Three-dimensional quasi-quantized Hall effect in bulk InAs
Rafa? Wawrzy?czak, Stanislaw Galeski, Jonathan Noky, Yan Sun, Claudia Felser, Johannes Gooth
TThree-dimensional quasi-quantized Hall effect in bulk InAs
R. Wawrzyńczak, ∗ S. Gałęski, J. Noky, Y. Sun, and C. Felser
Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany
J. Gooth † Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany andInstitut für Festkörper- und Materialphysik, Technische Universität Dresden, 01062 Dresden, Germany (Dated: February 10, 2021)The quasi-quantized Hall effect (QQHE) is the three-dimensional (3D) counterpart of the integerquantum Hall effect (QHE), exhibited only by two-dimensional (2D) electron systems. It has recentlybeen observed in layered materials, consisting of stacks of weakly coupled 2D platelets. Yet, it ispredicted that the quasi-quantized 3D version of the 2D QHE occurs in a much broader class of bulkmaterials, regardless of the underlying crystal structure. Here, we report the observation of quasi-quantized plateau-like features in the Hall conductivity of the n -type bulk semiconductor InAs. InAstakes form of a cubic crystal without any low-dimensional substructure. The onset of the plateau-like feature in the Hall conductivity scales with (cid:112) / k z F /π in units of the conductance quantum andis accompanied by a Shubnikov-de Haas minimum in the longitudinal resistivity, consistent with thepredictions for 3D QQHE for parabolic electron band structures. Our results suggest that the 3DQQHE may be a generic effect directly observable in materials with small Fermi surfaces, placed insufficiently strong magnetic fields. Electrons subjected to a magnetic field ( B ), are forcedto move on curved orbits with a discrete set of energyeigenvalues - the Landau levels (LLs). By increasing B ,the LLs move upwards in energy and cross the Fermi level E F one after another, leading to oscillatory behavior ofthe charge carrier density of states, which is exhibited intransport and thermodynamic quantities (quantum os-cillations) [1]. At sufficiently large B , where only a fewLLs are occupied, 2D electron systems enter the quan-tum Hall regime. It is characterized by a fully gappedelectronic spectrum and current-carrying gapless edgestates, in which the Hall conductance G xy becomes pre-cisely quantized in units of the inverse of the von Kl-itzing constant, i.e. half of the conductance quantum /R K = G / e / h , value based solely on the funda-mental constants: the electron charge e and the Planckconstant h [2]. This is the quantum Hall effect, tradi-tionally considered to be strictly limited to 2D electronsystems.In 3D systems, the exact quantization of the Hall con-ductance is deterred due to the dispersion of the LLbands in the third dimension, preventing the openingof the bulk gap, which is an important ingredient thatleads to quantized values of σ xy . Efforts to extend theQHE to 3D systems usually involve the introduction ofa characteristic length scale λ that transforms the origi-nal 3D system into spatially separated 2D quantum Halllayers, stacked along the magnetic field direction. Thisreduces the problem to the parallel conduction of decou-pled 2D electron systems, each of them being in the quan-tum Hall regime. The total Hall conductance of the 3Dsystem is then given by discrete values of e /h · N , where ∗ Email address: [email protected] † Email address: [email protected] N = t/λ is the numbers of 2D layers with t - the totalheight of the 3D system, in the direction of B and λ -the real space period of the stacked structure. There-fore, the 3D Hall conductivity σ xy = 1 / ( R k t ) scales with e /h · /λ = e /h · G k in spin-degenerate systems, where G k is the reciprocal lattice vector component along themagnetic field, corresponding to the periodicity of thestacking of the 2D systems.Material systems to realize this version of the 3D QHEare metals, semimetals and doped semiconductors, inwhich a periodic potential modulation is imposed bythe lattice structure [3, 4]; by spin and charge densitywaves [5–7]; or by standing electron waves in sufficientlythin samples [8]. Interestingly, in several Weyl and Dirac-semi metals a quasi-quantized Hall effect was observed,an effect in which quantum oscillations in the Hall voltageclosely mimic the Hall response of 2DEG systems [9–12].However, recently it was pointed out that, when ap-proaching to the extreme quantum limit (EQL), i.e. theregime where only the lowest spin-polarized LL is occu-pied, the Hall conductivity of a genuine 3D electron sys-tems without a periodic substructure can also be derivedfrom the conductance quantum scaled by a characteris-tic length scale [13]. This length scale is k /π and relatesto an intrinsic parameter, i.e. the momentum vector ofthe electron system k , which is given by the electronicband structure rather than by a spatial extent of thesystem’s dimensional reduction. The corresponding σ xy scales with e /h · k /π and is called quasi-quantized .In 3D semimetals with a Dirac-type band structure, ithas been shown that k = k F (1)when only a single Landau level is occupied, where k F is the Fermi wave vector component in the direction of a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b T (K)051015 xx ( m c m ) (a) B I xy z V xx V xy B (T)0200400600 xx ( m c m ) T = 2 K B z (b) 0 2 4 6 8 B (T) T = 2 K B y (c) 0 2 4 6 8 B (T) T = 2 K B x (d)0.0 0.5 1.0 1.5 2.01/ B (T )025 xx ( m c m ) T = 2 K B z (e) 0.0 0.5 1.0 1.5 2.01/ B (T ) T = 2 K B y (f) 0.0 0.5 1.0 1.5 2.01/ B (T ) T = 2 K B x (g) 01230°45°90°135°180°225° 270° 315°00.81.6 B f ( T ) (h) xy z B xy z B FIG. 1. (a) Temperature dependence of longitudinal resitivity ( ρ xx ) of InAs. The inset shows the measurement configuration.(b-d) Field dependence of ρ xx for the three main configurations. (e-g) Oscillatory part of ρ xx (1 /B ) , retrieved by subtraction ofa power law function. The red dots mark the positions of the SdH minima and maxima with a dashed line illustrating a linearfit of the Landau indices positions. (h) Frequency of the SdH oscillations as a function of the angle between magnetic field andthe direction normal to the plane of the sample. The insets show the two axes around which the rotations were performed.The data was symmetrized accordingly to the cubic symmetry of the crystal structure. B [10, 12]. In metals and doped semiconductors withparabolic bands, k in the lowest Landau band is givenby [13] k = (cid:114) k F . (2)In both cases the characteristic length scale is strictlyrelated to the Fermi wavelength λ F = 2 π / k F .Consistently, quasi-quantized plateau-like features of e /h · k F /π have recently been observed in the Hallresponse at the lowest Landau level of the 3D Diracsemimetals ZrTe [10, 12] and HfTe [11]. However,ZrTe and HfTe are in fact layered materials with ahighly anisotropic Fermi surface (aspect ratio 1:10) andhence on the verge of a 2D system. In addition, the k -scaling in the vicinity of the EQL of 3D materials hasso far only been shown for Dirac systems. It is thereforedesirable to go beyond these experiments and investigatethe Hall effect below the EQL of isotropic 3D electronsystems with parabolic bands.In this regard, many of III–V semiconductor materialsare particularly interesting ( e.g. InAs, InSb, GaAs), asthey are known for having a cubic lattice structure, highcharge carrier mobilities and almost parabolic bands.What is most important is that their Fermi level, withuse of chemical doping, or by spontaneous defects, can be placed just above the edge of the conduction band toreach the EQL in moderate magnetic fields. In fact, pre-vious measurements of the Hall resistivity on InAs andInSb bulk crystals revealed oscillatory features of the Hallcoefficient in the vicinity of the lowest Landau level [14–16]. In order to investigate this in the light of the recentdevelopments, we have chosen undoped n -type InAs asa material, which is accessible with one of the lowestachievable charge carrier concentrations to date.InAs takes form of single crystals exhibiting the cubiczinc-blende structure [17]. For our electrical transportexperiments, mm-size rectangular samples were cleavedout of a . mm-thick InAs wafer (wafers were acquiredfrom Wafer Technology Ltd. [18]). The long edges ofthe samples, defining x and y , are parallel to the two [110] -type directions of the crystal lattice and their height z is aligned with the [001] . We measured two com-ponents of the resistivity tensor, namely longitudinal ρ xx ∝ V xx /I and Hall resistivity ρ xy ∝ V xy /I (the in-set of Fig. 1a) of 2 samples (referred later as A an B ) asa function of temperature T and magnetic field B . Themeasurement employed low-frequency ( f = 77 . Hz)lock-in technique facilitated by means of Zurich Instru-ments MFLI lock-in amplifiers. An excitation currentof I e = 100 µ A (peak-to-peak value) was applied alongthe x axis of the samples, giving a current density of theorder of j ∼ mA/cm . B (T)050100150200 xy ( m c m ) (a) T = 2 K B z xyxx xx ( m c m ) B (T)0150300450 xx ( m c m ) (b) 2 K10 K20 K30 K 0.0 2.5 5.0 7.5 B (T)080160240 xy ( m c m ) (c) 2 K10 K20 K30 K FIG. 2. (a) Field dependence of ρ xy and ρ xx . (b) and (c)Temperature dependence of ρ xx ( B ) and ρ xy ( B ) , respectively,measured with B (cid:107) z . At T = 2 K, our samples exhibit ρ xx ∼ m Ω cm,with a nominal electron density n = 1 . × cm − and Hall mobility µ = 2 . × cm V − s − . All valuesabove are consistent with slight electron n -type charac-ter. Upon cooling in zero magnetic field, our samplesdisplay a metallic ρ xx ( T ) at high temperatures, followedby a thermally activated behavior below K (Fig. 1a).All investigated samples show similar electrical transportproperties. In the main text, we focus on data obtainedfrom sample A . Additional data of sample B can be foundin the Supplementary Material [19].To characterize the Fermi surface morphology o f ourInAs samples and exclude the influence of 2D surfacestates [20] on our transport experiments, we measuredShubnikov-de Haas (SdH) oscillations with the magneticfield applied along different crystallographic directions.The shape of the surface was reconstructed using theOnsager relation, which connects the period of the os-cillations with the maximal orbit of the Fermi surfacecross-section perpendicular to the magnetic field direc-tion [21]. Specifically, we rotated B in the z − x and z − y planes of our samples, while measuring ρ xx ( B ) at2 K at a series of angles (Fig. 1(b-d)). The oscillatorypart of ρ xx (Fig. 1(e-g)) was obtained by subtraction ofa smooth background using a power-law function. Thecross-section area of the Fermi surface was then deter-mined from linear fits to the positions of the minima andmaxima of the SdH oscillations presented as a functionof the inverse of magnetic field /B . We find that allextracted band structure parameters are independent ofthe field direction, which is in agreement with a single3D spherical Fermi pocket (Fig. 1h).The Shubnikov-de Haas frequency found in sample A is B F = 1 . ± . T, resulting in k F = 7 . ± . × − Å − , where the errors denote the standard devia- B (T)0306090 xx ( m c m ) T = 2 K 0306090 ( d xy / d B ) B ( a . u . ) FIG. 3. The derivative relation. Field dependence of ρ xx isgiven in absolute units and field dependence of ( dρ xy /dB ) B is scaled to fit the ρ xx . tion of the B F for different field directions (Fig. 1h). Thepreceding analysis indicates that for our InAs samples theoccupancy of only the lowest Landau level is achieved al-ready for the field of B EQL = 4
T regardless of the fielddirection. As it was shown by calculations [22], despite g ∼ , in case of InAs with such a small charge car-rier concentration (Fermi energy close to the conductionband edge), one should not observe the effects of spinsplitting on the scheme of LLs, as the Zeeman term forfields corresponding to the E F is negligibly small in com-parison with LL splitting. The only spin related featureobserved in the SdH oscillations is the strong additionalpeak in ρ xx at B ∼ T vanishing for currents parallelto the field direction. The appearance of this peak is theresult of crossing of the Fermi energy by one LL of thespin split pair: ν = 0 + [14, 23] ( ν marks the LL index)and has been established that its disappearance only for B (cid:107) x is a result of the suppression of spin-flip assistedscattering between the remaining two spin-polarized Lan-dau bands [14, 23]. For more detailed discussion of ρ xx at ν < see the Ref. [19].Having confirmed the isotropic 3D electronic Fermisurface in our InAs samples, we next turn to investigatethe Hall effect in the configuration with B applied in z -direction, that is aligned with the [001] crystallographicaxes of the crystal. As shown in Fig. 2a, we observesigns of plateau-like features in ρ xy that coincide in B with the minima of the SdH oscillations in ρ xx for allfield directions, an observation commonly related to theQHE. Both the SdH oscillations in ρ xx and the featuresin ρ xy are most pronounced at low temperatures, but stillvisible up to T = 15 K (Fig. 2(b-c)).Qualitative insights into the possible origin of the fea-tures observed in ρ xy can be obtained from comparingthe shapes of dρ xy ( B ) /dB and ρ xx . In canonical 2D QHEsystems, an empirical observation is that these are con-nected via dρ xx ( B ) = γB · dρ xy ( B ) /dB [24, 25], where γ is a dimensionless parameter of the order of . − . ,which measures the local electron concentration fluctua-tions [26, 27]. As shown in Fig. 3, dρ xy ( B ) /dB measured B (T)0100002000030000 xy ( m S / c m ) (a) T = 2 K =
23 2 e k zF h fixed fixed n FIG. 4. Field dependence of σ xy , the red, dashed line is theresult of fit to the Eq. 1, which implied the fixed chemicalpotential of charge carriers. The red, dotted lines mark thevalues of quasi-quantized Hall conductance for ν = 1 and ν =2 filling factors. The green dotted line marks the contributionfrom ν = 0 − last spin-split Landau level. The dotted blue lineshows the result of evaluation performed in conserved particlenumber regime. Both calculation routines are described in themain text. on our InAs samples show maxima and minima at thesame field positions as ρ xx . In particular, the derivativerelation shows strong resemblance with γ = 0 . (Fig. 3),which deviates from the values observed in 2D systems.The factor γ taking a very different value in bulk InAsthan previously observed in 2D systems might be ac-counted by the lack of fully understanding the nature ofthe derivative relation itself. In addition, no attempts ongeneralizing its meaning for 3D systems were undertakenso far. However, the good agreement produced by thederivative relation suggests that the observed plateau-like feature observed below the EQL of bulk InAs mightbe related to quantum Hall physics.A quantitative analysis of the Hall effect in 3D sys-tems based on the resistivity data is not straight forwardthough. As ρ xx is always finite in our experiments, theHall conductivity σ xy (cid:54) = 1 /ρ xy and therefore, the theoret-ical prediction for materials with a 3D parabolic band [13] σ xy = 2 e h π ∞ (cid:88) ν =0 C ν (cid:115) k F − (cid:18) ν + 12 (cid:19) eB (cid:126) , (3)is not immediately obvious from ρ xy . Here, C ν = 1 is the Chern number of the parabolic LL bands. Hence,we have calculated the Hall conductivity tensor element σ xy = ρ xy / ( ρ xx + ρ xy ) , shown in Fig. 4. Despite thefact that, k F is the only system specific parameter inEq. 3, which is retrieved here from the frequency of theSdH oscillations, the results of the calculations are invery good quantitative agreement with the experimentalHall conductivity up to the field B EQL above which onlythe lowest Landau level ( ν = 0 ) is occupied. Furtherinvestigations reveals that σ xy at ν = 1 scales with σ =2 e /h · (cid:112) / k F at the onset of the next Landau level, as expected by Ref [13].The presented model depends heavily on the chemicalpotential of the electrons ( µ ) being fixed, providing theshift of the LL energy to be linear in field. In 2D electronsystems, dimensionality and disorder cause the pinning ofthe Fermi level, giving rise to very well defined quantumHall plateaus. In contrast, ordinary 3D metals, such ascopper, usually have a constant electron density to avoidlarge charging energies. This forces the chemical poten-tial to vary as a function of magnetic field. In materialswith small Fermi surfaces such as undoped n -type InAs,the charging energies remain relatively small.While the opposite scenario of µ moving freely andcharge carrier concentration being constant, where σ xy = en/B (Fig. 4), gives estimates close to both calculationswith Eq. 3 and the measured conductivity it completelymisses the details of σ xy ( B ) . These are captured bytheoretical model employed in this work. However, themodel breaks down at the fields where the bottom ofthe last populated LL is approaching the chemical po-tential ( B EQL ) what should completely suppress σ xy , dueto Fermi level entering the gap. In reality the predictionand the observation bifurcate at the point where con-ductivity value is close to contribution expected for lastspin-split LL with index ν = 0 − (Fig. 4). Above thisfield the fixed- n scenario provides a good agreement, butin a very limited range. This suggest, that the investi-gated system seems not follow perfectly neither of the twocases (fixed µ or fixed n ), but balances in between, how-ever close enough to exhibit distinct features associatedwith the considered theoretical model for fixed µ .Our analysis shows that the model proposed inRef. [13] of the quasi-quantized Hall conductivity forisotropic 3D electron systems with a parabolic band isin good agreement with the experimental data close tothe EQL of bulk n -type InAs. While some localizationof the electronic states is required to ensure a constantFermi level, the mechanism proposed in Ref. [13] neitherdepends on the particular purity level of the sample norits shape and is rather an intrinsic property of the 3Delectronic structure. Compared to earlier attempts to ex-plain the generically observed features close to the EQLof doped III-V semiconductors [15, 16, 28], this makes themodel of the quasi-quantized Hall conductivity a morecomprehensive explanation of the experiments. However,we would like to emphasize that these different explana-tions do not necessarily contradict each other but someare rather complementary.In summary, we have show that the features observedin the Hall measurements of InAs are in qualitative andquantitative agreement with the existence of the quasi-quantized Hall effect in an isotropic 3D electron system.Our findings render the Hall effect in InAs a truly 3Dcounterpart of the QHE in 2D systems. The require-ments to observe the quasi-quantized Hall effect in 3Dmaterials are a low charge carrier density and a low, butfinite amount of defects. Hence, we propose that dopedsemiconductors and semimetals are ideal future candi-dates for its observation. 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Three-dimensional quasi-quantized Hall effect in bulk InAs B (T)0100002000030000 xy ( m S / c m ) Sample B xy T = 2 Kfixed FIG. S1. Field dependence of σ xy for sample B . The red, dashed line is the result of fit to the Eq. 3, which implied the fixedchemical potential of charge carriers. The red, dotted lines mark the values of quasi-quantized Hall conductance for ν = 1 and ν = 2 filling factors. The green dotted line marks the contribution from ν = 0 − last spin-split Landau level.
2. Longitudinal magnetoresistance beyond B EQL . In Figs. S1(e-g) showing the oscillatory part of ρ xx ( B ) , apart from the maximum marking the µ = 0 + spin-splitLL one can determine the presence of much smaller shoulder placed in slightly higher fields. Careful examination of ρ xx curves and their derivatives, in the vicinity of B EQL (Fig. S2), shows that for field directions not parallel to thedirection of electric current flow no signs of such a shoulder is distinguishable in the ρ xx . This would suggest thebump at the side of ν = 0 + peak being an artifact resulting from power-law background subtraction, which does notcapture increase in rate change of resistivity in field after crossing B EQL , what could be expected.On the other hand, ρ xx for B (cid:107) x does show a bump, easily noticeable thanks to the suppression of ν = 0 + peak inthat configuration. This peak appears in the fields outside of range covered by the theory employed in this work andat this point we cannot account for this feature. We hope that future measurements at sub-Kelvin temperatures, withuse of dilution refrigerator, could allow us, by suppressing thermal broadening of energy levels to resolve observedfeature in greater detail and address this open question. xx ( m c m ) T = 2 K B zB x d xx / d B ( a . u . ) B zB x