Strong spin-orbit interactions and weak antilocalization in carbon doped p-type GaAs heterostructures
Boris Grbic, Renaud Leturcq, Thomas Ihn, Klaus Ensslin, Dirk Reuter, Andreas D. Wieck
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Strong spin–orbit interactions and weak antilocalizationin carbon doped p-type GaAs heterostructures
Boris Grbi´c ∗ , Renaud Leturcq ∗ , Thomas Ihn ∗ , Klaus Ensslin ∗ , Dirk Reuter + , and Andreas D. Wieck + ∗ Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland, + Angewandte Festk¨orperphysik, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany
We present a comprehensive study of the low-field magnetoresistance in carbon doped p-typeGaAs/AlGaAs heterostructures aiming at the investigation of spin–orbit interaction effects. Thefollowing signatures of exceptionally strong spin-orbit interactions are simultaneously observed: abeating in the Shubnikov-de Haas oscillations, a classical positive magnetoresistance due to thepresence of the two spin-split subbands, and a weak anti-localization dip in the magnetoresistance.The spin-orbit induced splitting of the heavy hole subband at the Fermi level is determined to bearound 30 % of the total Fermi energy. The phase coherence length of holes of around 2.5 µ m ata temperature of 70 mK, extracted from weak anti-localization measurements, is promissing for thefabrication of phase-coherent p-type nanodevices. I. INTRODUCTION
Two-dimensional (2D) systems with strong spin-orbitinteractions (SOI) are promising for the realization ofspintronics devices due to the fact that in such systemsthe electron (hole) spin could be affected, not only bymagnetic, but also by electric fields [1, 2]. SOI are ex-pected to be very strong in p-type GaAs heterostructures,due to the high effective mass of holes [3] which makesthe ratio of the SOI energy and the kinetic energy largerthan in the conduction band. As a result of the SOI, theheavy hole subband in GaAs is split into two subbandseven in the absence of an external magnetic field.In magnetotransport experiments the existence of twospin-split subbands with different mobilities results in aclassical positive magnetoresistance. In addition, a beat-ing can be observed in Shubnikov-de Haas (SdH) oscilla-tions, because the Landau levels of the two non-equallypopulated subbands give rise to magnetoresistance oscil-lations with slightly different 1 /B -periodicities [4]. Whilethese two signatures can be observed in any two-subbandsystem, the spin splitting due to SOI can be unambigu-ously identified and characterized by measurements ofthe weak anti-localization effect.Weak localization is a quantum mechanical effectwhich arises from the constructive interference betweentime reversed partial waves of the charge carriers in dis-ordered materials. It leads to an enhanced probability ofcarrier backscattering and therefore to an enhanced lon-gitudinal resistivity. This interference effect is relevantfor diffusive orbits up to the length scale l ϕ , the phase-coherence length. The application of a magnetic fieldnormal to the plane of carrier motion breaks the timereversal symmetry, suppresses the weak localization, andtherefore leads to a negative magnetoresistance at lowmagnetic fields around B = 0 [5].In systems with strong SOI the spin dynamics of thecarriers is coupled to their orbital motion and the in-terference of time-reversed paths has consequences be-yond the weak localization effect. As the spin experi-ences a sequence of scattering events along its path, the spin orientation is randomized on a characteristic lengthscale l so . The stronger the SOI, the smaller is l so . At B = 0, the interference of time reversed paths leads toa reduction of the backscattering probability below itsclassical value [6], an effect called weak anti-localization,if l so ≪ l ϕ (strong SOI). It manifests itself as a positive(rather than a negative) magnetoresistance at small fieldsaround B = 0 [7].Weak anti-localization was experimentally observed byBergmann in thin metallic films [8]. As the strengthof SOI is increased, a transition from weak localiza-tion to weak anti-localization is observed. Weak anti-localization was subsequently observed also in semicon-ductor heterostructures [9, 10]. A smaller zero-fieldanti-localization resistance minimum superimposed on alarger weak localization peak was seen in the magnetore-sistance of an inversion layer of InP [9], and an n-typeGaAs/AlGaAs heterostructure [10]. A fully developedanti-localization minimum was observed by Chen et al. in the magnetoresistance of an InAs quantum well [11].Koga et al. demonstrated the transition from a zero-fieldweak localization maximum to a weak anti-localizationminimum by tuning the symmetry of an InGaAs quan-tum well (QW) wih a metallic top-gate [12].Weak anti-localization is expected to be particularlyexpressed in the case of p-type GaAs heterostructuresdue to the strong SOI in these systems. Experimentalstudies of weak anti-localization in Be-doped (100) p-typeGaAs heterostructures are reported in Refs. 13 and 14,and a detailed study of the low-field magnetoresistance inSi-doped (311) p-type GaAs heterostructures is presentedin Ref. 15.Here we report measurements of the classical magne-toresistance, SdH oscillations, and weak anti-localizationin C-doped p-type GaAs heterostructures. Weak anti-localization is typically more pronounced in diffusive,low-mobility samples, while for the observation of beat-ing in SdH oscillations higher mobility samples are re-quired. The fact that our sample is in the regime of inter-mediate mobilities enables us to simultaneously observeboth effects and to perform a complementary analysis ofspin–orbit interactions in the system. The observationof a fully developed anti-localization minimum around B = 0 clearly demonstrates the presence of very strongSOI. A phase-coherence time of the holes of around 190ps, corresponding to a phase-coherence length of 2.5 µ mis extracted from these measurements. We investigatethe temperature dependence of the phase-coherence timeof holes and find that it obeys a 1 /T dependence withreasonable accuracy. Limitations in extracting the spin-orbit scattering time are due to the fact that SOI is verystrong. It cannot be treated as a weak perturbation only,as discussed below. II. SAMPLE AND MEASUREMENT SETUP
We have studied the low-field magnetoresistance intwo C-doped p-type GaAs heterostructures with the two-dimensional hole gas (2DHG) buried 45 nm and 100 nmbelow the surface. The results obtained from both sam-ples are qualitatively the same. For the sake of clarity,we present here results obtained on the sample with the2DHG formed at the interface 100 nm below the sam-ple surface. The heterostructure consists of a 5 nm C-doped GaAs cap layer, followed by a 65 nm thick, ho-mogeneously C-doped layer of Al . Ga . As which isseparated from the 2DHG by a 30 nm thick, undopedAl . Ga . As spacer layer [16]. A rectangular Hall barwas fabricated by standard photolithography. Its widthis 100 µ m and the separation between adjacent voltageleads is 500 µ m. Ohmic contacts were formed by evapo-rating Au and Zn and subsequent annealing at 480 o C for2 min. Afterwards, a homogeneous Ti/Au topgate wasevaporated, which allows to tune the density in the rangeof 2 − × cm − . The average mobility in the sam-ple at T = 70mK is 160’000 cm /Vs at a density 3 × cm − . The high quality of the investigated sample hasbeen demonstrated by the observation of the fractionaland integer quantum Hall effects, as well as by measure-ments of highly resolved SdH oscillations [17].The Hall bar is fabricated along one of the two maincrystallographic directions in the (100) plane. Measure-ments at T = 4 . µ T. III. BEATING OF SHUBNIKOV-DE HAASOSCILLATIONS
We have mentioned before that the two spin-splitheavy hole subbands arising as a result of SOI lead toa beating of SdH oscillations. Figure 1(a) displays amagnetoresistance trace taken in the magnetic field rangebetween -0.1 T and 1.6 T showing SdH oscillations. TheFourier analysis of the magnetoresistivity ρ xx vs. 1 /B [see inset of Fig. 1(a) and discussion below] is used todeduce the densities N , of the two spin-split subbands.They are related to the two frequencies f , obtainedfrom the Fourier transform of the SdH oscillations via N , = ( e/h ) · f , [3, 4, 20].Three magnetic field regimes can be identified in theraw data, where SdH oscillations exhibit a different be-havior. For very low fields in the interval 0 . < B < . . < B < . < B < . f + f = f tot , which reflectsthe fact that the two subband densities sum up to thetotal density, is reasonably satisfied. For magnetic fieldsabove approximately 2 T (not shown) we observe mag-netoresistance oscillations related to the total density inthe system, and only the total density peak is present inthe Fourier spectrum.From the three peaks in the Fourier transform shownin the inset of Fig. 1(a) we read the densities of thetwo spin-split subbands N = 1 × cm − , N =1 . × cm − , and the total density N = 3 × cm − .This corresponds to a relative charge imbalance betweenthe two spin-split subbands ∆ N/N = 0 .
30. The strengthof the spin-orbit interactions can be quantified using thisrelative charge imbalance, if a cubic wave vector k de-pendence ∆ SO = 2 βk k is assumed for heavy holes in the(100) plane [3, 21]. The two subband’s Fermi wavevec- N(x10 cm -2 ) ∆ S O ( m e V ) ∆ SO =2 β k Iow3 E F (meV) B (T) ρ ( Ω ) xx FFT ( a . u . ) V TG =0 V; N=3.0x10 cm -2 (a) (b) FIG. 1: (a) Shubnikov-de Haas oscillations in the magnetore-sistance, with a top gate set to V TG = 0V, and the total den-sity 3 . × cm − ; Inset: Fourier transform of the shownmagnetoresistance, taken in the B-field range (0.4T, 1.5T)displaying three peaks. (b) Spin-orbit splitting energy of theheavy hole subband at the Fermi level as a function of theFermi energy in the system and the total density. tors k and k are different. This difference increases withincreasing spin-orbit interaction. From the general rela-tion k i = √ πN i we find k = 1 . × m − , and k =1 . × m − at the total density N = 3 × cm − .The energy splitting of the two spin-split subbands de-pends on the k -vector where this splitting is calculated.The values of the spin-splitting energy which we quotefurther in the text, are all obtained using the smaller ofthe two wavevectors, and therefore represent the lowerbound for the spin-splitting of the heavy hole subband.Using the masses of the carriers in the two spin-splitsubbands determined experimentally in Ref. 17 and thetwo subband densities, we calculate the spin-orbit cou-pling parameter from eq. (6.39) in Ref. 3 to be β =2 . × − eVm . This gives the spin-orbit inducedsplitting of the heavy hole subband ∆ SO ≈ . N = 3 × cm − . The Fermi energy for thisdensity is E F = 2 . SO /E F ≈ β increases with increasing den-sity by about 20%.The evolution of the spin-splitting energy ∆ SO uponchanging the total density in the system with the metallictop- gate is shown in Fig. 1(b). It can be seen that fordensities in the range 2 − × cm − the spin-splittingenergy is in the range of 0 . − . SO /E F , is quite large, increasing from 0.29 to0.35 with the Fermi energy increasing from 1.35 to 2 meV.This documents the presence of exceptionally strong SOIin C-doped p-GaAs heterostructure. IV. CLASSICAL POSITIVEMAGNETORESISTANCE
The longitudinal magnetoresistance of a system withtwo types of charge carriers with different mobilities isparabolic around zero magnetic field, whereas the Hallresistivity, contains a small cubic correction at low fieldsin addition to the usual term linear in B . This is a purelyclassical effect and follows from the standard Drude the-ory of conductivity [22]. If inter-subband scattering be-tween the two subbands is significant, a more complextheory based on the Boltzmann transport equation hasto be considered [23]. However, the qualitative behaviorof the low-field magnetoresistance remains very similarto that obtained using the simpler model neglecting in-tersubband scattering.In the transport theory of two-subband systems devel-oped by Zaremba [23], where inter-subband scattering isincluded, the longitudinal and transverse magnetoresis-tivity are given by ρ xx = m ∗ e · Re( 1Tr N ( K − iω c I ) − ) , (1) ρ xy = m ∗ e · Im( 1Tr N ( K − iω c I ) − ) , (2)where Tr stands for the trace operation, I is the 2 × N is a matrix defined as N ij = p N i N j ( N , N are the densities of the two subbands), and K isthe scattering matrix (cid:18) K − K − K K (cid:19) , where K , K are rates quantifying intra-subband scat-tering, while K is the inter-subband scattering rate.Previously, a strong positive magnetoresistance wasobserved in p-type (311) GaAs heterostructures [15].However, in that case the low-field magnetoresistancecould not be fitted satisfactorily with the two-subbandtheory, even when intersubband scattering was taken intoaccount. This finding was attributed to the strong mo-bility anysotropy in (311) samples. B (T) ρ ( Ω ) xx ρ ( Ω ) xx ∆ ρ ( Ω ) xy ∆ ρ ( Ω ) xy TG =0 VN=3.0x10 cm -2 V TG =1 VN=2.3x10 cm -2 data2-band fit K1=0.18x1011K2=0.45x1011K12=0.012x1011K1=0.33x1011K2=0.64x1011K12=0.086x1011 ρ xy - ρ xy (linear) ρ xy - ρ xy (linear) B (T) -0.2 0.1-0.1 0 0.15
B (T) -0.15 -0.1 0.10.05-0.05 0 0.15
B (T) -0.15 -0.1 0.10.05-0.05
FIG. 2: Left column: Fit of the low-field magnetoresistivitywith the two-band theory[23] (black lines represent the mea-surement, and thicker gray lines are fitted lines) in the follow-ing gate configurations: (a1) V TG = 0, N = 3 . × cm − ,(b1) V TG = 1 V, N = 2 . × cm − . Right column:Nonlinearity in the low-field Hall resistivity (black lines aremeasured data, gray lines are calculated curves, see text fordetailed explanations) in the following gate configurations:(a2) V TG = 0, N = 3 . × cm − , (b2) V TG = 1 V, N = 2 . × cm − . Figure 2 shows a strong positive magnetoresistancearound B = 0 in two gate configurations: (a1) V TG = 0, N = 3 . × cm − , and (b1) V TG = 1 V, N =2 . × cm − . The black lines correspond to measureddata, while the thicker gray lines show the fits follow-ing eq. (1) in the range | B | < .
15 T for V TG = 0, and | B | < . V TG = 1 V,where SdH oscillations are notyet developed. In the fitting procedure, the densities ofthe two subbands N , N are fixed parameters given fromthe Fourier analysis of the SdH oscillations, whereas thescattering rates K , K , K are fitting parameters.In the configuration V TG = 0, N = 3 . × cm − [Fig. 2(a1)] the scattering rates are K = 0 .
018 ps − , K = 0 .
045 ps − , and K = 0 . − . Theinter-subband scattering rate is much smaller than theintra-subband scattering rates. The corresponding sub-band mobilities are µ = 270 ′
000 cm /Vs and µ =110 ′
000 cm /Vs. These values explain why in SdH mea-surements oscillations arising from the subband withpopulation N are observed at lower magnetic fields thanthose from the subband with population N . In the sec-ond configuration with V TG = 1 V, N = 2 . × cm − [Fig. 2(b1)] the scattering rates are K = 0 .
033 ps − , K = 0 .
064 ps − , and K = 0 . − . Again theinter-subband scattering rate is about one order of mag-nitude smaller than the scattering rates of individual sub-bands. However, as the density is reduced, we observethat the scattering rates of the individual subbands in- crease by less than a factor of 2, while the intersubbandscattering rate increases by a factor of 7. Such a behaviorcan be related to the fact that the energy separation ∆ SO between the two spin-split bands decreases with densityand therefore it is easier for the carriers to scatter fromone subband to the other. By reducing the density, theparabolic feature in the magnetoresistance around B = 0becomes broader and shallower.We have also observed that an increase of the tem-perature causes a broadening of the magnetoresistanceminimum around B = 0, and also an increase of theintersubband scattering rate. The intersubband scatter-ing rate increases faster with increasing temperature thanthe intrasubband scattering rates. This indicates that thepresence of the two spin-split subbands in p-type samplesmight be relevant for the strong temperature dependenceof the resistivity even at mK temperatures.Beside the longitudinal magnetoresistance minimumaround B = 0, the presence of the two spin-split sub-bands also modifies the Hall resistivity around B = 0and introduces non-linear corrections (see eq. 2) [23]. Wehave calculated the Hall resistivity using the scatteringrates K , K and K obtained from the ρ xx -fits as in-put parameters. In order to make these small non-linearcorrections to the Hall resistivity visible, we subtractthe linear contributions from both the measured dataand the calculated ρ xy . The result for the measureddata (black lines) and the calculated ρ xy (gray lines) ispresented in Figs. 2(a2) (configuration V TG = 0, N =3 . × cm − ) and 2(b2) (configuration V TG = 1 V, N = 2 . × cm − ). We find reasonable agreement be-tween the data and the simulated non-linear correctionsof the Hall resistivity. V. WEAK ANTI-LOCALIZATIONMEASUREMENTS
Weak (anti)localization effects are observable in lowermobility samples in the diffusive transport regime, if thecarrier mean free path is much smaller than the phase-coherence length. In higher mobility samples where k F l m ≫ k F –Fermi wavevector, l m –mean free path) lo-calization effects are weaker and harder to resolve. Themeasured density and mobility values in the investigatedsample give k F l m ∼ − B -field rangearound B = 0 with very high accuracy.In the left column of Fig. 3, the raw magnetoresistivitydata are presented for the gate-configurations V TG = 0, N = 3 × cm − , µ = 160 ′
000 cm /Vs, k F l m = 200[Fig. 3 (a1)], and V TG = 1 V, N = 2 . × cm − , µ =130 ′
000 cm /Vs, k F l m = 120 [Fig. 3 (b1)]. In both caseswe observe a sharp anti-localization resistance minimumaround B = 0 with a magnitude much smaller than 1 Ω. -5 0 5 10-1-0.6-0.2-10 B (mT) -0.6-0.4 -5 0 5 10-10 B (mT) ρ − ρ c l a ss . ( Ω ) -0.8 ρ − ρ c l a ss . ( Ω ) ρ ( Ω ) ρ ( Ω ) V TG =1Vk F l m =120V TG =0Vk F l m =200 (b1) (b2)(a1) (a2) FIG. 3: Left column: Raw magnetoresistivity data at T =65 mK in the following gate configurations: (a1) V TG = 0, N = 3 × cm − , µ = 160 ′
000 cm /Vs, (b1) V TG = 1 V, N = 2 . × cm − , µ = 130 ′
000 cm /Vs. Right column:Quantum correction of the resistivity obtained after subtrac-tion of the classical two-band positive magnetoresistivity for(a2) V TG = 0, (b2) V TG = 1 V. It can be seen that the magnitude and the width of theanti-localization minimum become larger as the factor k F l m decreases due to a reduction of the sample mobilityand density at positive top gate.As discussed before a classical magnetoresistance min-imum is present around B = 0 due to the presence ofthe two spin-split subbands. In order to separate thequantum correction from the low-field magnetoresistiv-ity, we subtract the classical positive magnetoresistivity ρ class [thick gray lines in Fig. 2(a1) and 2(b1)] from thetotal resistivity ρ . The quantum corrections to the resis-tivity, ρ − ρ class , are plotted in the right column of Fig. 3for both gate configurations.It can be seen in Fig. 3(a2,b2) that in both cases a welldeveloped weak anti-localization minimum is present inthe low-field magnetoresistance. The fact that the nar-row weak anti-localization minimum is not superimposedon a wider weak localization peak confirms that spin-orbit interactions in the system are exceptionally strong[8, 12].In order to proceed with fitting the data with theHikami–Larkin–Nagaoka (HLN) theory [7], we need tocalculate the conductivity correction∆ σ ( B ) = [ σ ( B ) − σ (0)] − [ σ class ( B ) − σ class (0)] , (3)where σ is the longitudinal conductivity, obtained fromthe inversion of the measured resistivity tensor, and σ class is the classical longitudinal conductivity, obtained fromthe fitted ρ class . The obtained conductivity correction∆ σ ( B ) is plotted in Fig. 4. The dots represent the mea- -0.5 0-0.6-0.4-0.20 B(mT)data V TG =0Vfit τ φ = 190 psfit τ φ = 165 ps ∆ σ ( e / π h ) data V TG =1V T=70mK 0.5 FIG. 4: Fit of the anti-localization conductance peak withthe HLN-theory [eq. (4)]—full lines are fitted, points are ex-perimental data for the top-gate configurations V TG = 1 V, k F l m = 120 (grey), and V TG = 0, k F l m = 200 (black). sured data and the full lines are fits of the HLN-theoryfor the top-gate configurations V TG = 1 V, k F l m = 120(grey), and V TG = 0, k F l m = 200 (black). Strictly speak-ing the HLN-theory is valid in the diffusive regime, where B < B tr = ~ / (2 el m2 ). In the case of the investigatedsample we have B tr < . B ≪ B ϕ [7, 24]∆ σ ( B ) = − e πh (cid:18)
12 Ψ (cid:18)
12 + B ϕ B (cid:19) −
12 ln B ϕ B (cid:19) , (4)where Ψ( x ) is the digamma function, B ϕ = ~ / (4 Deτ ϕ ), D is the diffusion constant and τ ϕ is the phase-coherencetime. The only fitting parameter is B ϕ . Satisfactory fit-ting is obtained (full lines in Fig. 4) for both top-gateconfigurations and the phase-coherence time of holes isextracted. In the configuration V TG = 1 V, k F l m = 120(gray points) we obtain B ϕ = 5 . × − T, τ ϕ = 165 ps,and in the configuration V TG = 0, k F l m = 200 (blackpoints) we obtain B ϕ = 2 . × − T, τ ϕ = 190 ps.The corresponding phase coherence length l ϕ of holes,calculated according to the diffusive regime expression l ϕ = p Dτ ϕ , are 1 . µ m and 2 . µ m, respectively. Thesevalues show that the phase-coherence length of holes de-creases as the density in the sample is reduced. The val-ues are compatible with those obtained from measure-ments of Aharonov–Bohm oscillations in p-type GaAsrings [25]. They demonstrate that the fabrication ofphase-coherent p-type GaAs nanostructures is accessiblewith present nanofabrication technologies. However, thevalues of l ϕ in hole systems are approximately one or-der of magnitude smaller than in electron systems with / τ φ ( x / s ) -0.5 0 0.5 (cid:21) (cid:21) -0.4 (cid:21)(cid:21)(cid:21)(cid:21)(cid:21)(cid:21)(cid:21) -0 B(mT)data T=70mKfit τφ = 165 psdata T=190mKfit τφ = 53 ps ∆ σ ( e / π h ) (b) -10 -5 0 5 10212212.5213213.5
70 mK145 mK190 mK235 mK
B (mT) ρ ( Ω ) V tg =1VN=2.3x10 cm
11 -2 (a)
FIG. 5: (a) Temperature dependence of the anti-localizationresistivity minimum in the top-gate configuration V TG =1 V, k F l m = 120; (b) Temperature dependence of the in-verse phase-coherence time of holes; Inset: fit of the anti-localization conductance peak with the HLN-theory [eq. (4)]for temperatures 70 mK (black) and 190 mK (gray)—full linesare fitted curves, points are experimental data. comparable densities and mobilities [26, 27]. Such a ten-dency was also observed in recent measurements of de-phasing times of holes in open quantum dots [28]. Itsuggests stronger charge dephasing in hole than in elec-tron systems, presumably due to stronger carrier-carrierinteractions [29].Figure 5(a) shows the temperature evolution of the re-sistivity around B = 0 in the top-gate configuration V TG = 1 V, k F l m = 120. The anti-localization dipdepends strongly on temperature and disappears com-pletely above 300 mK, compatible with the temperatureevolution of the Aharonov–Bohm oscillations in p-typeGaAs rings [25]. In addition, the resistance at B = 0exhibits metallic behavior with the zero-field resistivityincreasing with temperature.The fitting of the anti-localization peak in the con-ductance is performed for each measured temperatureand the phase-coherence times are extracted. The in-set of Fig. 5(b) shows fits obtained for temperatures of70 mK and 190 mK, from which the phase coherencetimes τ ϕ = 165 ps and τ ϕ = 53 ps, respectively, are ex-tracted. It can be seen in Fig. 5(b) that the dephasing rate τ − ϕ depends almost linearly on temperature.Before we proceed with the evaluation of the spin-orbitscattering time τ SO from weak anti-localization measure-ments we estimate τ SO from SdH measurements. Theestimated spin-orbit induced splitting of the heavy holeband at a density of N = 2 . × cm − is ∆ SO =0 .
47 meV [Fig. 1(b)]. In semiconductor heterostructureswith inversion asymmetry the dominant spin-orbit relax-ation mechanism is the Dyakonov-Perel mechanism [30],which leads to the relation τ SO − = ∆ SO2 τ tr / ~ [30].Inserting the Drude transport scattering time τ tr = 26 ps,we estimate τ SO ∼ . τ SO ≪ τ tr . Therefore the SOI can-not be treated as a weak perturbation, which is thecommon assumption in theoretical calculations. An es-timate of the characteristic field B SO = ~ / (4 Deτ SO ),at which the effects of SOI become suppressed and theweak anti-localization positive magnetoresistance turnsinto a weak localization negative magnetoresistance [31]gives B SO ∼
30 mT, which is far beyond the transportfield B tr ∼ . B SO ∼
30 mT provides a qualitative understanding of thefact that we observe only a weak anti-localization dipwithout a weak localization peak in the measured mag-netoresistivity.We show the results of fitting the data in a wide mag-netic field range with the HLN-theory using the expres-sion [7, 31, 32]∆ σ ( B ) = − e πh (cid:20)
12 Ψ( 12 + B ϕ B ) −
12 ln B ϕ B − Ψ( 12 + B ϕ + B SO B ) + ln B ϕ + B SO B −
12 Ψ( 12 + B ϕ + 2 B SO B ) + 12 ln B ϕ + 2 B SO B (cid:21) (5)It should be mentioned that the HLN-theory wasoriginally developed for metallic samples where the El-liot SO skew-scattering mechanism is present. For thismechanism the spin-splitting energy is proportional to k . However, in most semiconductor heterostructuresthe Dyakonov-Perel spin relaxation is dominant [30].The theory by Iordanskii–Lyanda-Geller–Pikus (ILP) de-scribes the weak anti-localization correction for this typeof spin relaxation, and it involves both linear and cubicin k spin-splitting terms [32]. If the linear contribution isnegligible and cubic spin-splitting is dominant, the ILP-theory gives the same result as the HLN-theory in eq. (5)[31, 32]. Since the spin-orbit induced splitting of theheavy hole GaAs band is proportional to k [3, 21], it isappropriate to use the HLN eq. (5) for fitting the weakanti-localization in hole systems. Equation (5) containstwo fitting parameters, namely B ϕ and B SO .The sharpness of the weak anti-localization conduc-tance peak is determined by τ ϕ , whereas the tail of thepeak depends on τ SO [24, 31]. Therefore we explore inFig. 6(a) how the fit of the data with eq. (5) depends on -5 0 5 (cid:21) (cid:21) -1 (cid:21) -0.8 (cid:21) -0.6 (cid:21) -0.4 (cid:21) -0.200.2 dataHLN theory fit(-5mT, 5mT) τ φ =120ps , τ so =3ps(-0.5mT, 0.5mT) τ φ =210ps , τ so =40ps B(mT) ∆ σ ( e / π h ) T=70mKN=2.3x10 m
15 -2 -5 0 5-1-0.8-0.6-0.4-0.20 B(mT) ∆ σ ( e / π h ) -1.2 τ φ =120ps τ tr =26psdata τ so =3ps (fit) τ so =8ps τ so =1.6ps τ so =0.8ps(a)(b) τ tr =26psB tr =0.35mT FIG. 6: (a) Fits of the weak anti-localization conductancepeak with the HLN-theory including SOI [eq. (5)] in the range | B | < . | B | < | B | < τ SO (the full grey line represents thefit obtained in (a), while dashed lines correspond to differentvalues of τ SO as quoted in the figure). the magnetic field range. Fitting in the narrow range | B | < . B ∼ B tr , butabove that field the fit does not match the experimentaldata. The obtained fitting parameters are in this case τ ϕ = 210 ps and τ SO = 40 ps. On the other hand the fitof the data in the larger range | B | < B tr satisfactorily.The fit parameters in this second case are τ ϕ = 120 psand τ SO = 3 ps. While the obtained values for τ ϕ differby less than a factor of 2, and are comparable with thevalue obtained by fitting with eq. (4) which neglects thecontribution of τ SO , the obtained values for τ SO differby more than an order of magnitude. Although the fitin the range | B | < . B tr , is theoreti-cally more justified, it is clear that it underestimates theSO strength, because it gives an up-turn from weak anti-localization to weak localization which is not observedin the measured data. Also the value τ SO = 40 ps issignificantly larger than that estimated from the beat- ing of the SdH oscillations τ SO = 0 . | B | < τ SO = 3 ps and the value obtained from SdHoscillations. In Fig. 6(b) we investigate the influence ofchanging τ SO at fixed τ ϕ = 120 ps on the fitted curvesand find that the fitting procedure becomes less sensitivefor τ SO < τ SO .It should be mentioned that even admitting anisotropicspin relaxation and using the theory of Ref. 33 with threeinstead of two fitting parameters did not give better fitsto the data. Also, curves simulated using the theory de-veloped for the ballistic regime [34] could not satisfacto-rily match the data in the entire investigated magneticfield range.Due to the exceptionally strong SOI effects and thehigh mobility of holes in our p-type GaAs sample, itis in the regime where τ SO ≪ τ tr ≪ τ ϕ ( τ SO ∼ . τ tr = 26 ps, τ ϕ = 190 ps). In this regime SOI cannotbe treated perturbatively, as it is the case in the morecommonly studied regime τ tr ≪ τ SO ≪ τ ϕ , where asmall and sharp weak anti-localization resistance mini-mum is superimposed on a wider weak localization resis-tance peak. This might explain the difficulties in fittingour weak anti-localization data in a larger magnetic fieldrange with present theoretical models. Similar difficul-ties in fitting weak anti-localization data were observedfor an InGaAs/InP quantum well with strong SOI [35].It is also possible that the difficulties in fitting the weakanti-localization data arise from the fact that the low-field magnetoresistance contains some other contribution,in addition to the weak anti-localization, presumablydue to carrier–carrier Coulomb interactions [24]. Theseinteraction-corrections might be particularly strong in p-type GaAs systems due to the effective mass of holeswhich is significantly larger than in n-GaAs systems. VI. CONCLUSIONS
In conclusion, we have performed a detailed analysis ofthe low-field magnetoresistance in a carbon doped p-typeGaAs heterostructure. The presence of exceptionallystrong spin–orbit interactions in the structure is demon-strated by the simultaneous observation of a beating ofSdH oscillations, a classical positive magnetoresistanceand a weak anti-localization correction in the magne-toresistance. A spin–orbit induced heavy-hole subbandsplitting of around 30% of the Fermi energy is deducedfrom the beating of SdH oscillations. The classical posi-tive magnetoresistance, originating from the presence ofthe two spin-split subbands, has been fitted with a two-band model up to the fields where SdH oscillations arenot yet developed, allowing to estimate the inter- andintra-subband scattering rates. In a very narrow mag-netic field range around B = 0, a weak anti-localizationresistivity minimum is observed. The fact that this min-imum is not superimposed on a wider weak localizationpeak confirms that the sample is in the regime where τ SO ≪ τ tr ≪ τ ϕ , i.e., where spin–orbit interactions arevery strong and can not be treated perturbatively incalculations of quantum corrections of the magnetore-sistance. From weak anti-localization measurements thephase-coherence time of the holes is determined to bearound 190 ps at T = 70 mK. The temperature depen-dence reveals that the weak anti-localization resistanceminimum persists up to 300 mK and that τ − ϕ depends on temperature in an almost linear fashion. The extractedphase coherence length of holes of around 2.5 µ m at T =70 mK shows that the fabrication of phase-coherent p-type GaAs nanodevices is possible using present nanofab-rication technologies.We thank L. Golub and M. Glazov for stimulating dis-cussions. Financial support from the Swiss National Sci-ence Foundation is gratefully acknowledged. [1] G. Dresselhaus, Phys. Rev. , 580 (1955).[2] Y.A. Bychkov, E.I. Rashba, J. Phys. C: Solid StatePhysics , 6039 (1984); JETP Lett. , 78 (1984).[3] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems , Springer Tractsin Modern Physics, Volume , Springer-Verlag (2003).[4] W. Zawadzki and P. Pfeffer,
Semiconductor Science andTechnology , R1 (2004).[5] C.W.J. Beenakker, H. van Houten, in: Solid StatePhysics , edited by H. Ehrenreich and D. Turnbull,(Academic Press, New York, 1991), p. 1.[6] G. Bergmann, Solid State Commun. , 815 (1982).[7] S. Hikami, A.I. Larkin and Y. Nagaoka, Prog.Theor.Phys. , 707 (1980).[8] G. Bergmann, Phys. Rev. Lett. , 1046 (1982).[9] D.A. Poole, M. Pepper and A. Hughes, Journal ofPhysics C , L1137 (1982).[10] P.D. Dresselhaus, C.M.A. Papavassiliou, R.G. Wheeler,and R.N. Sacks, Phys. Rev. Lett. , 106 (1992).[11] G.L.Chen and J.Han and T.T.Huang and S.Datta andD.B.Janes, Phys. Rev. B , 4084 (1993).[12] T. Koga, J. Nitta, T. Akazaki and H. Takayanagi, Phys.Rev. Lett. , 046801 (2002).[13] S. Pedersen, C.B. S¨orensen, A. Kristensen, P.E. Lindelof,L.E. Golub,and N.S. Averkiev, Phys. Rev. B , 4880(1999).[14] Y. Yaish and O. Prus, E. Buchstab, G.Ben Yoseph, U.Sivan, I. Ussishkin and A. Stern, cond-mat , 0109460(2001).[15] S.J Papadakis, E.P. De Poortere, H.C. Manoharan, J.B.Yau, M. Shayegan, and S.A. Lyon, Phys. Rev. B ,245312 (2002).[16] A.D. Wieck and D. Reuter, Inst. Phys. Conf. Ser. ,51 (2000).[17] B. Grbi´c, C. Ellenberger, T. Ihn, K. Ensslin, D. Reuter,and A.D. Wieck
Appl. Phys. Lett. , 2277 (2004). [18] B. Grbi´c, PhD thesis No. 17248, ETH Zurich (2007).[19] J.J. Heremans, M.B. Santos, K. Hirakawa, M. Shayegan J. Appl. Phys. , 1980 (1994).[20] J. P. Lu, J.B. Yau, S.P. Shukla, M. Shayegan, L.Wissinger, U. R¨ossler, R. Winkler, Phys. Rev. Lett. ,1282 (1998).[21] L.G. Gerchikov and A.V. Subashiev, Sov. Phys. Semi-cond. , 73 (1992).[22] N.W. Ashcroft and N.D. Mermin, Solid State Physics ,HRW International Edditions, (1976).[23] E. Zaremba, Phys. Rev. B , 14143 (1992).[24] B.L. Altshuler, A.G. Aronov, A.I. Larkin, and D.E.Khmelnitskii, Sov. Phys. JETP , 411 (1981).[25] B. Grbi´c, R. Leturcq, T. Ihn, K. Ensslin, D. Reuter, andA.D. Wieck Phys. Rev. Lett. in print, arXiv:0704.1264.[26] T. Ihn, A. Fuhrer, M. Sigrist, K. Ensslin, W. Wegschei-der, and M. Bichler ,
Adv. in Solid State Phys. , 139(2003).[27] A. E. Hansen, A. Kristensen, S. Pedersen, C. B. Sørensen,and P. E. Lindelof, Phys. Rev. B , 045327 (2001).[28] S. Faniel, B. Hackens, A. Vlad, L. Moldovan, C. Gustin,B. Habib, S. Melinte, M. Shayegan, V. Bayot, Phys. Rev.B , 193310 (2007).[29] G. Seelig and M. B¨uttiker, Phys. Rev. B , 245313(2001).[30] M. I. Dyakonov and V. I. Perel’, JETP , 1053 (1971).[31] W. Knap et al. , Phys. Rev. B , 3912 (1996).[32] S.V.Iordanskii, Yu.B.Lyanda-Geller, and G.E.Pikus, JETP Lett. , 204 (1994).[33] N. S. Averkiev, L.E. Golub, and G.E. Pikus, JETP ,780 (1998).[34] L.E. Golub, Phys. Rev. B , 235310 (2005), M.M.Glazov and L.E. Golub, Semiconductors , 1209 (2006).[35] S. A. Studenikin, P.T. Coleridge, G. Yu, and P.J Pole, Semicond. Sci. Technol.20