Table Of Content

Anomalous Hall effect in field-effect structures of (Ga,Mn)As D. Chiba,1,2,3 A. Werpachowska,4 M. Endo,2 Y. Nishitani,2 F. Matsukura,2,1 T. Dietl,4,5 and H. Ohno2,1 1Semiconductor Spintronics Project, Exploratory Research for Advanced Technology, 1 Japan Science and Technology Agency, Sanban-cho 5, Chiyoda-ku, Tokyo 102-0075, Japan 1 2Laboratory for Nanoelectronics and Spintronics, 0 Research Institute of Electrical Communication, Tohoku University, 2 Katahira 2-1-1, Aoba-ku, Sendai, Miyagi 980-8577, Japan n 3Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan a 4Institute of Physics, Polish Academy of Sciences, PL-02 668 Warszawa, Poland J 5Institute of Theoretical Physics, University of Warsaw, ul. Hoża 69, PL-00 681 Warszawa, Poland 8 (Dated: January 11, 2011) The anomalous Hall effect in metal-insulator-semiconductor structures having thin (Ga,Mn)As ] i layers as a channel has been studied in a wide range of Mn and hole densities changed by the gate c electric field. Strongand unanticipated temperature dependence,includinga changeof sign, of the s - anomalous Hall conductance σxy has been found in samples with the highest Curie temperatures. rl Formoredisorderedchannels,thescalingrelationbetweenσxy andσxx,similartotheoneobserved t previously for thickersamples, is recovered. m . PACSnumbers: 72.25.Dc,73.40.Qv,75.50.Pp,85.75.-d t a m Alongwithanisotropicmagnetoresistance,theanoma- importance of yet unrevealed confinement effects. - d lous Hall effect (AHE) results from aninterplay between The studied thin layers of tensile-strained (Ga,Mn)As n spin-orbit interactions and spin polarization of electric o havebeendepositedbylow-temperaturemolecularbeam current specific to ferromagnets. It has been recently c epitaxy ontoa buffer layerconsistingof 4-nmGaAs/30- [ realized that for a certain region of conductivities, the 2 anomalous Hall conductivity σxy is a measure of the gnrmowAnl0o.7n5Gaas0e.m25iA-isn/su5l0a0t-inngmGIna0A.1s5G(0a001.8)5sAusb/s3t0ra-ntem. GUapAons Berry phase of carrier trajectories in the k space and, v growth,Hallbarshavinga channelof30or40-µmwidth thus, provides information on the hitherto inaccessible 2 and ∼200-µm length are patterned by photolithography 5 aspects of the band structure topology in the presence and wet etching. Subsequently, samples are annealed at 6 of various spin-orbit interactions [1, 2, 3, 4, 5]. Interest- ◦ 180 C for 5 min or introduced directly into an atomic 0 ingly, the effect appears to be qualitatively immune to . layer deposition chamber, where a 40-nm-thick oxide in- 1 disorder, except for the case of linear-in-k Rashba-type sulator is deposited at a substrate temperature of 120- 0 Hamiltoniansintwo-dimensionalelectronsystems,where ◦ 1 150 C. Finally, 5-nm Cr/ 100-nm Au gate electrode is the contribution to σ vanishes [6] unless the lifetime is 1 xy formed. Owing to the tensile strain,the easy axis is per- spin-dependent[7]. Furthermore,asurprisinglyuniversal : pendicular to the plane, so that the height of the Hall v empiricalscalingrelationbetweentheHallandlongitudi- Xi nal conductivities, σxy ∼σxγx, γ ≈1.6 has been found to vanolotmagaelohuysstHeraellsersespirsotavnidceesRdire.ctlythemagnitudeofthe be obeyed by a number of materials on the lower side of yx r a theirconductivityvalues[8],whereAnderson-Mottquan- Altogether 18 MIS structures,numbered from1 to 18, tum localization effects should be important. have been investigated. As tabulated in the supplemen- tary material[9], they differ in nominal Mn composition In this Letter, we report on Hall resistance studies as (3% ≤ x ≤ 17.5%), thickness (3.5 nm ≤ t ≤ 5 nm), a function of temperature and gate electric field carried and crystallographic orientation ([110] vs. [¯110]) of the out for metal-insulator-semiconductor (MIS) structures Ga1−xMnxAs channel, as well as contain three kinds of containing a thin conducting channel of ferromagnetic gate insulators (Al2O3, HfO2, and ZrO2). For this set of (Ga,Mn)As. We find out that in the σ range up to samples, and in the employed gate electric field range xx 102 S/cm, σxy obeys a scaling relation with a similar (5 MV/cm ≥ EG ≥ −5 MV/cm), σxx spans over 3 value of the exponent γ. However, for σxx >∼ 102 S/cm orders of magnitude, and the corresponding TC varies the scaling relation breaks down entirely. Surprisingly, between 35 and 165 K. The lateral homogeneity of the in this regime and below the Curie temperature TC, σxy grownwafers is provedby results displayedin Fig. S1 in tends to decrease rather abruptly with decreasing tem- EPAPS, whereas the data in Fig. S2 demonstrate that perature, and even reversesits signin some cases,in the themagnitudesofTCintheMISstructuresandthicklay- region where neither resistance R nor magnetization M ersarevirtuallyidentical. Sincetislargerthanthemean vary significantly with temperature. The effect has not free path but shorter than the phase coherence length of been observed in thicker films and appears to have no holesin(Ga,Mn)As,whoselowerlimitisprovidedbythe explanation within the existing theory, pointing to the conductance studies in 1D systems [10, 11], the density 2 of states preservesa 3D shape, whereaslocalizationphe- While, according to Fig. 2(b) the temperature depen- nomena acquire a 2D character. dence of R is rather standard, R shows a nonmono- xx yx Asanexampleofexperimentalfindings,Figs.1(a)and tonic temperature dependence attaining a maximum at 1(b) showthe temperature dependence ofremanentHall about 20 K below TC ≈120 K, followedby a decrease of resistance Ryx and sheet resistance Rxx at various gate Ryx towards zero with lowering temperature, culminat- electric fields for the MIS structure containing a 5-nm ing in a change of sign at EG > 1.5 MV/cm at 10 K, as thick Ga0.949Mn0.051As channel and the Al2O3 gate in- shown in detail in Fig. 2(c). The data on Ryx(T) are to sulator. In the inset, the results of the Hall measure- be contrastedwith the behaviorof magnetizationwhich, ments as a function of the external magnetic field H at according to the results displayed in Fig. 2(d), shows a 10Karepresented. Thesquarenessofthehysteresisloop monotonicincreaseuponcooling. Interestingly,whilethe indicates the perpendicular-to-plane orientation of the magnitude of TC in our MIS structures attains similar magnetization easy axis, whereas its counter-clockwise values as those in thicker films, a non-Brillouin charac- chirality, as in the case of an ordinary M vs. H loop, terofM(T)indicatesthatmagnetizationfluctuationsare demonstrates that the anomalous Hall coefficient is pos- rather strongly enhanced by the confinement. itive in this case. The vertical axis of Fig. 1(a) presents The findings for this and other samples showing sim- Ryx(T),asdeterminedbyhysteresisheightsatzeromag- ilar properties, collected in Fig. 3, demonstrate clearly netic field. As seen, TC of the channel layer increases thattherelationbetweentheHallresistanceandmagne- (decreases)by the applicationof negative(positive)gate tizationorcarrierpolarizationcanberathercomplex. In electric field which accumulates (depletes) holes in the particular, despite that R is virtually temperature in- xx channel, as witnessed in Fig. 1(b) by a corresponding dependentandtheholeliquiddegenerate,R and,thus, yx decrease (increase) of Rxx. Importantly, according to σxy decrease abruptly or even change sign on lowering Fig.1(a)alsothelow-temperaturevaluesofRyx increase temperature, as shown in Figs. 2 and 3. In this range, with Rxx changed by the gate electric field. Figure the temperature derivatives of |Ryx(T)| and M(T) ac- 1(c) shows σxy = Ryx/[t(Rx2x +Ry2x)] as a function of quire opposite signs, indicating that Hall measurements σxx =Rxx/[t(Rx2x+Ry2x)] under threedifferent valuesof for temperature dependent magnetometry should be ap- EG atT =10–30K<∼TC/2,wherethespontaneousspin pliedwithcare. Furthermore,therevealedchangeofsign splitting should be only weakly temperature dependent. of R calls into question the generalityof a simple scal- yx The dotted line is a guide for the eyes, which indicates ing formula. that the values of σxy fall on a single curve, verifying According to the recent theory of the intrinsic AHE anempiricalscalingbehaviorwithγ ≈1.6forthis structure. In12samplesofallsamples,γ isfoundtobewithin the range of 1.6±0.4, showing virtually no temperature dependence below TC/2 [9]. )120 I // [110] (a) anEdxspo-efraimrneonttarlepreosruteltdsbiellhuasvtrioartionfgRaynx(eTn)tiarerleypdreiffseenrteendt (Ryx 4800 -5.0 inFig.2(a)fortheMISstructurecontaininga4-nmthick ) 140 +5.0 MV/cm H ~ 0 Ga0.875Mn0.125As channel and the HfO2 gate insulator. (kRxx1102 (b) +5.0 MV/cm -5.0 8 0 20 40 60 80 100 120 140 H ~ 0 ) 1.0 T (K) (k)Ryx001...050 (a) --0+52 2M...055V MMM/cVVVm///cc cmmm (kRyx--1000....0505-100-500H1 0( 0mT5)0100 ()Ryx ---3211230000000 (c-)5.0-4.0-3.0-2.0-1.0 0 +0.5+1.0+1.5+2.0+2.5+3.0+3.5+4.0+41.5+05 .K0 MV/cm +5.0 MV/cm 10 K )40 (b) cm) 1250 KK 0.10 80 mT 0H ( mT) (kRxx20 (S/xy 1 2350 KK (T)M00..0068 (d) HH //// [[010010]] 0.04 (c) 0.02 0 0.1 0.00 0 10 20 30 40 50 60 70 10 100 1000 0 20 40 60 80 100 120 140 FIG.1: [Coloronline]T(a ()KH) alland(b)longitudinaxlx s(Sh/ecemt)resis- FIG.2: [Coloronline](a)HallaTn (dK)(b)longitudinalsheetresis- tances of 5-nm thick Ga0.949Mn0.051As (sample 3) at various tancesof4-nmthickGa0.875Mn0.125As(sample16)atvarious gate electric fields. Inset to (a) shows hysteresis loops of the gateelectricfields. (c)Hysteresisloopsatvariousgateelectric Hall resistance. (c) Relation between Hall and longitudinal fieldsdocumentingthesign change of theanomalous Hall ef- conductivities obeying a simple scaling law shown by dotted fect. (d)Temperaturedependenceofremanentmagnetization line. without gate bias. 3 ) 0 40 80 0 40 80 120 1.0 k 0.3 H ~ 0, 10 K -5 MV/cm R (yx 00..12 (a) (c) cm) 0.5 -5 MV/cm 0.0 S/ ( R (k)xx-1001.001 -74,. 0%, 3-2.6, nm 0,(b) 7.2%, 4.5 nm (d) xy-00..50 +5 MV/cm +6 MV/cm II //// [[1-11100]] +2, +4 MV/cm -5, 0 , +5 MV/cm 180 200 220 240 260 280 ) 1 R (kyx000...012 (e) (g) FGanIaGd0..8[¯17451:M0]Hn(0sa.al1lm25cpAolnesd1cuh7c)atniivnni-etplyslavonsrex.ixe cn(rcStyoe/sndctdmaalul)locotgnirvgaipt[y1h1icf0o]ror(4isea-nnmtmaptlietohn1isc6.k) )-0.1 8.9%, 4.5 nm 10.0%, 4.0 nm 100 k (f) (h) R (xx 10 result of this experiment does not provide a support for -5, 0, +5 MV/cm -4, 0, +4 MV/cm 1 the structural asymmetry model, it does not disprove it, 0 40 80 0 40 80 120 asanaccidentaldegeneracycannotbeexcluded(e.g.,due T (K) FIG. 3: [Color online] Hall Ryx and longitudinal Rxx sheet toacompensationofasymmetryinthevaluesoftheden- resistances at variousvalues of thegate electric fieldsin MIS sity of states and relaxation time). structures [(a) and (b) for sample 11, (c) and (d) for sample Another possibility is that dimensional quantization 12, (e) and (f) for sample 14, and (g) and (h) for sample 15] showing a strong temperature dependence of Ryx, similar to of the transverse motion leads to a significant recon- theone presented in Fig. 2. struction of the topology of the Fermi surface introduc- ing,in particular,a number ofadditionalsubbandcross- ings. This scenario may explain why the negative sign in (Ga,Mn)As [5], an appropriately strong tensile strain of σ appears on depleting the channel in one struc- xy or the bulk inversion asymmetry— the Dresselhaus ture[Fig.2(a)],whereasitshowsupinthe accumulation effect[14]—can result in a negative sign of the AHE for regime in another sample [Fig. 3(a)]. A detailed Fermi a sufficiently small magnitude of scattering broadening sheetbehaviorinthis region(e.g.,crossingvs. anticross- Γ. In fact, a negative sign of the R has been observed ing) may be sensitive to symmetry lowering and phase yx in films of (In,Mn)As [12, 13], (In,Mn)Sb [15, 16], and breaking mechanisms such as spin-orbit as well as tem- (Ga,Mn)Sb [17], where the bulk asymmetry is expected perature dependent inelastic and spin-disorder scatter- to be much stronger. However, the striking behavior of ing. At the same time, the confinement-induced upward R (T), including the change of sign revealed here, has shift of the hole energiesmay enhance the importance of yx not been anticipated theoretically [4,5]. It appears to be the Dresselhaus contribution determined by the admix- related to the reduced dimensionality, as it has not been ture of the conduction band wavefunctions. Within this observed in thicker (Ga,Mn)As films grown by us with scenario,thebehaviorofσ asafunctionoftemperature xy similar conductivity TC and strain [18, 19]. reflectsasubtleandspinpolarization-dependentbalance In an attempt to interpret our findings we note that betweenpositiveandnegativetermsoriginatingfromthe thin layers are expected to exhibit structure inversion intra-atom and bulk inversion asymmetry electric fields, asymmetry constituting an additional source of spin- respectively. orbit coupling. Despite that the gate electric field does In Fig. 5 we summarize our findings by reporting σ xy not affect significantly the temperature dependence of asafunctionofσ at10Kforvariousgateelectricfields. xx Ryx,asseeninFig.3,thepresenceoftwodifferentinter- Theresults[18,19,20]obtainedpreviouslyforthickfilms faces can account for the structure asymmetry and the are also shownfor comparison. Despite the fact that the corresponding lowering of the point symmetry from D2d particular samples may differ in magnetization values— toC2v. Inorderto explainthe datawithinthis scenario, which may likely affect the form of the scaling [21]— the structure asymmetry contribution to Ryx should be we see that over a wide range of conductivities up to negative and its amplitude take over the bulk terms at 102 S/cm the relation σ ∼ σγ is obeyed and implies xy xx sufficiently high magnetization values. γ ∼ 1.6, in agreement with the value γ = 1.5 found Since the in-plane anisotropy of the conductivity ten- for thick films of (Ga,Mn)As in the magnetic field [22]. sor would provide a direct proof that the symmetry is However,thescalingrelationbreaksdownratherseverely lowered to C2v, we have examined MIS structures origi- inthecaseofMISstructures,asdiscussedabove. Wealso nating from the same wafer but having the channels ori- note that γ determined at each temperature has almost ented along either [110] or [¯110] crystal axis. As shown notemperaturedependencebelowTC/2,accordingtothe in Fig. 4, similar values of Hall and longitudinal conduc- results shown in Fig. S2 of the supplementaly material. tance are observed in both cases. Though the negative However, on approaching TC we have found the scaling 4 in-Aids from MEXT/JSPS, the GCOE Program at To- 3.0%, 4.0 nm (Sample 1) H ~ 0, EG = -5 ~ +5 MV/cm 3.0%, 4.0 nm (Sample 2) hoku University, and the Research and Development for 5.1%, 5.0 nm (Sample 3) 101 5.1%, 5.0 nm (Sample 4) Next-Generation Information Technology Program from 5.1%, 5.0 nm (Sample 5) 5.2%, 4.5 nm (Sample 6*) MEXT, whereas the work in Warsaw was supported by 6.5%, 3.5 nm (Sample 7) 6.5%, 4.0 nm (Sample 8) the President of the PolishAcademy of Sciences for doc- 0 6.5%, 4.5 nm (Sample 9) m)10 6.5%, 5.0 nm (Sample 10) toralstudents (A.W.)andbyFunDMS AdvancedGrant 7.0%, 3.6 nm (Sample 11) S/c 77..25%%,, 44..50 nnmm ((SSaammppllee 1123**)) ofEuropeanResearchCouncilwithin"Ideas"7thFrame- (xy10-1 1 0 8..09%%,, 44..05 nnmm ((SSaammppllee 1154**)) work Programme of EC (T. D.). 12.5%, 4.0 nm (Sample 16*) 17.5%, 4.0 nm (Sample 18*) Previous data for thick films 10-2 1.5-7.1%, 200 nm (2 K) 54..00%-7,. 01%8 ,& 5 02 5n mnm ( 1(05 KK)) [1] G.SundaramandQ.Niu,Phys.Rev.B59,14915(1999). [2] T.Jungwirth,Q.Niu,andA.H.MacDonald,Phys.Rev. 10-3 Lett. 88, 207208 (2002). [3] T. Jungwirth, J. Sinova,J. Mašek, J. Kučera, andA.H. -1 0 1 2 3 10 10 10 10 10 MacDonald, Rev.Mod. Phys.78, 809 (2006). xx (S/cm) FIG.5: [Coloronline]RelationbetweenHallandlongitudinal [4] N.Nagaosa, J.Sinova,S.Onoda,A.H.MacDonald,and conductivities at various values of gate electric fields EG at N. P. Ong,arXiv:0904.4154. 10 K (30 K for Sample 1) for the studied MIS structures of [5] A. Werpachowska and T. Dietl, arXiv:0910.1907 (2009). Ga1−xMnxAs. The legend shows the values of the Mn con- [6] E. I.Rashba, Phys.Rev.B 70, 201309(R) (2004). tent x, channel thickness t, as well as the sample number. [7] J. I. Inoue, T. Kato, Y. Ishikawa, H. Itoh, G. E. W. The asterisks (*) mark the samples which were annealed be- Bauer, and L. W. Molenkamp, Phys. Rev. Lett. 97, foredepositionofinsulatorlayer. Half-filledsymbolsshowthe 046604 (2006). absolute value of negative σxy. The previous data for thick [8] T. Fukumura,H.Toyosaki, K.Ueno,M.Nakano,T. Ya- (Ga,Mn)Asfilmsareshownbyopensymbols(circles[20],tri- masaki, and M. Kawasaki, Jpn. J. Appl.Phys. 46, L642 anglesup[18],anddiamonds[19]). Thedottedlineshowsthe (2007). dependenceσxy ∼σxγx, γ =1.6. [9] See EPAPS Document No. E-PRLTAO-xxx for a table and figures summarizing sample properties and uniformity. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html. to be not obeyed (Fig. S4). [10] K.Wagner,D.Neumaier,M.Reinwald,W.Wegscheider, Wenotethattheempiricalscalingfoundforthickfilms and D.Weiss, Phys.Rev.Lett. 97, 056803 (2006). aswellasforthinlayerswithlowconductancecannotbe [11] L.Vila,R.Giraud,L.Thevenard,A.Lemaitre,F.Pierre, explained by the intrinsic mechanisms of the AHE, as J.Dufouleur,D.Mailly,B.Barbara, andG.Faini,Phys. the correspondingtheory predicts a decrease ofσ with Rev. Lett.98, 027204 (2007). xy [12] H. Munekata, T. Abe, S. Koshihara, A. Oiwa, M. Hira- both hole density [2, 5, 23] and scattering time [5, 24] sawa, S.Katsumoto,Y.Iye,C.Urano,andH.Takagi,J. in a wide range of relevanthole concentrationsand spin- Appl. Phys.81, 4862 (1997). splittings. ThissuggeststhatthephysicsofAHEisdom- [13] F. Matsukura, D. Chiba, T. Omiya, E. Abe, T. Dietl, inated by the proximity to the Anderson-Mott localiza- Y. Ohno, K. Ohtani, and H. Ohno, Physica E 12, 351 tion boundary [25, 26]. So far the influence of quantum (2002). interference effects on the anomalous Hall conductance [14] G. Dresselhaus, Phys. Rev.100, 580 (1955). have been studied considering the side-jump and skew- [15] S. Yanagi, K. Kuga, T. Slupinski, and H. Munekata, Physica E 20, 333 (2004). scattering terms within the single-particle theory [27]. [16] G.Mihály,M.Csontos,S.Bordács,I.Kézsmárki,T.Wo- The data summarized in Fig. 5 call for the extension of jtowicz,X.Liu,B.Jankó,andJ.K.Furdyna,Phys.Rev. the theory towards the intrinsic AHE with the effects of Lett. 100, 107201 (2008). disorder on interference of carrier-carrier scattering am- [17] M. Eginligil, G. Kim, Y. Yoon, J. P. Bird, H. Luo, and plitudes taken into account. B. D.McCombe, Physica E 40, 2104 (2008). In summary, we have found that the anomalous Hall [18] D. Chiba, M. Yamanouchi, F. Matsukura, T. Dietl, and H. Ohno,Phys. Rev.Lett. 96, 096602 (2006). effectofthemagneticsemiconductor(Ga,Mn)Asacquires [19] Y. Pu, D. Chiba, F. Matsukura, H. Ohno, and J. Shi, qualitatively new and not anticipated features when di- Phys. Rev.Lett. 101, 117208 (2008). mensionality and disorder are reduced. The revealed [20] F. Matsukura, H. Ohno, A. Shen, and Y. Sugawara, strikingtemperaturedependenceoftheHallconductance Phys. Rev.B 57, R2037 (1998). indicates a significant contribution of confinement phe- [21] S. H. Chun, Y. S. Kim, H. K. Choi, I. T. Jeong, W. O. nomena. Athigherlevelsofdisorder,thescalingrelation Lee, K. S. Suh, Y. S. Oh, K. H. Kim, Z. G. Khim, J. C. betweenσ andσ ,similartothatobservedpreviously Woo, et al., Phys. Rev.Lett. 98, 026601 (2007). xy xx [22] S. Shen, X. Liu, Z. Ge, J. K. Furdyna, M. Dobrowolska, for thicker samples and so-far unexplained, is recovered. and J. Jaroszynski, J. Appl.Phys. 103, 07D134 (2008). The authors thank S. Murakami and N. Nagaosa for [23] T. Dietl, F. Matsukura, H.Ohno,J.Cibert, and D.Fer- useful discussions, and L. Ye for experimental support. rand,inRecentTrendsinTheoryofPhysicalPhenomena The work in Sendai was supported in part by Grant- in High Magnetic Fields, edited by I. Vagner (Kluwer, 5 Dordrecht,2003), p. 197, cond-mat/0306484. 261 (1994). [24] T. Jungwirth, J. Sinova, K. Y. Wang, K. W. Edmonds, [26] T. Dietl, J. Phys. Soc. Jpn. 77, 031005 (2008). R. P. Campion, B. L. Gallagher, C. T. Foxon, Q. Niu, [27] V.K.Dugaev,A.Crépieux,andP.Bruno,Phys.Rev.B andA.H.MacDonald,Appl.Phys.Lett.83,320(2003). 64, 104411 (2001). [25] D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66,