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Non-equilibrium transport in electron solids Rossokhaty, Oleksandr 2016

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Non-equilibrium Transport in ElectronSolidsbyOleksandr RossokhatyB.Sc., Moscow Institute of Physics and Technology, 2007M.Sc., Moscow Institute of Physics and Technology, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2016c© Oleksandr Rossokhaty 2016AbstractElectron-electron interactions inside of two dimensional electron gases (2DEG)in out-of-plane magnetic field and at very low temperatures under certainconditions can lead to electron localization in Wigner crystals or even morecomplex periodic structures. These states are usually referred to as electronsolid phases and result in Reentrant Integer Quantum Hall Effect (RIQHE)in transport measurements. However, their microscopic description remainsunclear, as insulating phases with different microscopic structure demon-strate indistinguishable macroscopic transport properties. In this work thetransport of the electron solids is investigated away from equilibrium con-ditions. This approach allows to break an insulating state by application ofsignificant current bias to the 2DEG. As bias current increases, longitudinaland Hall resistivities measured for these states show multiple sharp break-down transitions. Whereas the high bias breakdown of fractional quantumHall states is consistent with simple heating, the nature of RIQH breakdownremains to be a subject of a considerable debate.A comparison of RIQH breakdown characteristics at multiple voltageprobes indicates that these signatures can be ascribed to a phase boundarybetween broken-down and unbroken regions, spreading chirally from sourceand drain contacts as a function of bias current and passing voltage probesone by one. It is shown, that the chiral sense of the spreading is not setby the chirality of the edge state itself, instead depending on electron- orhole-like character of the RIQH state. Although at high current bias theelectron temperature is unmeasurable with standard techniques, the datashows that electron solid states appear to stay temperature sensitive evenafter the RIQH effect is destroyed. A comparison of temperature dependenceand the spatial distribution of the Hall potential along the edge provides anevidence, that the bulk 2DEG remains insulating up to surprisingly high bi-ases. Finally a metastable stripe phase around ν = 9/2 is investigated undernon-equilibrium conditions in the sample with electron density, which is closeto the stripe reorientation critical point. The anisotropy of non-equilibriumstripe phase under high current biases shows a strong dependence of thenatural orientation of stripes on exact filling factor.iiPrefaceThis thesis describes work associated with the following publications:1. Low-temperature illumination and annealing of ultrahigh qual-ity quantum wellsM. Samani, A. V. Rossokhaty, E. Sajadi, S. Luscher, J. A. Folk, J. D.Watson, G. C. Gardner, and M. J. ManfraPhys. Rev. B (Rapid) 90, 121405 (2014)2. Electron-Hole Asymmetric Chiral Breakdown of ReentrantQuantum Hall StatesA. V. Rossokhaty, Y. Baum, J. A. Folk, J. D. Watson, G. C. Gardner,and M. J. ManfraPhys. Rev. Lett. 117, 166805 (2016)3. Versions of Ch. 5 and 6 are in preparation for publicationI helped to design and build the low temperature electrical filtering, wirecooling modules and electromagnetic radiation shielding. These changes tothe refrigeration system significantly lowered its base temperature, which iscrucial for experiments on reentrant states. I also built the fiber system inexperimental setup, which allowed direct illumination of the sample and itstemperature control during the cooldown process. This helped to developsample preparation recipe, described in experiment [1], which was used in allother experiments. All experimental data from experiment [1] was acquiredby M. Samani.I identified initial direction of all experiments under supervision of Prof.J.A. Folk. I performed all of the data collection, data analyses and developedthe interpretations/conclusions, shown in the main chapters of these thesis.The drafts were prepared by me, and these were edited together with thelisted coauthors.Semiconductor heterostructures and samples studied in this thesis wereprovided by the group of Prof. M.J. Manfra in Purdue University.Numerical simulations were performed by Yuval Baum.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Structure of this thesis . . . . . . . . . . . . . . . . . . . . . 82 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . 102.1.1 Low dimensional physics . . . . . . . . . . . . . . . . 102.1.2 Integer Quantum Hall Effect . . . . . . . . . . . . . . 132.1.3 Theory of the edge states . . . . . . . . . . . . . . . . 162.1.4 Non uniform samples . . . . . . . . . . . . . . . . . . 192.2 Coulomb interaction effects . . . . . . . . . . . . . . . . . . . 222.2.1 Fractional Quantum Hall Effect . . . . . . . . . . . . 232.2.2 Reentrant Quantum Hall States . . . . . . . . . . . . 252.2.3 Anisotropic states around LL half filling . . . . . . . 282.2.4 Correlated states at N=1 LL . . . . . . . . . . . . . . 303 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Sample details . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.1 Heterostructure design . . . . . . . . . . . . . . . . . 323.1.2 Cooldown protocol . . . . . . . . . . . . . . . . . . . 343.2 Low temperature system . . . . . . . . . . . . . . . . . . . . 36ivTable of Contents3.2.1 Dilution refrigerator . . . . . . . . . . . . . . . . . . . 363.2.2 Temperature control . . . . . . . . . . . . . . . . . . . 413.3 Electrical measurements . . . . . . . . . . . . . . . . . . . . . 434 Breakdown spatial resolution . . . . . . . . . . . . . . . . . . 444.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 454.2.1 Measurement details . . . . . . . . . . . . . . . . . . 454.2.2 Measurements at different contacts . . . . . . . . . . 474.2.3 Electron- vs hole-like reentrant states . . . . . . . . . 484.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.2 Numerical simulation . . . . . . . . . . . . . . . . . . 534.3.3 Comparison with experiment . . . . . . . . . . . . . . 564.3.4 RIQHE vs FQHE breakdown . . . . . . . . . . . . . . 604.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 High bias reentrant states . . . . . . . . . . . . . . . . . . . . 635.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Transport measurements . . . . . . . . . . . . . . . . . . . . 645.2.1 Electrical potential spatial distribution . . . . . . . . 665.2.2 Comparison with experiment . . . . . . . . . . . . . . 685.2.3 Longitudinal voltage . . . . . . . . . . . . . . . . . . 715.3 Temperature measurements . . . . . . . . . . . . . . . . . . . 725.3.1 Experimental technique . . . . . . . . . . . . . . . . . 725.3.2 Experimental results . . . . . . . . . . . . . . . . . . 745.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796 Bias induced anisotropy at half-filled Landau level . . . . . 816.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 836.2.1 Measurement details . . . . . . . . . . . . . . . . . . 836.2.2 Bias measurements at ν = 9/2 . . . . . . . . . . . . . 866.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947 Summary and future experiments . . . . . . . . . . . . . . . 95Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101vList of Figures1.1 Hall bar measurement geometry . . . . . . . . . . . . . . . . . 11.2 First experimental evidence of integer quantum Hall effect.Picture is adopted from [2]. . . . . . . . . . . . . . . . . . . . 21.3 CDW patterns [3]. (a) Stripe pattern. (b) Bubble pattern.(c) WC. One cyclotron orbit is shown. . . . . . . . . . . . . . 51.4 Characteristic Rxy ≡ dVxy/dI and Rxx ≡ dVxx/dI traces,measured at base temperature (14 mK) in filling factors ν =2− 3 (Iac=1 nA, Idc=0) . . . . . . . . . . . . . . . . . . . . . 62.1 Structure of electron energy levels in the presence of randompotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Structure of electron energy levels in the presence of the edges.After [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Propagation of the edge states in inhomogeneous sample withtwo filling factors ν = N and ν = N + 1. The outer thickarrow depicts first N co-propagating edge states around thewhole sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Conduction band edge and free charge density in the immedi-ate vicinity of the doping well located 75 nm below the edgeof the primary 30-nm GaAs quantum well. On the inset:the general structure of the conduction band edge and chargedensity profile for a whole quantum well. Adopted from [5]. . 343.2 Design of the dilution fridge low temperature side. . . . . . . 373.3 LT1167 instrumentation amplifier block diagram (figer takenfrom datasheet). . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Ground breaking schematics: a) voltage measurements; b)voltage source. Circles around the voltage probes depict cableshielding, connected to the measurement ground. . . . . . . . 403.5 On chip heater calibration. Comparison of R2c width depen-dence on mixing chamber temperature (circles) and heaterpower (triangles). . . . . . . . . . . . . . . . . . . . . . . . . . 42viList of Figures3.6 Schematics of electrical measurements: a). AC‖DC; b). AC⊥DC. 434.1 a) Schematic of a measurement combining AC (wiggly arrow)and DC (solid arrow) current bias through contacts 1 and5 (see numbering). R˜xx = dV86/dI, R˜+D = dV26/dI, andR˜−D = dV84/dI. Curved arrows indicate edge state chirality.b) Evolution of R˜xx with DC bias for the R2c reentrant andν = 5/2 FQH state, showing breakdown regions ‘A’, ‘B’, and‘C’ (IAC=5 nA). c) R˜xx and R˜xy (contacts 3 and 7) for fillingfactors ν = 2−3, showing the breakdown at very high DC bias(IAC=5 nA). (d,e) Simultaneous measurements of R˜+D (d) andR˜−D (e), taken together with R˜xx in b). Dashed lines indicateidentical parameters in each panel. . . . . . . . . . . . . . . . 464.2 (a) Simultaneous measurements showing the evolution of R˜+D,R˜xy, and R˜−D, with DC bias, in the middle of the R2c reentrantstate (IAC=5 nA); Evolution of (b) R˜−D, (c) R˜xy and (d) R˜+Dof R7a reentrant state with DC bias. . . . . . . . . . . . . . . 484.3 Diagonal measurement with the current flowing in two per-pendicular orientations with respect to the sample. It isclearly seen, that the strong/weak reentrant state is definedby the current contacts and any kind of measurement can beobtained on the same voltage probes. . . . . . . . . . . . . . . 504.4 Geometry of the domain Ω, where simulation is performed.Regions A and B denote the areas with different densities,corresponding to different σxy’s in the simulation. . . . . . . . 544.5 Classical simulation of dissipation due to current flow in asample divided into three regions: semicircles correspondingto the melted state near each contact (hellow hatched) withRxy = h/(2.5e2), and the bulk (dark blue) reentrant statewith Rxy = h/(2e2) (a) or Rxy = h/(3e2) (b). Hotspotsappear at different corners of the melted region in a) and b). 55viiList of Figures4.6 a) Hole-like reentrant state breakdown propagation in thesample; hexagonal, diamond and round marks depict the po-sition of the hot spots for three different sizes of broken re-gions (dark grey, light grey and hatched), corresponding todifferent bias currents; b) electrical potential distribution inthe sample with half-way broken reentrant state; comparisonof breakdown propagation, measured on different ohmic con-tacts along one side with common contact in c) frozen and d)melted region. The marks depict current value, correspondingto the hot spot locations, shown in a) . . . . . . . . . . . . . 574.7 Comparison of two possible measurement geometries: a)AC||DCand b) AC ⊥ DC; arrows label the source and drain con-tacts for DC (straight) and AC (wiggly) bias, and chiralityof the edge states (curved). Hatched areas denotes partially-extended melted regions for a hole-like reentrant state at in-termediate DC bias, corresponding to IDC ∼ 50nA in panels(c,d). Hotspots at the melted/frozen boundary indicated by?. Yellow line indicates potential along the edge. (c,d) R3aRxx maps in (IDC , B) plane for c) AC||DC and d) AC ⊥ DCmeasurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.1 (a) Characteristic R˜xy ≡ dVxy/dI and R˜xx ≡ dVxx/dI traces,measured at base temperature (13 mK) in filling factors ν =3 − 4 (Iacb =5 nA, Idcb =0); (b) evolution of R˜+D (Iacb =5 nA) atthe R3a reentrant state with dc bias current Idcb . Character-istic regions denoted ‘A’, ‘B’, and ‘C’ in the R2c breakdown;(c) schematic of the measurement; (d) traces of R˜+D and R˜−Dbreakdown with bias. Inset demonstrates R˜+D measurement2D map in region C with correlated pattern of ripples. . . . . 655.2 Breakdown propagation in the sample in a) electron and b)hole case, diamond and round marks depict the position ofthe hotspots for two different sizes of broken regions (greyand hatched), corresponding to different bias currents; com-parison of two diagonal measurements for the c) electron(R3a) and d) hole-like (R3c) reentrant states (data taken fromFig. 4.3a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67viiiList of Figures5.3 (a) schematics of the hot spot propagation with current bias;(b) schematics of the melted regions and current flow throughthe bulk in region C; (c)-(d) breakdown traces of electron-likereentrant state R7a with bias; (e) comparison of the break-down of hole-like reentrant state R2c, measured at differentvolatge probes, and 2-point measurement. . . . . . . . . . . . 695.4 Evolution of R2c with DC current bias: a) longitudinal volt-age Vxx and b) longitudinal differential resistance R˜xx (Iac =5nA). Note the very different color scale in (b) compared toprevious figures. c). Characteristic traces of Vxx and R˜xx atν = 2.569. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.5 (a) Characteristic R˜xy ≡ dVxy/dI and R˜xx ≡ dVxx/dI traces,measured at base temperature (13 mK) in filling factors ν =2 − 3 (Iacb =1 nA, Idcb =0); (b) evolution of R˜xx (Iacb =5 nA)at the R2c reentrant state and ν = 5/2 FQH state with dcbias current Idcb . Characteristic regions denoted ‘A’, ‘B’, and‘C’ in the R2c breakdown. Color scale is saturated in regionB. Inset demonstrates cross section of the graph, marked bydash-dotted line. . . . . . . . . . . . . . . . . . . . . . . . . . 735.6 Evolution of a) R˜xy and b) R˜xx at the R2c reentrant state,with temperature between 15 and 48 mK, Iac=1 nA, Idc=0;c) temperature dependence of R2c width at Idc=0; d) R˜xxevolution at Iac=1 nA, Idc=300 nA. . . . . . . . . . . . . . . . 755.7 Region C remains highly temperature dependent, despite sig-nificant Joule heating. (a) the full breakdown diamond ofR2c, analogous to Fig. 1b but after a refrigerator cycle to 4K.Sharp features in region C serve as effective thermometers,because they are extremely temperature sensitive. Interiorbox denotes area in (b,c), with hashed region at higher biasconsidered to be beyond breakdown. (b) Zoom in to boxfrom (a). Sharp features are qualitatively similar in low (200-300nA) and high (400-500nA) bias regions. (c) Analogousdata after warming the mixing chamber to 15.6 mK. Sharpfeatures are completely washed out, at both low and high bias. 765.8 Field sweeps over the spike on the boundary of insulatingarea near ν = 2.55, see Fig. 1b. Horizontal axis representsestimated lattice temperature controlled by chip carrier re-sistor, taking into account additional dissipation in sample’scontacts due to bias current (Iac=5 nA, Idc=300 nA) . . . . . 77ixList of Figures6.1 Characteristic R˜xx, R˜yy and R˜xy (for current driven along Yaxis) for filling factors ν = 4 − 14, showing the weak trans-port anisotropy (IAC=4 nA). Schematics show the prefferedorientation of the stripes at high filling factors. . . . . . . . . 846.2 Evolution of R˜yy with a)DC||Oy, d)DC||Ox and R˜xx withb)DC||Oy, c)DC||Ox. Dashed lines show approximate melt-ing boundary of the stripe phase. Schematics depict stripealignment and measurement orientation relative to DC bias,driven in a-b) hard and c-d) easy direction. Short (long) wig-gly lines depict orientation of stripes suppressed (enhanced)by bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.3 a) Schematic of a measurement with at metastable (top) andanisotropic (bottom) ν = 9/2 stripe phase; b) comparisonof R˜xx and R˜yy field scans at zero DC bias, solid (dashed)arrow mark field sweep down (up); 0 nA DC R˜xx and R˜yyfield scans annealed at 3 uA DC along c) Oy and d) Ox axisat every filling factor. . . . . . . . . . . . . . . . . . . . . . . 876.4 Schematic description of stripe orientation in the vicinity ofν = 9/2 relative to DC bias, driven in a) hard and d) easy di-rection. Short (long) wiggly lines depict orientation of stripesdestabilized (stabilized) by bias. Evolution of R˜yy with b)DC||Oyand f)DC||Ox. Evolution of R˜xx with c)DC||Oy and e)DC||Ox.Dashed lines show approximate melting boundary of the stripephase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.5 Bias scans of anisotropic states, stabilized by annealing ata,b) ν = 4.42; c,d) ν = 4.62. The direction of the DC bias aswell as the expected orientation and strength of the stripes isshown on the schematics. . . . . . . . . . . . . . . . . . . . . 916.6 Schematic phase diagram of the stripe phase in samples withelectron density around the critical point. . . . . . . . . . . . 926.7 Schematic description of the competition of two pinning stripepotentials in a) low electron density; b) close to the criticalpoint. Wiggly lines in the middle depict the anisotropy of theground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.1 R˜xx bias dependence of R2c at different temperatures a) 24.8 mKb) 22.9 mK c) 21.2 mK. Colorscale is saturates at 500 Ω. . . . 987.2 Rxy bias scans in the range a) ν = 4− 6 and b)ν = 6− 8. . . 100xList of Acronyms• QHE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Hall Effect• IQHE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Integer Quantum Hall Effect• RIQHE . . . . . . . . . . . . . . . . . . . . Reentrant Integer Quantum Hall Effect• FQHE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractional Quantum Hall Effect• LL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Landau Level• FQHE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractional Quantum Hall Effect• 2DEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Two Dimensional Electron Gas• MBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Beam Epitaxy• MOSFET . . . . . . . . . . . . . . . . . . . . . Metal Oxide Field Effect Transistor• CDW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Charge Density Wave• ELC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron Liquid Crystal• LED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light Emitting Diode• RRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Residual-resistance ratio• WF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wiedemann-FranzxiChapter 1Introduction1.1 OverviewIn 1879, eighteen years prior to the discovery of the electron and long beforethe discovery of its spin, Edwin Hall detected an effect, bearing his namenowadays. He measured a voltage, which appears across the sample due tothe displacement of moving charge carriers in external electric and magneticfields (Fig. 1.1). In the twentieth century, the Hall effect, well-known as aroutine technique for semiconductor characterization, revealed a non-trivialquantum nature of two dimensional systems – quantum Hall effect (QHE).Figure 1.1: Hall bar measurement geometryAlthough the classical Hall effect has been discovered and studied morethan a hundred years ago, QHE appeared to be absolutely unexpected. Evenafter the discovery of this phenomenon, no single even indirect predictionhave been found in the literature. Two dimensional electron systems ina perpendicular magnetic field remain to have unforeseen and surprisingbehaviors even thirty years after the discovery of the QHE. Today this isstill a field of an active research, since the solidity of this effect provides avast of phenomena, connected to it [6–8].The first experimental evidence of integer quantum Hall effect (QIHE)was discovered by Klaus von Klitzing in 1980 [2]. He detected appearanceof quantized plateaux in Hall resistance traces as a function of the externalout-of-plane magnetic field, measured in metal-oxide field effect transistor11.1. OverviewFigure 1.2: First experimental evidence of integer quantum Hall effect. Pic-ture is adopted from [2].(MOSFET) at low temperatures. Simultaneously with plateaux, longitudi-nal resistance demonstrated the deep minima, approaching zero. The valuesof those plateaux appear to be a universal combination of fundamental con-stants, independent of the specific sample:ρxy =VxyIxx=hνe2, (1.1)where ν is an integer number (Fig. 1.2) and the measurement is performed asshown on Fig. 1.1. This effect is called integer quantum Hall effect (IQHE).21.1. OverviewSpecifically, the filling factorν =hnseB(1.2)determines the electronic ground state of the system with 2D electron den-sity ns at a particular magnetic field B. The physical meaning of this pa-rameter is the ratio between areal densities of magnetic flux quanta (eB/h)and conduction electrons (ns). Filling factors 0 to 2 correspond to partialfilling of the first spin-degenerate Landau level (LL) with orbital quantumnumber N=0; from ν = 2 − 4 the first (N=0) level is completely filled andthe second (N=1) is partially occupied, etc.These two main features of the quantum Hall effect (existence of theplateaux and their universal values) are the consequences of fundamentalquantum properties of 2D systems. The plateaux are usually explainedby a single-particle localization on underlying impurities and result fromdiscreteness of LLs: if the filling factor is slightly increased from integerν, e.g., by lowering the magnetic field, electrons are promoted to the nextLL. However they are localized, as the minima of the underlying impuritiespotential are populated first. Since localized electrons do not contributeto the electrical transport, Hall resistance stays constant in some range ofmagnetic fields.The universal values of the plateaux are the result of a spectrum struc-ture of a 2D system of charged particles. In a perpendicular magnetic field,charge carriers are confined to quantized cyclotron orbits. Each filled LL ina real space appears to be a dense packing of those orbits, representing cou-pling between a magnetic flux and charge carriers. Thus the charge densityat each LL and, therefore, the value of ρxy depends on the internal structureof the system.If the system is a two dimensional electron gas (2DEG), the density ofstates at spin resolved LL is nL = B/Φ0, where Φ0 = h/e is the magneticflux quantum. In this case, classical Hall conductivity of the 2DEG withelectron density ns could be rewritten in the following form:σxy =nseB=nLeBnsnL= νe2h. (1.3)This explains the “magic” value of Hall conductance in (1.1).Interestingly, the same idea can be extended to a model of any non-interacting quasi particles. For instance, in high magnetic fields, i.e. lowfilling factors, the ground states of interacting electrons are described bya quantum liquid of Laughlin state at ν = 1/(2s + 1) [9] or composite31.1. Overviewfermions at ν = p/(2ps+1) [10], where p, s are integers. Composite fermionsexperience a reduced magnetic field (eB)∗ = eB/(2ps± 1), and thus fill LLswith a filling factor ν∗ = nelh/(eB)∗ as do electrons in case of the integerquantum Hall effect. These states account for a fractional quantum Halleffect (FQHE), realized experimentally as plateaux in ρxy and vanishingresistances ρxx at the appropriate fractional filling factors [7].Further quality improvement of the samples and lowering the tempera-tures revealed a large variety of exotic collective states in 2DEG caused bymany body interaction of electrons in high perpendicular magnetic fields,ranging from FQH liquids to charge-ordered electron solid states, such asWigner crystals or nematic stripe phases [3, 11–14]. In some cases electronsolid states show similar transport characteristics to their FQH cousins,though the underlying microscopic description of these phases are believedto be very different. When the electron solid is fully formed, Rxx vanishesand Rxy returns to the nearest integer value of ν, so such states are of-ten referred to as reentrant integer quantum Hall (RIQH) states. Thesetransport characteristics are explained by crystallization of the electrons inthe partially filled Landau level: when frozen they no longer contribute totransport.However, completely different phases develop around the half filling ofthe uppermost Landau level. Corresponding to Landau level filling factorsν = 9/2, 11/2, 13/2, and so on, these phases are characterized by a hugeanisotropy in the longitudinal resistance of the 2DEG which develops rapidlyon cooling below about 150 mK [11, 12]. This impressive signature hasbeen widely interpreted as strong evidence for the unidirectional, or stripedelectron solid phases.The nature of electron solid states can be understood if the followingphenomenological interaction potential is considered [15]:U(r) = A1r−Be−kr + Vd. (1.4)The first term in (1.4) represents Coulomb repulsion, the second one isresponsible for attraction, coming from electron screening and the last termrepresents the impact of random potential. Such kinds of interactions appearin many physical systems and were intensively studied theoretically. Similarto the normal crystal, under certain conditions, it becomes energeticallyfavorable for particles to reorganize into periodic crystal structure or particledomains of higher particle density. Stripe and bubble phases have beenshown to occur in such kinds of systems [15]. The onset of the differentstripe and bubble phases can be controlled by directly varying the relative41.1. Overviewstrength of the repulsive and attractive interactions at a fixed density or byholding the interactions fixed while the density of the system is changed.Figure 1.3: CDW patterns [3]. (a) Stripe pattern. (b) Bubble pattern. (c)WC. One cyclotron orbit is shown.First numerical calculations showed, that under conditions of QHE elec-tron solids can be formed at N ≥ 2 by breaking 2DEG in alternating regionsof neighboring integer filling factors. Depending on the occupation of lastpartially filled Landau level, these regions can have different patterns [14].Close to the integer filling factor (i.e. at low electron density at the lastLandau level) an ordinary Wigner crystal is formed. These states usuallyappear straight after conventional IQH plateaux and result in its wideningin experimental measurements. If the electron density is increased (i.e. thefilling factor increases under constant magnetic field), it becomes energeti-cally favorable for the system to reorganise in more complex structure withtwo electrons per site. Although putting two electrons close to each otherleads to energy loss, much higher gain is achieved as the lattice period dou-bled. Therefore, at high filling factors, electrons can form many types of,so called, super Wigner crystals or ”bubble” phase with two, three, four,etc. electrons per lattice site. Eventually, around the half filling of the high-est Landau level, the bubbles collapse, forming a unidirectional structure ofperiodic stripes.Observation of RIQHE at N=1 LL [13, 16–18] caused more accuratecalculation of the ground state at the second half filled LL [19, 20] andrevealed extremely rich phase diagram of multiple FQH and electron solidstates interspersed within a narrow range of magnetic field.51.2. Motivation1.2 MotivationThe nature of electron solid states at intermediate filling factors, 2 < ν < 4,is much less clear. Here, the N=1 Landau level is partially filled and compe-tition between Coulomb and magnetic energies intersperses multiple FQHand RIQH states within a narrow range of magnetic field [19] (Fig. 1.4a).One important puzzle is why four RIQH states appear between ν = 2 andν = 3, or between ν = 3 and ν = 4, but only two such states appear ateach higher filling factor. This might be explained by the stabilization ofthe ν = 7/3 and ν = 8/3 fractional states in the middle of reentrant states,splitting each in two parts. On the other hand, theory predicts existenceof multiple bubble phases with a different number of electrons per bubble.Since all those phases are insulating, all corresponding reentrant states willdemonstrate the same integer plateau in experiment. This fact makes dif-ferent bubble phases indistinguishable in transport measurements. So thequestion of the internal structure of reentrant states at different Landau lev-els remains under discussion. Experimental input is even more important,B (T)R2d R2c R2b R2a0.00.20.40.60.81.0Rxx  (kW)789101112134.4 4.6 4.8 5.0 5.5 5.4 5.6 5.8R xy (kW)Figure 1.4: Characteristic Rxy ≡ dVxy/dI and Rxx ≡ dVxx/dI traces, mea-sured at base temperature (14 mK) in filling factors ν = 2 − 3 (Iac=1 nA,Idc=0)61.2. Motivationas many theoretical tools used to understand RIQH states at high fillingfactor become less trustworthy below ν = 4 [3, 21].The microscopic structure of FQH and RIQH states may be explored viatheir thermodynamic characteristics, for example by measuring how robustthey are to destabilization at finite temperature. Numerous gap measure-ments of FQH states in the literature fall into this category [22], as domeasurements of the melting temperatures for electron solids [17, 18, 23].For example, the melting temperatures for ν = 2− 4 RIQH states scalewith Coulomb energy, indicating that they are stabilized by interactions (asexpected for electron crystals) [23]. But large differences in melting tem-perature below and above ν = 4 may indicate that the crystal structurechanges with filling factor [24]. The problem with this approach is that itdoes not provide a tool for probing a microscopic structure at base tem-peratures. From a practical point of view RIQH states from 2 < ν < 4are extremely fragile, limited to temperatures below 40 mK and the high-est mobility samples [17, 23, 25, 26]. Even small temperature change candestroy low temperature electron crystal phases prior to their separation inthe magnetic field.Another way to destabilize RIQH states is to drive a large current biasthrough the sample, giving rise to an electric field in the 2DEG plane. Likeother charge ordered systems across the field of strongly-correlated elec-tronics, RIQH states are susceptible to destabilization by an electric field,which depins the crystal from an underlying disorder potential or reformsit with an altered long-range order [27–29]. Experimental measurements ofthe bias-driven sliding dynamics of charge density waves date back to workon NbSe3 nearly four decades ago and remain an area of active research [30–32], as they provide a lot of information about the underlying microscopicstructure of the electron system.Transport signatures of depinning, whether for RIQH states or tran-sition metal oxides, include sharp transitions out of the insulating statefor increased bias, and excess resistance noise in the transition region oreven narrow-band oscillations [27, 28]. At the same time, the temperature-induced transition out of insulating RIQH states was observed to be far moreabrupt that would be expected for thermal activation in a gapped quantumHall liquid. This led to an understanding that the onset of conduction inRIQH states at moderate temperatures is a collective effect associated witha phase change of the electronic system that is effectively a melting of theelectron solid.[23]However, the unusual bias and temperature dependence of RIQH statesmust not be considered in isolation from each other. All previous interpre-71.3. Structure of this thesistations relied on the assumption, that the phase transitions happen simul-taneously in the entire sample. However, finite currents driven through asample in the quantum Hall regime necessarily create a temperature risedue to Joule heating, and that heating is in many cases strongly localized.What is the effect of heating due to current flow in an electron system closeto an abrupt phase change between insulating and solid states?Induced in-plane electric field creates interesting effects at the half filledlower Landau levels (ν = 9/2, 11/2, etc), where strongly anisotropic stripephase is believed to stabilize. It was previously shown to reorient underexternal parallel magnetic field. What is the effect of the electric field onthat state and what is the structure of the symmetry breaking potentialwhich aligns the stripe phase along specific crystallographic direction in allGaAs samples?1.3 Structure of this thesisFollowing this introduction, Chapter 2 starts with the theoretical basicsof two dimensional physics of interacting electrons. After brief descriptionof integer, fractional and reentrant quantum Hall effect theory, I focus onreview of the main theoretical models of microscopic structure of reentrantstates and induced anisotropic phases at higher Landau levels. The detailsof experimental procedures and setup comprise Chapter 3.In the rest of this thesis I present the measurements results of propertiesof different electron solid states under non-equilibrium conditions of highbias.Chapter 4 refers to the spatially-resolved measurements of RIQH break-down, which indicate that a phase boundary between broken and unbrokenregions spreads chirally from source and drain contacts as a function of biascurrent. As the phase boundary passes various contacts, its spreading gen-erates multi-stage breakdown signatures like those observed elsewhere. Thechiral sense of the spreading is not set by the chirality of the edge state itself,instead depending on electron- or hole-like character of the RIQH state.The temperature measurements of high bias RIQH states breakdown atν = 2− 4 are described in Chapter 5. The measurements of reentrant statespresented here suggest that the electronic temperature of the bulk stays closeto the cold GaAs lattice even when Joule heating is significant, in markedcontrast to fractional states, which melt more easily with bias. Althoughthe temperature range where the reentrants states are stable does not allowto accurately estimate electron temperature near the high bias breakdown,81.3. Structure of this thesispresented thermal analysis show, that the observed effects can not resultfrom a simple melting of the electron crystal by Joule heating. The possibleexplanation of the observed effects in reentrant states is given in terms ofnon-uniform break-down, discussed in previous chapter.Chapter 6 describes experimental results of current induced anisotropymeasurements of the stripe phases in the range of filling factors ν = 9/2 −15/2. The experimental findings are compared to the experiments from othergroups. These data suggest an existence of multiple regions of magneticfields around each half filling of LL with different alignment orientation ofthe stripe phase. The model of double-well potential is suggested to explainthe experimental data, providing a new vision at the mechanism, responsiblefor alignment of anisotropic phases.Conclusions and discussion of future possible experiments form Chap-ter 7.9Chapter 2Literature review2.1 Quantum Hall Effect2.1.1 Low dimensional physicsTwo dimensional electron gas is the most versatile playground for probingfundamental interactions in solid state physics. This term is usually re-ferred to the systems where the movement of electrons is confined in onedirection to the extent that the energy splitting between the levels of spatialquantization becomes significant. In real samples the confinement is usuallyachieved at the interfaces of materials with different band gaps, creating apotential barrier for electrons in one direction. The simplest realization ofthis approach is the interface of semiconductor and insulator. A metal gateon top of the insulator allows to apply the electric field perpendicular to thesurface, providing a lot of control on the energy levels of the electrons at theinterface.First MOSFETs opened an access to the experimental probing of lowdimensional physics. Further technology improvement of epitaxial growthallowed to create high quality heterostructures with pre-engineered compli-cated band structure of electrons. The most advanced are the MBE grownheterostructures based on the interface of GaAs/AlGaAs semiconductor.This technique provides a lot of control on the thickness of the thin filmsgrown, as well as, “sandwiching” them in any desired manner. Implementa-tion of the side δ-doping technology in 1975 [33] allowed to achieve extremelylow disorder in crystal lattice and the record mobilities of electrons. Dis-covery of fractional quantum Hall effect in 1986 [7], made GaAs/AlGaAsheterostructures the main playground for probing low dimensional physics.Consider the simplest case of a single quantum well when a 2DEG isconfined between two infinite walls at the distance a from each other. Thismodel is a good approximation of a regular quantum well, since the differenceof GaAs and AlGaAs band gaps is much larger compared to the energysplitting of spatially quantized levels. In this case, free motion of particlesis limited to two degrees of freedom, while in the third one the momentum102.1. Quantum Hall Effectcan have only certain discrete values. Similarly to pure 2D case, the densityof state at each spatially quantized sub-band is constant and given bydnsdE=gsm∗pih¯2, (2.1)where m∗ and gs are the effective mass and spin degeneracy of an electronrespectively.Following conventional notation, we set the z axis of coordinate sys-tem aligned along the confinement direction. The kinetic energy of non-interacting electrons isEk(nz) =h¯k2x2m∗+h¯k2y2m∗+ E(nz), (2.2)where kx,y are the wave vectors in the plane of 2DEG, andE(nz) =(pihnz)22m∗a2. (2.3)is the energy of spatially quantized sub-band nz.If this system is cooled to temperatures below the energy splitting ofspatial quantization ∆E(nz) = E(nz + 1) − E(nz), electrons condense tothe lowest energy and occupy states at the first energy sub-band (unlesselectron density is higher, than the degeneracy of states at this level andfew sub-bands are populated). This leads to the situation when electronmomentum in the z direction is fixed and the system has only two degreesof freedom.Application of the magnetic field B, perpendicular to the plane of the2DEG, confines the electrons to cyclotron orbits. If an in-plane electricfield E ‖ x is applied on top of the magnetic field, in addition to circularmotion at cyclotron frequency ωc = eB/m∗, electrons start to drift at speedv = E/B perpendicular to the electric and magnetic fields. Since averageelectron velocity has a component perpendicular to the magnetic field, thetransport properties of 2DEG are described by the conductivity tensor σˆ:jx = σxxEx + σxyEy,jy = σyxEx + σyyEy,(2.4)where σxy = −σyx and σxx = σyy for an isotropic space.Using the equation of classical free electron motion in an electromagneticfield in the limit of relaxation time τ :mdvdt= ev×B + eE− m∗vτ(2.5)112.1. Quantum Hall Effectand Ohm’s lawj = nsev, (2.6)the components of conductivity can be expressed in the following form:σxx =nse2τm∗11 + ω2c τ2,σxy =nse2τm∗ωcτ1 + ω2c τ2.(2.7)The quantity µ = eτm∗ is usually referred to as the mobility of the 2DEG.It is proportional to the electron scattering rate and is the measure of thesample quality. In an ideal 2DEG, τ →∞ and (2.7) takes form:σxx = 0,σxy =nseB.(2.8)In the quantum limit, perpendicular magnetic field splits the electronspectrum (2.2) into completely discrete Landau levels:E(N,nz) = h¯ωc(N +12) +(pihnz)22m∗a2. (2.9)Here ωc =eBm∗ is the cyclotron frequency and N=0,1,2... is the orbitalquantum number of the Landau quantization. Degeneracy of each Landaulevel can be written in the formγ =S2pil2B, (2.10)where S is the area of the 2DEG and lB = (h¯eB )1/2 is the magnetic length.Consider the density of electrons at fully filled Landau level, expressed interms of magnetic flux quantum Φ0 =he ,nL =γS= (2pil2B)−1 =BΦ0. (2.11)Equation (2.11) means, that in case of completely filled Landau level, eachelectron encircles one magnetic flux quantum. Therefore, one can define thefilling factorν =nsnL=nsheB, (2.12)as the number of filled Landau levels at zero temperature.122.1. Quantum Hall EffectBesides the orbital quantization, out-of-plane magnetic field removes spindegeneracy of electrons. Zeeman effect splits each Landau level in two spinpolarized branches. Consequently, the state of the 2D electron in the mag-netic field is set by three quantum numbers: orbital number N = 0, 1, ...,spin s = ±1/2 and the number of spatial quantization sub-band nz = 1, 2, ...:E(N, s, nz) = h¯ωc(N +12) +(pihnz)22m∗a2+ sgµBB. (2.13)Here g is electron’s g-factor and µB = eh/2m0 – Bohr’s magneton. EachLandau level corresponds to the cyclotron orbit of radius rN ≈√NrB. Eachorbit can be occupied by two electrons with opposite spins, correspondingto spin resolved branches. As the energy of the Zeeman splitting is muchsmaller than the cyclotron energy, filling factors 0 to 2 correspond to par-tial filling of the first spin-degenerate Landau levels with orbital quantumnumber N=0; from ν = 2 − 4 the first (N=0) level is completely filled andthe second (N=1) is partially occupied, etc.Consider the sample in a fixed perpendicular magnetic field. As wasmentioned above, at zero temperature electrons occupy the lowest possibleenergy levels. Therefore, if the electron density is increased, more and morestates are being filled at the last Landau level, increasing the filling factor.Once this level is completely filled, the electrons start to occupy the upperone, etc. Since the occupation of each level is defined by the filling factor,this experiment is identical to magnetic field sweep at the fixed electrondensity in the sample. Formula (2.11) shows that the density of states ateach Landau level depends on the magnetic field. As the magnetic field islowered, the number of states at each level is decreased and electrons areprompted to occupy the higher energy states.2.1.2 Integer Quantum Hall EffectThe important consequence of discrete energy levels is that at integer fillingfactors the finite energy, equal to the Landau level splitting ∆E = h¯ωc, isneeded to add the next electron to the system. In such a state the 2DEG doesnot screen an external electric field, i.e. demonstrates insulating behaviour.If, for instance, the density is varied by the voltage application on a metaltop gate, the number of electrons in 2D channel at integer filling factor willstay constant in the range of voltages ∆Vg = h¯ωc/e until next electron canbe added. According to (2.8) and (2.11) the conductivity of the 2DEGσxy =νnLeB= νe2h(2.14)132.1. Quantum Hall Effectwill demonstrate the quantized plateau in this range.In an ideal system, the application of voltage between the metal gateand electron system implies, that the voltage source forces electrons to movefrom the metal into the 2DEG. This charge difference on the plates creates avoltage between them. If no electron can enter the 2DEG (equivalent to theinsulating state), the voltage in the range ∆Vg = h¯ωc/e can not be applied bythe source. However, in real systems the localized states are always present(at uncontrolled impurities, for example). When the intermediate voltageis applied, the extra electric field, not screened by the 2DEG, penetratesthrough the channel and is screened elsewhere in the sample.Although this hypothetical experiment leads to the transverse conduc-tivity quantization, it can not be considered as the explanation of IQHE.For example, it implies zero width of a plateaux in the magnetic field sweep(σxy has quantized value exactly at one point, corresponding to integer fillingfactor), which is not the case in real samples.Discreetness of electron spectra in perpendicular magnetic field, dis-cussed above, is important, but not enough for explanation of Hall conduc-tivity quantisation. Another important property, required for observationof the plateaux, is the presence of a random potential.The “magic” conductance value at the plateau, coinciding with classicalHall resistance at completely filled Landau level, is rather a hint than acoincidence. According to (2.14), σxy is defined by the filling factor and isindependent on the magnetic field (providing the filling factor is the same).In other words the contribution to net Hall voltage, appearing in completelyfilled Landau level, is independent on the magnetic field, as both quanti-ties, the transverse conductivity and electron density, depend linearly onB. Consequently, once the Landau level is filled, it contributes equally tothe net conductivity of the 2DEG, at any value of the magnetic field. Ifthe filling factor variation leads to population of localized states by extraelectrons, the net conductance is independent on the total number of theelectrons in the system, as it is defined only by the number of underlyingfilled Landau levels.Random fluctuations of electron’s potential result from uncontrolled im-purities and defects in a crystal structure, which are always present in realsamples. The role of random potential is twofold. At first, it broadens Lan-dau levels, which creates finite density of states in the some range of energiesbetween Landau levels. This allows for gradual change of the Fermi level bysweeping the external parameters such as magnetic field or electron density.Secondly, random potential creates a number of localized states in certainrange around integer filling factor. The former is important since only de-142.1. Quantum Hall Effectlocalized electrons contribute to electronic transport. Therefore, to achievea quantized conductance only small fraction of delocalized states has to beoccupied [34–36] (Fig. 2.1).EDELOCALIZEDDELOCALIZEDDELOCALIZEDEDELOCALIZEDDELOCALIZEDDELOCALIZED1/2 hωC3/2 hωC5/2 hωCEDELOCALIZEDDELOCALIZEDDELOCALIZEDEFEFEFRxy= 2e2/h Rxy= 3e2/hRxy= 2.5e2/hFigure 2.1: Structure of electron energy levels in the presence of randompotentialWhen the filling factor of the 2DEG is changed from ν = i to i + 1(by changing electron density, for instance), the Fermi level gradually movesfrom one Landau level to the next one. Underlying random potential leadsto spatial variation of local filling factor of the 2DEG. First electrons occupylocalized states at the bottom of the upper Landau level. Consequently, theHall conductance remains unchanged, given by (2.14):σxy = ie2h. (2.15)Further filling leads to occupation of delocalized states. At this stage allextra electrons change the local filling factor at the parts of the sample,where the electron motion is not finite. Since only the density of delocalizedelectrons at the Fermi level is important for transport properties, at thispoint the transverse resistance deviates from quantized plateau. By the time,when all delocalized states are occupied, the density of “transport” electronsequals to (i+ 1)nL. Therefore, σxy hits the next level of quantization.152.1. Quantum Hall Effect2.1.3 Theory of the edge statesThe importance of the edges in QHE was realized soon after its discovery.In [37] IQHE was explained in terms of a supercurrent due to the long-rangephase rigidity of the wave functions around the closed loop. Then this pic-ture was supplemented by discussion of the edge states which form at theboundary of the sample [4]. In 2DEG, placed in a strong perpendicularmagnetic field, the effect of the edge can be described in terms of increasingenergy of Landau levels due to additional confinement near the boundary(Fig. 2.2). In semiclassical limit this can be understood as the quantizedenergy levels lifting when the edge potential confines electron motion, pre-venting them from circling over cyclotron orbit.Figure 2.2: Structure of electron energy levels in the presence of the edges.After [4].Generalized theory of the edge states was given in [38]. The absence ofbackscattering results from the cyclotron motion of electrons: when scat-tered from the edge their trajectories are curved back by the magnetic field.The skipping orbits of such kind result in continuous unidirectional electrondrift within a distance on the scale of the magnetic length from the edge. On162.1. Quantum Hall Effectthe other hand, electrons feel electric field dUdy due to bending of the bandsclose to the edge. This field is perpendicular to the boundary of the 2DEG,causing the centers of cyclotron orbits to drift along the edge. This propertyof the edges eliminates electron backscattering, turning edge channel into aperfect one dimensional conductor.To describe this picture on the language of quantum mechanics, let’sconsider an ideal two-dimensional strip of width w. Let us set the Ox axisof coordinate system along the strip and Oy perpendicular to it. The spinlesselectron Hamiltonian of this system isH =12m∗((px − eBy)2 + p2y) + V (y), (2.16)where we set the vector potential A = (−By, 0, 0) and edge electric potentialV (y). The wavefunction is separable in the form ψj,k = eikxfj(y), whichleads to the eigenvalue problem:[− h¯22m∗∂2∂y2+m∗2ω2c (y0 − y)2 + V (y)]f = Ef, (2.17)where parameter y0 = −kl2B. In the bulk V (y) ≡ 0 and (2.17) turns into theproblem of free electron in the magnetic field with quantized Landau levelsEjk = h¯ωc(j + 1/2). Near the edges V (y) > 0 around y = 0 and y = wand Landau levels are dependent on y. Edge effect lifts the degeneracyof the Landau quantization. At zero temperature all states are occupiedbelow the Fermi level and vacant above it. As a result, electrons at theFermi level are confined in lateral direction to certain distances from theedge, corresponding to the points where Landau level crosses the Fermienergy. This confinement creates N channels on each side of the sample,corresponding to N filled Landau levels below the Fermi level in the bulk.Longitudinal electron velocity can be expressed in terms of y0:vjk = h¯−1dEjkdk= h¯−1dEjkdy0dy0dk, (2.18)wheredEjkdy0is proportional to the slope of Landau level. This slope hasdifferent signs on opposite sides of the sample, representing the chiral motionof edge electrons around the perimeter of the sample.This natural absence of electron backscattering near sample’s edges inthe magnetic field allows to describe quantum Hall system in terms of idealconductors [39]. Consider the strip connecting two electron reservoirs atchemical potentials µ1 and µ2, which serve as a source and a sink of carriers172.1. Quantum Hall Effectand energy. Let us calculate the conductance of this conductor, assumingµ1 > µ2. Since backscattering is forbidden, reservoir emits carriers intocurrent-carrying states up to its chemical potential. Once the carrier reachedanother reservoir, it is absorbed. Since below µ2 all the states, movingin both directions, are occupied, the net current is produced only by theelectrons in energy interval between µ2 and µ1. The injected current in onechannel isj = evdndE∆µ. (2.19)Employing (2.18) and the density of states for 1D system dn/dk = 1/2pi,(2.19) turns intoj = e1h¯dEdkdndE∆µ =eh∆µ. (2.20)Thus, the current fed into an edge state by a reservoir is the same as thecurrent fed into a quantum channel in a zero-field perfect conductor. Theresulting two-terminal resistance for a single edge state:R =VI=∆µehNe∆µ=he2. (2.21)Consequently, in ideal sample every Landau level below the Fermi energycreates a pair of edge channels, one on each side of the sample, which actsas a resistor with quantized conductance. At integer filling factor ν = N ,every pair acts as a parallel resistor, i.e. the total 2 point resistance of thesample isR =he21N. (2.22)This result has important consequences. First, it provides alternative ex-planation for the conductance quantization in quantum Hall regime: everyLandau level below the Fermi level creates ideal one dimensional channel.Secondly, in order to change the direction of motion, electron has to moveover macroscopic distance through the bulk to the opposite edge. There-fore, similar to an ideal 1D wire, the potential is constant along the edgechannel, and one would measure vanishing longitudinal resistance betweenany contacts at one side of the sample.It is important to mention, that described effect does not depend on theedge shape: the edge state is naturally formed everywhere, where the Landaulevel crosses the Fermi energy. This means, that (2.22) is independent onimpurities at the edge: addition of an extra impurity changes only the shapeof the edge state. This provides an explanation of the universality of theHall effect: its total independence on the shape and size of the sample [40],182.1. Quantum Hall Effectonly topology of the measurement contacts with respect to the source/drainones matters.Interaction effects are also important in real samples. The detailed the-ory was developed in [41, 42] and led to introduction of compressible andincompressible stripes concept: competition between Coulomb interactionand electrostatic energy of electron in the effective edge electric field leadsto 2DEG density redistribution. This becomes possible even at zero temper-ature due to a large number of unoccupied states in the vicinity of Landauand Fermi level crossing. Thus the sample breaks into compressible andincompressible strips, parallel to the edge. Although, the microscopic de-tails of such systems are very different from the simple Landauer picture,described above, the accurate calculation shows, that (2.22) stays correcteven in the interacting electrons case.2.1.4 Non uniform samplesPreviously, only samples with uniform density where considered. Here theeffects of sample inhomogeneities are considered. Specifically, this paragraphis focused on macroscopic variations of a random potential, leading to differ-ent electron densities at different parts of the sample. For the purpose of thisthesis, sharp density changes are considered, i.e. it is assumed that there isinfinitely narrow finite density jump at the boundary between two uniformregions with different electron density. Since number of the edge states inthe sample varies with the filling factor, some of them must be reflectedat the boundary. The sharpness of the density jump allows to avoid thediscussion of the question, how the edge states behave inside the boundaryunder the density gradients and simplify this problem to several independentedge channels, some of which are reflected by the density jump, while suchboundary is transparent for the others. In a real world such conditions canbe achieved by the voltage, applied to the macroscopic gate, covering partof the sample. Since in real structures the proximity effects at the edges arecomparable to the depth of 2DEG, this assumption is feasible.Scattering of the edge states has been a topic of intensive research for along time [43]. For example, it was shown, that in a system with a narrowconstriction, the electrical potential varies along the edge, and, longitudinalvoltage deviates from zero. Therefore, diagonal measurements could probedifferent values at different contacts.Here this theory is applied for the case of one reflected edge state ateach side of the sample, modeling the sample with non-uniform density.Obtained results are important for interpretation of experimental findings192.1. Quantum Hall EffectHLH+H-L+L-n=N+1n=N+1n=NFigure 2.3: Propagation of the edge states in inhomogeneous sample withtwo filling factors ν = N and ν = N + 1. The outer thick arrow depicts firstN co-propagating edge states around the whole sample.in the following chapters. The schematics of the considered system is shownon figure 2.3. Source and drain ohmics are surrounded by the region of higherdensity and, consequently, have different number of propagating edge statesin the “melted” (grey areas at Fig. 2.3) and “frozen” regions (the meaningof such notation will be explained later). Assume, that the “frozen” bulk isat integer filling and, therefore, N completely filled Landau levels create Npropagating edge states. In “melted” regions the density is higher, but thefilling factor is still integer (ν = N + 1). This adds one extra edge state ateach “melted” area.According to [38], the current flowing through the ohmic is given by thechemical potential and the number of incoming and outgoing edge states:Ihe= Noutµout −Ninµin. (2.23)Applying (2.23) to source, drain and four corner contacts, shown on Fig. 2.3,202.1. Quantum Hall Effectone gets the following system of equations:Ihe= (N + 1)µH − (N + 1)µ−L ,Ihe= (N + 1)µ−H − (N + 1)µL,0 = NµH −Nµ+H ,0 = Nµ+H + µL − (N + 1)µ−H ,0 = NµL −Nµ+L ,0 = Nµ+L + µH − (N + 1)µ−L .(2.24)Lets define the drain chemical potential µL = 0, then µH = V . Then,(2.24) takes form:Ihe= (N + 1)(V − VN + 1)= NV,µ−H =NµHN + 1=NN + 1V,µ−L =µHN + 1=VN + 1,µ+H = µH = V,µ+L = µL = 0.(2.25)If one defines diagonal voltages V + and V − with asV + = µ+H − µ+L ,V − = µ−H − µ−L ,(2.26)and solve (2.26), the diagonal resistances could be defined asR+ =V +I=he21N,R− =V −I=he21NN − 1N + 1,(2.27)or, expressed with the resistance of the melted state:R+ =he21N + 1(1 +1N),R− =he21N + 1(1− 1N).(2.28)212.2. Coulomb interaction effectsThe physical meaning of (2.28) becomes clear after it is expressed in termsof longitudinal resistanceRxx =µ+H − µ−HI=he21N + 11N(2.29)and Hall resistance of the melted regions, adjacent to the source and draincontacts RH =he21N+1 :R+ = RH +Rxx,R− = RH −Rxx.(2.30)In other words the diagonal voltages include not only the potential differencebetween the edges of the sample, but the potential drop along the edge.2.2 Coulomb interaction effectsSo far only the systems of non-interacting electrons were considered. Al-though single particle approximation can provide theoretical explanation forIQHE, it suggests only plateaux of σxy appearing at the integer multiples ofe2/h. However, it was a great surprise, when the quantization of transverseconductivity was found at e2/3h and 2e2/3h [7, 44]. Another mystery wasthe width of the integer plateaux, which become wider in higher mobilitysamples (µ > 106cm2 ·V/s). These experiments could not be explained onlywith the impurity effects in non-interacting 2DEG. Consequently, it is evi-dent, that under certain conditions, the Coulomb interaction becomes moreimportant than random variation of underlying electron potential.First theoretical investigations demonstrated, that Coulomb interactioncan cause stabilization of a large variety of qualitatively different states, re-vealing rich field of many-body physics. From mathematical point of view,the fundamental difference from the non-interacting picture, is that the vari-ables of an interacting Hamiltonian can not be split. Therefore, the wavefunction of the system can not be described as a product of single electronwave functions. In order to find the ground state of 2D electron system,one has to solve the differential equation in 4N+1 dimensional space, whereN is the number of electrons in a system. Different numerical techniques,such as Hartree-Fock approximation, are applied to address this problem,but in many cases the accuracy of those calculations is questioned, since itsuncertainty often is comparable to the difference between energy levels ofthe system. Another problem was that neither of ground states, found bythose calculations, could explain quantization of the transverse conductivity222.2. Coulomb interaction effectsat fractional filling factors (what is now called FQHE). Therefore, this effecthas to be explained by a fundamentally different ground state.The purpose of this section is to provide a brief theoretical review of twomain families of the interacting electron ground states in the quantum Hallregime and to give a qualitative explanation of their microscopic structures.As will be shown later, stabilization of each state is defined by the temper-ature and interaction energy. Similar to the atoms, depending on the ratiobetween temperature and interaction, the electrons can form a liquid – astructure with no long range order, and electron crystals (completely peri-odic phases, sometimes referred to as electron solids). Both of those types ofstates are interesting in the scope of this thesis, as they lead to quantizationof Hall conductivity in some range of filling factors, however, the underlyingphysical mechanisms of these effects are completely different.2.2.1 Fractional Quantum Hall EffectFollowing the chronological order of realising the microscopic structure ofmany-body states in quantum Hall regime, first the experimental signaturesof fractional Quantum Hall Effect (FQHE) are described. The detailed mi-croscopic theory of fractional states is beyond the scope of this manuscript.Instead, a very general qualitative picture is provided, allowing to build aconceptual understanding of the formation mechanism of FQHE plateaux.Since experimental evidence of FQHE appeared before the theoreticalexplanation, we first describe the basic experimental facts, which such atheory has to explain:1. This effect takes place only at certain densities, defined by the strengthof out-of-plane magnetic field.2. Experimental manifestation of FQHE is very similar to IQHE, be-sides the plateaux correspond to a fractional level of quantization, i.e.(1/3)e2/h3. The effect appears only in high quality samples. It is very fragile andis affected by even small amount of impurities4. Fractional quantization is observed only at low temperatures (below1 K) and high magnetic fields5. In addition to the simple 1/3 and 2/3 large family of quantized plateaux,corresponding to other fractions with odd denominators, such as 2/5,232.2. Coulomb interaction effects2/7,..., has been observed. Those states are even more fragile andtemperature sensitiveThe similarity between fractional and integer quantum Hall effects, hintson the similar underlying microscopic mechanisms (otherwise, one wouldneed to find a completely different theory to explain quantized plateaux).It was shown, that Laughlin’s wave function of ν = 1/3 fractional state [9]ψ ={∏j<kf(zj − zk)m}exp(−14∑l|zl|2), (2.31)where zj = xj + iyj , was a good approximation to the ground state of aCoulomb interacting system. Therefore, its properties could be understoodfrom those of Laughlin’s ground state. (2.31) describes a bound state with anumber of zeros, corresponding to the number of flux quanta in the system.The fermionic nature of this system requires (2.31) to be an antisymmetricfunction, i.e. m must be an odd integer. This means, that the ratio betweenthe number of particles and magnetic flux (in units of the flux quantum),being an inverse of the filling factor, is an odd integer as well.However, Laughlin’s wave function explains the ground state only atthe filling factor, which is exactly 1/q. Consequently, (2.31) itself can notaccount for an existence of finite width plateaux. In order to build a theoryof fractional states, analogous to IQHE, one has to construct the systemof non-interacting particles, localized on a random potential around thefractional filling factor [45]. When the area of the system is slightly changed,the number of flux quanta changes, and, therefore, Laughlin’s ground statehas to be modified. It could be shown [46], that the wave function changesby means of quasiparticles, which carry fractional charge e∗ = e/q. Atsufficiently low densities of quasiparticles, i.e. sufficiently close to ν = 1/q,the interaction between particles becomes low and all the correlated statesare destroyed by the potential fluctuations. In this case random potentiallocalizes quasiparticles and their contribution to transport properties of thesample vanishes. This results in the quantized plateau around ν = 1/q.Since the system is particle-hole symmetric, similar speculations could beapplied to the ν = 1− 1/q fractional states.This approach can be generalized for the case of other fractions, suchas n/(2kn + 1). In [47] the concept of composite fermions (CF) was intro-duced. This approach allows to combine weakly interacting quasiparticlesby attaching to each electron 2k flux quanta in the direction opposite to theexternal magnetic field. As a result, every CF effectively feels smaller mag-netic flux. For the Laughlin state ν = 1/q, the residual flux q − 2k = 1, as242.2. Coulomb interaction effectsq is an odd number. In other words, formally ν = 1/q state corresponds toIQHE of CFs at ν∗ = 1. A quantized plateaux corresponds to every integerfilling factor n of CF with 1/n flux quanta per particle, i.e.q − 2k = 1n, (2.32)which gives the equation for FQHE plateaux sequence:ν =1q=11/n+ 2k=n2kn+ 1, (2.33)where k,n are positive integers.Similarly, CFs could help to understand recently discovered FQHE ateven denominators [48]. A two-dimensional electron system in an externalmagnetic field, with Landau-level filling factor ν = 1/2, can be transformedto a system of CFs in zero effective magnetic field [49]. Nonetheless, the mi-croscopic descriptions of some other even-denominator states are still underdebate [50] and vary from exotic particles with non-Abelian statistics [51]to Bose-Einstein condensation of two-electron pairs with opposite spins [52].Recently, the requirement of complete spin polarization in the ground statewas also reconsidered. It was found that a partially filled lowest LL maycontain Skyrmions [53]. But independently on the microscopic structure ofthe ground state, the general approach is always the same: an interactingelectron system is described by means of complex weakly interacting quasi-particles in the vicinity of the fractional filling factor. If those new particlesdo not contribute to net current (localized on random potential, for exam-ple), the Hall resistance resides on the plateau, corresponding to the classicalHall value at the fractional filling factor.2.2.2 Reentrant Quantum Hall StatesThe primary parameter, defining the state of interacting system, is the ratiobetween carrier potential and kinetic energies. In 2DEG the strength ofinteraction is measured by rs, the ratio of Coulomb energyEC ∼ e2a∼ √ns (2.34)to Fermi energyEF =pih¯2nsm∗∼ ns. (2.35)252.2. Coulomb interaction effectsTherefore,rs ∼ 1√ns(2.36)increases, when the carrier density is decreased.Similar to atoms in a crystal lattice, the ground state of a 2DEG in astrong interaction limit (high rs) in the absence of the disorder is expected tobe a Wigner solid [54]. In such ideal system the critical density, above whichthe crystal melts even at T=0, was estimated to correspond to rs ∼ 37 [55].However, in real systems disorder competes with Coulomb repulsion, result-ing in even lower critical densities. Although the current level of disorder inthe best quality samples does not allow to reach the critical values of rs, it ispossible to create carrier densities where deviation of a strongly correlatedsystem from Fermi liquid can be observed even at B = 0 [56, 57].Under conditions of QHE, the perpendicular magnetic field quenches thekinetic energy of electrons to cyclotron motion, leading to higher values ofrs for the same densities. An electron Wigner crystal was predicted to bea fundamental state even in realistic samples in the limit of high magneticfields [58]. Moreover, the main parameter in this regime, defining the groundstate of the system, is not the electron density, but the filling factor. In realsamples electron solids are believed to be pinned to the underlying disor-der [59], producing an insulating state of the sample. Experimentally thisstate is very different from localization of free electrons or low interactingquasipartiles on random potential for the case of IQHE and FQHE. In con-trast to quantum Hall states, where free carriers are delocalized near theedges, resulting in non-dissipative transport Rxx → 0 as T → 0, the resis-tivity of pinned electron solid is expected to demonstrate pure insulatingbehavior with diverging Rxx →∞ at T → 0.Lots of attention has been drawn to reentrant insulating phases (RIP)in the chase of 2D electron Wigner crystal in high perpendicular magneticfields. Experimentally RIPs where observed at the edges of fractional [60–62]and integer plateaux [63]. However, the role of the competition between ran-dom potential and electron-electron interaction, defining electrons groundstate, is still under debate [64]. RIP’s transport characteristics [65, 66] arevery similar to charge ordered structures in other systems and considered tobe an experimental demonstration of the weak WC pinning [67, 68].If the cyclotron motion is preserved and LL mixing is small, underlying,completely occupied LLs affect only the background screening of the systemat high filling factors and can be excluded from the problem. Consequently,the ground state of such a system is defined only by few electrons left at thelast partially filled LL. Namely, the partial filling factor of the last Landau262.2. Coulomb interaction effectslevel ν∗ = ν − [ν] at ν > 1, plays the same role as real filling factor for thecase of ν < 1. This means, that the system behaviour in higher filling factorsshould be approximately periodic with ν. For example, in the vicinity ofinteger fillings, effective charge carrier density in the problem is low andthe system becomes highly interactive, creating conditions for electron solidformation, similar to WC in high magnetic fields. Consequently, the systemin the vicinity of integer plateaux could turn into insulating state beyond thesingle electron localization on random potential. This effect is believed tobe responsible for widening of IQH plateaux, observed in ultra-high mobilitysamples.After discovery of RIQHE at the Landau levels with high (N > 2) orbitalnumbers [27], the theory of electron solids was reconsidered. First exper-iments demonstrated, that reentrant states (RS) and integer plateaux atelevated temperatures are interspersed by the free electron states. Further-more, temperature measurements showed, that RSs collapse around partialfilling factors ν∗ = 1/4 and ν∗ = 3/4, which can not be explained by lowdensity Wigner crystallization. However, reenterance of Hall resistance tothe values of integer quantization hinted on the pinning mechanism of elec-tron localization at the last LL. Therefore, the electron crystallization theorywas expanded to charge density waves (CDW) [3, 14]. The basic conceptof CDW lies in the idea, that cyclotron motion alters electron-electron re-pulsive interaction and the features of a real single particle potential couldat certain electron densities make energetically favourable to double the pe-riod of WC lattice for the price of putting two electrons per site [3]. Suchapproach extends a conventional single electron WC state to the sequenceof the ordered electron structures with n = 1, 2, ... electrons per site.Qualitatively this result could be understood from analysis of a simple1D toy model, adopted from [14]. Consider an electron gas, interacting in aboxlike potentialu(x) = u0Θ(2R− |x|), (2.37)and compensated by a surrounding average uniform background potential.Let’s also assume, that the number of states “inside” each box is limited.In such a system the lowest energy state corresponds to the equidistantpositions of boxes, minimizing repulsion between neighbours. Let’s nowconsider how the energy of this gas depends on the number of the particlesper site. It costs zero energy for the particle to get “inside” the box, however,the repulsion energy decreases when the spacing between sites is increased.Therefore, such model system tends to maximize the lattice period and filleach box with the maximum possible number of electrons.272.2. Coulomb interaction effectsTurning back to the case of a reentrant quantum Hall system, the elec-trons appear to be broken in clusters with equal number of electrons, whichplay the role of the “box”. The number of electrons in the cluster is lim-ited by the the local Landau level capacity. Thus, the ν∗ = 1/4 RS wouldcorrespond to Wigner-like lattice with electrons grouped in the clusters atmaximum local density, corresponding to ν∗ = 1. For the particle-hole sym-metric case at ν∗ = 3/4 the “bubbles” turn into ν∗ = 0 puddles in the seaof ν∗ = 1. Random potential in the sample, disturbs an ideal order in thisstructure, leading to formation of electron clusters or “bubbles”, as theywere historically referred to. Consequently such microscopic structures areusually called the “bubble phases”.Of course, electrons in 2DEG experience short range repulsive interac-tion. Thus, adding an extra electron to the bubble increases the energy ofthe system. However, in contrast to the case of vanishing magnetic field,where the repulsion between two electrons increases as 1/r, the overlap oftwo electrons, orbiting around the cyclotron motion does not strongly de-pend on the distance between them, when the centers of the orbits are closer,than 2rc. Therefore, the model of the “box”-like potential physically resultsfrom electron cyclotron motion.In order to explain RIQHE, one needs not only to constract a funda-mental electron solid state away from integer filling factor, but also demon-strate the mechanism of its localization. Similar to ordinary Wigner crystal,the question of the CDW pinning was investigated theoretically [69, 70].Fukuyama et.al. demonstrated, that competition between the random im-purity potential and the elastic energy for phase fluctuations leads resultsin two limiting cases. In the strong pinning case the phase is pinned to eachimpurity site. However, in the weak pinning case the pinning mechanism ismore subtle. The system is pinned by effectively breaking up into domains,suggesting a mechanism from intrinsic inhomogeneity of the sample.2.2.3 Anisotropic states around LL half fillingCompletely new anisotropic states were discovered next to the reentrantstates [11, 12] around half filling of the Landau level. In contrast to all otherexperimental manifestations of quantum Hall effect, the bulk of the sample isnot insulating in these states and, thus, transverse resistance does not quan-tize. Additionally, the longitudinal resistance experiences high anisotropywith respect to crystallographic axis of the sample.Although the behavior of these new states was very different, the maincandidate for their microscopic description is a correlated electron solid.282.2. Coulomb interaction effectsFirstly, temperature measurements show sharp melting of such states withdistinct onset temperature for the anisotropy. Above this critical point, theanisotropy vanishes and longitudinal resistance corresponds to conventionalIQHE between integer plateaux. Secondly, the IV characteristic of thosestates are extremely nonlinear [12].From a theoretical point of view, the unidirectional CDW states, whichbreak rotational symmetry of the system, can be formed around half fillingsof the Landau level. They can be considered as a limit of the isotropicbubble phase, which is collapsed into the periodic alternating stripes withresidual filling factors ν∗ = 1 and ν∗ = 0. An edge state has to be formedat the boundary of each stripe due to the density jump, creating a naturalset of Hall bars with low resistance along the edge and exponentially highresistance across the boundary. Therefore, along the stripe direction, onecan consider this system, as a set of Hall bars, connected parallel to eachother. Therefore, the resistance along this directionR‖xx =rxxNS→ 0, (2.38)where NS ∼ w/rc is the number of stripes, formed at the edge of size w,and rxx is the longitudinal resistance of one Hall bar at current filling factor.On the opposite, for the direction, perpendicular to the stripe ordering, thestripes can be considered as a chain of Hall bars connected in series. In thiscase:R⊥xx = rxxNS →∞. (2.39)Despite an easy general explanation, the theory of the electron stripephase at half filling of higher LLs appeared to be complicated in details.First intriguing question was: how do the stripes choose the orientation di-rection? From the first site, in isotropic sample the orientation of the stripephase has to be spontaneously chosen, therefore, one would observe ran-dom anisotropy orientation not only in different samples, but in differentcooldowns as well. However, in early experiments the stripe phase was al-ways found to be oriented along one crystallographic direction. The resultwas confirmed by many experimental groups for different LLs in many sam-ples and many multiple cooldowns. This observation led to a lot of research,trying to identify the rotational symmetry breaking parameter. Experimen-talists tried to connect that to different factors, varying from tiny built-inpotential, resulting from crystal imperfections due to the growth anisotropyof the sample [71], to the form factor of the cleaved sample.It was found, that the stripe phase can be reoriented by many effectssuch as external parallel magnetic field [72, 73], mechanical deformation [74]292.2. Coulomb interaction effectsor density effects [75]. However, the underlying microscopic mechanism ofstripe alignment is still a topic of significant debates. Recent experimentsrevealed, that the orientation of stripes at the same LL can be different,depending on the specific filling factor, providing an evidence of complexmicroscopic structure of every macroscopic anisotropic state [76].Another puzzle of the electron solids in general and stripe phases in par-ticular is associated with the melting temperature. According to the the-ory, stripes at higher LLs melt at temperatures on the order of 1 K [3, 14],which overestimates the real experimental values by about an order of mag-nitude [11, 12]. In order to deal with this problem, the effects of randompotential were considered theoretically. It was shown, that potential varia-tion disturbs CDW and lowers cohesive energy, but the amount of this effectis still an open question. Alternatively, electron phases without long rangeorder, called electron liquid crystals (ELC), were considered [77]. It wasshown, that the depending on interaction strength and amount of disorder,introduced into the system, CDW can experience several phase transitions,first melting into the glassy state without long range order, and only afterthat, ELC melts into isotropic state, gaining the rotational symmetry [78].Unfortunately, experimental investigation of these states is way behindthe theory. Extreme fragility of electron solids limits the affordable ex-perimental techniques and makes accessing the microscopic structure ofsuch state a very challenging experimental task. In addition to transportanisotropy measurements as a response to an external disturbance by someparameters, a very few experiments, trying to directly measure microscopicfeatures of the electron solids, have been performed. For instance, the highfrequency resonance of stripe and bubble phases where investigated. Tsuiet.al observed disappearance of cyclotron resonance, approaching integerquantum Hall plateaux, as well as in reentrant states at high Landau leveland reentrant insulating states at the boundaries of fractional plateaux inhigh magnetic field limit. In [79] Kukushkin et.al. used photoaccoustic ex-citation of the stripe phases at ν = 9/2 in order to measure the period ofthe stripes. Although they got a period of stripes around 2.9rC , which ispretty close to the theoretical estimate of 2.7rC , calculated by Koulakovet.al. in [3], these techniques do not allow to distinguish different types ofmicroscopic phases.2.2.4 Correlated states at N=1 LLIn previous paragraphs two main types of interacting electron states wereconsidered. They correspond to two different limits of magnetic field, i.e.302.2. Coulomb interaction effectsthe limit of high and low magnetic confinement. In the former, the stronginteraction is given by a large reduction of kinetic energy in high magneticfield. While in the latter case, the effective electron density is significantlyincreased due to the “condensation” of the majority of electrons under thecyclotron gap in completely filled Landau levels.Filling factors ν = 2 − 4, corresponding to the orbital number N = 1,represent the “grey” area, where rS is small and electron interaction couldnot be considered as the main part of electrical potential. On the otherhand, it is not small enough to allow one to approximate this system withfree electrons (this case corresponds to rS  1). Although, CDWs, whichare believed to be responsible for the reentrant states at higher filling factors,were not predicted to exist at N = 1 LL [80], not only is RIQHE observedexperimentally in the ν = 2−4 range, but there exists four instead of conven-tional two reentrant states for each region between integer plateaux [16, 17].This suggests a strong competition between many different electronic states,making the phase diagram of the system extremely complicated [81].Electron ground state phase diagram was assessed by several differentnumerical methods [19–21]. In spite of theoretical predictions of differentCDW phases with one and two electrons per bubble at the opposite sides ofthe fractional plateaux, no direct experimental observation of microscopicelectronic structure under conditions of RIQHE was demonstrated so far.On the other hand, there is a growing experimental evidence, that particle-hole symmetry do not hold for the case of RIQHE at the second Landaulevel [23, 82], which creates even more doubts about the theoretical results.The widely used Hartree-Fock approach, for example, has many limitations.This technique is exact only in the limit of high Landau level occupation,which is not very accurate over the whole range of filling factors, where theRIQHE is observed. Secondly, the competition with the FQHE enhancesfluctuations which might be critical for electron ordering. Eventually, thisapproximation does not take into account LL mixing, which was shown tosignificantly affect the energy gaps of FQH states [83]. Thus, despite morethan a decade of intensive research, the microscopic structure of 2D electronsat the second Landau level is still far from being understood.31Chapter 3Experimental setupIn this section technical details of the experimental setup are given. Al-though it was based on a commercial dilution refrigerator, the system hasbeen modified in several ways in order to achieve lower electron temperaturesand maximal sample quality. A detailed description of the heterostructuresand sample preparation recipes is followed by the a description of the mea-surement scheme.3.1 Sample detailsMost measurements were performed on a square sample with Van-der-Pauwmeasurement geometry. Electrical contact to the 2DEG was achieved by dif-fusing indium contacts placed at the corners and midpoints of the ≈5x5 mmwafer. In few samples, however, a Ni/Ge/Au recipe was used to provideelectrical contact with the 2DEG. This technique was typically used forlithographically defined smaller devices. In this case few more steps of elec-tron lithography were made. The first one defined the etch trenches, whichlimit the mesa of the device. At the second step ≈100x100µm square win-dows for contacts were defined. Subsequent evaporation and lift-off resultedin deposition of Ni/Ge/Au layers in appropriate places, which diffused insidethe 2DEG during annealing at 430◦C and formed ohmic contacts.3.1.1 Heterostructure designThe strength of the observed RIQHE states strongly depends on the het-erostructure design of the wafer, of which the sample is made. Molecularbeam epitaxy (MBE) is the primary technique for growth of the state of theart heterostructures; however, it takes a lot of technical ingenuity to achievethe record mobility and strength of RIQHE.Historically, electron mobility was the main figure of merit for the qual-ity of the 2DEG. In addition to the trend in the strength of FQH states inthe first Landau level, measured in low mobility samples at the first stagesof FQHE research, this belief was backed up by the simple reasoning, that323.1. Sample detailsmobility is limited by internal disorder, which prevents stabilization of cor-related states. Although, the growing evidence of poor correlation betweenthe mobility and quality of FQH states (size of the 5/2 gap, for instance)has been obtained in the last decade [84], electron mobility on a scale of107cm2/V s is still a necessary, though insufficient, condition for observationof RIQHE.The key factor, defining the mobility of the sample, i.e. the random po-tential, experienced by electrons, is the intrinsic disorder incorporated intothe sample’s crystal structure during MBE growth. In first place it is definedby the number of uncontrolled impurities, which is directly connected to thepurity of the chamber and vacuum level during the growing process [5]. An-other significant source of electron scattering centers and disorder comesfrom the ionized doping centers, which lead to direct variation of the elec-tron’s potential inside the channel. The quantity of dopants, defining thedensity of 2D electrons, can not be decreased, but their effect can be sig-nificantly lessened. Many improvements in this scope can be achieved byadjusting the design of the heterostructure.Although, the simplest heterostructure design to produce a high-mobility2DEG is a single heterojunction at the interface of GaAs/Al0.35Ga0.65As,the highest mobility 2DEGs grown today are all quantum well structures,realized as a thin layer of GaAs between two AlGaAs barriers. An obviousstep to lower the amount of disorder, introduced by the dopants - is to movethem away from the channel. Introduction of structures with a δ-dopingsilicon layer result in a significant increase in mobility and is a commonpractice nowadays. In the case of quantum well, the doping layers can beplaced on both sides of the electron channel, significantly increasing 2Delectron density. In reality, the highest mobility heterostructures, which arealso the ones typically used to study fragile quantum Hall states in the secondLandau level, are more complicated than what has just been described andinvolve one of several variations of short-period superlattice doping [85].Measurements were performed on a 300 A˚ symmetrically doped GaAs/AlGaAsquantum wells with low temperature electron density ns ∼ 3 × 1011 cm−2and mobility above 15 × 106 cm2/Vs. The typical conduction band struc-ture of our samples is shown in figure 3.1a. Silicon doping is incorporatedinto the AlAs barriers on both sides of the quantum well (Fig. 3.1b). Al-though AlAs acts as a potential barrier in the main Γ valley of GaAs, thefeatures of the structures of other bands generate a significant amount ofcharge to move from Si atoms into the AlAs layer. This has an importantconsequence: the charge in these channels, although it does not appear asa parallel channel in low bias conductance, can move under illumination at333.1. Sample detailsFigure 3.1: Conduction band edge and free charge density in the immediatevicinity of the doping well located 75 nm below the edge of the primary30-nm GaAs quantum well. On the inset: the general structure of theconduction band edge and charge density profile for a whole quantum well.Adopted from [5].intermediate temperatures and low fields, screening the potential of ionizeddonors from the 2DEG. This effect reveals another important step in samplepreparation: illumination and cooldown protocol.3.1.2 Cooldown protocolAs was noted above, in order to prepare a high quality 2DEG with strongreentrant states, it is important to freeze the charge in AlAs side channelsin such a way that it provides the best screening of the electric field fromionized dopants. First this requires illumination of the sample at a tem-perature range below 1 K. This results in deionization of the majority of Sidonors. At this point, electrons start to tunnel from shallow donor levelsinto the quantum well, surface states and localized states at the AlAs layer.343.1. Sample detailsAdditionally, the elevated electron temperature provides the mechanism forcharge equilibration by means of thermally activated conductivity in AlAslayers. On the other hand, thermal fluctuations prevent the charges fromstaying at equilibrium distribution. Thus the sample has to be warmed to ahigh enough temperatures to enable charges to completely redistribute andequilibrate donor potential, as well as, recover the electron population ofthe quantum well before the system is frozen. However, warming to veryhigh temperatures makes the system sensitive to external electric noise andcharge fluctuations. Therefore, the annealing procedure must be undergoneunder strict temperature control, and thorough parameter optimization isrequired.Illumination of the sample was usually performed by a red LED, whichshone on the sample during the natural cooldown of the system from nitro-gen to base temperatures. In addition to the lack of control over light powerdensity, this design has the drawback of uncontrolled sample temperaturedue to Joule heat dumped into the LED. To overcome this problem a mul-timode optical fibre was used to shine light onto the sample, coming from ared LED located outside the refrigerator. With a power of 140 mW appliedto the LED (80 mA at 1.7 V), 6±2µW of optical power reached the bottomof the fibre, as estimated from the 15 mK of the mixing chamber temperaturerise from base temperature during illumination. Geometrical considerationssuggest an optical intensity of 120 ± 50nW/mm2 at the sample location.Under continuous illumination, the sample temperature, monitored by localthermometer, was approximately 600 mK.A detailed investigation of the effects of low temperature illuminationand annealing on FQH characteristics of a GaAs/AlGaAs quantum wellsample is presented in [1]. For the experiments, presented in this thesis,the following optimized protocol was used for sample preparation. Fromroom temperature the sample was cooled down to roughly 30 mK and thenilluminated for 30 minutes. Next, the light was turned off and, using alocal heater the sample was annealed at a temperature between 2.2 K for 16minutes. After annealing the sample was left in the dark with all contactsgrounded to cool down to the system’s base temperature.This approach has several advantages. Firstly it enables the separationof shining and heating stages, providing direct control over the ionizationprocess. From experimental point of view, the external light source is stable,and the sample temperature is independent and has a controlled light powerand wavelength. Secondly, annealing after the illumination has been turnedoff, eliminates extra uncontrolled sample heating due to the light absorbedin the sample. This allows for precise sample temperature control by a local353.2. Low temperature systemheater and thermometer. Another benefit of this technique is that the 2DEGcan be restored to exactly the same state in situ, without decondensing therefrigerator’s dilution unit. This feature considerably decreases the turnaround time of the sample reset, when its mobility is severely broken by, forinstance, electrical shock.3.2 Low temperature system3.2.1 Dilution refrigeratorCommercial 3He/4He dilution refrigerator from Oxford Instruments with abase temperature of 13 mK was used to cool down the sample. Figure 3.2demonstrates a cross-sectional view of the low temperature custom partbelow the mixing chamber plate. The design of this unit is the most crucialin refrigeration of the sample, since it provides thermal connection betweenthe sample and the dilution unit, where the main cooling power originates.The sample in a standard ceramic chip carrier is fixed inside the socketat the end of the cold finger, attached directly to the mixing chamberplate. The cold finger is covered by a hand made radiation shield ther-mally sunk into the mixing chamber. In addition to the sample screeningfrom the background thermal radiation, this shield is made rigid, provid-ing mechanical stability, and centering the sample in the magnet bore. Itsstructural strength comes from a paper-phenolic tube, but it is also coveredby the vertical stripes of high RRR thin copper foil with insulating adhe-sive. These stripes are electrically connected to the mixing chamber andguarantee that the metal shield cools down close to the mixing chambertemperatures. Moreover, electrical resistance between different foil piecesprevents its heating due to Addy currents when the magnetic field is var-ied. The second radiation shield is a part of the Oxford commercial setup.It covers the handmade shield and the mixing chamber and is thermallyconnected to a 1K pot. The entire low-temperature unit below the mixingchamber has to fit inside the 3-inch superconducting solenoid bore therefore,space around the cold finger is quite limited. This requires an extra efforts toensure that all cylindrical shields are coaxial and no touches appear betweenthem.The chip carriers were custom manufactured in order to avoid the pres-ence of any magnetic materials in the vicinity of the sample. Thus, magneticfield disturbance and extra heating due to magnetization of Ni nuclei in theregular industrial contact plating were minimized. Similarly, the socketNi/Au-plated pins were replaced by the custom-made ones, lasercut from a363.2. Low temperature systemBMixing chamberFiberAttenuatorWire coolingSolenoidDeviceRadiation ShieldFigure 3.2: Design of the dilution fridge low temperature side.373.2. Low temperature systemBe/Cu plate.Although the mixing chamber can reach temperatures below 20 mK,cooling the electron system is not straight-forward. The problem is thatat this temperature range the electron and phonon systems are weakly cou-pled. On the other hand, electrons from wires diffuse into the 2DEG throughlow resistive electrical contacts. Thus, cooling electrical wires is essential forcooling two dimensional electron system. For the same reason induced elec-trical noise in the wires creates another source of electron heating in highresistive samples. This noise also has to be filtered as much as possiblebefore the wires are cooled to the base temperature.Few tricks were incorporated into the design of the cold finger in order tosolve the problems described above. Electrical noise filtering was performedin several stages. First, the wires passed through a set of RC-filters mountedon PCB, screwed to the top of the mixing chamber plate. Although thosefilters were exposed to the external thermal radiation, they were warmer,but still close to the base temperature. This significantly decreased thermalfluctuations inside the filter resistors and made RF filtering much moreeffective. At the second stage of attenuation, the wires were cooled to thebase temperatures inside an attenuator plate. This plate was made of lowresistive wires, wrapped in copper foil and soldered to a copper plate pressedto the mixing chamber plate. Such design provided good thermal connectionto the mixing chamber and realized effective cooling of the wires. At the laststage, the wires undergo the final step of cooling on a separate copper plateseparately connected to the mixing chamber and eventually reached the chipcarrier socket. More details on the cold finger design are given in [86].Special attention was paid to grounding. In order to separate the systemfrom external noise coming from electronics through the electrical ground,the sample circuit was made to be completely floating. All measurement andcurrent source were separated from the world by high quality instrumenta-tion amplifiers LT1167 (Fig. 3.3), powered by a battery. An extra referencecontact at the output of preamp enabled the separation of measurement andinstrumentation ground. Similarly, if connected in the opposite direction,such an amplifier could provide ground separation for the input current sig-nal. Fig. 3.4 shows schematics for the voltage measurement and voltagesource used in our measurement setup.Because this experiment depends crucially on removing heat from thesample, the back of the chip was additionally soldered with indium to thegold surface of the chip carrier, which was thermally connected to the mixingchamber by wire-bonding the gold bottom to 16 of the cold finger’s electricalwires.383.2. Low temperature systemFigure 3.3: LT1167 instrumentation amplifier block diagram (figer takenfrom datasheet).Although dissipation due to high current biases in the bulk of the 2DEGwas quite small in the described experiments—10’s of picowatts at most—the total Joule power dissipated in the sample was much larger (almost2 nW at 400 nA) because the two-probe resistance between the source anddrain (mostly rxy) is much larger than rxx. Here the question is addressed:how can 2 nW be dissipated at the sample without raising its temperaturesignificantly above the mixing chamber temperature?Almost all heat dissipation in the quantum Hall regime occurs in thesource and drain contacts. With half of the power, 1 nW, dissipated in asingle contact, the rise in contact temperature is set by the thermal resistancefrom the contact, through a bond wire and a chip carrier connector, to well-cooled measurement wires at the mixing chamber temperature. This thermalresistance can be estimated from the measured cooling power when heat wasapplied to a resistor mounted directly to the back of the chip carrier. Anyheating applied to the chip carrier had to be carried away by the 16 leadsbonded to the carrier backplane. 6 nW of power applied to the heater on393.2. Low temperature systemVin+Vin-VoutVcc+Vcc-+ VoutVcc+Vcc-+~a) b)Figure 3.4: Ground breaking schematics: a) voltage measurements; b) volt-age source. Circles around the voltage probes depict cable shielding, con-nected to the measurement ground.the back of the chip carrier was seen to warm the sample by 11 mK, giving athermal resistance through each lead of 29 mK/nW. That is, 1 nW into thesource or drain would warm it to a temperature around 40 mK. Is it possibleto keep 2DEG cold if source and drain contacts are at 40 mK?In the filling factor range 2 < ν < 3, source and drain ohmic contacts areseparated from the 2DEG by ∼ 10kΩ contact resistance, and longitudinalconductivity per square through the 2DEG is  10−4Ω−1 (σxx = ρxx/ρ2xy).Together, these numbers give a Wiedemann-Franz (WF) thermal conductiv-ity from the source or drain into the bulk of the 2DEG that is vanishinglysmall,  10−10 nW/mK or less at 15 mK. By comparison, the interior 3mm×3 mm region of the 2DEG is coupled to phonons with a thermal con-ductivity that is many orders of magnitude larger, between 10−6 and 10−2nW/mK depending on 2DEG conductivity[87, 88]. This massive discrep-ancy between WF coupling and coupling to phonons ensures that electronsin the bulk of the 2DEG (though not the ones right next to the source anddrain contacts) are locked to the phonon temperature of the chip rather thanto the electron temperature of the leads.With the source and drain at 40 mK, and the chip backplane at 15 mK,the phonon temperature will then be determined by the relative areas ofsource/drain contacts (∼ 0.25 mm2 for each) compared to the area of contactto the backplane (∼ 25 mm2). Even taking into account that phonon heatflux through the boundary at 40 mK is 10 times higher than at 15mK, thephonon temperature will stay close to the (cold) backplane temperature.Note that through these calculations we have assumed a homogeneous403.2. Low temperature systemphonon temperature throughout the chip, despite the fact that heating islocalized at source and drain contacts while cooling is spread across thebackplane. This approximation can be justified by comparing lateral phononthermal resistivity per square through the 0.3 mm-thick GaAs chip itself(2×108 K/W, estimated from Ref. [89]), compared to the thermal boundaryresistance 3.5× 108 mm2K/W estimated from Ref. [90]. Together these givea characteristic thermal “spreading” length of 1.3 mm from each contact,close to half of the chip dimension.3.2.2 Temperature controlSample temperature measurement was performed by several sensors to covera wide temperature range. At higher temperatures (≥50 mK) a calibratedRuOx resistor, bolted to the mixing chamber, was sensed. A second ther-mometer/heater pair, located on the back of the chip carrier, was calibratedusing the mixing chamber thermometer/heater and enabled much fasterheating and cool down times for temperature-dependent measurements.The second thermometer was sanded down from a commercial resistor,suggested in [91], to roughly 0.3 mm thick film. After that, the sensor wasglued to the back of a ceramic chip carrier using GE varnish, which alsoprovided the sensor with electrical insulation. After that, electrical contactto the embedded metal contacts was made by bonding them to the contactpads of the chip carrier. It is important to mention that removal of oxidefilm from the metal contacts is needed to make a reliable bonding at lowtemperatures.Another important detail, mentioned by Samkharadze et.al., was theneed to insulate the carbon part from the air to prevent calibration driftduring thermal cycling. In our setup the carbon sensor was not completelycovered due to technical reasons. Therefore, it had to be recalibrated afterevery thermal cycling above liquid nitrogen boiling temperature.Calibration of the second thermometer was done against RuOx mea-surements above 50 mK and extrapolated down to the base temperature.However, at low temperatures, electrons are weakly thermally coupled tothe crystal lattice, therefore, their temperature can differ significantly. Theelectronic temperature in the sample was monitored using temperature-dependent features in the magnetoresistance, which could be calibrated athigher temperatures where thermal equilibration is more reliable and thenextrapolated to lower temperatures. In Ref. [23], for example, it was shownthat the width of a reentrant state grows with temperature down to 6 mK.In order to obtain the heater calibration, R2c temperature dependence413.2. Low temperature system1086420Heater,Power,(nW)60402003530252015Temperature,(mK)R2c,width, x103DnOn,Chip,HeaterMixing,ChamberFigure 3.5: On chip heater calibration. Comparison of R2c width depen-dence on mixing chamber temperature (circles) and heater power (triangles).was compared with its dependence on the heater power (Fig. 3.5). The for-mer was measured by application of constant power to the mixing chamber.After the time, significant to stabilize the system (normally ∼ 3 hours forthe lowest temperatures), there was no detectable difference between twomagnetoresistance traces, taken one after the other. This allows us to as-sume, that the sample was thermally equilibrated with the mixing chamberand that its temperature was equal to the RuOx calibrated resistor tem-perature at the mixing chamber. If the on-chip heater was used to warmthe sample, there were no significant mixing chamber temperature changesobserved within 10 minutes.In the second step of calibration, constant power was applied to theon-chip heater and a few traces at different biases were taken one afteranother. The first and last ones were taken at zero bias. Their differenceallowed to estimate chip carrier temperature change to be less than 1.5 mKafter application of high biases. This difference can be also observed infigure 3.5 as zero heater power point corresponds to higher temperatures.423.3. Electrical measurementsConsequently, the base temperature of the sample depends on the bias andintroduces uncertainty of about 2 mK in our temperature measurements.3.3 Electrical measurementsElectrical measurements of differential resistances Rxx ≡ dVxx/dI and Rxy ≡dVxy/dI were performed using a low frequency lock-in technique with an ACcurrent bias, Iac, between 1 and 5 nA. An additional DC bias current Idc upto 5 uA was used to induce the in-plane electric field, Ey = Vxy/w, in theplane of the sample and transverse to the current direction (w = 5 mm isthe sample width). The standard measurement scheme for driving DC andAC currents in the same direction is shown in figure 3.6a.a) b)ACDC DCAC10MW1GW10MW 500MW500MWFigure 3.6: Schematics of electrical measurements: a). AC‖DC; b).AC⊥DC.For the measurements, where AC and DC currents were applied in per-pendicular directions, a scheme with floating AC current was used (Fig. 3.6b).In this case the AC ground was broken by means of the acoustic transformerwith a high common mode rejection level. This enabled the construction ofafloating AC circuit and the application of a small probing electric fieldperpendicular to the main Hall one generated by DC current.In some cases, pure DC current was applied to the sample and appropri-ate DC voltage responses measured. In this case, the data were differentiatedto obtain the values of differential resistances, and the size of the currentstep was an equivalent of the AC bias.43Chapter 4Breakdown spatial resolution4.1 MotivationReentrant states breakdown under high currents, driven through the sample,is a well known phenomenon, observed shortly after the discovery of theRIQHE [27]. These experiments are normally explained in terms of slidingdynamics of depinned charge density waves [31], or alignment of electronliquid crystal domains by the induced Hall electric field [29]. For examplein [29] authors investigated the sample’s state anisotropy with respect tothe direction of high DC bias based on the macroscopic measurement oflongitudinal resistance between corners of a square sample. They concluded,that the reentrant states become anisotropic in high bias, based on resistancedifference, measured in the direction parallel or perpendicular to the DC biascomponent. Additionally, the scaling of the critical breakdown current withthe size of the sample, led them to a conclusion, that the induced electricfield is the primary parameter, defining the state of the 2DEG.An important assumption in this interpretation of the experimental datais that the main effect of elevated DC bias current is to induce a homogeneouschange in macroscopic samples, which may range from 100’s of microns toseveral mm’s depending on the specific realization. However, when currentflow through the sample is strongly inhomogeneous (as is always the casein the quantum Hall regime) one’s ability to approximate bulk propertiesfrom non-local measurements is immediately suspect. For example, finitecurrents driven through a sample in the quantum Hall regime necessarilycreate temperature rise due to Joule heating, and that heating is in manycases strongly localized.In this chapter it is shown, that large current biases, driven through aRIQH state, induce a sharply inhomogeneous breakdown. In other words,the electronic system becomes spatially fractured into macroscopic regionsthat are either melted (conducting) or frozen (insulating). This fracturingis monitored through a comparison of multiple voltages, probed simultane-ously at different contact pairs. Notably, this phase-separated breakdown isentirely absent from quantum Hall liquid states, consistent with the distinc-444.2. Experimental resultstion between collective phase transitions in RIQH electron solids vs activatedtransport in gaped FQH liquids. Considering numerous RIQH states, fromν = 2 all the way up to ν = 8, and multiple contacts arrangements, theRIQH breakdown is shown to propagate clockwise or counterclockwise fromthe source and drain contacts with a sense that depends on the electron- orhole-like character of the particular RIQH state. Supported by numericalsimulations, the data are explained by a phase boundary between frozen andmelted regions of an underlying electron solid that spreads around the chip,following the location of dissipation hot spots induced by local changes inRxy.4.2 Experimental results4.2.1 Measurement detailsMeasurements were performed on a 300 A˚ symmetrically doped GaAs/AlGaAsquantum well with low temperature electron density ns = 3.1 × 1011 cm−2and mobility 15×106 cm2/Vs.[5] Electrical contact to the 2DEG was achievedby diffusing indium beads into the corners and sides of the 5×5 mm chip(Fig. 4.1a). Electronic temperature, Te, at low bias was monitored usingtemperature-dependent features in Rxx, and confirmed to follow Tmix downto 13 mK.[23] FQH characteristics were optimized following Ref. [1], in adilution refrigerator with base temperature Tmix ∼ 13 mK.Differential resistances R˜ ≡ dV/dIb for various contact pairs were mea-sured by lockin amplifier with an AC current bias, IAC = 5 nA, at 71 Hz.An additional DC current bias IDC was added to the AC current in manycases, providing 2D maps of various resistances as both bias voltage andfilling factor are changed (see e.g. Fig. 4.1b).At zero DC bias, characteristic R˜xx and R˜xy traces over the region 2 <ν < 3 show an example of fractional quantum Hall states at filling factorsν = 2 + 1/5, 2+1/3, 2+1/2, 2+2/3 and 2+4/5, with four RIQH stateslabelled R2a-R2d following conventional notation (dashed lines in Fig. 4.1c).At high current bias (solid lines in Fig. 4.1c) the reentrant states disappear,with R˜xy moving close to the classical Hall resistance, while most fractionalstates remain well-resolved: quantized in R˜xy with vanishing R˜xx.The breakdown process is most clearly seen in 2D maps of IDC and mag-netic field, typically showing multiple regions with sharply delineated bound-aries. Fig. 4.1 presents several such maps for the R2c reentrant state. Con-sidering first the measurement of R˜xx (Figs. 4.1b), the breakdown transitionsfollow a pattern that is similar to those observed by other groups[29, 82];454.2. Experimental resultsRxy (h/e2 ) Rxx  (kW)0 uA DC0.5 uA DC5/25/2RxxRxxAABBCCABC250150200100500-50 00.20.40.60.8a).5.105.004.90Rxx (kW)I DC (nA)B (T)5.85.65.45.25.04.84.64.40.500.450.400.350.300.251.00.80.60.40.20.0c).B (T)R2b R2aR2d R2c5.105.004.90 5.105.004.90B (T) B (T)d). e).10.49.68.8I DC (nA)b).5/23R+DR-DR-DR+D12345678250200150100500-50Rxy (kW)5/2Figure 4.1: a) Schematic of a measurement combining AC (wiggly arrow)and DC (solid arrow) current bias through contacts 1 and 5 (see numbering).R˜xx = dV86/dI, R˜+D = dV26/dI, and R˜−D = dV84/dI. Curved arrows indicateedge state chirality. b) Evolution of R˜xx with DC bias for the R2c reen-trant and ν = 5/2 FQH state, showing breakdown regions ‘A’, ‘B’, and ‘C’(IAC=5 nA). c) R˜xx and R˜xy (contacts 3 and 7) for filling factors ν = 2− 3,showing the breakdown at very high DC bias (IAC=5 nA). (d,e) Simultane-ous measurements of R˜+D (d) and R˜−D (e), taken together with R˜xx in b).Dashed lines indicate identical parameters in each panel.464.2. Experimental resultsthese transitions naturally divide the map into three distinct subregions,labelled ‘A’, ‘B’, and ‘C’.Region A is characterized by very low R˜xx: here the electron solid stateis presumably pinned and completely insulating. The sharp transition toregion B corresponds to a sudden rise in Rxx, while for higher bias (regionC) the differential resistance drops again to a very small value. Unlike theflat zero of region A, however, R˜xx in region C fluctuates up and down asa function of IDC and B, giving an undulating pattern of ripples in the 2Ddata.It is worth noting that the sharp transitions in the RIQH state break-down are entirely absent from the neighbouring ν = 5/2 state, a distinctionseen for all RIQH states compared to all fractional states. Data for differ-ent cooldowns and different RIQH states were slightly different in the de-tails, but qualitative characteristics were consistent for every well-developedRIQH state in every cooldown.4.2.2 Measurements at different contactsThe observation of sharp delineations in the resistance of a bulk sample,measured between voltage probes separated by 5 mm, at first glance wouldseem to imply that the entire sample must suddenly change its electronicstate for certain values of bias current and field. If this were true, onewould expect simultaneous jumps in resistance monitored at any pair ofvoltage probes, although at a quantitative level, of course, the resistancejumps might be by differing amounts. Considering the two ‘diagonal’ pairsof voltage probes, marked R˜+D and R˜−D in Fig. 4.1a, one sees immediatelythat this is not the case. R˜+D (Fig. 4.1d) exhibits the same transitions as R˜xx(Fig. 4.1b), but for R˜−D (Fig. 4.1e) the A-B transition is entirely missing.Qualitative differences between diagonal measurements R˜+D and R˜−D areexpected when samples are inhomogeneous, especially in the quantum Hallregime[43]. R˜+D and R˜−D contacts are distinguished by the chirality of quan-tum Hall edge states: moving from source or drain contacts following theedge state chirality, one first comes to the R˜+D contacts, then to R˜xy contactsin the middle of the sample, and finally to the R˜−D contacts. This distinc-tion is often used to isolate the conductance (via R˜+D) of a mesoscopic gatedregion in the middle of a quantum Hall sample, whereas R˜−D would includecombinations of gated and bulk filling factors[38].Given the stark difference in breakdown characteristics for R˜+D and R˜−Dmaps, one could ask what happens halfway in between, that is, for the middle474.2. Experimental resultsRxyRxyh/3e213121110987-200 -100 0 100 200IDC (nA)Rxy (kW)1.761.721.68 1.70 1.761.721.68B (T) B (T) B (T)3210I DC (uA)Rxya).b). c). d).3.73.53.3Rxy (kW)R-DR+DR+DR-D1.74R-D R+DFigure 4.2: (a) Simultaneous measurements showing the evolution of R˜+D,R˜xy, and R˜−D, with DC bias, in the middle of the R2c reentrant state(IAC=5 nA); Evolution of (b) R˜−D, (c) R˜xy and (d) R˜+D of R7a reentrantstate with DC bias.set of contacts, R˜xy. As shown in Fig. 4.2a, the A-B transition (signifiedby a sharp rise resistance) simply moves to higher bias following the edgestate chirality, from R˜+D to R˜xy to R˜−D. An analogous progression was seenfor every reentrant state, even up to much higher filling factor. Figs. 4.2b-dshow the breakdown progression for R7a, albeit extending up to much higherbias currents.4.2.3 Electron- vs hole-like reentrant statesAlthough the sequence for R7a is at first glance qualitatively similar to thatseen for R2c, the direction of the sequence is opposite: whereas R˜+D breaks484.2. Experimental resultsdown before R˜−D for R2c (Figs. 4.1d,e and Fig. 4.2a), R˜+D breaks down afterR˜−D for R7a (Figs. 4.2b-d). This distinction was found to be linked to theelectron- or hole-like (e/h-like) character of the reentrant state, consider-ing every reentrant state observed in this experiment, independent of thenumber of occupied Landau levels. In other words, all reentrant states cor-responding to a less-than-half-filled Landau level [R2ab,R3ab,R4a,R5a,etc.]broke down at lower bias for R˜−D, higher bias for R˜+D, whereas all states corre-sponding to a more-than-half-filled Landau level [R2cd,R3cd,R4b,R5b,etc.]broke down at lower bias for R˜+D, higher bias for R˜−D.Although slight differences in quantum Hall traces for various contactpairs are common, the qualitative distinctions between R˜+D and R˜−D did notdepend on the specific contact used in the measurement, but only on thechiral distance of voltage probes with respect to source/drain current leads.For example, rotating the contact configuration by 90◦ with respect to thechip axes leaves the R˜+D/R˜−D distinction intact, shown in Fig. 4.3 for R3reentrant states. As before, R˜+D and R˜−D are defined by the chirality ofedge state transport away from the source or drain, together with the e/hcharacter of the reentrant state.494.2.Experimentalresults3.93.83.73.63.53.420010000.320.280.24Rxy (h/e2)20010003.93.83.73.63.53.43.93.83.73.63.53.4200100020010003.93.83.73.63.53.4B (T) B (T)DC Bias (nA)DC Bias (nA)R-D R-DR+DR+DR+DR-DR+DR-Da) b)Figure 4.3: Diagonal measurement with the current flowing in two perpendicular orientations with respect to thesample. It is clearly seen, that the strong/weak reentrant state is defined by the current contacts and any kind ofmeasurement can be obtained on the same voltage probes.504.3. Discussion4.3 Discussion4.3.1 ModelThe fact that R˜+D and R˜−D switch roles for electron- versus hole-like reentrantstates provides an important hint as to the origin of this effect. Edge statechirality is fixed by the magnetic field direction, and would not be expectedto suddenly reverse when crossing half-filling for each Landau level. Instead,proposed explanation is based on localized dissipation in the quantum Hallregime, a phenomenon that is known to give rise to “hotspots” any time asignificant bias is applied to a quantum Hall sample.Consider current injected into a sample in the integer quantum Hall(IQH) regime, where ρxx is exponentially close to zero but Rxy is large.Driving a current Ibias through such a sample requires a potential differenceRxyIbias between source and drain, and this potential drops entirely at thesource and drain contacts (no voltage drop can occur within the sample sinceρxx → 0). Specifically, the voltage drops where the current carried along afew-channel edge state is dumped into the metallic source/drain contact –a region of effectively infinite filling factor.In the language of chemical potentials, current-carrying edge states mustbe filled to a chemical potential well above equilibrium due to their limiteddensity of states, but the chemical potential in the drain contact can remainvery close to equilibrium. The voltage drop always occurs where currentmoves from a region of lower filling factor (smaller density of states, largerchemical potential) to a region of higher filling factor.Moving now to a sample in the reentrant IQH regime (ρxx → 0 as be-fore), hotspots again appear at any location where current flows from aregion of higher to lower Rxy. But now the local value of Rxy is stronglytemperature dependent, with a sharp melting transition in both longitudi-nal and transverse resistances as previously observed in Ref. [23]. The R2areentrant state, for example, has Rxy = 0.5(h/e2) in the low temperature,low bias limit [Fig. 1b], but at higher temperature or bias the state melts toRxy ' 0.425(h/e2). In the limit of very low current bias applied to a samplein the R2a reentrant state, the entire sample is effectively at ν = 2 and onlythe two hotspots associated with IQH are observed. As the bias increases,however, the regions around those two hotspots melt to Rxy ' 0.425(h/e2),and an extra two hotspots will appear, where the current passes from thebulk (at Rxy ' 0.5(h/e2)) into the melted region (at Rxy ' 0.425(h/e2))near the contact.The localized heat dissipation, however, is not enough for existence of the514.3. Discussionlocalized area with elevated temperature, i.e. high temperature gradients.Another requirement is low heat conductance, which prevents the dissipatedenergy from spreading over the large areas. The temperature distributioninside the 2DEG is set by the Wiedemann-Franz law, defining the lateralheat conductivity, and electron-phonon coupling, which provides physicalmechanism for a heat flow through the crystal lattice, perpendicular to thesurface of the chip. Competition of those two heat flows defines the sizeof the hot spot. Although the precise calculation of the heat dissipationrequires complicated simulations, the order-of-magnitude estimate of thespot size could be given found from the comparison of Wiedeman-Franz andelectron-phonon heat conductance. The first one could be directly calculatedfrom the 2DEG conductivityκel = LσxxT, (4.1)where σxx = ρxx/(ρ2xx + ρ2xy) and L = 2.44 × 10−8WΩK−2 is known asLorentz constant. After the breakdown of the reentrant state, measuredlongitudinal resistance of the sample was on the scale of a 100 Ω, leading toσxx = 10−6 Ω−1 for R2c reentrant state (Rxy ∼ 10kΩ).Electron-phonon heat conductance could be estimated following [92].The power, dissipated into phonon system, is given as following:dP =CτdT, (4.2)where 1/τ = αT 3 is the phonon scattering time and C = βT is the heatcapacity of electrons. For the free electron case α was experimentally mea-sured to be 2.9× 109s−1K−3 [88] and β = pik2Bm∗S3h¯2, where k2B,m∗, S are theBoltzmann constant, the effective electron mass in GaAs and the hotspotarea respectively. Therefore, in the general case,κph =dPdT= αβT 4. (4.3)Equilibration of lateral and perpendicular thermal conductivities pro-vides an estimate for the hotspot area:S =3h¯2Lσxxpiαk2Bm∗T 3, (4.4)At 20 mK this formula gives the area of heat dissipation localizationabout (30µm)2, justifying the concept of the hotspots.Therefore, a conceptual picture that explains the data at a qualitativelevel is following:524.3. Discussion1. Starting from low bias, conventional quantum Hall hotspots near thesource and drain contacts locally melt the reentrant state, giving riseto two extra hotspots at the boundary between melted and unmeltedreentrant phases, either upstream or downstream from the source anddrain depending on electron- or hole-like character of the reentrantstate.2. As bias is increased, the dissipation at each hotspot increases also.The melted region (that is, the area where the local temperature isabove the melting temperature) spreads, shifting the location of thereentrant hotspots upstream or downstream away from the contacts.The location of the reentrant hotspots stabilizes approximately whenthe dissipation per unit area drops to the point that the local tem-perature in the melted region is just above the melting temperature.Here we assume that the cooling rate from the electron system to thephonons (assumed to be cold) scales with area.3. The transition between regions A and B in 2D maps such as Figs. 4.1b,4.1d and 4.1e—that is, the transition from lowRxx and integer-quantizedRxy,D to high Rxx with associated jump up or down in Rxy,D—occurswhen the boundary between melted and frozen regions passes the rel-evant voltage probe.4.3.2 Numerical simulationAlthough the chemical potential analysis is appealing, it is not necessary tounderstand the location of quantum Hall hotspots. Even a classical electro-statics simulation of current flow between regions with different Rxy (andRxx close to zero) is sufficient to indicate the same result: dissipation al-ways occurs at the locations where current passes from regions of high Rxyto lower Rxy.To backup this argument with qualitative theoretical justification, bro-ken (A) and unbroken (B) regions are approximated as the ones with dif-ferent transverse conductivity σxy. Within this approximation, the classicalsolution of Kirchhoff equations is considered in a two dimensional domainΩ (Fig. 4.4). For any point (x, y) ∈ Ω, Kirchhoffs laws dictate:∇ · j = 0,∇× E = 0, (4.5)where j and E are the electric current density and field respectively. In-troducing the electric potential, E = −∇φ, the equation for E is triviallyfulfilled. Finally, assuming the local relation j = σE, we get:534.3. DiscussionA AB−l/2 −x0 0 x0 l/2−l/2-y0−w/2w/2y0l/2Figure 4.4: Geometry of the domain Ω, where simulation is performed.Regions A and B denote the areas with different densities, corresponding todifferent σxy’s in the simulation.∇ · (σ∇φ) = 0, (4.6)where σ has the general form (see eq. 2.4):σ =[σxx −σxyσxy σxx]Next, the boundary conditions have to be defined. In presented simula-tion two cases, corresponding to electron-like and hole-like reentrant state,are considered:1. σAxy = 2.5e2/h, σBxy = 2e2/h2. σAxy = 2.5e2/h, σBxy = 3e2/hIn both cases σxx = 0.02e2/h for any (x, y) ∈ Ω. A current I is injecteduniformly through the red boundary (contact) and it is collected from theblue one.Two extra hotspots are clearly seen in the dissipation map in Fig. 4.5a,based on a numerical solution of the problem, described above1. Current1Numerical simulations in this paragraph were performed by Yuval Baum, WeizmannInstitute of Science544.3. Discussionn*<1/2n*>1/2a)b)R-DR+DR-DR+DFigure 4.5: Classical simulation of dissipation due to current flow in a sampledivided into three regions: semicircles corresponding to the melted statenear each contact (hellow hatched) with Rxy = h/(2.5e2), and the bulk(dark blue) reentrant state with Rxy = h/(2e2) (a) or Rxy = h/(3e2) (b).Hotspots appear at different corners of the melted region in a) and b).554.3. Discussionflows predominantly along the edges due to the inequality Rxy  Rxx, sohotspots appear at intersections of the boundary between Rxy regions andthe sample edge. Note that only one corner of each melted region has ahotspot, but whether that corner is where the edge state enters or leaves themelted region depends on the relative value of Rxy as compared to the bulk[compare Fig. 4.5a to 4.5b].The difference between Figs. 4.5a and 4.5b justifies at a qualitative levelwhy the boundary between melted and frozen reentrant phases should prop-agate in opposite directions for electron-like and hole-like reentrant states.In order to make quantitative connection between precise hotspot locationand the spreading direction for the melted region, one would need to un-derstand the detailed feedback process by which the boundary of meltedphase changes as dissipated power increases—a thermal analysis is beyondthe scope of the present chapter and require future theoretical research.4.3.3 Comparison with experimentNow let’s compare experimental details against the theoretical conclusions,which could be derived from suggested model, in order to test it for theconsistency.Breakdown propagation along the edgeAs further confirmation that a melted pool of free carriers around source/draincontacts spreads into the bulk of the sample at a frozen reentrant state alongthe edge, the spreading of potential jumps around the sample are mappedout in detail for the R2c reentrant state (Fig. 4.6). This is the hole-likereentrant state with R˜xy corresponding to i=3 (R˜xy = h/3e2 = 8.6 kΩ) inthe frozen state, and ν=2.56 (R˜xy = h/2.56e2 = 10 kΩ) in the melted state.Consequently, for R2c reentrant state the potential jump at the hot spot∆ = Ihe2(1ν− 13). (4.7)Similar to previous notation sample contacts are enumerated clockwise withthe current sourced into contact 1 and drained from contact 5, assumed tobe the ground potential.Fig. 4.6a and b denote three plausible extents of the melted region withtheir associated hotspots, as well as, the edge state potentials expected forone of this situations. Figure 4.6c demonstrates the potential difference be-tween multiple contact pairs all starting from contact 8 on the low-potential564.3. Discussiona)12345678b) m=Ih/3e2+Dm=Dm=0m=Ih/3e2+2D12345678c)d)11109R xy (kW)11109R xy (kW)-200-200-100 100-100 10020000IDC (nA)S1S2S38-28-38-42wire6-46-36-22wire200IDC (nA)Figure 4.6: a) Hole-like reentrant state breakdown propagation in the sam-ple; hexagonal, diamond and round marks depict the position of the hotspots for three different sizes of broken regions (dark grey, light grey andhatched), corresponding to different bias currents; b) electrical potentialdistribution in the sample with half-way broken reentrant state; comparisonof breakdown propagation, measured on different ohmic contacts along oneside with common contact in c) frozen and d) melted region. The marksdepict current value, corresponding to the hot spot locations, shown in a)574.3. Discussionside of the sample, compared to contacts 2, 3, and 4 on the high-potentialside. The relevant data for contact pairs 6-2, 6-3, and 6-4 are shown inFig. 4.6d.For the first situation (S1), corresponding to the dark grey melted regionwith the hexagonal hotspots (Fig. 4.6a), the potential drops along the upperedge state occurred before (following edge state chirality) contacts 2,3,4,and the potential drop along the lower edge state occurred before contacts6 or 8, so none of the voltage contact pairs 8-2, 8-3, 8-4, 6-2, 6-3, or 6-4 have recorded a resistance jump relative to the zero bias measurement.For situation 2, corresponding to the light grey melted region and diamondhotspot, the potential drop in the upper edge state occurs after contact 2,but before contacts 3 or 4; along the lower edge the potential drop occursin between contacts 6 and 8. Considering pairs 8-2, 8-3 and 8-4 in case ofS2, we, therefore, expect a resistance jump only in 8-2, whereas potentialjumps have occurred for all three pairs 6-2, 6-3, and 6-4. The magnitude ofthe jump in 6-2 is twice what it is in 6-3 or 6-4, because that pair includesjumps along both the upper and lower edge states (see Fig. 7b). However,in S3, when the hot spot passes contact 3, putting both contacts in the pair6-3 in the melted state, the 6-3 deviates from the 6-4 trace and jumps tothe twice higher level, similar to the measurement of the 6-2 pair.The observed resistances corresponding to various contact pairs (Figs. 4.6cand 4.6d) follow precisely the sequence, described above. Bias currents cor-responding to each situation are marked with the corresponding hotspotsymbol in the graphs, that is, bias current ∼50nA corresponds to the hexag-onal hotspot symbol marking to situation 1.Figures 4.6c and d also include a trace (dashed line) for the two-wireresistance of the sample, that is, the potential measured across source anddrain contacts. A resistance of 2.6 kOhms is subtracted from the trace totake into account wire resistance in the cryostat as well as imperfect ohmiccontacts. At zero DC bias, when the entire sample is in the frozen reentrantstate corresponding to R˜xy(i = 3) = h/3e2, the two wire resistance mustbe R = 8.6 kΩ (after subtracting wire and ohmic contact resistances). Butonce a finite melted region has spread away from source and drain contacts,resulting in e.g. situation 1 with the hexagonal hotspots in Fig. 4.6a, thetwo-wire resistance increases to include the two additional potential dropsat the hotspots. Thus the two-wire resistance records the first appearanceof a melted region spreading beyond source and drain contacts, whereas thevoltage probes 2 and 6 only register the movement of the hotspot past thesetwo voltage probes.On the other hand, the drop of two-wire resistance to classical Hall584.3. Discussionvalue, indicates that two melted regions become electrically connected, i.e.this is the indication of the breakdown of the insulating bulk. Notably,this transition coincides with the breakdown transition, measured at thestrongest 8-4 pair. This fact indicates, that the insulating bulk state breaksdown before the hotspots reach contacts 4 and 8 and region C in figures 4.1d,ecorresponds to the conductive bulk state. Detailed investigation of region Cis a subject of the following chapter.Anisotropic measurementAlthough the following chapter focuses on anisotropic measurements, hereexperimental data is compared to the findings from other groups. Specifi-cally, Ref. [29] compared Rxx, measured parallel or perpendicular to a largeDC current bias, by rotating the Rxx voltage probes and AC current biascontacts by 90◦ with respect to the DC bias contacts [Fig. 4.7a,b]. It wasobserved that the low-Rxx region A extended to much higher bias for the(AC ⊥ DC) orientation, compared to the conventional (AC||DC) orienta-tion. While Ref. [29] focused on R4 states exclusively, we found analogousbehaviour for all reentrant states measured, from filling factor 2 through 8(see e.g. R3a in Figs. 4.7c,d).However, instead of complicated picture, involving reorientation of elec-tron liquid crystal domains, such behaviour can be simply explained by thehotspot-movement mechanism outlined above. Figs. 4.7a and 4.7b schemat-ics include dashed lines to show the melted-frozen boundary at an intermedi-ate bias, with associated hotspots marked with ?’s; the boundary propagatesfrom the DC (not AC) current contacts, because the measurement is donein the limit of vanishing AC bias. The local potential along the edge of thesample drops sharply when passing the DC source and drain (the conven-tional IQH hotspots), but a second smaller potential drop occurs at each ?.For the bias-induced distribution of melted and frozen phases indicated inFig. 4.7, the small potential drop occurs between the Rxx voltage probesin Fig. 4.7a, but not in Fig. 4.7b, so large Rxx in region B (e.g. 50 nA)is recorded only in the AC||DC configuration. For the small DC biases,when the hot spot is in between voltage probes, the potential drop is notdetected by the AC measurement, as the probes are connected only to theedge states, associated with the underlying completely filled Landau levels.594.3. Discussiona) AC||DCRxxEB (T) B (T)I DC (nA)3.883.843.803.76250200150100500-503.883.843.803.76c) d)Rxx  (kW)0.80.40.0RxxEb) AC   DCFigure 4.7: Comparison of two possible measurement geometries: a)AC||DC and b) AC ⊥ DC; arrows label the source and drain contacts forDC (straight) and AC (wiggly) bias, and chirality of the edge states (curved).Hatched areas denotes partially-extended melted regions for a hole-like reen-trant state at intermediate DC bias, corresponding to IDC ∼ 50nA in panels(c,d). Hotspots at the melted/frozen boundary indicated by ?. Yellow lineindicates potential along the edge. (c,d) R3a Rxx maps in (IDC , B) planefor c) AC||DC and d) AC ⊥ DC measurement.4.3.4 RIQHE vs FQHE breakdownAs was pointed out earlier, described inhomogeneity of the breakdown wasobserved only for the reentrant states, while in fractional quantum Hallregime, the sample demonstrates similar activated transport behaviour atany location of the sample. This result is of the great importance, sinceit provides a direct observation of the differences between local thermody-namic properties of the 2D electron system in different correlated states.On the mundane level this can be explained by very sharp resistance tem-perature dependence of the RIQH around critical temperatures [23]. Asheat conductance is proportional to the resistance, the more σxx drops withtemperature, the higher heat localization is.604.4. ConclusionsHowever, such a dramatic local temperature dependence clearly implies adifference in the thermodynamical properties of the RIQH and FQH groundstates and creates a probe of microscopic structure of those states at thehot spot. The most popular theoretical picture of the bubble phases has nodirect experimental confirmation yet, as it requires an investigation of thecorrelated state local order. However, all experimental techniques, currentlyused, allow to study reentrant state microscopic structure only in indirectway, driving a conclusion about microscopic properties from macroscopicmeasurements, averaged over the whole area of the sample (for instance,microwave techniques). Therefore, the effect, which provides technique toprobe the local microscopic properties of the electron system under condi-tions of RIQHE is important finding.4.4 ConclusionsIn conclusion, it was demonstrated that bias-induced breakdown of theRIQH effect is strongly inhomogeneous across macroscopic (mm-scale) sam-ples, at least for large ranges of intermediate DC biases. As bias increases,the RIQH breakdown propagates away from source and drain contacts witha chiral sense that depends on the electron- or hole-like character of thereentrant state, leading to different critical breakdown biases for differentpairs of voltage probes.This phenomenon may result from opposite hotspot locations for the twotypes of reentrant states, giving rise to melted (no longer reentrant) regionsnear source and drain contacts that spread in opposite directions as biasincreases. Suggested model was qualitatively confirmed by the numericalsimulation and comparison with the data. It was experimentally shown,that hotspots propagate chiraly along the edges of the sample, sequentiallypassing contacts, following edge state chirality. The voltage, probed at dif-ferent contacts is in agreement with the potential differences, derived fromthe electrical potential distribution. This effect creates a nice tool for dis-tinguishing different conducting states of the 2DEG, surrounding the ohmiccontacts.Experimental data was also compared with similar results from othergroups [29]. Despite the general agreement, theoretical explanations sug-gested before fail to explain new experimental results. This experimenthighlights an example of counterintuitive phenomena that can appear intransport measurements of correlated electronic states. It also shows thedanger in interpreting macroscopic measurements at a microscopic level, in614.4. Conclusionssamples, where electronic phase transitions are sharp and phase segregationmay occur.The observed macroscopic phase separation was demonstrated to be adistinct signature of correlated reentrant states, as it was not observed in ac-tivated transport during the breakdown of fractional quantum Hall phases.This difference most likely results from a sharp temperature dependence ofthe resistance around breakdown of RIQHE, observed before. The prop-erties of the breakdown are directly connected to the thermodynamics ofthe microscopic electron state, and bias-breakdown measurements thereforeoffer a valuable insight into the dynamics of RIQH states in general.62Chapter 5High bias reentrant states5.1 MotivationOne of the main candidates for microscopic description of high bias RIQHEbreakdown is a destabilization of the electron solid by Hall electric field,which is thought to depin the crystal from an underlying disorder potentialor reform it with altered long-range order [27–29]. These hypothesis arebased on the analogy with other charge ordered systems across the fieldof strongly-correlated electronics. Experimental measurements of the bias-driven sliding dynamics of charge density waves date back to work on NbSe3nearly four decades ago, and remain an area of active research [30–32].Transport signatures of high bias RIQHE breakdown, in contrast tosmooth free electron traces at higher temperatures, include sharp transi-tions out of the insulating state, and excess resistance noise in the transitionregion [27, 28]. A simple free-electron picture fails to explain this experimen-tal fact. On the other hand RIQH states, especially in high magnetic fields(2 < ν < 4), are extremely fragile, limited to temperatures below 40 mKand the highest mobility samples [17, 23, 25, 26]. However, high currentsunder conditions of QHE (high perpendicular magnetic field and ultra-lowtemperatures) unambiguously dissipate a lot of heat in the electronic sys-tem, questioning the existence of the high bias electron solid state undersuch conditions.Furthermore, spatially resolved measurements of the RIQHE breakdowndemonstrated a macroscopic phase separation within the sample at interme-diate biases (see Ch. 4). This led to a conclusion, that the RS breakdownresults from partial electron solid melting by localized hot spots rather thanfrom the sliding dynamics. However, at intermediate currents, some parts ofthe fractured sample demonstrate reentrant behaviour, implying that elec-trons there stay below crystal melting temperature. What is the microscopicstructure of the intermediate bias states (region C) at low temperatures im-mediately after breakdown from insulating reentrant state?In this chapter, the measurements of high bias breakdown process, probedsimultaneously at various contact pairs, are analyzed. This allows to under-635.2. Transport measurementsstand details of electrical potential distribution along the edge of the sam-ple, providing an evidence for the insulating bulk state breakdown, whilesome regions of the 2DEG stay frozen. Additionally, the first temperaturemeasurements of high bias RIQH states in the second (N=1) Landau levelpresented at the second part of this chapter offer new insights into the natureof electron solids under non-equilibrium conditions. Experiment is focusedon finely-featured phase transitions, which appear as a function of bias cur-rent, before the state melts completely at very high bias. It is found thatthese transport signatures retain extremely fine bias and temperature sen-sitivity even down to 15 mK, under conditions where simple estimations ofself-heating by bias current would imply electron temperatures at the hotspots well above 100 mK.Together these data hint on the correlated electron nature of non-insulatingreentrant states under intermediate biases. Presumably they stay cold, ashighly localized dissipation allows to draw most of Joule power away fromthe 2DEG through the bottom of the sample. Besides, it is demonstrated,that the bulk breakdown mechanism is different from a simple melting, sincethe heat dissipation is likely localized near the edges of the 2DEG. Finally,alternative conducting bulk states, suggested in literature, and their possibleexperimental manifestations are reviewed at the end of the chapter.5.2 Transport measurementsLongitudinal and transverse voltages, probed at the edges of the samplein quantum Hall regime, depend on whether its bulk is insulating or not.At the same time those parameters of electronic transport are independenton the specific location of measurement contacts, i.e. matters only theirtopological position with respect to the bulk. This immediately leads toa conclusion, that the change of the bulk state has identical effect on thetransport measurements at any contacts of the sample. Contrary, if thesample is inhomogeneous, as was shown in the previous chapter, completelydifferent voltages could be simultaneously sensed at various contact pairsof the same sample. This provides a criteria for distinguishing transportfeatures caused by bulk from those, which are just a result of a local potentialchange around the specific ohmics. In the following paragraph I focus oninvestigation of the RIQHE bias breakdown features, measured at differentcontacts in order to drive a conclusion about state of the 2DEG in the bulkof the sample.The measurements where done at the same chip, as in Ch. 4. A sample645.2. Transport measurementsRxx  (kΩ)Rxx250150200100500c).3.863.823.78I DC (nA)B (T)a).B (T)b).R+DR-D123456781.21.00.80.60.40.20.08.07.06.0ABC8.68.27.87.47.0R3b R3aR3cR3dR+(kΩ)D3.4 3.6 3.73.5 3.8RD (kΩ)+8.27.87.47.03002001000IDC (nA)d).2003003.76 3.80B (T)I DC (nA)RD+RD-R xy (kW)B = 3.82 TFigure 5.1: (a) Characteristic R˜xy ≡ dVxy/dI and R˜xx ≡ dVxx/dI traces,measured at base temperature (13 mK) in filling factors ν = 3−4 (Iacb =5 nA,Idcb =0); (b) evolution of R˜+D (Iacb =5 nA) at the R3a reentrant state with dcbias current Idcb . Characteristic regions denoted ‘A’, ‘B’, and ‘C’ in theR2c breakdown; (c) schematic of the measurement; (d) traces of R˜+D andR˜−D breakdown with bias. Inset demonstrates R˜+D measurement 2D map inregion C with correlated pattern of ripples.655.2. Transport measurementslow bias measurement of the RIQHE for filling factors ν = 3 − 4 is shownon the figure 5.1a. Fig. 5.1b demonstrates the characteristic 2D map of R˜+Dmeasurement of R3a reentrant state in the measurement geometry, shownat Fig. 5.1c. Figure 5.1d demonstrates a characteristic R˜+D, R˜−D traces at thecenter of the R3a reentrant state. In addition to the chiral behaviour ofthe break-down described in previous chapter both traces simultaneouslytransfer into the region C with Hall resistance showing ripples around theclassical Hall value. The 2D map of that region shown on the inset tothe graph ensures that the ripples are consistent and totally reproducible inbias and magnetic field. Since two traces coincide in region C, this behavioursuggests, that transition to region C is caused by the change of the electronicbulk state, rather than local potential redistribution due to redistributionof various regions in inhomogeneous samples.5.2.1 Electrical potential spatial distributionFirst let’s consider electrical potential distribution in the sample. An elec-tron experiences a potential jump (the hotspot) at every boundary betweenmelted and frozen regions, where the local filling factor increases along theelectron flow. Figures 5.2a and 5.2b denote a propagation of the meltedregions and appropriate hotspots with bias current for electron-like and hole-like reentrant states. Following the previous labeling, the sample contactsare marked clockwise, with current sourced into contact 1 and drained fromcontact 5, assumed to be a ground potential.In general case of the RIQHE at filling factor ν, reentering to the integerplateau i, the electrical potential jump∆ = Ihe21i− 1ν, (5.1)where ν, i are the filling factors, corresponding to the current and nearest toit integer value of filling factor and I is the bias current through the sample.To reconstruct the value of electrical potential in all regions of the samplefor the case of electron-like reentrant state (Fig. 5.2a) let’s start from thecontact 6. Its potential equals to the potential of the ground, which is set tobe equal zero, as it is electrically connected to the drain contact by the edgestate. In this case contact 8, which is “after” the hot spot in the direction,opposite to the propagation of the edge state, has potential ∆. On theother hand, contact 2 has the potential Ih/ie2, corresponding to the integerquantized resistance of the unbroken reentrant state. Hence, the potential ofthe last contact 4 is lowered by ∆ due to the potential jump at the hot spot.665.2. Transport measurementsa) electrons b) holes0Ih/3e2Ih/4e2+DD0Ih/4e2+2DDc) d)8.07.02001000 4-8 2-68.07.02001000 2-6 4-8IDC (nA) IDC (nA)R xy (kW)R xy (kW)1234567812345678B=3.84 T B=3.52 TR3a R3cIh/3e2-DFigure 5.2: Breakdown propagation in the sample in a) electron and b)hole case, diamond and round marks depict the position of the hotspots fortwo different sizes of broken regions (grey and hatched), corresponding todifferent bias currents; comparison of two diagonal measurements for thec) electron (R3a) and d) hole-like (R3c) reentrant states (data taken fromFig. 4.3a)Following similar logic, one can reconstruct electrical potential distributionfor the case of the hole-like reentrant state. Specific values of the electricpotential at all 4 corner contacts for the case of hole-like R3c reentrant stateare shown in Fig. 5.2b.Now, consider Hall voltages, which could be possibly measured betweendifferent pairs of contacts. Assume, that the potential difference within ev-ery microscopic region is small. Therefore, the ohmic state can be approx-imated by “melted” or “frozen”, defined by the state of the 2DEG aroundit. Hence, transverse resistance between two given ohmic contacts changesonly when the 2DEG around at least one of them changes its state. In otherwords, there are only three possible voltages, which could be probed at anyHall contact pair for each reentrant state. The corresponding values of thetransverse resistances, calculated from the potential differences at different675.2. Transport measurementscontact states with use of (5.1), are summarised in the following table (F/Mlabels frozen or melted state of the corresponding ohmic):High/lowpotential F MF hie2hνe2M hνe2hie2± 2∆IAlthough the detailed discussion of the transport signatures of the RIQHEbreakdown in high bias is the subject of following paragraphs, such a sim-ple speculation allows to explain, for instance, such a non-obvious fact, astransverse resistance below classical Hall value, measured for electron-likereentrant states at intermediate biases (Fig. 5.2c). In homogeneous sampleit would mean, that the number of electrons, contributing to the transportis higher, than one would expect from the filling factor of the system. How-ever, for the case of partially broken reentrant state such ”undershoot” ofresistance results from two potential jumps at two hot spots, one at eachside of the sample. Similar behaviour, although reversed, is observed for thehole-like reentrant states (Fig. 5.2d). Transverse resistance trace overshootsthe free electron Hall one due to reversed direction of the breakdown prop-agation, resulting in different potential distribution (compare figures 5.2aand 5.2b).5.2.2 Comparison with experimentIn order to support the theory, described above, with more qualitative ex-planations, the specific values of the features in experimental traces areconsidered in details. Figures 5.3c,d contrast the breakdown of the R7areentrant state, measured at different contacts. The dashed vertical linesdepict current thresholds, corresponding to boundaries between regions A,B and C, defined earlier. In region A the hotspots are located ”before” allcontacts (Fig. 5.3a), so every voltage pair probes integrally quantized value.This area corresponds to non-broken reentrant states and was previouslylabeled as region A.Region B was defined as the intermediate state, when the sample isfractured into several macroscopic broken and non-broken regions. As thebias is increased, the transverse resistance, measured at different probesdeviates from the integer plateau and gets to the classical Hall value oreven lower, depending on the state of the voltage probes being measured.For the measurement geometry shown on Fig. 5.2a, the voltage probes aresymmetric with respect to the propagation of the hot spot. Thus, in ideal685.2. Transport measurementsd)e)R xy (kW)IDC (nA)R2cc)3.73.63.53.43.330020010003.73.63.53.43.33002001000R xy (kW)RD+RxyRD-RD+RxyRD-R7a R7aB=1.731 T B=1.802 TIDC (nA)A B CCBA0Ih/3e2D12345678Ih/3e2-D0Ih/3e2D12345678Ih/3e2-Db)a)1211109200150100500-50IDC (nA)B=5.08 TRD+RxyRD-2ptFigure 5.3: (a) schematics of the hot spot propagation with current bias; (b)schematics of the melted regions and current flow through the bulk in regionC; (c)-(d) breakdown traces of electron-like reentrant state R7a with bias;(e) comparison of the breakdown of hole-like reentrant state R2c, measuredat different volatge probes, and 2-point measurement.695.2. Transport measurementssample, one should see a direct transfer from integer plateau to the valuetwice lower, then the classical Hall resistance. However, in real samples, thehotspots do not simultaneously reach the contacts on each side, therefore,the trace demonstrates the step in the middle (light grey area on figures5.3c-d). Since, the conductance of the melted area is high, contact potentialis independent on the position of the hotspot, thus, measured Hall resistanceis constant in bias independently to the exact position of hotspot. This canbe easily seen for the case, where the contact asymmetry is large enough tocreate a significant difference between the chiral position of two hot spots.Fig. 5.3c depicts R˜−D trace with the step at current biases between 50 and100 nA. It is clearly seen, that measured resistance is constant in this rangecurrents. This situation is schematically shown on Fig. 5.3a for contacts 3and 7 and the hotspot position, marked by the diamond. Similar behaviouris observed for the hole-like reentrant states, shown on figure 5.3e. At theregion, corresponding to two measurement contacts surrounded with themelted reentrant state, the transverse resistance jumps twice as high, asthe difference between integer plateau and classical Hall value in completeagreement with (5.1).Another interesting observation is that under certain conditions the sam-ple could still be in the inhomogeneous state, corresponding to region B,while all the measurement contact pairs are situated in the melted regionsand transverse resistance is not quantized anywhere within the sample. Anexample of such situation is marked by dark grey area on Fig. 5.3c-d andshown schematically on Fig. 5.3a (the hot spot position is marked by star).Despite the sample does not demonstrate integrally quantized Hall resis-tances in transport measurements, the boundary between regions B and Cappears to be at higher biases. It can be identified, as the bias current,where all measurements simultaneously experience a jump to the classicalHall resistance level (for example, jump around 290 nA on Fig. 5.3c-d). Thefact that this transition happens at exactly the same bias for any contactpair (including source/drain ones, Fig. 5.3e) means, that the Hall resistancechange in region C is caused not by the local movement of the hotspot (whichhas already passed all contacts), but by the transition of the bulk from insu-lating to conducting state. The sample in such state is shown schematicallyin Fig. 5.3b. In contrast to the 4-wire voltage probes, the 2-point measure-ment is set by the potential difference between opposite edges of the sample.When the hot spots are formed, source and drain contacts automatically ap-pear in the melted region, which means, that their breakdown to classicalHall value could be caused only by shorting two opposite edges through thebulk of the sample.705.2. Transport measurements5.2.3 Longitudinal voltageFinally, I want to draw attention to another interesting experimental fact,shown at Fig. 5.4. For hole-like reentrant state R2c longitudinal DC voltageVxx(Idcb ) remains large into region C, even though the differential resistanceR˜xx (measured simultaneously) drops to a very small value.2.572.592.55νIb   (nA)dcIb   (nA)dcVxx  (μV)Rxx(k)Ωcν = 2.569R xx (k )ΩVxx  (μV)a b1.51.00.50-0.540030010006040200  3002001000-20 20 60 40030020010002.01.00.0~~Figure 5.4: Evolution of R2c with DC current bias: a) longitudinal voltageVxx and b) longitudinal differential resistance R˜xx (Iac = 5nA). Note thevery different color scale in (b) compared to previous figures. c). Charac-teristic traces of Vxx and R˜xx at ν = 2.569.Previously regions A, B and C were shown to correspond to a chiralmovement of the boundary between regions of melted and frozen reentrantstates in the sample (see Ch. 4). Turning to R˜xx measurement, region Acorresponds to both ohmic contacts appearing in the frozen insulating area.In this case, no current is leaking out of the edge states between the volt-age probes, therefore, the measured longitudinal voltage is zero. On thecontrary, region B corresponds to the case, where the melted boundary is715.3. Temperature measurementsbetween two R˜xx probes. In this case the measurement represents the po-tential drop at the boundary, showing high signal. Once the bulk starts toconduct, the voltage, measured between R˜xx probes experiences a drop (ataround 200 nA, Fig. 5.4c), remaining, however, at high level, demonstratingnearly current independent trace.Such a diode-like I-V curve is rather surprising. As was mentioned inprevious section, the bulk breakdown happens before the hot spots leavethe region between R˜xx voltage probes. The constant longitudinal voltagemeans, that the potential jump at the boundary of two regions remains thesame, which requires constant current to flow through the hotspot at theedge. Thus, when the bias is increased in region C, all extra current flowsnearly dissipationlessly through the bulk, maintaining constant potentialdifference at the density jump near the edge. This means, that relativelylarge current could flow through the bulk of the sample in region C, whichquestions an existence of correlated electron state. In order to bring somelight to this puzzle, I performed temperature measurements, which are thetopic of the next paragraph.5.3 Temperature measurements5.3.1 Experimental techniqueTemperature measurements were performed on the same sample and in thesame cooldown. Chip temperature was controlled using the mixing chamberheater or a thermometer/heater pair on the back of the chip carrier for fasterresponse. Differential resistances R˜xx ≡ dVxx/dIb and R˜xy ≡ dVxy/dIb weremeasured by lockin amplifier with an AC current bias, Iacb = 5 nA, at 71 Hz.Characteristic R˜xx and R˜xy traces at Idcb = 0 show well-developed frac-tional quantum Hall states at ν = 2 + 1/5, 7/3, 5/2 (∆5/2 ∼ 0.45 K), 8/3and 2+4/5 (Fig. 5.5a). The diamond of R˜xx features surrounding the centerof the reentrant state, as before, demonstrates three distinct subregions, la-belled ‘A’, ‘B’, and ‘C’ in Fig. 5.5b. The differential resistance is very smallfor higher bias, in region C, where R˜xx fluctuates up and down as a functionof Idcb and B. Resulting pattern of ripples in the 2D data becomes sharper astemperature is reduced. These features are completely reproducible for con-secutive Idcb sweeps, and are, therefore, not oscillations or noise of the typereported in [28]. Images with features as sharp as those seen in Figs. 5.5band 5.6 were obtained only below 15 mK and only for very well developedreentrant states, especially R2c and R2a but not R2b in Fig. 5.5a (in fact,725.3. Temperature measurements13121110987 5.85.65.25.04.64.41.00.80.60.40.20.05/22+1/52+2/32+1/32+4/5B (T)10020030040002.52.542.62 2.58νRxx (k )Ω0 1baI b   (nA)dcRxx (k)ΩRxx(k)ΩR2c R2aR2bR2dBACTmix=14mKRxx(k)W 21025.2 06.2nRxxRxyFigure 5.5: (a) Characteristic R˜xy ≡ dVxy/dI and R˜xx ≡ dVxx/dI traces,measured at base temperature (13 mK) in filling factors ν = 2−3 (Iacb =1 nA,Idcb =0); (b) evolution of R˜xx (Iacb =5 nA) at the R2c reentrant state andν = 5/2 FQH state with dc bias current Idcb . Characteristic regions denoted‘A’, ‘B’, and ‘C’ in the R2c breakdown. Color scale is saturated in region B.Inset demonstrates cross section of the graph, marked by dash-dotted line.735.3. Temperature measurementsR2b was never observed to be well-developed). Similar data were also seenfor every well-developed RIQH state between ν = 2 and ν = 4, taking intoaccount many samples and many cooldowns, though high bias features wereoften less sharp and/or intricate than those shown here.The electronic temperature in the sample was monitored using temperature-dependent features in the magnetoresistance, which could be calibrated athigher temperatures where thermal equilibration is more reliable and thenextrapolated to lower temperatures. In Ref. [23], for example, it was shownthat the width of a reentrant state grows with temperature down to 6 mK.Figs. 5.6a and 5.6b show temperature dependence of the R2c reentrant state.Following a similar approach to [23] we confirm that, in the absence of sig-nificant current bias, our sample temperature follows that of the mixingchamber down to its base, 13 mK (Fig. 5.6c).When DC bias is added, the shape of the reentrant feature changesdramatically, and one must be careful to separate the direct effects of theinduced electric field from those of Joule heating. Figure 5.6 contrasts Rxxfor the reentrant state at Idc=0 (Fig. 5.6b) and Idc=300 nA (Fig. 5.6d), overa range of temperatures. One clue that the electric field is responsible formost of the reentrant state breakdown in region C comes from the exis-tence of extremely sharp features in the magneto-fingerprint for low latticetemperatures. When the temperature is high enough to melt the reentrantstate, on the other hand, the magnetoresistance traces even at high bias areessentially featureless.5.3.2 Experimental resultsThe remainder of this chapter focuses on region C (Fig. 5.5b). The 2Ddata shows numerous fine features that shift in bias and magnetic field.The rippled pattern around ν = 2.58 and above 150 nA, for example, maycorrespond to pinning/depinning transitions of an electron crystal state.Narrow-band noise previously reported in a high-bias phase analogous toregion C, but for ν > 4 [28], is sometimes attributed to a bubble phase slidingover a pinning potential. One difficulty with this explanation, however, isthat the frequency associated with the bubble lattice spacing is many ordersof magnitude away from what was observed.Because region C appears only at high bias, after passing the highly-dissipative region B, it may be tempting to associate it with a RIQH statethat has been destroyed by bias-induced heating. It is clear, that C doesnot represent the remains of a completely molten RIQH electron crystal.Such a melting transition is observed for even higher biases (dashed bound-745.3. Temperature measurementsh/3e20.340.360.380.401.61.20.80.40.02.5 2.6 2.5 2.6-1012342.5 2.6Rxx(kW)Rxx(kW)nT (mK)a bc d15 20 25 30020406080R2c  width,x103DnRxy(e /h)2Figure 5.6: Evolution of a) R˜xy and b) R˜xx at the R2c reentrant state, withtemperature between 15 and 48 mK, Iac=1 nA, Idc=0; c) temperature de-pendence of R2c width at Idc=0; d) R˜xx evolution at Iac=1 nA, Idc=300 nA.755.3. Temperature measurements2.592.582.572.562.552.54600400DC Current Bias (nA)a6005004003002002.5552.562000νν2.562.5552.565DC Current Bias (nA)νbcTmix=14 mKTmix=15.6 mKR xx (Ω)10 80R xx (Ω)10 80R xx (Ω)40 120Tmix=14 mK~~~Boundary of region CA B CFigure 5.7: Region C remains highly temperature dependent, despite signif-icant Joule heating. (a) the full breakdown diamond of R2c, analogous toFig. 1b but after a refrigerator cycle to 4K. Sharp features in region C serveas effective thermometers, because they are extremely temperature sensi-tive. Interior box denotes area in (b,c), with hashed region at higher biasconsidered to be beyond breakdown. (b) Zoom in to box from (a). Sharpfeatures are qualitatively similar in low (200-300nA) and high (400-500nA)bias regions. (c) Analogous data after warming the mixing chamber to 15.6mK. Sharp features are completely washed out, at both low and high bias.765.3. Temperature measurements102030051525Temperature (mK)n(   -2.545)x10000=300 nADCI1 2Rxx (kW)022 232120191817Figure 5.8: Field sweeps over the spike on the boundary of insulating areanear ν = 2.55, see Fig. 1b. Horizontal axis represents estimated lattice tem-perature controlled by chip carrier resistor, taking into account additionaldissipation in sample’s contacts due to bias current (Iac=5 nA, Idc=300 nA)ary in Fig. 5.5b), where R˜xx becomes again nearly flat with temperature-independent resistance.In general, Joule heating of electrons is a severe problem at ultra-lowtemperatures because the coupling between electrons and the crystal latticefalls off as T 5. This is due to a collapse in electronic heat capacity andphonon density of states [88, 92, 93]. On the contrary, region C appears tobe anomalously cold (Fig. 5.6d), significantly colder than might be expectedby balancing the measured dissipation in the sample with electron-phononcooling of the 2DEG into the lattice [87].Figure 5.7 illustrates an extreme sensitivity of high bias data to small in-creases in lattice temperature. R˜xx data at Tmix = 14 mK exhibits numeroussharp features that shift smoothly in bias and magnetic field (Figs. 5.7a,b).Increasing Tmix from 14 to 15.6mK (Fig. 5.7c) melts all of these features775.4. Discussionwithin region C, though a small amount of rippling remains and a sharptransition is still observed at the high-bias boundary (transition to hatchedregion in the figure).To study this more quantitatively, I focus on a sharp peak in Rxx atIdc=300 nA that appears as the reentrant state breaks down (refer to insetin Fig. 5.5b). Fig. 5.8 shows that this peak moves with temperature, justlike the edge of the reentrant state at zero bias.[23] This strong temperaturedependence indicates that the electronic system remains tightly coupled toTmix even at extremely low temperature, and even with hundreds of nA ofbias current applied.Since described high bias features exist in a very narrow region of tem-peratures below 30 mK, conventional methods of electron temperature mea-surement, such as Arrhenius plots of resistivity, are not relevant in regionC where the insulating state is already broken. However, an order-of-magnitude estimate of electron overheating by dissipation in region C canbe extracted from the observation that qualitative characteristics (sharpfeatures inside region C) appear similar in the low and high bias regions inFig. 5.7b (14 mK), whereas the character at both low and high bias changesdramatically after only a 1.6 mK increase in mixing chamber temperature(Fig. 5.7c). In other words, raising the bias from 200 to 400 nA apparentlyaffects the electron gas less than raising the mixing chamber from 14 to15.6 mK.5.4 DiscussionA dramatic contrast between differential (AC) and static (DC) resistancesat high bias, as well as, the analysis of the breakdown, measured at differentcontacts indicate a nearly-frictionless sliding mechanism that is activatedabove a critical electric field in the bulk of the sample. On the other hand,low electron temperature immediately suggests electron solid as a micro-scopic description of the bulk state.Correlated electron systems under conditions of RIQHE are predictedto have many different phases, especially Wigner-like solids (bubble phases)and nematic/smectic liquid electron crystals (ELC) [78]. As was alreadymentioned in previous chapters, such phases are susceptible to the externalHall electric field. For example, under certain conditions electron crystalmay simply depin from the underlying potential, resulting in a sliding of theentire electron crystal structure instead of a single electron transport [31]. Inthis case the charge is transferred between the opposite edges of the sample,785.5. Conclusionshowever, cold electrons form correlated state. The assumption of cold bulkelectrons is possible, as the power is dissipated locally at the boundarybetween melted and frozen regions, where the potential jump takes place.Although in high mobility systems the number of scattering centers isextremely low, in the case of reentrant state their pinning might be muchmore effective due to the fact that they are coupled to the electron crystal.In high biases, when the insulating state is broken, under certain conditionsa “plastic flow” can occur [15]. In this case transport is given by few randomelectrons occasionally hopping between pinned clusters. This creates anothermechanism for the current flowing through the bulk of the sample, while themajority of electrons is localized within pinned correlated electron state.However, it has to be mentioned, that in this case delocalized electronsdissipate the energy at every jump, i.e. straight into the bulk. Thus, it isunlikely, that such state could stay cold under high currents.Alternatively, after the insulating state is broken down, the bulk currentdistribution can be non uniform. In the most extreme case the current flowis localized to one narrow channel. While specific shape and conductanceof this current is set by surrounding insulating cold bulk, the Joule heat,dissipated locally, heats few electrons inside the hot current flow. Eventhough no mechanism is known, allowing for such strong non-uniformity,the question of current distribution in the bulk after breakdown remainsopen.5.5 ConclusionsThe breakdown of the reentrant states under high current bias was previ-ously explained by the partial melting of the correlated electron solid stateby Joule heat, dissipated into electron system. In this chapter I investi-gated the structure of high bias states, which are stabilized after RIQHEis completely destroyed. Compared with the temperature measurements,the spatial distribution of the Hall potential along the edge, measured atvarious contact pairs, allows to extract an information about electron prop-erties in the bulk. Although Joule heating is the primary effect, causingRIQH breakdown, the 2DEG in the bulk of the sample turns into conduct-ing state before the correlated electron state is destroyed. It was shown,that after the insulating bulk state is broken, the system retains extremelyhigh temperature sensitivity down to 15 mK.These data suggest the depinning of electron crystal phase from the un-derlying lattice as the main candidate for microscopic description of the795.5. Conclusionsnon-equilibrium RIQHE states. Such scenario is qualitatively consistentwith existing predictions for sliding dynamics[15], however, a more quanti-tative analysis of this mechanism awaits further theoretical consideration.80Chapter 6Bias induced anisotropy athalf-filled Landau level6.1 MotivationAnisotropic transport around half filling of the higher Landau levels (N > 1),usually referred to as a stripe phase, was one of the first experimental ev-idences of the stabilization of a correlated electron solid states under con-ditions of quantum Hall effect [11, 12]. On a microscopic level transportanisotropy is believed to result from a unidirectional charge density wave(CDW). Similar to its isotropic sibling, in the anisotropic state, electronsform alternating stripes at neighbouring integer filling factors [3, 14].Broken rotational symmetry is a fundamental property of a stripe phase,which implies that the electron system must have a preferred direction fororientation of the stripes. Over the last few decades, strong experimentalevidence has been collected, showing that the alignment of stripes is con-nected to the crystallographic orientation. The stabilization of the stripephase in GaAs samples results, in most cases, in a low Rxx probed along”easy” 〈110〉 crystal orientation and high Rxx along the orthogonal ”hard”axis. Despite the large amount of experimental data, the origin of the partic-ular favored stripe orientation remains poorly understood. The theory ”bydefault” does not predict the existence of a preferred direction for CDW.Different groups tried to connect the alignment of the stripes to anisotropicroughness of MBE growth [71] or to the asymmetry of the electron potentialin the direction perpendicular to the 2DEG [94]. However, experimentally,no complete correlation between those features and the stripe orientationwas found [95, 96]. Furthermore, theoretical studies suggest, that underperiodic potential modulation, unidirectional CDW may align both paral-lel or perpendicular to the modulation direction, depending on a potentialstrength [97].In the search of the native underlying potential responsible for such align-ment, stripe phases were shown to rotate by 90◦ under external directional816.1. Motivationperturbations. For example, transport anisotropy can be switched by in-plane magnetic field [72, 73], current bias [29], surface acoustic waves [79],or by application of an external strain to the chip [74]. In all these experi-ments, the directional perturbation is believed to create a symmetry break-ing potential which is competing with the intrinsic one. Above the criticalvalues of the external parameter, the native potential becomes weaker andthe stripe orientation is set by the direction of the external influence [98].Most of early experiments on the anisotropic CDW states were limitedto the electron densities below 3× 1011 cm−2. However, different transportanisotropy not only was demonstrated at the samples with higher density,but the rotation of the stripes was detected, while the density of the 2DEGwas varied within the same sample [75]. Additionally, weakly anisotropicmetastable states were observed around half filling of higher Landau lev-els [99]. All these experimental facts led to the conclusion that the intrinsicalignment potential is more complex than the unidirectional rotational sym-metry breaking potential. It was suggested that the alignment of the stripesis actually set by two competing orthogonal components, tending to alignCDW along 〈110〉 or 〈11¯0〉. The relative strength of these potentials changeswith electron density, leading to the stripe reorientation at certain density.Around the critical point, the two potentials have similar effects, allowingfor metastability and anisotropy suppression.However, it was recently demonstrated that in high mobility samplestransport anisotropy can depend on the filling factor [76], revealing an evenmore complicated structure of the CDW states around half filling of Landaulevel. By application of in-plane magnetic field, the authors induced a statewith opposite anisotropies away from the half filling and in the center of thestripe phase. Such switching of the anisotropy axis within a single Landaulevel hints to a strong dependence of the native symmetry breaking potentialon the filling factor.In this experiment a metastable stripe phase around ν = 9/2 is investi-gated under non-equilibrium conditions in the sample with electron densityns ∼ 3 × 1011 cm−2, which is close to the stripe reorientation density. De-spite the metastability, the anisotropy of the non-equilibrium stripe phaseunder high current biases allows to restore the orientation of the funda-mental alignment potential in the sample. These experiments suggest theexistence of multiple microscopic phases within each anisotropic state andprovide a new vision at the CDW alignment mechanism.826.2. Experimental results6.2 Experimental results6.2.1 Measurement detailsMost of the data, demonstrated in this chapter, was taken at the samesquare sample with Wan-der-Pauw contacts and in the same cooldown as inprevious experiments. However, the main details of the experimental resultswere reproduced in multiple cooldowns as well as in an another sample,although the quantitative values, such as anisotropy factor could vary. Thetemperature of the sample was estimated by the carbon thermometer, gluedto the back of the chip carrier and confirmed to be below 20 mK all timeduring the experiment. All resistances in this chapter were probed usingstandard lock-in technique with 4 nA AC at 81 Hz.The GaAs sample was cleaved along 〈110〉 and 〈11¯0〉 crystallographicplanes, defining naturally the longitudinal resistance orientation relative tothe crystal lattice. Independently on the direction of bias current applied,the R˜xx and R˜yy resistances denote differential resistance measured alongcorresponding edge of the sample (refer to 3.1.3 for the electrical measure-ment setup details).Characteristic longitudinal and Hall resistance traces are shown on fig-ure 6.1. The integer quantum Hall plateaux are well developed showingsharp steps between quantized values. Longitudinal resistances Rxx andRyy measured along two orthogonal edges of the square chip demonstrateweak anisotropy for all peaks up to ν = 11/2. This allows to assign easy andhard orientations to the chip (refer to the schematics at Fig. 6.1). The lastanisotropic state around ν = 9/2 demonstrates much stronger anisotropy,albeit easy/hard directions are swapped compared to all higher filling fac-tors.The screening from underlying completely filled Landau levels can varythe strength of inherent potential. Hence the anisotropy switching of thestrongest stripe phase in the sample with electron density close to the crit-ical is not that surprising. Nonetheless, before turning to the discussionof this effect, which is the main subject of the current chapter, I want todemonstrate that at half fillings of higher Landau levels (ν ≥ 11/2), the or-dinary stripe phases are formed despite the fact that anisotropy is stronglysuppressed.In the samples with lower densities with strong anisotropic state aroundν = 9/2, the high current biases were shown to stabilize or destabilize thestripe phases depending on their direction with respect to the orientationof the stripes [29]. Specifically, if the current is driven along the stripes,836.2. Experimental results65432102.52.01.51.03.02.52.01.51.00.50.0B (T)R xy (kW)Rxx, yy  (kW)RyyRxxXYFigure 6.1: Characteristic R˜xx, R˜yy and R˜xy (for current driven along Yaxis) for filling factors ν = 4 − 14, showing the weak transport anisotropy(IAC=4 nA). Schematics show the preffered orientation of the stripes at highfilling factors.i.e. in the easy direction, the stripe phase is enhanced and its anisotropyremains unchanged up to high currents, until the melting of the entire statedue to the Joule heating. On the opposite, if the current bias is drivenperpendicular to the stripes in the hard direction, the anisotropy disappearsat much lower biases. This effect is usually explained by destruction of thelong range stripe order, leading to vanishing transport anisotropy, while themelting of the correlated state by bias is roughly independent on the currentdirection.Similar behavior was confirmed in our sample for the anisotropic statesin the range of ν = 11/2 − 15/2. Figure 6.2 shows longitudinal resistancemeasurements for all four possible AC/DC orientations, relative to the crys-tal axis. Small schematic on the side of each graph demonstrates the relativeorientation of AC/DC currents and alignment of stripes with respect to thesample. At zero DC bias R˜yy measures high resistance state independently846.2.ExperimentalresultsIDC (uA) IDC (uA)νν6.606.556.506.456.403210 32103210 32106.606.556.506.456.406.606.556.506.456.406.606.556.506.456.40a  RXX b  RYYc  RXX d  RYY0.60.40.20.0R (kW)Y XRxxYXRyyYXRyyY XRxxFigure 6.2: Evolution of R˜yy with a)DC||Oy, d)DC||Ox and R˜xx with b)DC||Oy, c)DC||Ox. Dashed linesshow approximate melting boundary of the stripe phase. Schematics depict stripe alignment and measurementorientation relative to DC bias, driven in a-b) hard and c-d) easy direction. Short (long) wiggly lines depictorientation of stripes suppressed (enhanced) by bias.856.2. Experimental resultson the orientation of DC component (Fig. 6.2a,d). Similarly R˜xx probes lowsignal in accordance with easy measurement orientation along the stripes(Fig. 6.2b,c). However, in this sample remains of a metastable state couldbe observed even at high filling factors at very low biases (IDC < 100nA). Itis represented by weak zero bias anomalies on figure 6.2 and is very likely tobe responsible for the low R˜xx/R˜yy anisotropy at field sweep traces (Fig. 6.1).Once even a small DC bias is applied, the metastable state is destroyedand high transport anisotropy revives. If the bias current is driven alongthe stripes (Fig. 6.2c-d) the anisotropic state survives up to the biases above3 uA. This is opposed to the case with current driven in the perpendiculardirection (Fig. 6.2a-b), where no significant anisotropy is detected aboveapproximately 1 uA DC.This behavior is in complete agreement with the measurements, reportedby other groups [29]. Therefore, Ox axis can be unambiguously identifiedas the preferred stripe orientation in a low electron density limit. Followingthe conventional notation, hereafter the direction showing high longitudinalresistance is referred to as “hard”, while low resistive direction is called“easy” orientation.6.2.2 Bias measurements at ν = 9/2Now I turn to the description of measurement results at ν = 9/2. Figure 6.3adepicts schematically the stripe orientation preferred by 9/2 state in thissample. The metastable state is thought to have two alignment potentialsof approximately equal strength, therefore, the stripes are randomly orientedalong Ox or Oy axis (Fig. 6.3a top). However, the electron system can relaxinto anisotropic state, for example as a result of annealing [99], aligningall stripes along one orientation. This case is depicted at the bottom ofFig. 6.3a. Note, that at 9/2 state transport anisotorpy is opposite to the onemeasured at half filling of higher Landau levels. Therefore stripes are shownto be oriented along Oy axis on the bottom schematics of Fig 6.3a.Figure 6.3b shows zero DC bias traces along two perpendicular crys-tallographic directions. The longitudinal resistance is clearly anisotropic,although anisotropy is weak. Another sign of nearly metastable state comesfrom hysteretic behavior between up and down field sweeps [99]. However,zero DC bias traces undergo a dramatic change after a 3 uA DC bias cur-rent is applied for a short period of time (∼ 10 s) at each point (Fig. 6.3c,d).After the application of the bias, transport properties of 9/2 state becomestrongly dependent on the exact filling factor. In the vicinity of half filling,the anisotropy disappears and the longitudinal resistance does not depend866.2. Experimental resultsν4.74.64.54.44.34.84.64.44.2 4.84.64.44.22.01.51.00.50.02.01.51.00.50.02.01.51.00.50.0ννR xx (kW)R (kW)R (kW)badcRyyRxxRyyRxxRyyRxxIDC || YXYIDC || XXYXYXYanisotropyanisotropymethastabilityanisotropyanisotropyanisotropyFigure 6.3: a) Schematic of a measurement with at metastable (top) andanisotropic (bottom) ν = 9/2 stripe phase; b) comparison of R˜xx and R˜yyfield scans at zero DC bias, solid (dashed) arrow mark field sweep down(up); 0 nA DC R˜xx and R˜yy field scans annealed at 3 uA DC along c) Oyand d) Ox axis at every filling factor.(within the uncertainty of few hundred Ohms) neither on measurement di-rection, nor on the orientation of the DC bias applied at each point. Awayfrom ν = 9/2, on the contrary, anisotropy is strongly enhanced in compari-son to field sweeps on Fig 6.3b.Since this effect does not depend on the direction of current bias, it sug-gests that the annealing due to Joule heating is the primary effect causingthe switching of the state. Interestingly, the new anisotropic state is per-pendicular to the conventional orientation of stripes previously identified forhigher Landau levels. This means that the ground state of the stripe phaseaway from 9/2 is orthogonal to the conventional alignment of the stripes at876.2. Experimental resultslower electron densities or higher filling factors (Fig 6.2).These experimental findings are consistent with the metastability mea-surements of the stripe state at high density samples, reported in litera-ture [99]. Two different anisotropic directions at ν = 9/2 state were re-cently reported in by other group [76]. However, the main experimentalfinding of this work is that the critical in-plane magnetic field, which re-aligns the CDW phase, strongly depends on exact filling factor within oneanisotropic state. Although the fact that the stripe phase behavior dependsdramatically on filling factor is intriguing, it does not provide any informa-tion about the intrinsic ground state of the CDW. For example, these datacould be explained by transition between Coulomb dominant and modula-tion dominant (modulation is given by in-plane field in this case) regimes,suggested theoretically [97].Anisotropic states around ν = 9/2 were previously investigated in bias atsignificantly lower electron densities [29]. This resulted in the stabilizationof the conventional anisotropic state and no metastability was reported.However, as was already confirmed at the beginning of this chapter forother stripe states at higher Landau levels, such measurement allows toput the system away from the metastable state and directly probe intrinsicalignment potential. In contrast to effects of the in-plane magnetic field,which just creates the primary modulation potential above the critical point,driving the current affects the pinning of the stripes in an additive way. Inother words, if the current is driven parallel to natural orientation of thestripes, the anisotropy of the state is strongly enhanced under high biases.This allows to investigate the initial state of the system by its response tothe external perturbation instead of investigation of the system affected byan external field. In order to study the fundamental stripe state aroundν = 9/2 in our sample, the transport anisotropy around was probed awayfrom equilibrium at different filling factors.Figures 6.4a,d demonstrate the direction of DC bias with respect to thesample as well as Ox and Oy directions of AC current used during the R˜xxand R˜yy measurements correspondingly. The 2D maps on Fig. 6.4b,c showR˜xx and R˜yy under DC current applied along the Oy axis. Similar data forthe case of DC ‖ Ox is shown on figure 6.4e,f. One can easily identify threeanisotropic high bias regions on Fig. 6.4: in the middle of the stripe state(ν ∼ 4.45− 4.6) and two regions, one on each side (ν ∼ 4.375− 4.45 andν ∼ 4.6− 4.675).886.2.Experimentalresults4.704.654.604.554.504.454.404.354.654.604.554.504.454.404.353.02.52.01.51.00.50-0.5 3.02.52.01.51.00.504.704.654.604.554.504.454.404.354.654.604.554.504.454.404.353.02.52.01.51.00.50-0.5 3.02.52.01.51.00.50IDC (uA) IDC (uA)νν1.00.80.60.40.20.0R (kW)b  RXXc  RYYe  RXX f  RYYadY XY XFigure 6.4: Schematic description of stripe orientation in the vicinity of ν = 9/2 relative to DC bias, driven in a)hard and d) easy direction. Short (long) wiggly lines depict orientation of stripes destabilized (stabilized) by bias.Evolution of R˜yy with b)DC||Oy and f)DC||Ox. Evolution of R˜xx with c)DC||Oy and e)DC||Ox. Dashed linesshow approximate melting boundary of the stripe phase.896.2. Experimental resultsFor the purpose of this chapter, I investigate the anisotropy of high bias(above 100 nA) states. Consider first the side regions at ν ∼ 4.375− 4.45and ν ∼ 4.6− 4.675. If the DC bias is applied along Oy axis (Fig, 6.4a-c), thetransport demonstrates strong anisotropy (R˜xx  R˜yy) even at IDC > 1 uA.At the same time high bias anisotropy already vanishes at 500 nA of DC biasapplied in perpendicular direction along Ox (Fig, 6.4a-c). In order to makethis distinction clearer R˜xx and R˜yy bias traces at ν = 4.42 (Fig. 6.5a,b)and ν = 4.62 (Fig. 6.5c,d) are contrasted for two perpendicular orientationsof DC component. These filling factors correspond to the center of thenew anisotropic states stabilized by annealing (Fig. 6.3) and coincide withthe filling factor of inverse anisotropy states under parallel magnetic fieldin [76].Fig. 6.5a,c compares R˜xx/R˜yy bias scans at different sides from the halffilling when DC drive enhances the stripes. At both filling factors R˜xx probeshigh resistance, corresponding to the “hard” measurement orientation. No-tably, significant anisotropy survives up to 2 uA of bias in both cases as well.If the bias is applied along Ox axis (Fig. 6.4b,d), the “hard” and “easy”orientation remain the same, however no anisotropic transport is detectedabove approximately 400 nA. These results are completely consistent withprevious findings: the bias driven perpendicular to the stripes destabilizesthe stripe phase, killing the long range order. This leads to an anisotropyvanish at the currents well below the critical melting current. Presumableorientation of stripes is shown on the insets on figure 6.5. As before the short(long) wiggles mark suppression (enhancement) of the stripe by bias drivenperpendicular to (along) the direction of their initial alignment potential.Notice that anisotropy of a low bias state (< 100nA) could be signifi-cantly different from the one above ∼ 200nA. This effect is most clear onFig. 6.4b,c and could be attributed to a bias reorientation threshold. At lowcurrents, the effects of current alignment are negligible and the sample stateis mostly independent on bias. However, above ∼ 200nA the zero bias stateis destroyed and the anisotropy demonstrates a sharp change.Turning back now to Fig. 6.4, let’s focus on the center of the stateν = 4.45− 4.6). In this range stripe phase is enhanced by DC current flow-ing along Ox axis (Fig. 6.4d-f). Longitudinal resistance anisotropy is clearlyseen at high biases, albeit the isotropic state is observed around zero DC cur-rents, which is consistent with the measurements on Fig. 6.3. Nonetheless,there is no measurable transport anisotropy under the bias above 200 nAwhen DC current is driven along Oy (Fig. 6.4a-c). This effect might resultfrom a very weak pinning potential. Therefore the stripe phase is immedi-906.3. Discussion32101.21.00.80.60.40.20.01.00.80.60.40.20.01.21.21.00.80.60.40.20.01.00.80.60.40.20.01.23210R (kW)R (kW)R (kW)R (kW)IDC (uA) IDC (uA)RyyRxxIDC || Yν=4.62XYRyyRxxIDC || Xν=4.62XYRyyRxxIDC || Xν=4.42XYRyyRxxIDC || Yν=4.42XYa bc dmetastablemetastableFigure 6.5: Bias scans of anisotropic states, stabilized by annealing at a,b)ν = 4.42; c,d) ν = 4.62. The direction of the DC bias as well as the expectedorientation and strength of the stripes is shown on the schematics.ately destructed by bias beyond the threshold and anisotropic state is neverstabilized under such conditions.Experimental data show, that the orientation of natural alignment po-tential changes with the filling factor: it tends to align the stripes along Oxaxis in the center region and in perpendicular direction on the sides.6.3 DiscussionThe schematic phase diagram on figure 6.6 summarizes experimental data.Initial orientation of the stripe phase, as well as the direction of AC and DCbias components, are schematically shown on the left side of each graph.The anisotropic phase is divided into three regions with different anisotropyin the center and on the sides. The dark (light) background of each statedepicts high (low) longitudinal resistance under high bias, corresponding to916.3.DiscussionEASYEASYSTRONGSTRONGHARDHARDWEAKDCACACACACACACDCDCDC 9/29/2IDC IDCXYEASY HARDWEAKWEAKSTRONGa bc d0 0ACACRxxRxxRyyRyyFigure 6.6: Schematic phase diagram of the stripe phase in samples with electron density around the critical point.926.3. Discussionthe measurement along hard (easy) axis. The icon of the stripes demon-strates the direction of the alignment, while the wiggle length shows the en-hanced by bias, regular or suppressed stripe phase correspondingly.Horizontal(vertical) orientation of stripes symbol depicts alignment along Ox (Oy).The data shows that for the center state the bias stabilizes regular CDWorientation (along Ox), which coincides with the self-organized orientationof the stripe phase at higher Landau levels. This results in the existence ofanisotropy up to high biases (strong state) if the DC current is driven alongOx (Fig. 6.6c,d), while the same anisotropy survives to much lower biases(weak state) if the DC orientation is switched to Oy axis (Fig. 6.6a,b). Itis important to mention, that initial orientation of stripe phase around ν =9/2, stabilized during the field sweeps at zero bias, was identified along Oyorientation. However, the fact that it vanishes at zero bias after annealing,as well as high bias measurements, hint at the isotropic state results fromapplication of bias, while the inherent alignment potential is oriented alongthe Ox axis. Analogous behavior is observed for the high bias states on thesides of 9/2. These states seem to align along Oy axis in bias, retaining lowbias orientation of the anisotropy, which appears after annealing. Althoughthe stripe orientation can be varied at zero bias, the experimental datasuggests that the natural potential is strongly dependent on the filling factor.I suggest the following model explaining experimental results. The CDWpinning potential possesses two important features. First, it consists of twocompeting components, tending to align the stripes along two orthogonal ori-entations. Second, those components are strongly dependent on the fillingfactor. Figure 6.7a shows qualitative profile of two potentials and their rela-tive strength for the case of conventional low density sample. The anisotropicregion is marked by color background. In this case, the Ox alignment poten-tial is much lower than Oy, therefore the stripes are always aligned in thatorientation. However, the increase of density leads to decrease of differencebetween Ox and Oy potentials, making them overlap in a much narrowerregion (Fig. 6.7). This leads to a much smaller energy difference between thetwo states with different orthogonal orientations, which allows the existenceof the metastable state around 9/2. Conversely, it is the anisotropic statewith the opposite Oy anisotropy that gets stabilized on the sides. The sys-tem with relative pinning potential distribution shown on the figure 6.7b hastwo different ground states with opposite alignment in the middle and onthe sides of the stripe phase. Although the close energy difference betweentwo states is small, this anisotropy does not realign by itself. However, theground state affects the properties of the 2DEG in non-equilibrium stateunder high bias currents.936.4. Conclusions9/2 9/2 ννU Uba anisotropic anisotropic anisotropic4.64.4 4.64.4Figure 6.7: Schematic description of the competition of two pinning stripepotentials in a) low electron density; b) close to the critical point. Wigglylines in the middle depict the anisotropy of the ground state.6.4 ConclusionsIn conclusion, the bias induced anisotropy of the stripe phase around 9/2was investigated in the high electron density sample. This puts the electronsystem under conditions of strong competition between different potentials,tending to align the state along two perpendicular orientations. Althoughthe origin of those states is still unclear, it was demonstrated that the changeof the transport anisotropy is most likely caused by a change of the structureof ground pinning potential, rather than by an electron phase change. Theorientation of the stripes was observed to be extremely sensitive to the fillingfactor, leading to a different high bias transport anisotropies on the sidesand in the center of the state. The data are consistent with recent reportsfrom other groups, providing another experimental evidence for the strongdependence of pinning potential on the electron filling factor [76].An alternative explanation to the model of competing pinning potentialscould be the stripe realignment due to the change of electron-electron inter-action strength, suggested in [97]. Such interpretation suggests that there isa single pinning potential, however the electron-electron interaction changeswhen the density or filling factor is varied. Around the critical point, thesystem experiences transition between Coulomb dominant and modulationdominant regimes which makes the stripes to switch the alignment directionbetween orthogonal and parallel to the external modulation.94Chapter 7Summary and futureexperimentsIn this thesis, I have studied the transport properties of the electron solidstate, which are set away from equilibrium by application of high bias. Al-though this technique requires driving high currents through the sample atlow temperatures, it was demonstrated that under certain conditions coldcorrelated electron states can survive. This allows one to probe the coldelectron solid states away from equilibrium, which is very important in thisparticular case. Since correlated electron solid states are normally insulat-ing at zero bias, different microscopic phases are indistinguishable from eachother in transport. On the other hand, electron solids form at temperaturesbelow 100 mK (in the most extreme cases of N=1 Landau levels, the melt-ing temperature is even lower, less than 50 mK in the best quality samples).Such fragile properties of correlated electron solid states leaves very few ex-perimental techniques available for their probing. Electrical transport is themost suitable one.Although probing of an electron system away from equilibrium providesa lot of information about its state, the interpretation of experimental resultsis difficult. This is especially true in the case of the transport measurementsin macroscopic sample, since various inhomogeneities in the 2DEG lead tostabilization of different local states which are averaged in the resultingsignal.In the first part (Ch. 4) of this work I focus on the investigation of highbias states at filling factors, corresponding to the conditions of RIQHE.Specifically, the mechanism of insulating state breakdown was investigatedin details. It was shown, that the sample is strongly inhomogeneous duringbreakdown However, investigation of spatial distribution of the electricalpotential enables the probing of some microscopic properties of the system.For example, one could distinguish electron- and hole-like reentrant states.The breakdown was found to propagate chirally from source and drain con-tacts with a sense that depends on the electron- or hole-like character ofthe reentrant state. This phenomenon appears to result from a thermal95Chapter 7. Summary and future experimentsrunaway effect due to phase segregation and dissipation hotspots; it wasobserved only in (correlated) RIQH states, not fractional or integer states,pointing to their qualitatively distinct thermodynamic properties. This ex-periment shows the danger in interpreting macroscopic measurements at amicroscopic level, especially where electronic phase transitions are sharp.On the other hand, it demonstrates the power of using breakdown charac-teristics as a probe into the thermal properties of correlated electron states.Looking ahead, it would be particularly interesting to investigate similarbreakdown phenomenology in combined Corbino/hall bar geometries, wherehotspot-induced breakdown could be included or avoided as desired. Sim-ilarly, adding multiple small contacts within the interior of a sample couldenable melted and frozen phases to be probed separately and the nature ofthe transition region to be measured directly.Surprisingly, the electron temperature under high bias was shown to beunexpectedly low after RIQHE breakdown. This means that high currentscould be driven through the sample without significant heating, suggestingthe formation of a correlated non-insulating state after breakdown. Thesecond part of this thesis (Ch. 5) was devoted to the investigation of con-ducting high bias states, arising after breakdown of the normal insulatingreentrant states. By analyzing the distribution of electrical potential alongthe sample’s edge, it was experimentally demonstrated that non-insulatingcorrelated bulk electron states exist even after the conventional RIQHE iscompletely destroyed in the entire sample. This shows that investigationof the non-equilibrium states might help not only to access the thermody-namic properties of the system, but to probe properties of the underlyingpinning potential. Furthermore, the theory predicts a wide variety of dy-namic phases formed in electron crystals under external force. This makesphase transitions between different correlated dynamic electron states a goodalternative for microscopic explanation of the cold non-insulating states un-der high bias. Although there are several reports from literature aboutobservation of electrical noise after the RIQHE breakdown, the data fromChapter 5 demonstrate a lot of small reproducible features in the electri-cal signal in this region. The signal results in correlated pattern of ripplesin DC bias/magnetic field coordinates. Since this state is cold, it is hardto connect such behaviour with a simple depinning of the electronic crys-tal. Therefore, driving high bias current through the sample might be anefficient instrument for probing dynamic electron solid states.Finally, the ground pinning potential of the anisotropic phases was in-vestigated by means of anisotropic stripe phase enhancement by currentbias (Ch. 6). This effect was used as a probe for the direction of the intrinsic96Chapter 7. Summary and future experimentsalignment potential at different filling factors. The data suggest that at highdensity samples the strong dependence of transport anisotropy on the fillingfactor is set by the change of stripe orientation, rather than the phase transi-tion between different CDW states with orthogonal anisotropy. Investigationof the high bias state anisotropy around filling factor ν = 9/2, supportedthe conclusion that the fundamental aligning potential in the center of thatstate does not depend on the number of the Landau level, even though thestate demonstrates orthogonal anisotropy in zero bias. This experimentalfact suggests, that the Coulomb interaction plays a crucial role in the stripealignment mechanism. For instance, instead of rotation of alignment poten-tial, stripe reorientation could be caused by the change of optimal directionfrom parallel to orthogonal orientation relative to the modulation potential.Eventually, I would like to suggest several experiments as a possible di-rection for future research. First of all, I would like to draw the reader’sattention to the zero bias anomaly appearing at very low current biases inmany non-integer filling factors all over the range of ν = 2−8. It was demon-strated in Ch. 6 that around the half filling of the Landau level this anomalyis very likely to result from metastability of the stripe phase. However, sim-ilar data were observed at the edges of the reenrant states at N=1 Landaulevels. Figure 7.1 shows an example of the zero bias peak at the edges ofan R2c reentrant state at several temperatures. Since those states are be-lieved to be isotropic and no reliable experimental evidence of anisotropictransport in RIQHE has been reported so far, it is hard to connect theseeffects to the metastability of a unidirectional CDW phase all over the en-tire region of filling factors. On the other hand, theory predicts instabilityof the CDW state with respect to the melting into smectic and nematicphases [77]. In [29] the authors hypothesize, that the isotropy of reentrantstates in macroscopic samples is given by the formation of the electron liq-uid crystal phase without long range order, rather than the bubble phase.In this case local alignment of the stripe domains along the Hall electricfield could lead to enhancement of the insulating state and, consequently,zero bias peak. The key feature of metastability is hysteretic behavior withrespect to the direction of the magnetic field sweep. Investigation of tem-perature dependence and hysteresis in the magnetic field could, therefore,help to favor one theoretical explanation over the other.Another interesting future experiment might be related to the break-down of reentrant state at higher Landau levels. Figure 7.2 shows a 2D mapof Rxy in the range of ν = 4 − 8. Reentrant states at N=2 Landau leveldemonstrate ordinary diamond-like collapse in bias (Fig. 7.2a). However,at N=3 (Fig. 7.2b) the diamonds show splitting. In other words, reentrant97Chapter 7. Summary and future experiments2.602.592.582.572.562.552.542.5360040020002.602.592.582.572.562.552.542.5360040020002.602.592.582.572.562.552.542.536004002000DC Current Bias (nA)a)b)c)nnnFigure 7.1: R˜xx bias dependence of R2c at different temperatures a) 24.8 mKb) 22.9 mK c) 21.2 mK. Colorscale is saturates at 500 Ω.98Chapter 7. Summary and future experimentsstates R6a and R7a, for instance, at a bias current of around 2.5 uA split intotwo quantized parts, while in the middle (ν ∼ 6.8 and ν ∼ 7.8 respectively)Rxy deviates from the quantized integer plateau. Such splitting was alwaysobserved only for N=3 Landau level in several samples and cooldowns andnever for N=2. However, the diamond splitting was not reproducible in ev-ery cooldown of the same sample. Although, such non-reproducibility couldbe attributed to the absence of complete control on the sample preparationprocess, more samples need to be tested in several cooldowns to obtain re-liable experimental evidence for such splitting. If this effect is real and notcaused by inhomogeneities in the sample, such splitting is very interesting.Effectively, this means that reentrant states corresponding to N=3 Landaulevels, in contrast to N=2 ones, are less stable at the center than on thesides. The microscopic description of reentrant states at higher Landau lev-els predicts a sequence of bubble phases with a different number of electronsper bubble within one reentrant state. Numerical calculations [19, 21] con-firmed the stabilization of a different number of bubble phases within R4,5and R6,7 reentrant states. Therefore, such splitting could be the first directexperimental observation of the different microscopic bubble phases.99Chapter 7. 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