Spin-orbit coupling and transport of strongly correlated two-dimensional systems
SSpin-orbit coupling and transport of strongly correlated two-dimensional systems ∗ Jian Huang
Department of Physics and Astronomy,Wayne State University, Detroit, MI 48201, USA
L. N. Pfeiffer and K. W. West
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 (Dated: May 11, 2019)Measuring the magnetoresistance (MR) of ultraclean
GaAs two-dimensional holes in a large r s range of 20-50, two striking behaviors in relation to the spin-orbit coupling (SOC) emerge in responseto strong electron-electron interaction. First, in exact correspondence to the zero-field metal-to-insulator transition (MIT), the sign of the MR switches from being positive in the metallic regimeto being negative in the insulating regime when the carrier density crosses the critical density p c of MIT ( r s ∼ ρ to the MR decreases with reducingcarrier density (or the in-plane wave vector), it exhibits an upturn in the close proximity just above p c where r s is beyond 30, indicating a substantially enhanced SOC effect. This peculiar behaviorechoes with a trend of delocalization long suspected for the SOC-interaction interplay. Meanwhile,for p < p c or r s >
40, in contrast to the common belief that a magnet field enhances Wignercrystallization, the negative MR is likely linked to enhanced interaction.
Electron systems governed by strong Coulomb inter-action manifest unique charge transport behaviors char-acterized by the collective modes such as those in theWigner crystal (WC) [1], fractional quantum Hall states(FQHS) [2], and the zero-field metal-to-insulator transi-tion (MIT) [3]. There is an important interaction-driveneffect associated with the spin-orbit coupling (SOC)which has attracted a lot of interests due to its promis-ing spintronic applications. SOC studies often utilizesystems that lack inversion symmetries, both bulk andstructural inversion asymmetries (BIA and SIA), whichgive rise to Dresselhaus [4] and Rashba SOCs [5]. As aresult, the spin degeneracy of the energy bands is liftedeven in a zero magnetic field ( B ). The band splitting inresponse to controllable quantities, especially electron-electron interaction, is a fundamental effect that is notwell understood. SOC-driven effects in weakly interact-ing systems are usually perturbations, i.e. those tested p -type two-dimensional (2D) InGaAs, GaAs/AlGaAs het-erostructures with carrier densities ∼ − × cm − ,where r s = E ee /E F = a/a B ∼ − k F l (cid:29) E ee ∼ e /(cid:15)r -Coulomb energy, E F = nπ (cid:126) /m ∗ -Fermi en-ergy, a -average charge spacing, a B -Bohr radius, k F -Fermiwave vector, and l -mean free path. However, in a stronglycorrelated system, the SOC-interaction interplay modi-fies the exchange interaction and leads to more prominenteffects [6–8]. For example, a diverging density of states(DOS) known as the Van Hove sigularity is expected inthe limit of small wavevectors k (cid:107) (or very low carrier den-sity). However, such effects have not been studied previ-ously due to lack of access to strongly interaction-drivensystems. Large r s = E ee /E F = a/a b (cid:29) ∗ email:[email protected] tions for which the usual level of disorder renders anAnderson localization (or percolation transition). Thisstudy utilizes ultra-high purity two-dimensional (2D)holes in GaAs heterostructures and demonstrates en-hanced SOC-driven effects in a truly interaction-drivenlimit.In p -type (accumulation-type) GaAs heterostructures,SOC results in splittings of both the light hole (LH) andheavy hole (HH) bands. While the BIA-induced effectremains constant, the SIA (Rashba) contribution can betuned cexternally, i.e. via a metal gate. For high car-rier densities, BIA-induced splitting of the HH band isdominated by a k (cid:107) contribution ( k (cid:107) is the in-plane car-rier momentum). However, the situation becomes com-plicated when the carriers are sufficiently dilute. Apartfrom weakening the external electric field, reducing thegate bias modifies the confinement potential which affectsthe HH-LH separation and thus influences the Rashbacoefficient ( α ). Meanwhile, the effect of the LH-HH mix-ing (noparabolic dispersion) is expected to rise. RecentlyMR measurements on the p -dependence shows a moder-ately increasing splitting with reducing k (cid:107) [9], suggestingan enhancement of α possibly related to the LH-HH mix-ing. However, due to limited density range, such effectsare not explored into the strongly correlated metal-to-insulator (MIT) transition regime [7] where larger en-hancement is anticipated for small enough k (cid:107) [10].We adopt undoped ultra-dilute two-dimensional (2D)holes in GaAs HIGFETs (hetero-junction-insulated-gatefield-effect-transistors) in which the carrier density p canbe continuously tuned from 0.2 to 2 × cm − (or r s from 60 to 25). This allows a first probe to the MRacross MIT with the critical density p c being only 4 × cm − (or r s ∼
39 if an effective mass m ∗ = 0 . m is as-sumed). The zero-field temperature ( T ) dependence ofthe conductance shows non-activated transport, exclud- a r X i v : . [ c ond - m a t . s t r- e l ] J un G a A s A l G a A s n + V gate source drainV s V sd (a) E V E c (b) n+AlGaAs GaAs R b E F E F FIG. 1. (a) Schematics for the HIGFET, contacts, andmeasurement setup. (b) Band structure under gate voltagebias. ing the domination of an Anderson-localization, which isa crucial indicator for an interaction-driven nature. Fora decreasing p from well above p c down to the proximityof p c , the measured p -dependence of the MR captures asubstantial resistivity variation (or correction) ∆ ρ thatcorresponds to a nearly three-fold rise around 6 × cm − . Moreover, crossing into an insulator at p c , thepositive sign of MR switches rapidly to negative, in con-trast to the usual expectation of a WC stabilized by a B field.The devices adopted are 2D holes in undoped p -channel (cid:104) (cid:105) HIGFET [11–13] patterned into 1 mm × ∼ . GaAs hetero-interface. The Ohmic contactsare made with AuBe alloy annealed at 420 o C and thecontacts to the top gate layer of n + -GaAs are made withCr/Au alloy without annealing. The top gate, a 30 nmheavily doped GaAs layer, and the 2D hole layer form acapacitor with 600 nm of dielectric AlGaAs layer in be-tween. The 2D holes are, as shown in Fig. 1 (b), onlycapacitively induced by gating the n + layer beyond thethreshold voltage V c ≈ -1.3 V with respect to 2D channel. p is tuned from 7 × cm − to 1 . × cm − [13] viavarying V gate and is determined through quantum Hallmeasurements with a field sweep rate of 100 Gauss/minin a dilution refrigerator.We first present the T -dependence of the conductiv-ity ( σ ) and resistivity ( ρ ) results for a range of p from2 to 18 × cm − in a zero-field. It is well knownthat hopping conductance is often assumed for all di-lute systems. This means that, with small E ee ∼ m eVor 10 K and E F ∼ m K, disordered systems usuallygive in to unscreened (or poorly screened) disorders andundergoes Anderson Localization [14]. As a result, oneobserves exponentially suppressed conductance by cool-ing: σ ∼ e − ( T ∗ /T ) ν . ν = 1 is for Arrhenius case athigher T and ν = 1 / / ξ is exceeded by the average chargespacing a = 1 / √ πp at low enough p . Thus, ultrahigh pu- rity systems are required so that interaction-driven na-ture persists even at the onset of a WC [17], r s ∼ p ∼ × cm − (or 1 × cm − for electrons).As shown in Fig. 2(a) in log-log scales, the apparentMIT occurs around p c ∼ × cm − below whicha striking non-activated power law behavior [13, 18], σ ∼ ( T /T ∗ ) γ , is observed in multiple tested samples. p c corresponds to r s ∼
39 [17], the anticipated onset pointof a WC. Notice that the critical density obtained pre-viously in more disordered systems varies substantiallyfrom sample to sample, i.e. 8 × to 3 × cm − (or 3 < r s <
15) for electron devices and 8 × to8 × cm − (or 5 < r s <
25 for holes. However, the p c found in HIGFETs is significantly lower and varies littleamong many different samples: 3.8 to 4 . × cm − .The nonactivated power-law T -dependence on the insu-lating side contrasts the disorder-dominated hopping sce-nario. The exponent α varies from 1 to 2 with decreasing p . This power law T -dependence, which has been re-ported for both GaAs [18] and SiGe [19] systems, likelybelongs to a strongly correlated liquid (i.e. a melted WCor glass) since the experimental temperature is above theWC melting temperature T m .An important fact about the power-law T -dependenceis its vulnerability to even a slight increase of disorder.Long-range disorders are introduced through LED illu-mination to the sample within the same cooling cycle.The photons energy is approximately the band gap in GaAs and
AlGaAs . There is at least 24 hours of waittime after the illumination before measurement resumesand the uniformity of the carriers are verified via theHall measurement. The amount of disorder introducedis roughly controlled by the time duration of the lightexposure, which is usually 0.5-2s, at a constant 0.05 µ Acurrent excitation. This causes p c to rise [20]: i.e. p c isalmost doubled after 0.5s of illumination and tripled after1.5 s illumination. Meanwhile, as shown in the log − log σ FIG. 2. (a) Conductivity σ vs. T in log scales for various p from 0.8 to 18 × cm − . (b) Comparison of the resistivity ρ vs. T (in log scales) for selected p ’s before and after disorderis introduced. Dotted lines are guide for the eye. FIG. 3. (a) ρ xx vs. B for p = 1 . × cm − at T = 45mK. (b) SdH oscillations seen in a zoom-in view of (a) at lowfields. Inset: Fourier spectrum of the shown ρ xx ( B ). plot in Fig. 2(b) for p = 1 . − . × cm − , ρ ( T ) un-dergoes a qualitative change from the power-law (priorto the illumination) into an exponential law ρ ∼ e − ( T ∗ /T ) of hopping, with a characteristic energy T ∗ ∼ − ρ jumps by orders of magnitude. This qualitativechange of behaviors indicates that the power-law repre-sents a different state likely due to strong correlation.We performed further work to confirm this by studyingthe exponent γ of the power-law and found it scales witha dimensionless parameter a/d with a being the chargespacing, and d the distance to the gate. This is due tothe onset of a dipolar screening [21] and thus confirmsthe interaction-driven nature which is foundational to thefollowing MR results.The longitudinal MR, ρ xx (B), shown in Fig. 3(a), isfor p ∼ . × cm − (or r s ∼
30) with a carrier mobil-ity of µ ∼ ,
000 cm / ( V · s ) and ρ xx (0) = 2 .
35 kΩ / ∼ . h/e . h/e is the quantum resistance. Shubnikov-deHaas oscillations (SdH) [Fig. 3(b)] are observed between0.05 to 0.25T before a substantial (re-entrant insulating)peak develops at B = 0 .
37T proceeding the filling fac-tor 1 (at ∼ ρ xx (1 /B ), shownin Fig. 3(c), resolves only a single frequency f peak at0 . f ∼ . p less than 2 × cm − . Therefore, the heavy hole (HH)band is approximately degenerate. The correspondingsubband density p = ( g s e/h ) f = 6 × cm − with( g s = 2), leading to a total p of 1 . × cm − .The correction ∆ ρ to the MR for the degenerate HHband differs from the scenario of amply split subbandsthat possess not only unequal charge carrier masses (dueto the warped dispersion caused by SOC), but also thenon-proportionally different densities [22, 23]. Signifi-cantly split HH subbands exhibit different mobility µ i (band index i = 1 ,
2) and the interband scattering givesrise to a correction ∆ ρ ( B ) /ρ (0) ∝ ( µ − µ ) [24]. How-ever, for the (nearly) degenerate case, the two-band clas- sical Drude term vanishes, so does the inter-band scat-tering since µ = µ . In addition, even the single-bandDrude term is out, if isotropic scattering assumed, be-cause the Lorentz force is cancelled by the Hall field [24].Therefore, ∆ ρ ( B ) /ρ ( B = 0) here is mainly due to thequantum corrections, especially by SOC, as long as aFermi Liquid or, more generally, T < T F is valid. Asshown below, large ∆ ρ in the r s (cid:29) p (or increasing interaction) ina non-monotonic fashion as the system undergoes MIT.Fig. 4(a) shows the MR for a series of p from 0.2 to1 . × cm − , approximately 1-5% of the carrier den-sities used in Refs. [25, 26]. ρ reaches h/e (or 25.8 kΩ)around p c ∼ × cm − where the sign of MR switches.For p > p c , the positive MR measures of the strength ofSOC which produces corrections ∆ ρ ( B ) = ρ ( B ) − ρ (0)(in the unit of h/e ) dependent on p [Fig. 4(b)]: larger∆ ρ ( B ) for lower p , in the opposite trend of decreasingSOC with lower k (cid:107) [27]. The derivative dρ/dB shownin Fig. 4(d), taken within 30 mT, captures this increas-ing WAL effect with decreasing p peaked around 6 × cm − . Negative MR develops rapidly below p c and be-comes stronger with decreasing p . Fig. 4(c) shows thesame ∆ ρ result in (b) normalized by ρ (0). Taking thederivatives dρ/dB normalized by ρ (0), a local maximumis again captured as shown in Fig. 4(e).As k (cid:107) is significantly reduced, the band splitting ∆may exhibit a crossover from both the linear (BIA) andthe cubic (SIA) dependence for larger k (cid:107) to the linear- k (cid:107) dependence for smaller k (cid:107) [23]. Unlike the depletioncases, decreasing the gate voltage V g reduces the electricfield according to E = ( V g − V fb ) /d = −∇ V (where V fb is the flab-band voltage, d is the barrier thickness, and V is the approximately triangular confinement potential FIG. 4. (a) Low field magnetoresistance ρ xx ( B ) for various p = (from top down) 0 .
24, 0.35, 0.45, 0.66, 0.85, 1.22, and1 . × cm − ; (b) ∆ ρ = ρ ( B ) − ρ (0) vs. B for the samefixed p shown in (a); (c) Results in (b) normalized by ρ (0).(d) dρ/dB vs. B . (e) Results in (d) normalized by ρ (0). with a width w ). The electric field is expressed as E = 2 E g ep(cid:15) (1)where E g is the triangular well height, e the electroncharge, and (cid:15) = 13 is the dielectric constant. For a 50%decrease in p from 1.55 to 0 . × cm − , E is re-duced by approximately ∼ V g = − .
62V to V g = − . p =1.55 and 0 . × cm − respec-tively. The well width w = (2 (cid:15)E g /ep ) / is widened by1.4 times, resulting in a 30%-decrease in the z-direction(growth direction) wave-vector k z = π/w of the Airy-like wavefunctions. The Dresselhaus coupling parameter β ∝ k z is halved. If we consider the k (cid:107) splitting, it hasa p . dependence since p ∼ k (cid:107) / π (2D). Since β is alsodecreasing, the nearly linear decrease of [1 /ρ (0)] dρ/dB (dotted line in Fig. 4(e)) with decreasing p is reasonable.For this p range, the splitting ∆ of the HH band dueto the k (cid:107) Rashba term should be considered. It has beenshown for the accumulation-type heterostructures thatthe confinement electrical field E is ∝ p [28]. Using theAiry wave-functions confined to a triangular well, theRashba coefficient α ∝ p − / [23], which is enhanced2.5 times when p is reduced from 1.55 to 0 . × cm − . Because the observed overall correction is linearlydecreasing for this p range, it suggests a lesser Rashbaeffect than the Dresselhaus for higher p .Notice that for p = 1.55, 1.22, and 0 . × cm − ,the system is already strongly correlated with r s = 20 ∼
28. It turns out the usual fitting of MR correctionsbased on the non-interacting model [29] already becomesproblematic. Our fitting attempts produces 7 , 1.5, and0.6 ps respectively for τ so , however, with large errors( R = 0 . ∼ . k F l , which is between1 and 10, is much smaller than the cases in ref. [26] and∆ ρ/ρ (0), amounting to 15% at B = 50 m T, is beyondperturbation. Clearly, a model appropriately incorporat-ing interaction is needed. The theory on ∆ σ ( B ) due tointeraction by Altshuler et al [6] might be more relevant,even though it does not produce non-monotonic correc-tions for the large r s situations discussed below.Remarkably, the decreasing trend in [1 /ρ (0)] dρ/dB with lowering of p is replaced by a striking 40% increase(or 300% in dρ/dB ) around 6 × cm − . It signifiesa substantially rising SOC coupling coefficient, particu-larly, the Rashba coefficient α . In principle, as the k (cid:107) dependence fades at low p , the linear- k (cid:107) (or p / ) de-pendence eventually gives in to the rising α ∝ p − / ,resulting in the upturn of [1 /ρ (0)] dρ/dB . This is qual-itatively in agreement with the data. There are otherrelevant effects that should also be recognized. Lower-ing p to extremely dilute limits shrinks k (cid:107) to the HH-LHanti-crossing point [30] where the consequent effect on α is not fully known. Meanwhile, growing effective inter-action with reducing p raises r s beyond 30 and it givesrise to effects via exchange interaction [10] through an enhancement factor λ SO . Theory predicts an approx-imately linear relation λ SO ∼ r s for moderate r s val-ues less than 20 [10]. Another interaction-driven effectderives from the enhancement of the effective mass m ∗ which has been known to exist in the close vicinity ofthe p c of the MIT [31]. Due to the non-parabolic dis-persion relation at low p and the HH-LH mixing, m ∗ is a complicated quantity. Actual comparison would re-quire a measurement of m ∗ which is difficult to achievewith cyclotron resonance (due to the small energy) or theSdH (due to the large Coulomb energy). Nevertheless,the non-monotonic rising is consistent with an enhancedSOC which influences the MR opposite to the trend oflocalization as p crosses below p c . Therefore, SOC sup-ports MIT [7].∆ ρ ( B ) → p = p c corresponds to the diminishingWAL eventually overcome by the localization. A Wignercrystal (WC) regime is arrived at p < p c or r s >
40. Therole of SOC depends on the actual carrier state which islikely a WC liquid because of the following: Due to thetiny energy scales, E F ∼ −
400 mK, a WC is fragileto the influences of disorder, thermal [32] and quantumfluctuations (on a scale of ∼
480 nm / √ T ). Consequently,the melting temperature T m is usually reduced well belowthe classical estimate: ∼ E ee / ∼ mK . A recentstudy demonstrates a dynamical pinned WC marked byenormous pinning strength and extremely sharp dc-VIthreshold [33]. The T -dependence of pinning suggest a T m ∼
30 mK which is lower than the temperature forthe MR measurement. Meanwhile, since those data alsosupport a second-order phase transition, an intermediatephase [34] is relevant for the MR results for
T > T m .The quantum scenario still holds because T is less than T F = E F /k B , plasma frequency Ω = √ r s E F , and E ee .Thus, the hydrodynamic flow model for the semiquantumcase [35, 36] does not apply.The negative MR for p < p c is intuitively surprisingsince an external B -field should stabilize a WC. On theother hand, the B -field raises the Zeeman energy gµ B B which increases the overall carrier energy, favoring delo-calization. Though, the Zeeman effect is small for higher p (or weakly interacting) cases, it becomes prominent asthe g -factor is greatly enhanced with increasing interac-tion (or lower p ) [37]. For a reducing p (below p c ), boththe absolute Coulomb energy E ee = e √ πp/(cid:15) and the E F = ( π (cid:126) /m ∗ ) p decrease. Moreover, for lower enough k (cid:107) , 2D DOS diverges (towards the van Hove singularity),resulting in lower kinetic energy states. Thus, interactioneffect, including the g -factor, is further enhanced. As theZeeman energy grows with increasing B , a larger nega-tive MR for lower p is a possibility. Relevant transporttheory is needed.In summary, interaction-driven effects are realized inhigh purity ultra-dilute systems possessing unique trans-port behaviors distinct from the activated transport inAnderson insulators (or percolation) governed by disor-der. SOC-driven band splitting is small in the dilutelimit. Yet, there exists a substantial non-monotonic risein the WAL in the metallic side, consistent with a de-localization effect. The switching of the signs of MR isin excellent correspondence to the zero-field MIT. Therising SOC coupling parameters are due to the large in-teraction including the scrambled exchange energy. Thenegative MR in the insulating side seems to be consistentwith a rising carrier energy through the Zeeman term fa- cilitated by the reducing absolute Coulomb energy.This work is supported through NSF-1105183 andNSF-1410302. 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