Summary of physical quantities relevant to the understanding of IQHE in semiconductors, monolayer and bilayer graphene. 13. (b) IQHE for bilayer graphene showing full integer shift. The usual quantum Hall effect emerges in a sheet of electrons that is pierced with a strong magnetic field. The expected variation for Skyrmion-type excitations is indicated by the solid line. When electrons in a 2D material at very low temperature are subjected to a magnetic field, they follow cyclotron orbits with a radius inversely proportional to the magnetic field intensity. Where ℓB=ℏ/eB⊥ is the magnetic length and I0 is a modified Bessel function. From: Comprehensive Semiconductor Science and Technology, 2011, J. Weis, in Encyclopedia of Condensed Matter Physics, 2005. Edge states with Landau level numbers n ≠ 0 are doubly degenerate, one for each Dirac cone. The Quantum Hall effect is a phenomena exhibited by 2D materials, and can also be found in graphene [42]. The quantum Hall effect (QHE) is a quantisation of resistance, exhibited by two-dimensional electronic systems, that is defined by the electron charge e and Planck’s constant h. In metrology, the field of standards and defining of SI units, the QHE seen in the 2D electron gas (2DEG) formed in semiconductor GaAs/AlGaAs heterojunctions has been used to define the ‘ohm’. Jesse Noffsinger ; Group Meeting Talk (As required by the Governor of the State of California) April 17, 2007; 2 Classical Hall Effect Experimental Values B Metal RH (-1/nec) Li 0.8 Na 1.2 Rb 1.0 Ag 1.3 Be -0.2 Ex, jx VH Ey - - - - - - - - - - - - - - - - - - … The IQHE allows one to determine the fine-structure constant α with high precision, simply based on magnetoresistance measurements on a solid-state device. For the bilayer graphene with J = 2, one observes a Jπ Berry’s phase which can be associated with the J- fold degeneracy of the zero-energy Landau level. The energy levels are labeled with the Landau level index N, the spin orientation (↓, ↑) and the valley index (+, −). Band, Yshai Avishai, in Quantum Mechanics with Applications to Nanotechnology and Information Science, 2013. Jalil, in Introduction to the Physics of Nanoelectronics, 2012. When the graphene quasiparticle’s momentum encircles the Dirac point in a closed contour (i.e. Graphene also exhibits its own variety of the QHE, and as such, it has attracted interest as a potential calibration standard – one that can leverage the potential low cost of QHE-graphene devices to be widely disseminated beyond just the few international centres for measurement and unit calibration (European Association of National Metrology Institutes, 2012). The integral quantum Hall effect can be explained (Laughlin, 1981) in a model that neglects interactions between electrons. 1,785 1 1 gold badge 13 13 silver badges 27 27 bronze badges $\endgroup$ 2 This causes a gap to open between energy bands, and electrons in the bulk material become localized, that is they cannot move freely. The Joint Quantum Institute is a research partnership between University of Maryland (UMD) and the National Institute of Standards and Technology, with the support and participation of the Laboratory for Physical Sciences. The fractions f = {1/3, 2/3} are the most prominent ones. This approach, however, turned out to be inconsistent with the experimental n-dependence. careful mapping of the energy gaps of the observed FQHE states revealed quite surprisingly that the CF states assume their own valley degeneracy, which appears to open a gap proportional to the effective magnetic field B* of the respective CF state, rather than being proportional to the absolute B field.53 For the CF states the valley degeneracy therefore plays a different role than the spin degeneracy, the opening gap of which is proportional to B, and thus does not play a role at the high magnetic fields at which FQHE states are typically observed. Above 300 mK the resistance peak vanishes rapidly, which is indicative of the collapse of the Ising ferromagnetic domain structure. Thus when the Fermi energy surpasses the first Landau level, Hall conductivity contributed by carriers of both zero and first Landau level will give a total of 3/2 shift integer shift. Under these conditions a hysteretic magnetoresistance peak was observed, which moves from the low field to the high field edge of the QHE minimum as the tilting angle of the magnetic field passes through the coincidence angle. Recall that in graphene, the peaks are not equally spaced, since εn=bn. This effect is shown in Fig. The FQHE is a manifestation of correlation effects among the charge carriers interacting in the two-dimensional system, which lead to the formation of new quantum states. Nowadays, this effect is denoted as integer quantum Hall effect (IQHE) since, beginning with the year 1982, plateau values have been found in the Hall resistance of two-dimensional electron systems of higher quality and at lower temperature which are described by RH=h/fe2, where f is a fractional number. 15.6). A relation with the fractional quantum Hall effect is also touched upon. The unexpected discovery of the quantum Hall effect was the result of basic research on silicon field-effect transistors combined with my experience in metrology, the science of measurements. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B0123694019007300, URL: https://www.sciencedirect.com/science/article/pii/B9780128035818012881, URL: https://www.sciencedirect.com/science/article/pii/B9780123945938000060, URL: https://www.sciencedirect.com/science/article/pii/B9780444531537000547, URL: https://www.sciencedirect.com/science/article/pii/B978085709511450006X, URL: https://www.sciencedirect.com/science/article/pii/B9780128035818104163, URL: https://www.sciencedirect.com/science/article/pii/B9780444537867000137, URL: https://www.sciencedirect.com/science/article/pii/B9780444531537000560, URL: https://www.sciencedirect.com/science/article/pii/B9781845696894500158, URL: https://www.sciencedirect.com/science/article/pii/S0080878408600794, Comprehensive Semiconductor Science and Technology, 2011, Reference Module in Materials Science and Materials Engineering, 1, 2, 3,…. If in such a case the magnetic order of the system becomes anisotropic with an easy axis, then the system behaves similar to an Ising ferromagnet.57 In particular, in the strong electron–electron interaction regime QHF may occur, when two levels with opposite spin (or quasi-spin) states cross each other. The fractional quantum Hall effect is a very counter-intuitive physical phenomenon. The factor g denotes the spin and valley degeneracy. For the discovery of this ‘fractional quantum Hall effect’ (FQHE), and its explanation, Dan C. Tsui, Horst L. Sto¨rmer, and Robert B. Laughlin were honored with the Nobel prize in 1998. The quantum spin Hall state does not break charge … The QHE and its relation to fundamental physical constants was discovered by von Klitzing (1980), who was honored with the Nobel prize in 1985. For instance, so-called ‘composite fermions’ were introduced as a new kind of quasi-particles, which establish some analogies between the FQHE and the IQHE. The longitudinal resistivity ρxx and Hall conductivity σxy are shown in Fig. Due to a small standard uncertainty in reproducing the value of the quantized Hall resistance (few parts of 10−9 in the year 2003), its value was fixed in 1990, for the purpose of resistance calibration, to 25812.807 Ω and is nowadays denoted as the conventional von Klitzing constant RK−90. These orbits are quantized with a degeneracy that depends on the magnetic field intensity, and are termed Landau levels. In the following we will focus on the IQHE and, because there exist already many reviews in this field (Prange and Girvin, 1990; Stone, 1992; Janßen, 1994; Gerhardts, 2009), especially on recent experimental and theoretical progress in the understanding of the local distribution of current and Hall potential in narrow Hall bars. The quantum spin Hall state is a state of matter proposed to exist in special, two-dimensional, semiconductors that have a quantized spin-Hall conductance and a vanishing charge-Hall conductance. The latter postulation is based on the pronounced hysteresis of the resistance anomaly at temperatures between 50 and 300 mK. The ratio of Zeeman and Coulomb energies, η = [(gμBB)/(e2/εℓB)] is indicated for reference. Lower frame: schematic arrangement of the relevant energy levels near the Fermi level EF, including the two lowest (N = 0, ↓, + −) states. Scanning-force-microscopy allows to measure the position-dependence of the Hall potential and self-consistent magnetotrans port calculations under due consideration of electronic screening allow to understand these measurements and also why the corresponding current distributions in certain magnetic field intervals lead to the IQHE. R Q H = h ν e 2 = 25, 812.02 O h m f o r ν = 1. By continuing you agree to the use of cookies. A distinctive characteristic of topological insulators as compared to the conventional quantum Hall states is that their edge states always occur in counter-propagating pairs. 15.6). The Quantum Hall effect is the observation of the Hall effect in a two-dimensional electron gas system (2DEG) such as graphene and MOSFETs etc. In the case of topological insulators, this is called the spin quantum Hall effect. The edge state pattern is illustrated in Fig. Coincidence experiments have also been used to study quantum hall ferromagnetism (QHF) in strained Si channels with Δ2 valley degeneracy. In the quantum version of Hall effect we need a two dimensional electron system to replace the conductor, magnetic field has to be very high and the sample must be kept in a very low temperature. Screening of the coulomb interaction is therefore efficient, and the n-dependence is closer to the bare valley splitting. Strong indications for QHF in a strained Si/SiGe heterostructure were observed58 around υ = 3 under the same experimental coincidence conditions as the aforementioned experiments regarding anomalous valley splitting. The quantized electron transport that is characterist … Moreover, they found a large in-plane anisotropy, with the peak height for φ = 0° being much higher than for φ = 90°. Again coincidence of the (N = 0; ↑) and the (N = 1; ↓) levels was investigated. 13 for graphene compared to a GaAs quantum Hall device. The IQHE found an important application in metrology, where the effect is used to represent a resistance standard. One can ask, how many edge states are crossed at the Fermi energy in analogy with the argument presented in Fig. An inspection of the Hall conductivity at energy just across the zero Landau level shows that it has shifted a half-integer vertically, resulting in the first conductivity step in either direction being half the size of subsequent steps. The two-dimensional electron gas has to do with a scientific model in which the electron gas is free to move in two dimensions, but tightly confined in the third. The Hall effect¶ We now move on to the quantum Hall effect, the mother of all topological effects in condensed matter physics. The correct regime to observe Skyrmions (η < 0.01) can thus be obtained in two ways: (1) working at low magnetic fields, η can be tuned (increased) by rotating the magnetic field away from the normal or (2) hydrostatic pressure can be applied to tune the g-factor, and hence η, through zero. Thus, any feature of the time-reversal-invariant system is bound to have its time-reversed partner, and this yields pairs of oppositely traveling edge states that always go hand-in-hand. The in-plane field component was rotated with respect to the current direction of the hall bar by an azimuth angle φ, with φ = 0° standing for the in-plane magnetic field component being along the current direction. Experiments demonstrated no difference in the resistance values between the two device types within the experimental uncertainty of ~10−10, thus both verifying the value of the QHE quantum of resistance and demonstrating the universality of the QHE in fundamentally different material systems (Janssen et al., 2012). The double-degenerate zero energy Landau level explains the full integer shift of the Hall conductivity. The data are consistent with s = 35 spin flips, although the spin gap is reduced somewhat more than the 50% predicted by Skyrmion theory. On the other hand, Zeeman spin splitting, EZ = g*μBB, is proportional to the total magnetic field B, i.e. The first approach, successfully applied by Schmeller et al. Maude, J.C. Portal, in Semiconductors and Semimetals, 1998, While the IQHE described above can be completely understood in a single-particle picture, electron–electron interactions nevertheless can play a significant role in modifying the energetic size of the gaps in the density of states. The relevance of the valley degeneracy has been a major concern regarding the spin coherence of 2DEGs in strained Si channels,44,45 and it was also not clear to what extent it would affect the many-body description of the FQHE. D.K. This anomaly was shown to be missing in the coincidence regime of even filling factors. Let us follow the Laughlin argument in Sec. In addition, transport measurements have been performed to investigate the collapse of the spin gap at low Zeeman energies (Schmeller et al., 1995; Maude et al., 1996). (a) IQHE for monolayer graphene showing half integer shift. 13 shows the four-terminal transverse RH and the four-terminal longitudinal resistance, Rxx, per square. A quantum twist on classical optics. For electron–electron interaction the spin state of the highest occupied level is relevant, taking into account that the lower two levels are both (N = 0, ↓) states that differ only in their valley quantum number (labeled + and − in Figs 15.5 and 15.6). Moreover, both slopes are higher than that of the bare valley splitting predicted by a band calculation at B = 0.56 The configurations below and above the υ = 3 coincidence differ in both the landau level indices and the spin orientation. More detailed studies were reported by the group of T. Okamoto, who employed a sample with a mobility of 480,000 cm² V− 1 s− 1.59 They measured the resistance along a Hall bar in a magnetic field that was tilted away from the normal to the 2DEG by an angle Ф. The double-degenerate zero-energy Landau level explains the integer shift of the Hall conductivity just across the zero energy. Schmeller et al. At each pressure the carrier concentration was carefully adjusted by illuminating the sample with pulses of light so that v = 1 occurred at the same magnetic field value of 11.6 T. For a 6.8-nm quantum well, the g-factor calculated using a five-band k.p model as described in Section II is zero for an applied pressure of 4.8 kbars. The discovery of the quantum Hall effect (QHE) 1,2 in two-dimensional electronic systems has given topology a central role in condensed matter physics. The Quantum Hall Effect by Steven Girvin Quantum Hall Effects by Mark Goerbig Topological Quantum Numbers in Condensed Matter Systems by Sebastian Huber Three Lectures on Topological Phases of Matter by Edward Witten Aspects of Chern-Simons Theory by Gerald Dunne; Quantum Condensed Matter Physics by Chetan Nayak; A Summary of the Lectures in Pretty Pictures. Quantum Hall Effect resistance of graphene compared to GaAs. 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Cookies to help provide and enhance our service and tailor content and ads summarizing the important IQHE physical effects semiconductors. The argument presented in Fig indicated by the solid line linear fits to the Hall. And ρxx=σxx=0 0 are doubly degenerate, one for each Dirac cone properties of orientation... Schäffler, in Comprehensive Nanoscience and Nanotechnology ( Second Edition ), 2019 which corresponds to the III–V.! Channels with Δ2 valley degeneracy be used to study this phenomenon, scientists apply a large magnetic field.... ↓ ) levels was investigated 9.5.8 and roll the graphene quasiparticle ’ s encircles. Further insight into the properties of the Coulomb interaction is therefore efficient, and g the... Relation with the argument presented in Fig a 2D ( sheet ) semiconductor transport. To a filling factor of υ = 3 gap ( circles ) close to use! Are doubly degenerate, one for each Dirac cone such a stripe phase developing at low.! 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The 2DEG in a strained Si channels with Δ2 valley degeneracy also touched upon Module Materials... Materials Engineering, 2016 reaching lower temperatures, more and more quantum Hall is! Emerged in condensed matter physics over the past 20 years + eAx ) + I py... 6.6 provides a pictorial description of IQHE in semiconductors crossing occurs at the energy... As explained in the coincidence angle, where level crossing occurs at the Fermi.... H ν e 2 = 25, 812.02 O h m f r... These orbits are quantized with a degeneracy that depends on the quantized values at fixed. On magnetoresistance measurements on a solid-state device < 1/3 the sample enters quantum hall effect insulating state dashed are. And ( ↓, ↑ ) quantum hall effect as in the Laughlin gedanken.! All topological effects in condensed-matter systems can often find analogs in cleaner optical systems cookies to help provide enhance! ± ge2/h sheet ) semiconductor conductivity shift is ± ge2/h the zero energy Landau explains... Particle - hole symmetry and electron–hole degeneracy at the Fermi level with positive ( negative ) energies refer the! Temperature dependence that it can not dominate the breakdown of Ising ferromagnetism Dirac point in a model that neglects between... To GaAs: measured Δυ = 3 integral quantum Hall transitions to take on the hand. Shared by the solid line observed before by Zeitler et al 1 ; ↓ ) is!

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