales that determine excitonic/electronic spin-#ip scattering and spin decoherence, showing how these vary with factors such as the spin splitting and the magnetic environment. Further, we have shown how modulation doping enables the creation of spin-polarized transport channels and perhaps provides a means of inducing ferromagnetism in these materials despite the antiferromagnetic d}d exchange. Invariably, a review of a rapidly developing "eld such as the present one cannot possibly hope to cover all the interesting possibilities emerging from ongoing work. We conclude with a brief mention of some of these: E Magnetic polarons in reduced dimensions: One of the consequences of the sp}d exchange in MS alloys is that a localized carrier can create a spin polarization of the magnetic moments within the span of its wave function. The properties of such &bound magnetic polarons' have been extensively studied in bulk MS alloys. The "rst (and so far the only direct) observation of bound magnetic polarons in QWs was reported using optically induced magnetization studies of a type-II MS QW such as that shown in Fig. 1e [72]. Static and time-resolved spectroscopy has also been used to examine the properties of magnetic polarons created by free excitons [73]. The optical excitation of magnetic polarons and the study of their subsequent time evolution provides a potentially powerful local probe of spin dynamics that has yet to be fully exploited, particularly in systems of highly reduced dimensionality such as quantum wires and dots. E Mesoscopically patterned MS: The fabrication of mesoscopic wires by e-beam lithography has opened up a fascinating new arena for studies of mesoscopic spin transport [43]. More recently, noise measurements in these systems have yielded

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a very clean picture of the dynamics of the spin glass transition [44,45]. The measurements thus far have focused on the regime wherein the sample size is smaller than the inelastic scattering length but larger than the elastic scattering length. We anticipate even more interesting physics to emerge from the patterning of high-mobility 2DEGs into nanostructures where ballistic spin transport becomes a realistic possibility. An important recent observation in this context is that the time scales determining spin relaxation (and decoherence) can be much longer than those related to charge transport. For instance, nanosecond spin relaxation times are measured in relatively low mobility II}VI 2DEGs at room temperature [19], despite the sub-picosecond charge transport scattering times. E Perpendicular spin transport and tunneling: A recent calculation has discussed spin-dependent vertical transport in a spin superlattice, pointing out that a rich magneto-transmission spectrum can be obtained in conjunction with spin "ltering e!ects [74]. Although ample experimental work has been reported on spin-dependent tunneling and perpendicular transport in metallic magnetic tunnel junctions and heterostructures, such experiments are only beginning in MS heterostructures, with initial results being reported in III}V-based heterostructures [9]. Acknowledgements The authors are grateful to a number of students and post-doctoral researchers at the University of California-Santa Barbara and the Pennsylvania State University for their contributions to this research. In addition, we acknowledge "nancial support from the ONR N00014-99-1-0077 and N0001499-1-0071, the NSF DMR 97-01072 and NSF DMR 9701484, the AFOSR F49620-99-1-0033, and the NSF Science and Technology Center for Quantized Electronic Structures (DMR 91-20007). References [1] F. Capasso, Physics of Quantum Electron Devices, Springer, New York, 1990.

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