A.D. Mirlin

B. Kramer

C.W.J. Beenakker

D.S. Fisher

F.G. Brito

H. Ikada

J. C. Egues

J. Jaroszyński

J.C. Egues

J.K. Furdyna

J.M. Kikkawa

K. Chang

N. A. Mortensen

O. Ray

P. Farinas

R. Fiederling

R. Landauer

S.A. Wolf

Y. Guo

Y. Ohno

We study conductance and spin-polarization fluctuations in one-dimensional wires with spin-5/2 magnetic impurities (Mn). Our tight-binding Green function approach goes beyond the mean field thus including s-d exchange-induced spin-flip scattering. In a certain parameter range, we find that spin-flip suppresses conductance fluctuations while enhancing spin-polarization fluctuations. More importantly, spin-polarization fluctuations attain a universal value 1/3 for large enough spin-flip strengths. This intrinsic spin-polarization fluctuation may pose a severe limiting factor to the realization of steady spin-polarized currents in Mn-based one-dimensional wires

Mortensen, Asger

Egues, J.C.

Online Research Database In Technology

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Asger; Egues, J.C.Published in:Physical Review B (Condensed Matter and Materials Physics)Link to article, DOI:10.1103/PhysRevB.66.153306Publication date:2002Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Mortensen, A., & Egues, J. C. (2002). Universal spin-polarization fluctuations in one-dimensional wires withmagnetic impurities. Physical Review B (Condensed Matter and Materials Physics), 66(15), 153306. DOI:10.1103/PhysRevB.66.153306Universal spin-polarization fluctuations in one-dimensional wires with magnetic impuritiesN. A. Mortensen*Mikroelektronik Centret, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmarkand O rsted Laboratory, Universitetsparken 5, DK-2100 CopenhagenO , DenmarkJ. C. Egues†Departamento de Fı´sica e Informa´tica, Instituto de Fı´sica de Sa˜o Carlos Universidade de Sa˜o Paulo,13560-970 Sa˜o Carlos, Sa˜o Paulo, Braziland Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland~Received 12 July 2002; published 4 October 2002!We study conductance and spin-polarization fluctuations in one-dimensional wires with spin-5/2 magneticimpurities ~Mn!. Our tight-binding Green function approach goes beyond the mean field thus including s-dexchange-induced spin-flip scattering. In a certain parameter range, we find that spin-flip suppresses conduc-tance fluctuations while enhancing spin-polarization fluctuations. More importantly, spin-polarization fluctua-tions attain a universal value 1/3 for large enough spin-flip strengths. This intrinsic spin-polarization fluctuationmay pose a severe limiting factor to the realization of steady spin-polarized currents in Mn-based one-dimensional wires.DOI: 10.1103/PhysRevB.66.153306 PACS number~s!: 72.25.2b, 73.63.NmI. INTRODUCTIONSpin-related effects in solid state heterostructures give riseto a rich variety of fascinating physical phenomena. Thesespin-dependent properties also underlie a potential techno-logical revolution in conventional electronics.1 This para-digm is termed ‘‘Spintronics.’’ A particularly interestingtheme within this emerging field is spin-polarized transportin semiconductor heterostructures. This topic has attractedmuch attention after the fundamental discovery of exceed-ingly long spin diffusion lengths in doped semiconductors2followed by the seminal spin injection experiments in Mn-based heterojunctions.3Theoretically, a number of works have addressed issuesconnected with spin-polarized transport. These include, forinstance: spin filtering,4 spin waves,5 and quantum shotnoise,6—all in ballistic semimagnetic tunnel junctions—andmesoscopic conductance fluctuations in Rashba wires.7 Spin-dependent phenomena in connection with localization effectsshould bring about exciting interesting physics.Here we investigate conductance and spin-polarizationfluctuations for transport through one-dimensional wireswith spin-5/2 magnetic impurities, e.g., Mn-based II-VI al-loys such as ZnSe/ZnMnSe/ZnSe. The experimental feasibil-ity of these wires has already been demonstrated.8,9 In thesesystems, the conduction electrons interact with the localizedd electrons of the Manganese via the s-d exchangecoupling.10 UCF in Mn-based submicron wires was first ex-perimentally studied in Ref. 8. We describe transport withinthe Landauer formalism11 and calculate the relevant trans-mission coefficients via noninteracting tight-binding Greenfunctions.12We treat the s-d interaction beyond the usual mean-fieldtheory thus accounting for spin flip scattering. In a certainparameter range we find that spin-flip scattering suppressesconductance fluctuations13 ~below the UCF value for strictlyone-dimensional wires! while enhancing the correspondingspin-polarization fluctuations. More importantly, we showthat the spin-polarization fluctuations attain a universal value^(dz)2&51/3 for strong spin-flip scattering. This large spin-polarization fluctuation may pose a fundamental obstacle toattaining steady spin-polarized currents in Mn-based wires.II. HAMILTONIAN MODELWe consider a one-dimensional tight-binding chain ~seeFig. 1!, of N spin s55/2 magnetic impurities coupled to idealleads ~sites n,1 and n.N). We separate the electronic andimpurity-spin degrees of freedom and treat the latter classi-cally ~static scatterers!. The two-component electron wavefunction, c5(c↑ ,c↓) is then governed by the Schro¨dingerequation with a HamiltonianH5S H0 00 H0D 1S H↑↑ H↑↓H↓↑ H↓↓D . ~1!FIG. 1. One-dimensional tight-binding chain with N magnetic s55/2 impurities ~mutually uncorrelated, each spin equally distributedamong the six spin states! coupled to ideal impurity-free leads ~sites n,1 and n.N).PHYSICAL REVIEW B 66, 153306 ~2002!0163-1829/2002/66~15!/153306~4!/$20.00 ©2002 The American Physical Society66 153306-1Here H0 is spin independent, with elements12$H0%nm52gdnm2gdnm112gdnm211Vndnm , ~2!where Vn is the potential at site n and g5\2/2ma2, with abeing the ‘‘lattice constant.’’ In the leads H0 itself gives riseto the usual dispersion relation «(k)52g(12cos ka).In the following, s5↑[1/2 and s5↓ [21/2. We re-strict ourselves to zero magnetic field so that the block ma-trices Hss have elements given by$Hss%nm5dnmJzsSn ,z ~3!which is a Heisenberg-like interaction of the spin of the elec-tron (s) with the z-component spin of the impurity S5(Sx ,Sy ,Sz). The off-diagonal block matrix H↑↓5H↓↑† con-tains the interaction of the electron spin with the x and ycomponents of the impurity spins which leads to spin-flip:$H↑↓%nm5dnm@JxSn ,x2iJySn ,y#/2. ~4!We consider a sufficiently weak coupling between the impu-rity spins so that they can be considered mutually uncorre-lated, i.e., no magnetic ordering. The z-component of eachspin is equally distributed among the six spin states and the xand y components are uniformly distributed with the con-straint that S25Sx21Sy21Sz25s(s11); see Fig. 1.III. TRANSPORT PROPERTIESWe study transport in the low-temperature linear responselimit within the Landauer formalism11:g5e2h (ss8Tss8~«F!. ~5!Here T is a 232 matrix with the elements Tss8 being thetransmission probability of an electron from a state with spins8 in one lead to a state with spin s in the other lead. FromEq. ~5! we now define the degree of spin polarizationz[I↑2I↓I↑1I↓5T↑↑1T↑↓2T↓↑2T↓↓T↑↑1T↑↓1T↓↑1T↓↓, ~6!which we will focus on in this paper.Green function method. The transmission matrix T is re-lated to the retarded Green functionG~«!5@«12H˜ 2S~«!#21 ~7!via the Fisher-Lee relation14Tss8~«!5@\v~«!#2u$Gss8~«!%N1u2, ~8!where v5\21]«/]k is the group velocity in the leads. In Eq.~7! the 2N32N matrix H˜ is the Hamiltonian truncated to theN lattice sites with magnetic impurities. The effect of cou-pling to the leads is contained in the 2N32N retarded self-energy matrix with elements12$Sss8~«!%nm52geik(«)adss8dnm~d1n1dNn!. ~9!N51 case. A chain with a single impurity is a simpleillustrative example where analytical progress is possible.After performing the straightforward matrix inversion in Eq.~7! we findz~«!52V1JzSzV121«~4g2«!1~JS !2, ~10!where J5(Jx ,Jy ,Jz)T. In zero magnetic field ^Sz&50 and^Sz2&535/12. This implies that ^z&50 both with and withoutspin-flip, whereas the fluctuations are finite. The analyticalaveraging is of course complicated by the presence of Sz inthe denominator, but for isotropic coupling Jx5Jy5Jz5J0,we have (JS)25J02s(s11) so that Sz only shows up in thenumerator, i.e.,^~dz!2&53512V12J02@V121«~4g2«!1J02s~s11 !#2. ~11!In the absence of spin-flip (Jx5Jy50) the fluctuations areenhanced due to the replacement of s(s11)→Sz2,s(s11)in the denominator ~the final expression for the fluctuationsis much more complicated! and this means that spin-flip willlower the fluctuations of z . Of course this trend is strictlyvalid for N51, but in a limited parameter range this trend isstill true for larger N values.Finite N case. For a finite number of impurities the prob-lem is not analytically tractable and we study the problemnumerically by generating a large ensemble ~typically 105members! of spin configurations. For each spin configurationwe calculate Eqs. ~7! and ~8! numerically. In our simulationswe use the following parameters: «F5g , Jz5g/2, Vn50~i.e., we neglect spatial disorder!, and varying spin-flip cou-pling strengths 0<Jx5Jy<g .FIG. 2. Distributions P(z), P(Tss), and P(Tss8) for differentspin-flip scattering strengths Jx5Jy in the case of N510. The dash-dotted line in the lowest panel indicates the uniform limit P(z)51/2 @note the magnification of P(z) by of factor of 5# attained forstrong enough spin-flip scattering.BRIEF REPORTS PHYSICAL REVIEW B 66, 153306 ~2002!153306-2IV. RESULTS AND DISCUSSIONSFigure 2 shows the distributions P(z), P(Tss), andP(Tss8) for N510 and increasing strengths of the spin-flipcoupling Jx5Jy . The distribution P(z) is symmetric aroundz50, which implies that on average there is no spin filtering,^z&50. The distribution P(z) first gets narrower for spin-flip in the @0,0.15g# range ~not shown! and then broadens asspin-flip further increases. For sufficiently strong spin-flipscattering the distribution approaches that of the uniformlimit in which P(z)51/2. In this limit P(Tss) and P(Tss8)coincide, and so do all average transmission probabilities^Tss8&. As we discuss below, the initial narrowing and sub-sequent broadening of P(z) with spin flip gives rise to aminimum in the fluctuation of z ~Fig. 3!.Universal spin-polarization fluctuations. In the limit of ashort spin-flip length l s!L we in general find a uniformdistribution P(z)51/2 ~Fig. 2!. This uniform distributionyields the universal value ^(dz)2&51/3 for the spin-polarization fluctuations. Figure 3 clearly shows that this uni-versal value is attained for increasing spin flip strengths andis indeed independent of N. Interestingly, Fig. 3 also shows aminimum at around Jx5Jy50.15g . This minimum can beattributed to two competing energy scales: the longitudinal(;Jz) and the transverse (;Jx ,Jy) parts of the s-d ex-change interaction @Eqs. ~3! and ~4!, respectively#. A simple‘‘back-of-the-envelope’’ calculation shows that these twocompeting scales are equal for Jx5Jy5Jz /A2s(s11)/3g50.208g . The vertical dashed line in Fig. 3 indicates thisvalue. Observe that ^(dz)2& becomes larger for increasing N.This happens because P(z) broadens for larger N’s ~the tra-versing electrons see a wider region with random spins!. Thisis similar to the broadening due to increasing spin flipstrength.We should mention that the distribution P(z), and conse-quently ^(dz)2&, change dramatically for «F,Jz . In this re-gime, P(z) becomes U shaped ~not shown! because of thedominant filtering due to the ‘‘end states’’ with S j ,z565/2.This qualitatively different P(z) yields a monotonically de-creasing ^(dz)2& as a function of spin-flip strength. Here theuniversal ^(dz)2&51/3 value is approached from above forlarge spin-flip strengths (Jx5Jy;g).Suppression of conductance fluctuations. Whereas thefluctuations in the spin polarization z remain finite in thestrong spin-flip scattering regime ~Fig. 3!, we find that thefluctuations of the conductance g are strongly suppressed inthis limit. This is illustrated in Fig. 4 which shows the aver-age conductance and its fluctuations as a function of spin-flipscattering for N510, 20, and 30. Note that ^(dg)2&1/2 ismuch more sensitive to spin-flip than ^g&. In addition, for allN we essentially have ^(dg)2&1/2.^g& for Jx5Jy→0 and^(dg)2&1/2&^g& for Jx5Jy→g . Figure 4 clearly shows theconductance fluctuations get suppressed for increasing N.The horizontal dashed line shows the UCF value (0.73/250.365, see, e.g., Ref. 15! for a one-dimensional wire in themetallic regime. The spin-related conductance fluctuationsdo not approach a finite value for increasing spin-flip scat-tering. It actually seems to go to zero. This is in contrast tothe spin-polarization fluctuations ~Fig. 3!, which attain a uni-versal value ^(dj)2&1/251/A3 for strong spin-flip scattering.Incidentally, we observe that ^(dg)2&1/2 and ^(dz)2&1/2 alsopresent contrasting behavior for increasing N ~and «F.Jz):the former is suppressed while the latter is enhanced ~cf.Figs. 3 and 4!.Spin disorder as spatial disorder. To some extent, the s-dsite interaction considered here plays the role of spatial dis-order in the system with a mean free path l J . Let us con-sider first the case with no spin-flip ~i.e., Jx5Jy50). In thiscase, the term JzsSn ,z acts as a ‘‘random’’ spin-dependentpotential along the chain ~here the site potential has someinternal structure!. As shown in Fig. 4 the conductance fluc-tuations for zero spin-flip scattering are larger than, slightlyabove, and slightly below, the UCF value for N510, 20, and30, respectively. For increasing N we go from the metallicregime (L5Na!l Jz) with vanishing fluctuations and aGaussian P(g) strongly peaked near g;2e2/h to thestrongly localized regime (L@l Jz) where it is well knownthat P(g) is strongly peaked near g;0 with a log-normaldistribution so that fluctuations can be comparable to theFIG. 3. Average fluctuations ^(dz)2&1/2 as a function of spin-flipstrength for N510, 20, and 30. The dashed horizontal line indicatesthe universal value 1/A3 obtained from the uniform limit P(z)51/2. The vertical dashed line indicates where the spin-flip rate iscomparable to uJzsu/\ .FIG. 4. Average conductance ^g& and its fluctuations ^(dg)2&1/2as a function of the spin-flip scattering strength for N510, 20, and30. The conductance fluctuations are much more sensitive to spin-flip than the average conductance: the former is strongly suppressedfor increasing spin-flip rates.BRIEF REPORTS PHYSICAL REVIEW B 66, 153306 ~2002!153306-3mean value.16 This is in accordance with numerical studieswith different continuous distributions of the ‘‘on-site’’ po-tential ~e.g., Gaussian or uniform distributions!.17 In Fig. 4the ‘‘small’’ mean values ^g&, for N510, 20, and 30, indi-cate the onset of localization with fluctuations comparable tothe mean value. As N becomes larger conductance fluctua-tions are as expected suppressed.16,18Role of spin-flip scattering. Spin-flip clearly suppressesconductance fluctuations ~Fig. 4!. This can be understoodfrom Eq. ~4! being a complex number with a random phasewhich makes spin-flip act as a source of ‘‘decoherence’’ ~thetotal wave function is, of course, fully coherent!. Further-more, spin-flip mixes all the Sn ,z components on each sitethus smoothing the potential seen by the traversing electronand hence reducing conductance fluctuations. This is true forboth «F.Jz @except for the window (0,0.15g) in whichP(z) narrows# and «F,Jz .‘‘Truly’’ universal fluctuations. Why is ^(dz)2&1/2 univer-sal even for short spin-flip lengths l s!L ~strong spin-flipscattering! while ^(dg)2&1/2 is clearly suppressed below theusual UCF value in this limit? It is well known that conduc-tance fluctuations are suppressed in the incoherent limit.15More specifically, in one-dimensional wires with l w!L , l wis some ‘‘dephasing length,’’ the suppression factor is AL/l w~see Ref. 12!. Interestingly, we can likewise understand thesuppression of ^(dg)2&1/2 seen in our simulations by viewingspin-flip scattering as producing ‘‘dephasing’’ with l w;l Jx ,y.19 For the spin-polarization fluctuations, however, thepicture is slightly different: here we divide our system intoNL5L/lw segments. To each of these we can associate anaverage spin polarization ^z i&50(i:1..NL) and a correspond-ing spin-polarization fluctuation ^(dz i)2&. Neither ^z i& nor^(dz i)2& are additive quantities like ^g& and ^(dg)2& ~‘‘ex-tensive versus intensive’’ properties!. Sensible global aver-ages for the whole system are then z¯[(1/NL)( i^z i&50 and(dz)2[(1/NL)( i^(dz i)2& . We should expect (dz)25^(dz i)2&[^(dz)2& if the system is ergodic. Hence univer-sal spin-polarization fluctuations are not suppressed for largespin-flip scattering in contrast to conductance fluctuations.IV. CONCLUDING REMARKSSpin-flip scattering in Mn-based wires reduces conduc-tance fluctuations while enhancing spin-polarization fluctua-tions in a limited parameter range. Remarkably, spin-polarization fluctuations reach a universal value 1/3 for largespin-flip scattering in which the conductance fluctuationsvanish. This universal value should manifest itself in time-and polarization-resolved photoluminescence measurements.More important, these sizable spin fluctuations may limit thepossibilities for steady spin injection in these systems.The authors thank A.-P. Jauho for a critical reading of themanuscript. We also acknowledge K. Flensberg ~N.A.M.!and U. Zu¨licke and M. Governale ~J.C.E.! for useful discus-sions. This work was supported by the Swiss NSF, DARPA,ARO, and FAPESP/Brazil ~J.C.E.!.*Electronic address: nam@crystal-fibre.com†Electronic address: egues@if.sc.usp.br1 S.A. Wolf et al., Science 294, 1488 ~2001!.2 J.M. Kikkawa and D.D. Awschalom, Nature ~London! 397, 139~1999!.3 R. Fiederling et al., Nature ~London! 402, 787 ~1999!; Y. Ohno,et al., ibid. 402, 790 ~1999!.4 J.C. Egues, Phys. Rev. Lett. 80, 4578 ~1998!; K. Chang and F.M.Peeters, Solid State Commun. 120, 181 ~2001!; Y. Guo et al.,Phys. Rev. B 64, 155312 ~2001!.5 P. Farinas, Phys. Rev. B 64, 161310 ~2001!.6 F.G. Brito et al., J. Magn. Magn. Mater. 226, 457 ~2001!.7 B.K. Nikolic and J.K. Freericks, cond-mat/0111144 ~unpub-lished!.8 J. Jaroszyn´ski et al., Phys. Rev. Lett. 75, 3170 ~1995!.9 O. Ray et al., Appl. Phys. Lett. 76, 1167 ~2000!; H. Ikada et al.,Photonics Spectra 10, 373 ~2001!.10 J.K. Furdyna, J. Appl. Phys. 64, R29 ~1988!.11 R. Landauer, IBM J. Res. Dev. 1, 223 ~1957!; Philos. Mag. 21,863 ~1970!.12 S. Datta, Electronic Transport in Mesoscopic Systems ~CambridgeUniversity Press, Cambridge, 1995!.13 For ‘‘intermediate’’ spin-flip scattering strengths ~see Fig. 4! wefind the usual UCF value for the conductance fluctuations. Inthis parameter range, our numerical calculation seems to be con-sistent with the experimental results of Ref. 8, in which no s-dinduced effects were observed on the mean amplitude of theUCF.14 D.S. Fisher and P.A. Lee, Phys. Rev. B 23, 6851 ~1981!.15 C. W. J. Beenakker and H. van Houten, in Solid State Physics,edited by H. Ehrenreich and D. Turnbull ~Academic Press, NewYork, 1991!, Vol. 44, pp. 1–228.16 C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 ~1997!.17 B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469 ~1993!.18 A.D. Mirlin, Phys. Rep. 326, 259 ~2000!.19 Again, this is a sensible view since spin-flip is a complex numberwith a random phase in our model; see Eq. ~4!.BRIEF REPORTS PHYSICAL REVIEW B 66, 153306 ~2002!153306-4

Universal spin-polarization fluctuations in one-dimensional wires with magnetic impurities

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Universal spin-polarization fluctuations in one-dimensional wires with magnetic impurities

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