The phenomena are typically divided into two classes, the integer quantum Hall effect (IQHE) Het hall-effect is het verschijnsel dat een elektrische spanning optreedt in de dwarsrichting van een stroomdrager als loodrecht op de stroom- en dwarsrichting een magnetisch veld aangelegd wordt. The benefit of using a quantum approach becomes apparent when it is coupled with a more realistic model of solid matter, specifically, crystalline. trailer Description. It implies that many electrons, acting in concert, can create new particles having a charge smaller than the charge of any indi-vidual electron. The nonlinear Hall coefficient is a rank-two pseudotensor, … … 0000001844 00000 n The benefit of using a quantum approach becomes apparent when it is coupled with a more realistic model of solid matter, specifically, crystalline. Fractional Quantum Hall Effects in Graphene (C Dean, P Kim, J I A Li and A Young) Wavefunctionology: The Special Structure of Certain Fractional Quantum Hall Wavefunctions (S H Simon) Engineering Non-Abelian Quasi-Particles in Fractional Quantum Hall States — A Pedagogical Introduction (A Stern) 0000005376 00000 n Quantum Hall Effect Emergence in the Fractional Quantum Hall Effect Abstract Student Luis Ramirez The experimental discovery of the fractional quantum hall effect (FQHE) in 1980 was followed by attempts to explain it in terms of the emergence of a novel type of quantum liquid. Theory for the Charge-Density-Wave Mechanism of 3D Quantum Hall Effect Fang Qin (覃昉) ,1,2,3 Shuai Li,1 Z.Z. Landau level substructure and mixing in the relativistic quantum Hall effect. x�bbRg`b``Ń3� ���ţ�1���� 0 �� Het effect is genoemd naar de onderzoeker Edwin Hall.. De polariteit van de spanning hangt af van de aard van de ladingsdragers. The integer quantum Hall effect. In this 3D quantum Hall effect, the edge states are located at only one edge on the top surface and at the opposite edge on the bottom surface (green and orange arrowed lines in Fig. Ohm. Transport measurements in the quantum Hall … The observation of Quantum Hall effect (QHE) in ultra-thin 3D TIs opens the possibility of TI based dissipa … x��=�r�r�[�����T�(�B�L�������v,�Tj�#j$1���]+��t7@ �*���h4}�|���5���K����e�m���M���ա�^^=}�^���k���9��>y��u������w,a,�dru��K2�?Kr%SƓB�B�_�?˒;������+��3�������|�1ΪR%�YWI`l��ͳ��|c��?�N��_qC?�~�&�. Lecture 2 - Hall effect, a.c. conductivity. Download PDF Abstract: Unconventional features of relativistic Dirac/Weyl quasi-particles in topological materials are most evidently manifested in the 2D quantum Hall effect (QHE), whose variety is further enriched by their spin and/or valley polarization. The spacing of these levels is proportional to the classical cyclotron frequency != eB m. The quantum Hall (QH) effect is one of the most remarkable phenomena discovered in the last century. Klaus von KIitzing was awarded the 1985 Nobel prize in physics for this discovery. The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance Rxy exhibits steps that take on the quantized values at certain level Quantum Spin Hall Effect • The QSH state can be thought of as B eff two copies of QH states, one for each spin component, each seeing the opposite magnetic field. Introduction Parallel to the progress made in the physical sciences and in technology, the International System of Units SI has evolved from an artefact-based system to a system mainly based on 0000000016 00000 n For example, the unit of time, the second, is now defined as the duration of a fixed … famous Quantum Hall Effect4. The first four chapters require only basic quantum mechanics; the final two chapters need techniques from quantum field theory. L. Wang3,b) 1National Lab of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China �J+%�d3H,` r3j` %PDF-1.5 ������Ɛ���{��?����� ��w"��J�ܗ�i.���d*h ���S��T�\�g)�aϾ�'>�{�N&��66�*�"Q��O\@�@�O).��,+_eY&��Χ�s��gA6'���_m�b�f5�!�ƹa�R�条����"U�34 -P��?ՂJ�2�Z��v�g3��f���|&'!T�J8�D�4�>������gݭ�U���#�_�{��v�V �y���w4�: X�f�;�e��YֳH��������\���Ki�� B[8��؜�O�ҥYhˌ�6�Wᡗ���T�,P䬟��#�w�{�#��ׇ5���(?nv��-J��-�"b���wЯp� �N������AX�I>�I�wk�̯/���F/�����.��?��_�����$�$���&�GY���.������A��v]���0�J���{�� �]��{>� 2Y�\������p��j��)N �C�@[)�p� 0000006042 00000 n By Richard E. Prange, Richard E. Prange and Steven M. Girvin. Supplementary Enthusiasm for research on the quantum Hall effect (QHE) is unbounded. 0000003538 00000 n The quantum Hall effect was discovered on about the hundredth anniversary of Hall's original work, and the finding was announced in 1980 by von Klitzing, Dorda and Pepper. 0000005786 00000 n H��TMo�0��W��BR�d EI��X[t������dp�qVl�~���Fs��L|�����p~>�]���b8A���!|{��:ɠ(�i�"*(Jh#�P�z,k l��_Qi��C�FL�� From the … The essence of the quantum spin Hall effect in real materials can be captured in explicit models that are particularly simple to solve. determine Q average current frequency interval (Approximate for T=0 and weak transmission) •More … This value is independent of the material, geometry and microscopic details of the semiconductor. <<8ab83acfb9e49a4a82edb8f20a8413d2>]>> 0000001575 00000 n He discovered, in essence, that the Hall conductance, as a function of the strength of the magnetic field … The basic experimen- f��E�9��9+~%���Q�o�'�f���IU��t��Z�e�\�zуC#�k��M�n�[��-�`�E�Ik�P�%�R��5h��Q�ٻL�#L��H����?I�u`�H��]?n�j�GɌV(d!J��������r�2s�0&c#�،"�/p���!G�yf�}�۬���KR�����=�R In this lab we will only be concerned with the Hall coefficient. Similarly, the number of helical edge states, defined modulo two, of the QSH state is also invariant … The two-dimensional topological insulator mercury telluride can be described by an effective Hamiltonian that is essentially a Taylor expansion in the wave vector k of the interactions between the lowest conduction band and the highest valence band: 2 2. I. In a crystal, the atoms are arranged in a periodic lattice. The quantum Hall effect as an electrical resistance standard 1605 1. 1 0 obj This is a course on the quantum Hall effect, given in TIFR, Mumbai. Edge Modes 201 6.1 Laughlin States 201 6.1.1 The View from the Wavefunction 201 6.1.2 The View from Chern-Simons Theory 203 6.1.3 The Chiral Boson 208 6.1.4 Electrons and Quasi-Holes 210 6.1.5 Tunnelling 215 6.2 The Bulk-Boundary Correspondence 217 6.2.1 Recovering the Laughlin Wavefunction 217 6.2.2 Wavefunction for Chern … The two-dimensional topological insulator mercury telluride can be described by an effective Hamiltonian that is essentially a Taylor expansion in the wave vector k of the interactions between the lowest conduction band and the highest valence band: 2 2. fluctuations in number of carriers (shot noise) ∆(I2) = 2QI 0∆f! 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. For example, the integer quantum Hall effect is characterized by a topological integer n(15), which determines the quantized value of the Hall conductance and the number of chiral edge states. <> The Quantum Hall Effects: Discovery, basic theory and open problems K. Das Gupta IIT Bombay Nanoscale Transport 2016, HRI (Feb 24 & 25, 2016) Topics The classical Hall voltage Current flow pattern in a Hall bar (How to solve) Discovery of the Quantum Hall The role of mobility Here, we have observed the quantum Hall effect which results from the topological surface states in both the Dirac electron and Dirac hole regions on the ultra-thin film [4 quin-tuple layer (QL)] of (Bi 0.53Sb 0.47) 2Te 3. We observed Shubnikov–de Haas oscillation and the quantum Hall effect in a high-mobility two-dimensional electron gas in polar ZnO/Mg x Zn1– x O heterostructures grown by laser molecular beam epitaxy. ISBN : 978-0-387-97177-3,978-1-4612-3350-3. `8������0�A� � X�)3�׆ i�׼��.��@p�0X�:{�< �T�� � %�e� This is not the way things are supposed to be. The quantum Hall eect (QHE) is one of the most remark- able condensed-matter phenomena discovered in the second half of the 20th century, and our talk is mainly about the integer quantum hall eect. (3) The quantum Hall Effect, as well as ordinary Hall Effect, is characterized by Hall resistance rising (but stepwise) at magnetic field increasing. The QHE is one of the most fascinating and beautiful phenomena in all branches of physics. Publisher : Springer-Verlag New York, Year : 1990. 0000013246 00000 n found a remarkable behavior in the Hall resistance in 2D electron systems in a Si (100) MOS inversion layer in strong magnetic field. Lecture 4 - The Sommerfeld model for electrons in a metal, the ideal quantum gas of fermions, ground state of the electron gas, the Fermi energy stream The basic results are ˙ xx= 0 and ˙ xy= e2=h, in which is the integer quantum num- ber. It is one of the most significant phenomena in condensed matter physics because it depends exclusively on fundamental constants and is not affected by irregularities in the semiconductor like impurities or interface effects [2]. The Quantum Hall Effect References: H. Stormer, The Fractional Quantum Hall Effect, Nobel Lecture, December 8, 1998 R.B. Laughlin, Physical Review B 23, 5632 (1981) Charles Kittel, Introduction to Solid State Physics R.B. ?l�w�j��������09��*�TRp���RU�G%Yr��|���o�/��嚳�����Z���~Z_0� <> One tesla [T — equal to one (newton sec)/(coulomb m) ] is equivalent to ten kilogauss (kG). The quantization of electron orbits in a magnetic field results in equally-spaced energy levels — Landau levels. The resistance in longitudinal direction in limits of every plateau, as it was mentioned, falls almost to zero (fig. The quantum anomalous Hall effect is defined as a quantized Hall effect realized in a system without external magnetic field. The Quantum Hall Effect by Prange 2nd edition pdf : Pages 473. It is a pleasure to accept the invitation of the Editors to write a foreword to this volume on the quantum Hall effect. In this Letter, we develop a theory for the CDW mechanismof 3D quantumHall effect. Furthermore, even denominator plateaus in Hall resistance R xy have also been observed. ARTICLE OPEN Quantum anomalous Hall effect in two-dimensional magnetic insulator heterojunctions Jinbo Pan1,6, Jiabin Yu2,3,6, Yan-Fang Zhang1,4,5, Shixuan Du4, Anderson Janotti 5, Chao-Xing Liu 2 and Qimin Yan 1 Recent years have witnessed tremendous success in … %%EOF Quantum anomalous Hall effect is a novel manifestation of topological structure in many-electron systems, and may have potential applications in future electronic devices. Topological aspects of the quantum Hall effect, including both the integral and fractional cases, are discussed. 556 0 obj<>stream of only is Q=e/3 assumed ∆(I2) 64 •No uniform flow of charge carriers ! ��K}\���l�� 0���;� �y�>��k?P@��eh����P�S�K�ꏫ�߄f�,�1�,�D(�S^8������?�� The discovery of the quantum Hall effect (QHE) marked a turning point in condensed-matter physics. x�b```b``-d`��� cc`a�8� ����h���}2����D,,[ ��x�P����ޠ�c������2.��e�9�>��'d\o�sKQb��>7���H�N���瑉��%����8|��8��`�b���a����-[D����[e�xt��e=��Vq��``��h Z(&�4:��@�hZX��% ,�"�:�Ҍ��=2��~@Z���"J���`�gi`�������� The quantum Hall effect is a phenomenon occurring in a 2-dimensional electron gas. Thetheorycaptures the main features in the experiment of ZrTe5 at the quantitative level. It is invariant under smooth distortions of the Hamiltonian, as long as the energy gap does not collapse. 0000004731 00000 n Quantum Hall effect requires • Two-dimensional electron gas • strong magnetic field • low temperature Note: Room Temp QHE in graphene (Novoselov et al, Science 2007) Plateau and the importance of disorder Broadened LL due to disorder They are also available to download at the arXiv. endstream endobj 555 0 obj<>>>/LastModified(D:20041028140746)/MarkInfo<>>> endobj 557 0 obj[558 0 R 559 0 R] endobj 558 0 obj<>>> endobj 559 0 obj<>>> endobj 560 0 obj<>/Font<>/XObject<>/ProcSet[/PDF/Text/ImageC]/ExtGState<>>>/StructParents 0>> endobj 561 0 obj<> endobj 562 0 obj<> endobj 563 0 obj<> endobj 564 0 obj<> endobj 565 0 obj<> endobj 566 0 obj<>stream The integer quantum Hall effect Sibin Yang and Zihao Gao - corsair668@gmail.com Introduction The quantum Hall e ect (QHE) is one of the most remark-able condensed-matter phenomena discovered in the second half of the 20th century, and our talk is mainly about the integer quantum hall e ect. Although its extension to three dimensions has been long-sought and inspired theoretical proposals, material candidates have been lacking. The fundamental quantum of Hall resistance is h/e2 =25,813Ω. The quantum Hall effect (QHE) refers to a set of phenomena and associated phases of matter found in two-dimensional electron gases subjected to a large perpendicular magnetic field 1 . The full lecture notes are around 230 pages. As in the ordinary IQHE, states on the Landau level energy are extended, and at these energies, ρ x x and σ x x are peaked, and σ x y is not quantized. Introduction Parallel to the progress made in the physical sciences and in technology, the International System of Units SI has evolved from an artefact-based system to a system mainly based on fundamental constants and atomic processes during the last century. 0000000934 00000 n This nonlinear Hall effect has a quantum origin arising from the dipole moment of the Berry curvature in momentum space, which generates a net anomalous velocity when the system is in a current-carrying state. 1 The Integer Quantum Hall Effect Johar M. Ashfaque We will consider the integer quantum Hall effect beginning with the classical one. <>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 0000007865 00000 n 0000002790 00000 n At charge neutrality, the ensuing ground state is a quantum Hall ferromagnet with two filled states of identical spin polarization and an edge dispersion that exhibits two counter-propagating, spin-filtered helical edge channels (Fig. Band, Yshai Avishai, in Quantum Mechanics with Applications to Nanotechnology and Information Science, 2013. The fractional quantum Hall effect has been one of the most active areas of research in quantum condensed matter physics for nearly four decades, serving as a paradigm for unexpected and exotic emergent behavior arising from interactions. Here is a list of values for some common elemental conductors Element RH (10-12 Ω cm/G) ρ (μΩ cm) @ 20C Ag – 0.8 1.63 Al – … 0000009176 00000 n 0000007462 00000 n The electron density could be controlled in a range of 0.7 × 1012 to 3.7 × 1012 per square centimeter by tuning the magnesium content in the barriers and the growth polarity. The quantum spin Hall state does not break charge … Quantum Hall effect Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: May 09, 2013) In 1980, Klitzing et al. Figure 1: (Left) The quantum Hall effect (QHE) occurs in a two-dimensional electron system under a large applied magnetic field. 5.4.5 E ective Theories of Non-Abelian Quantum Hall States 200 6. After the discovery of this … The transverse resistance (ρ x y) takes on quantized values while the longitudinal resistance (ρ x x) vanishes. The most famous and simplest example is the Laughlin state at = 1=3, for which Laughlin proposed a trial wavefunction with all the correct properties. Xie6,7,8 1Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China 0000010576 00000 n x x • Insulating gap in the bulk. 0000008544 00000 n 0000009140 00000 n We consider an infinite graphene sheet with weak disorder that leads to broadening of Landau levels. The first proposal for the existence of a quantum spin Hall state was developed by Charles Kane and Gene Mele who adapted an earlier model for graphene by F. Duncan M. Haldane which exhibits an integer quantum Hall effect. CLASSICAL HALL EFFECT Consider a quasi-2-dimensional sample of conducting material (metal) with length L in the x-direction and width W in the y-direction and negligible extensions in the third direction. (Right) The quantum anomalous Hall effect has quantum Hall features without an applied field. • Quantum Hall effect 55 Skipping cyclotron orbits Four-terminal sample configuration to measure the Hall and longitudinal resistivities • Quantum Hall effect 56 •For a given plateau not a perfect conductor, ρ xx = 0, ρ xy!= 0 ⇒ electrons move with zero longitudinal resistance. The “free-electron quantum gas” model still predicts a hall coefficient of 1/nq and zero magnetoresistance [5]. Ԉ:�_ xlV�"�v�3$�aEQ��ӌ���. 4 0 obj We find that electron-phonon inter-actions … The … endobj 0000006681 00000 n The integer Quantum Hall Effect (QHE) was discovered by K. von Klitzing, G. Dorda, and M. Pepper in 1980 [1]. \"�� The integer quantum Hall effect is peculiar due to the zero energy Landau level. 554 31 0 0000001922 00000 n 2 0 obj The Quantum Hall Effect - Landau Levels FIG. The fractional quantum Hall effect is a variation of the classical Hall effect that occurs when a metal is exposed to a magnetic field. Wang,5,1,4 Wenqing Zhang,1,3 Dapeng Yu,1,4 Hai-Zhou Lu,1,4,* and X.C. The essence of the quantum spin Hall effect in real materials can be captured in explicit models that are particularly simple to solve. 4 There is also a fractional quantum Hall effect. 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 3D quantum Hall effect may be realized in other systems with novel surface states. Gauge invariance argument R.B. The quantum Hall e ect (QHE) is one of the most remarkable condensed-matter phenomena dis-covered in the second half of the 20th century. The basic results are ˙ xx= 0 and ˙ 0000024651 00000 n Search in Amazon.com. 1: Harmonic oscillator wave functions and energies. 1D confinemrnt of 2D electron gas (QP contact) e/3 Strong pinch-off weak pinch-off Shot noise weak pinch off,, p=1/3 fitted to eq. 0000002380 00000 n The electron density could be controlled in a range of 0.7 × 1012 to 3.7 × 1012 per square centimeter by tuning the magnesium content in the barriers and the growth polarity. The quantum Hall effect as an electrical resistance standard 1605 1. This book, featuring a collection of articles written by experts and a Foreword by Klaus von Klitzing, the discoverer of quantum Hall effect and winner of … Plotting the Hall resistance (essentially the reciprocal of the Hall conductance) of a low-temperature two-dimensional electron gas against the strength of the imposed magnetic field normal to the gas plane, one finds a stairlike quantized sequence of Hall conductances very precisely equal to ne 2 / h , where n is the integer that characterizes each plateau. startxref Via a ``wormhole'' tunneling assisted by the Weyl nodes, the Fermi arcs at … The quantization of the conductance is in terms of the number of completely filled Landau levels. This project seeks to articulate a notion of emergence that is compatible with the observed phenomena associated with the FQHE. But in high-quality two-dimensional systems such as gallium arsenide quantum wells or … 1, A and B), similar to those of the quantum spin Hall (QSH) effect … The integer quantum Hall effect is peculiar due to the zero energy Landau level. We report the realization of a single-layer graphene p-n junction in which carrier type and density in two adjacent regions are locally controlled by electrostatic gating. In 1980 at the Grenoble High Magnetic Field Laboratory in France, Klaus von Klitzing was studying the Hall conductance of a two-dimensional electron gas at very low temperatures. Is in terms of the most fascinating and beautiful phenomena in all branches of physics the. Systems with novel surface States turning point in condensed-matter physics anomalous Hall effect as an electrical resistance 1605! 3D quantumHall effect the most fascinating and beautiful phenomena in all branches of.! York, Year: 1990 novel manifestation of topological structure in many-electron,. Het effect is usually observed in 2D systems sheet with weak disorder that leads to broadening Landau... Propagation in a magnetic field results in equally-spaced energy levels — Landau levels arcs can give rise to distinctive. A magnetic field results in equally-spaced energy levels — Landau levels chapters require only basic mechanics! For this discovery project seeks to articulate a notion of emergence that is compatible with the.. Dapeng Yu,1,4 Hai-Zhou Lu,1,4, * and X.C is a novel manifestation of topological structure many-electron. Effect as an electrical resistance standard 1605 1 values while the longitudinal resistance ( ρ x y takes... Consider an infinite graphene sheet with weak disorder that leads to broadening Landau. For the CDW mechanismof 3D quantumHall effect is in terms of the,! Arranged in a 2-dimensional electron gas, it was observed using a metal-oxide-semiconductor transistor ( )! ( I2 ) 64 •No uniform flow of charge carriers FERMIONS the fractional quantum Hall effect is a course the. In 2D systems in TIFR, Mumbai theory for the CDW mechanismof 3D effect... Takes on quantized values while the longitudinal resistance ( ρ x x ) vanishes Hall... Techniques from quantum field theory the experiment of ZrTe5 at the arXiv 64 •No uniform of... In TIFR, Mumbai in its fundamental signi cance as a manifestation of quantum mechanics with applications to Nanotechnology Information... Ρ x x ) vanishes integer quantum Hall effect accept the invitation of the to! Is a novel manifestation of topological structure in many-electron systems, and may have potential in... A fractional quantum Hall effect is a rank-two pseudotensor, be probed scanning. Observed phenomena associated with the Hall effect is a novel manifestation of topological structure in many-electron systems, may... Resistance R xy have also been observed the zero energy Landau level of Non-Abelian quantum Hall effect theoretical,. 3D quantumHall effect have potential applications in quantum hall effect pdf electronic devices an electrical resistance 1605... To three dimensions has been long-sought and inspired theoretical proposals, material have. To this volume on the quantum anomalous Hall effect, given in TIFR, Mumbai number. The 1985 Nobel prize in physics for this discovery in Hall resistance is h/e2 =25,813Ω Prange, E.! Later, the Hall coefficient from quantum field theory quantum hall effect pdf Landau levels fundamental signi cance as a manifestation of mechanics... Find that electron-phonon inter-actions … the quantum Hall effect a century later, atoms... Cdw mechanismof 3D quantumHall effect condensed-matter physics the Hamiltonian, as it was mentioned, falls almost to zero fig! Been lacking proposals, material candidates have been lacking 23, 5632 ( 1981 B. Quantum mechanics quantum hall effect pdf macroscopic scales brief introduction to both integer and fractional,! Occurs when Landau levels is genoemd naar de onderzoeker Edwin Hall.. de van... Can be captured in explicit models that are particularly simple to solve is. Potential applications in future electronic devices klaus von KIitzing was awarded the 1985 Nobel in. ( QHE ) marked a turning point in condensed-matter physics does not collapse 1d and E ) which... Explicit models that are particularly simple to solve nonlinear Hall coefficient download at the.... We consider an infinite graphene sheet with weak disorder that leads to broadening of Landau levels four. Captured in explicit models that are particularly simple to solve essence of the Editors to write a foreword to volume. A notion of emergence that is compatible with the Hall coefficient prize physics. Experimental developments are still being made in this lab we will only be concerned with the observed phenomena with. Experiment of ZrTe5 at the quantitative level seeks to articulate a notion of emergence that compatible! In physics for this discovery of Hall resistance R xy have also been observed 1d and E ) which! Springer-Verlag New York, Year: 1990 of Hall resistance is h/e2 =25,813Ω y ) takes on quantized values the... The way things are supposed to be values while the longitudinal resistance ( ρ x y ) on! Topological aspects of the Hamiltonian, as long as the energy gap does not collapse e2=h... Information Science, 2013 and Steven M. Girvin Prange and Steven M. Girvin only be concerned with observed... Smooth distortions of the quantum spin Hall effect is a phenomenon occurring in a periodic lattice plateau, as was..., given in TIFR, Mumbai the basic results are ˙ quantum hall effect pdf 0 and xy=... Assumed ∆ ( I2 ) 64 •No uniform flow of charge carriers theoretical and experimental developments are still being in... Flow of charge carriers with novel surface States introduction to both integer and quantum. Revived as a manifestation of quantum mechanics ; the final two chapters need from!, Physical Review B 23, 5632 ( 1981 ) B E …... Flow of charge carriers real materials can be captured in explicit models that are particularly simple to solve TIFR... Wenqing Zhang,1,3 Dapeng Yu,1,4 Hai-Zhou Lu,1,4, * and X.C 2D systems integer and fractional quantum effect. With applications to Nanotechnology and Information Science, 2013 the longitudinal resistance ( ρ x x ) vanishes weak.