We report on the observation of weak localization in arrays of (Ga,Mn)As
nanowires at millikelvin temperatures. The corresponding phase coherence length
is typically between 100 nm and 200 nm at 20 mK. Strong spin-orbit interaction
in the material is manifested by a weak anti-localization correction around
zero magnetic field.Comment: 5 pages, 3 figure

B. L. Altshuler

D. Neumaier

D. Weiss

J. Sadowski

K. Wagner

S. Geißler

U. Wurstbauer

W. Wegscheider

English

arXiv

We report on the observation of weak localization in arrays of (Ga,Mn)As nanowires at millikelvin temperatures. The corresponding phase coherence length L(phi) is typically between 100 and 200 nm at 20 mK. Strong spin-orbit interaction in the material is manifested by a weak antilocalization correction around zero magnetic field

Neumaier, Daniel

Wagner, Konrad

Geißler, Stefan

Wurstbauer, Ursula

Sadowski, J.

Wegscheider, Werner

Weiss, Dieter

University of Regensburg Publication Server

Weak Localization in Ferromagnetic (Ga,Mn)As NanostructuresD. Neumaier,* K. Wagner, S. Geißler, U. Wurstbauer, J. Sadowski, W. Wegscheider, and D. WeissInstitut fu¨r Experimentelle und Angewandte Physik, Universita¨t Regensburg, 93040 Regensburg, Germany(Received 2 March 2007; published 13 September 2007)We report on the observation of weak localization in arrays of (Ga,Mn)As nanowires at millikelvintemperatures. The corresponding phase coherence length L is typically between 100 and 200 nm at20 mK. Strong spin-orbit interaction in the material is manifested by a weak antilocalization correctionaround zero magnetic field.DOI: 10.1103/PhysRevLett.99.116803 PACS numbers: 73.43.Jn, 72.25.Dc, 73.43.QtQuantum corrections to the resistance like weak local-ization are suppressed by a sufficiently strong perpendicu-lar magnetic field B [1]. Hence, the question arises whethersuch effects can be observed in ferromagnets having anintrinsic magnetic induction. While few experimentalworks explored this problem [2,3], a definite experimentalanswer is still lacking. Hence, the advent of the newferromagnetic semiconductor (Ga,Mn)As with signifi-cantly smaller internal field compared to conventionalferromagnets offers a new opportunity to address suchquestions. The spin 52-Mn-ions on regular sites of thezinc-blende lattice of the GaAs host act as acceptors,thus providing both holes and magnetic moments. Theferromagnetic order between the Mn-ions is mediated bythese holes [4]. By now, ferromagnetism in (Ga,Mn)As iswell understood, allowing to predict Curie temperatures[4], magnetocrystalline anisotropies , as well as the aniso-tropic magnetoresistance effect [6]. In this respect,(Ga,Mn)As is one of the best understood ferromagneticmaterials at all [7] and hence suitable as a model system tostudy quantum corrections to the conductivity.Interference effects originating from the charge carriers’wave nature are barely explored and understood in ferro-magnets in general and in (Ga,Mn)As in particular. To thisclass of effects belong universal conductance fluctuations(UCF) [8], the Aharonov-Bohm (AB) effect [9], weaklocalization (WL) [1], weak antilocalization (WAL) [1],and electron-electron interactions (EEI) [10]. In(Ga,Mn)As, the phase coherence length was extractedfrom UCFs in nanowires giving typical values between90 and 300 nm at 20 mK [11,12]. This raises the questionwhether WL corrections—or WAL effects—can be ob-served in ferromagnetic (Ga,Mn)As, a material in whichthe spin-orbit (SO) interactions for holes in the valenceband is quite strong.Below, we report the observation of WL and WAL inferromagnetic (Ga,Mn)As wires and films, thus demon-strating that WL is not destroyed by the ferromagnets’magnetization. The effect of WL in disordered electronicsystems—investigated intensively in the past for nonfer-romagnetic materials [13]—is due to quantum interferenceon time-reversed paths. This leads to an enhanced proba-bility of backscattering. As an applied perpendicularB-field suppresses the WL, the magnetoconductance ispositive [1]. In the presence of SO interaction, the spinpart of the wave function needs to be taken into account.The two partial waves on time-reversed closed paths ex-perience a spin rotation in opposite direction causing (par-tially) destructive interference [1]. So, SO interactionsleads to reduced backscattering and reverses the sign ofthe WL, hence called weak antilocalization. A typicalsignature of WAL is a double dip in the magnetoconduc-tance trace [1].For the experiments, two wafers having a 42 nm and a20 nm thick (Ga,Mn)As layer were used. Both were grownby low-temperature molecular beam epitaxy deposited onsemi-insulating GaAs(001) [14]. The nominal Mn concen-tration of the 42 nm layer was 5.5%, of the 20 nm layer 5%.The Curie temperature TC of the as grown layer was 90 K(42 nm) and 55 K (20 nm), respectively. The samples’remanent magnetization was always in-plane. Some ofthe samples were annealed at 200 C increasing bothcarrier density and TC [15]. To investigate phase coherentproperties Hall-bar mesas, individual nanowires and arraysof wires were fabricated employing optical and electronbeam lithography. For nanowire fabrication, we used ascanning electron microscope equipped with a nanonicpattern generator and subsequent reactive ion etching. Aucontacts to the devices were made by lift-off technique.The characteristic parameters of the samples investigatedare listed in Table I.TABLE I. Length L, width w, and thickness t of the samples.Some of the samples were annealed at 200 C. Resistivity  andcarrier concentration n were taken at T  300 mK.Sample 1a 2 2a 3 4L (m) 60 7.5 7.5 7.5 0.37w (nm) 7200 42 42 35 35t (nm) 20 42 42 42 42Number of wires N 1 25 25 12 1tanneal at 200 C (h) 8.5    51      n (1026=m3) 1.7 3.8 9.3 3.8 3.8 (105 m) 13 3.5 1.8 3.5 3.5TC (K) 95 90 150 90 90PRL 99, 116803 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending14 SEPTEMBER 20070031-9007=07=99(11)=116803(4) 116803-1 © 2007 The American Physical SocietyMagnetotransport was measured in a dilution refrigera-tor. To avoid heating, we used a low frequency (19 Hz) andlow current (25 to 200 pA) four probe lock-in technique.As we see no effects of saturation for the different experi-ments (UCF, WL, and conductivity decrease) at low T, weassume that the effective electron temperature is in equi-librium with lattice and bath temperature even at 20 mK.To search for WL effects in (Ga,Mn)As wires, we mea-sured the resistance of N parallel wires to suppress UCFsby ensemble averaging. A corresponding micrograph ofsample 2 with 25 wires is shown in Fig. 1(a). The sample’sconductance as a function of a perpendicular B field isshown in Fig. 1(b) [16]. First, we start with a description ofthe dominant features observed in experiment. The pro-nounced conductance maxima around B 0 are due to theanisotropic magnetoresistance (AMR) effect [6]. For an in-plane magnetization, the conductance is higher than for anout-of-plane orientation of M [17]. The positive slope of Gfor higher B is ascribed to increasing magnetic order [18]or to weak localization in bulk (Ga,Mn)As [19]. For tem-peratures larger than 65 mK, the different GB tracesare shifted but without noticeable change of shape, and theAMR peak height scales with the high-field backgroundconductance. The decreasing G for decreasing T inFig. 1(b) stems from the usual low T behavior of theresistance in (Ga,Mn)As which is plotted in Fig. 1(c).With decreasing T, the resistance rises and is ascribed toEEI. Similar low T behavior has been reported previouslyfor conventional ferromagnets, too [3,20]. According totheory [21], the EEI conductivity correction for 1D sys-tems goes with T1=2. The corresponding conductancecorrection   T  50 mK of our sample 2,taken at B  0 and at B  3 T, is plotted in Fig. 1(d) vsT1=2. The resulting straight lines for both B values dem-onstrate the expected T dependence, prove that the correc-tion is independent of B, and hence suggest that EEI isaccountable for the conductance decrease at low T. Thenovel features which are in the focus of this Letter appearat still lower temperatures. At about 50 mK, two downwardcusps at  0:4 T start to become noticeable and havedeveloped to a prominent feature at 20 mK.FIG. 1. (a) Electron micrograph of sample 2. (b) Conductanceof sample 2 for different temperatures measured in a perpen-dicular magnetic field [29]. The magnetic field range where themagnetization is rotated from in-plane to out-of-plane is gray-shaded. (c) Increase of resistance with decreasing temperature.This increase stems from EEI as proven by the T1=2 powerlaw for 1D-systems at B  0 and B  3 T in (d). Here, G istaken relative to the conductivity at 50 mK.-3 -2 -1 0 1 2 3GB(T)FIG. 2 (color). WL contribution for three wire and one 2Dsample obtained after subtracting the 120 mK trace as back-ground conductance. To compare the different samples, the totalG was divided by the number of parallel wires. In case of the2D-sample, 1a G was divided by 15 to fit into the graph. Again,the gray shaded B-range corresponds to the regime where thesamples’s magnetization follows the external field and changesdirection. The red lines are best fits to Eq. (1), discussed in thetext. The fit parameters were L  190 nm and LSO  113 nmfor sample 2a, L  150 nm and LSO  93 nm for sample 2,and L  160 nm and LSO  93 nm for sample 3. Fitting the2D sample requires a different formalism which is beyond thescope of the present Letter.PRL 99, 116803 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending14 SEPTEMBER 2007116803-2To separate the peculiar low T conductance featuresfrom the ‘‘high temperature’’ background, G G20 mK  G120 mK of four samples was takenand plotted in Fig. 2. The factor  takes the T depen-dence of G into account and is given by  G20 mK=G120 mK, taken at 3 T. We note, though,that putting   1 does not change G qualitatively asthe conductance change is only 10%. To compare thedifferent samples, G was normalized by the number ofparallel wires, N. All traces in Fig. 2 show a characteristicbroad conductance minimum for jBj< 1 T and a localmaximum at B 0 T. Such GB line shapes are char-acteristic for WAL in systems with spin-orbit interaction.Here, we have assumed that the T-dependency of the AMRcontribution does not change for T < 65 mK.To extract the characteristic lengths from the WL cor-rection, we compare the data of Fig. 2 with existing theory.In Fig. 3(a), we particularly compare the WL correction ofsample 3, with the standard expression for WL correctionin 1D. Since the width w and thickness t of our wires aresmaller than the phase coherence length L, which isexpected to be of order 100 nm at 20 mK [11], w t <L 	 L holds and the 1D assumption is justified. Thecorresponding equation for the conductance correctionreads [22,23] G  gs e2h12L1L2 13w2L4H1=2 32L1L2 43L2SO 13w2L4H1=2; (1)where gs is the spin degeneracy. Here, LSO DSOpis thespin-orbit length that characterizes the strength of spin-orbit coupling, L  Dp , and LH  @=eBp is the mag-netic length. Equation (1) is fitted to the WL data inFig. 3(a) for sample 3. As the valence band is spin split,the holes are highly (but not fully) spin polarized [24]. Toaccount for spin polarization, we approximate gs either by1 (fully spin polarized) or by 2 (spin degenerate) as adjust-able parameter. While the fit for gs  1 matches the con-ductance minima at 400 mT as well as the conductancecorrection G, the fit for gs  2 is less satisfying. Theparameters used for the fit were L  160 nm, LSO 93 nm for gs  1 and L  90 nm, LSO  50 nm forgs  2, respectively. Also, the WL data of the othersamples can be modeled by Eq. (1) and gs  1; the corre-sponding fits and parameters are given in Fig. 2.The size of the weak (anti)-localization contribution inFig. 2 and 3(a) is quite nicely fitted by two parameters, Land LSO. L can also be extracted from UCFs measured onindividual 1D-wires [11,12]. To study UCFs, we fabricateda single wire, w  35 nm wide and L  370 nm long,from the same material as sample 2 and 3 (sample 4 inTable I). A corresponding electron micrograph is shown aslower left inset in Fig. 3(b). GB was measured in aperpendicular B-field from 3 T to 3 T for T between20 mK and 1 K (for details see [11]). Corresponding datataken at 20 mK show pronounced, reproducible UCFs,displayed in Fig. 3(b). The root mean square amplitudeGrms hG hGi2ip of these fluctuations is connectedwith L, and the wire length L by Grms  e2=hL=L3=2fL=LSO [25]. The function fL=LSO takesspin-orbit interaction into account. For L=LSO  1:7, weobtain fL=LSO  0:56 [25]. Extracting L from Grms,taking only the fluctuations between 400 mT in Fig. 3(b)into account, results then in L  118 nm. The value ofL, extracted from UCFs, is thus in surprisingly goodagreement with the ones used to fit the WL correction.Hence, our analysis suggests that LSO ranges between93 nm and 113 nm in our devices.While WAL was observed, e.g., in nonmagnetic p-typeAl;GaAs=GaAs quantum wells [26] or in (In,Ga)Asquantum wells [27], the observation of WAL-signature inferromagnetic (Ga,Mn)As comes as a surprise. A recentFIG. 3 (color). (a) WL correction of sample 3 fitted with thestandard 1D WL theory [Eq. (1)] for spin degeneracy gs  1(red) and gs  2 (blue). (b) UCFs measured in an individual 1D-wire made from the same material (sample 4). An electronmicrograph of the wire is shown in the lower left inset. Thegray shaded regime again corresponds to the magnetic fieldrange where M changes direction. The upper inset shows thelow-field UCFs in a expanded magnetic field scale. The tem-perature dependence of G, extracted from the low-field fluctu-ations, is shown in the lower right inset.PRL 99, 116803 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending14 SEPTEMBER 2007116803-3theory suggests the processes, leading to WAL in non-magnetic systems, to be totally suppressed in ferromagnets[28]. The suppression of WAL in ferromagnets is due to thestrong magnetic polarization which excludes contributionsfrom the so-called singlet Cooperon diagrams, responsiblefor WAL. As a consequence, the quantum correction to Gis expected to be exclusively negative in ferromagnets,leading to positive magnetoconductance. This clearly con-tradicts our experimental observation.While the fits in Figs. 2 and 3(a) are in good agreementwith experiment for jBj< 400 mT, the concordance atlarger B is less perfect. The WL or WAL correction is, asa function of increasing B, more abruptly suppressed thanexpected from theory. There is a striking correlation withthe magnetic field dependence of the AMR effect. Themagnetic field region where the AMR occurs is highlightedby gray shading in Figs. 1(b), 2, 3(a), and 3(b). Within thisB-field range, the magnetization is rotated from in-plane toout-of-plane. Once the magnetization is out-of-plane, theWL correction drops quickly. At the same magnetic field,the magnetic length matches wire width and thickness,LH  w, t. Hence, the discrepancy between fit and experi-ment might be associated with dimensional crossover (1Dto 3D), if jBj exceeds 400 mT and Eq. (1) might beinapplicable. In the gray shaded B-field range, the fluctua-tions of an individual wire show a reduced correlation fieldBC. Corresponding data are displayed in Fig. 3(b), magni-fied in the upper inset. Similar behavior was observed inprevious experiments on samples with in-plane easy axis[11,12] and ad hoc ascribed to the formation of domainwalls in [12]. Though we can not exclude such a scenario,we note that BC is not a well defined quantity in the regimewhere the (magnetic) configuration changes.The observation of WAL, contrary to theoretical ex-pectation, the abrupt suppression of the WL correctiononce the magnetization is saturated, as well as the anoma-lous BC in the low B-regime suggest that some impor-tant ingredients are still missing to describe interferencephenomena in (Ga,Mn)As. This is not too surprising asneither the field dependent change of the magnetizationdirection nor the 32 -spin of the involved hole states wastaken into account. Especially, the latter could add a num-ber of additional interference diagrams not yet treatedtheoretically.In summary, we have shown that quantum inferenceeffects strongly affect the low-temperature conductanceof ferromagnetic (Ga,Mn)As. By resolving a clear weaklocalization signature, we demonstrate that interferencedue to scattering on time-reversed paths can exist also inferromagnetic materials with internal magnetic induction.The strong spin-orbit interaction in (Ga,Mn)As is mani-fested by a weak antilocalization contribution at low B.We thank K. Richter, I. Adagideli, J. Fabian, T. Dietl,A. Geim, and J. Nitta for valuable discussions. Financialsupport by the Deutsche Forschungsgemeinschaft (DFG)via SFB 689 is gratefully acknowledged.*daniel.neumaier@physik.uni-regensburg.de[1] G. Bergmann, Phys. Rep. 107, 1 (1984).[2] M. Aprili et al., Solid State Commun. 102, 41 (1997).[3] M. Brands et al., Phys. Rev. B 72, 085457 (2005).[4] T. Dietl et al., Science 287, 1019 (2000).[5] M. 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As the maximum(saturation) value of Jz in our samples is only of order40 mT, we approximate B0z  Bz  B.[17] H. Ohno, Science 281, 951 (1998).[18] E. L. Nagaev, Phys. Rev. B 58, 816 (1998).[19] F. Matsukura et al., Physica E (Amsterdam) 21, 1032(2004).[20] T. Ono et al., J. Magn. Magn. Mater. 226, 1831 (2001).[21] P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57,287 (1985).[22] B. L. Altshuler and A. G. Aronov, JETP Lett. 33, 499(1981).[23] F. Pierre et al., Phys. Rev. B 68, 085413 (2003).[24] J. G. Braden et al., Phys. Rev. Lett. 91, 056602 (2003).[25] V. Chandrasekhar, P. Santhanam, and D. E. Prober, Phys.Rev. B 42, 6823 (1990).[26] S. Pedersen et al., Phys. Rev. B 60, 4880 (1999).[27] T. Koga et al., Phys. Rev. Lett. 89, 046801 (2002).[28] V. K. Dugaev, P. Bruno, and J. Barnas´, Phys. Rev. B 64,144423 (2001).[29] In sample 2 and 2a, the two voltage leads were fabricatedon opposite sites. So, also the Hall conductance wasmeasured. To remove the Hall conductance from thedata, only the symmetric part of the trace was taken. Asthis leads to the same result then seen on sample 1a and 3,where the voltage leads are on the same side, physics is notchanged by this procedure.PRL 99, 116803 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending14 SEPTEMBER 2007116803-4

Weak localization in ferromagnetic (Ga,Mn)As nanostructures

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