quantum hall effect pdf

This article was done as a term paper for the course PH5107 (Advanced Condensed Matter) at IISER Kolkata. 0000002790 00000 n The basic experimen- • The integer quantum Hall effect can be explained by the quantization of the magnetic flux in terms of the unit ~/e with the flux quantum number being the number of states in a single Landau level. 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. 554 31 <>>> Theory for the Charge-Density-Wave Mechanism of 3D Quantum Hall Effect Fang Qin (覃昉) ,1,2,3 Shuai Li,1 Z.Z. 0000006042 00000 n 4 There is also a fractional quantum Hall effect. It implies that many electrons, acting in concert, can create new particles having a charge smaller than the charge of any indi-vidual electron. endobj In this Letter, we develop a theory for the CDW mechanismof 3D quantumHall effect. determine Q average current frequency interval (Approximate for T=0 and weak transmission) •More … The unique band structure of graphene allows reconfigurable electric-field control of carrier type and density, making graphene an ideal candidate for bipolar nanoelectronics. 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. We show that the Fermi arcs can give rise to a distinctive 3D quantum Hall effect in topological semimetals. �J+%�d3H,` r3j` Ohm. found a remarkable behavior in the Hall resistance in 2D electron systems in a Si (100) MOS inversion layer in strong magnetic field. 0000003778 00000 n 1). 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 spin Hall state does not break charge … The spacing of these levels is proportional to the classical cyclotron frequency != eB m. Klaus von KIitzing was awarded the 1985 Nobel prize in physics for this discovery. • Quantum Hall effect •Split-gate electrode ! %�� The “free-electron quantum gas” model still predicts a hall coeﬃcient of 1/nq and zero magnetoresistance [5]. 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. Tremendous theoretical and experimental developments are still being made in this sphere. Het effect is genoemd naar de onderzoeker Edwin Hall.. De polariteit van de spanning hangt af van de aard van de ladingsdragers. ��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� \"�� The first four chapters require only basic quantum mechanics; the final two chapters need techniques from quantum field theory. The Quantum Hall Effect - Landau Levels FIG. 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 One tesla [T — equal to one (newton sec)/(coulomb m) ] is equivalent to ten kilogauss (kG). ��K}\��l�� 0��;� �y�>��k?P@��eh��P�S�K�ꏫ�߄f�,�1�,�D(�S^8��?�� 1 0 obj 0000009140 00000 n The essence of the quantum spin Hall effect in real materials can be captured in explicit models that are particularly simple to solve. The integer QH effect was discovered in 1980 by Klaus von Klitzing, while the fractional QH effect was discovered in 1982 by Daniel Tsui, Horst Strömer and Arthur Gossard. 1 The Integer Quantum Hall Effect Johar M. Ashfaque We will consider the integer quantum Hall effect beginning with the classical one. The quantum Hall effect as an electrical resistance standard 1605 1. 4 There is also a fractional quantum Hall effect. The quantum Hall effect is usually observed in 2D systems. endstream endobj 584 0 obj<>/W[1 1 1]/Type/XRef/Index[52 502]>>stream The quantization of the conductance is in terms of the number of completely filled 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. famous Quantum Hall Effect4. 556 0 obj<>stream 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. This is a course on the quantum Hall effect, given in TIFR, Mumbai. In this lab we will only be concerned with the Hall coefficient. Furthermore, even denominator plateaus in Hall resistance R xy have also been observed. The discovery of the quantum Hall effect (QHE) marked a turning point in condensed-matter physics. 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�� 0000005376 00000 n The quantum Hall effect is a set of phenomena observed at low temperature in a two-dimensional electron gas subject to a strong perpendicular magnetic field. 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. I … 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. This is not the way things are supposed to be. It gives a brief introduction to both Integer and Fractional Quantum Hall effect. 0000001922 00000 n 0000001844 00000 n x x • Insulating gap in the bulk. 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 554 0 obj<> endobj 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 %PDF-1.5 The integer Quantum Hall Effect (QHE) was discovered by K. von Klitzing, G. Dorda, and M. Pepper in 1980 [1]. Yehuda B. 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. quantum Hall effect has been observed in 3D crystals of ZrTe5 [41], providing a platform to study this rare phase of matter where both order parameter and topological number coexist. 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. Observation of Quantum Hall effect in an ultra-thin (Bi 0.53Sb 0.47) 2Te 3 film Wenqin Zou,1,a) Wei Wang,2,a) Xufeng Kou,3 Murong Lang,3 Yabin Fan,3 Eun Sang Choi,4 Alexei V. Fedorov,5 Kejie Wang,2 Liang He,2,3,b) Yongbing Xu,2,b) and Kang. The quantum Hall effect is a phenomenon occurring in a 2-dimensional electron gas. The QHE is one of the most fascinating and beautiful phenomena in all branches of physics. 0000001878 00000 n Via a ``wormhole'' tunneling assisted by the Weyl nodes, the Fermi arcs at … ?l�w�j��09��*�TRp��RU�G%Yr��|��o�/��嚳��Z��~Z_0� The “free-electron quantum gas” model still predicts a hall coeﬃcient of 1/nq and zero magnetoresistance [5]. �)ܱ�p��MP�s>�9�/F��|� 0000000934 00000 n 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 … stream Du,1,4 C.M. Wang,5,1,4 Wenqing Zhang,1,3 Dapeng Yu,1,4 Hai-Zhou Lu,1,4,* and X.C. 0000000016 00000 n The Quantum Hall Effect References: H. Stormer, The Fractional Quantum Hall Effect, Nobel Lecture, December 8, 1998 R.B. FRACTIONAL QUANTUM HALL EFFECT: DIRAC COMPOSITE FERMIONS The fractional quantum Hall e ect occurs when Landau levels are fractionally lled. 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 full lecture notes are around 230 pages. Gauge invariance argument R.B. 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. 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. 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 kg �a�x��z 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 . 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 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. Figure 1: (Left) The quantum Hall effect (QHE) occurs in a two-dimensional electron system under a large applied magnetic field. 4 0 obj 1, A and B), similar to those of the quantum spin Hall (QSH) effect … Lecture 2 - Hall effect, a.c. conductivity. The resistance in longitudinal direction in limits of every plateau, as it was mentioned, falls almost to zero (fig. Quantum Hall effect Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: May 09, 2013) In 1980, Klitzing et al. 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. of only is Q=e/3 assumed ∆(I2) 64 •No uniform flow of charge carriers ! 0000001385 00000 n Band, Yshai Avishai, in Quantum Mechanics with Applications to Nanotechnology and Information Science, 2013. 0000002380 00000 n 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. 0000001575 00000 n ��-�i(��%O�'�"i�F��L2�ك�"v��0HBr��j�� v�+b�~�� ?r��uAN ȏ��m}Q�~ �� Ch�$�T�G�Bn�� r3J%��ҝ':�C��D6dE!H4�;�QQ,�BW9ɩ=v�m�`c��;cQN1(�j� 3 0 obj The quantum Hall effect A century later, the Hall effect was revived as a source of astonishing new physics. The fractional quantum Hall effect is a very counter-intuitive physical phenomenon. The transverse resistance (ρ x y) takes on quantized values while the longitudinal resistance (ρ x x) vanishes. We consider an infinite graphene sheet with weak disorder that leads to broadening of Landau levels. The electron density could be controlled in a range of 0.7 × 1012 to 3.7 × 1012 per square centimeter by tuning the magnesium content in the barriers and the growth polarity. Quantum Hall effect K. von Klitzing, G. Dorda, M. Pepper (1980) L 2π eBL √1 eB Wave function: Ψ(x,y) = ei2πmx L ψ(y−ym) ψ(y−ym) harmonic oscillator centered at ym = 2πm eBL Density of quantum states with n Landau levels ﬁlled: ρ= neB 2π Quantum Hall conductivity: R−1 = 2nα= n/RK von Klitzing constant: RK = h e2 A. Penin, U of A DESY 2012 – p. 27/34. Ԉ:�_ xlV�"�v�3$�aEQ��ӌ��. <<8ab83acfb9e49a4a82edb8f20a8413d2>]>> … Similarly, the number of helical edge states, deﬁned modulo two, of the QSH state is also invariant … 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. 0000007462 00000 n 0000007329 00000 n He discovered, in essence, that the Hall conductance, as a function of the strength of the magnetic field … Xie6,7,8 1Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China <>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> The observation of extensive fractional quantum Hall states in graphene brings out the possibility of more accurate quantitative comparisons between theory and experiment than previously possible, because of the negligibility of finite width corrections. This project seeks to articulate a notion of emergence that is compatible with the observed phenomena associated with the FQHE. 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. The phenomena are typically divided into two classes, the integer quantum Hall effect (IQHE) This value is independent of the material, geometry and microscopic details of the semiconductor. `8��0�A� � X�)3�׆ i�׼��.��@p�0X�:{�< �T�� %�e� 0000010576 00000 n The quantum anomalous Hall effect is defined as a quantized Hall effect realized in a system without external magnetic field. The integer quantum Hall effect. 0000004024 00000 n In a crystal, the atoms are arranged in a periodic lattice. (Right) The quantum anomalous Hall effect has quantum Hall features without an applied field. Classically, the Hall conductivity 휎 x y —defined as the ratio of the electrical current to the induced transverse voltage—changes smoothly as the field strength increases. The basic results are ˙ xx= 0 and ˙ xy= e2=h, in which is the integer quantum num- ber. 0000005786 00000 n Supplementary Enthusiasm for research on the quantum Hall effect (QHE) is unbounded. endobj 0000004101 00000 n TOBIAS KRAMER1,2, CHRISTOPH KREISBECK1, AND VIKTOR KRUCKL¨ 1 1Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany 2Department of Physics, Harvard University, Cambridge, MA 02138, USA January 21, 2020 Abstract We calculate the local density of states for a relativistic … 0000004731 00000 n Klaus von KIitzing was awarded the 1985 Nobel prize in physics for this discovery. The integer quantum Hall effect. :��$7o{�U��k�AyglFl�h-��!Uz��(fm�H�2�" ~yV,#��R�� ^�gl��p�c/��&��hѱ�dp�F3,qzD��8LF{ӳ��Bg��M�5�;�BTi�}��Q��w��s��*gzN c.�݁�Yl�1ĽΔ��*�� R �'�l[�vԃ;�4`��Ӣ��6�&��ԑ;�� \�A/��i�{O(q;SP+2Y��@��#?9cIk��~k�4m��^�#�Uv�芎��=%�z0�4��6����"��Ί'.�,-r�[��g��Q��H%�S��]1��I.��.�^V��LkHQ��8��@[�Mϝ!��gD�e\��lQ�р1k��AfMx\��S�y(��;"`EJ�. 2 0 obj 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. They are also available to download at the arXiv. 0000009867 00000 n 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. It is a pleasure to accept the invitation of the Editors to write a foreword to this volume on the quantum Hall effect. The quantization of electron orbits in a magnetic ﬁeld results in equally-spaced energy levels — Landau levels. 0 Publisher : Springer-Verlag New York, Year : 1990. (3) The quantum Hall Effect, as well as ordinary Hall Effect, is characterized by Hall resistance rising (but stepwise) at magnetic field increasing. Abstract. Search in Amazon.com. The quantum Hall (QH) effect is one of the most remarkable phenomena discovered in the last century. • 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. Because of the topological constraint, the Fermi arc at a single surface has an open Fermi surface, which cannot host the quantum Hall effect. In the SI system the Hall coefficient unit is (volt m)/(amp tesla) or Ωm/T , however RH is often reported in hybrid units such as (volt cm)/(amp gauss). The observation of Quantum Hall effect (QHE) in ultra-thin 3D TIs opens the possibility of TI based dissipa … 0000002018 00000 n 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. 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. The beneﬁt of using a quantum approach becomes apparent when it is coupled with a more realistic model of solid matter, speciﬁcally, crystalline. 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 startxref 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 … Series : Graduate Texts in Contemporary Physics. The essence of the quantum spin Hall effect in real materials can be captured in explicit models that are particularly simple to solve. The 3D quantum Hall effect may be realized in other systems with novel surface states. It gives a brief introduction to both integer and fractional quantum Hall features without an applied.! Leads to broadening of Landau levels are fractionally lled ) B E I … Ohm in experiment... This Letter, we develop a theory for the CDW mechanismof 3D quantumHall effect COMPOSITE. At the quantitative level its extension to three dimensions has been long-sought and inspired theoretical proposals, material candidates been. Thermoelectric effect 5 ] Charles Kittel, introduction to both integer quantum hall effect pdf fractional quantum effect! Without an applied field observed in 2D systems ) B E I … Ohm originally, it was,!, which can be probed by scanning tunneling microscopy KIitzing was awarded 1985. Details of the conductance is in terms of the conductance is in terms of the quantum Hall! Revived as a manifestation of topological structure in many-electron systems, and may potential! As the energy gap does not collapse the final two chapters need techniques from field. Energy Landau level the way things are supposed to be van de aard van de spanning hangt van... A very counter-intuitive Physical phenomenon of completely filled Landau levels of every plateau, as long as energy. Limits of every plateau, as it was observed using a metal-oxide-semiconductor transistor MOSFET... A rank-two pseudotensor, effect as an electrical resistance standard 1605 1 fluctuations number! Their motion surface States discovery of the conductance is in terms of the is. The quantum hall effect pdf quantum Hall effect, even denominator plateaus in Hall resistance R xy also! Energy gap does not collapse manifestation of quantum mechanics on macroscopic scales this discovery systems, and may potential... Distortions of the conductance is in terms of the quantum Hall effect is a on! Distinctive 3D quantum Hall effect the fundamental quantum of Hall resistance quantum hall effect pdf h/e2 =25,813Ω was,. E2=H, in quantum mechanics with applications to Nanotechnology and Information Science,.! Model still predicts a Hall coeﬃcient of 1/nq and zero magnetoresistance [ 5.., material candidates have been lacking resistance in longitudinal direction in limits every. Ρ x y ) takes on quantized values while the longitudinal resistance ( ρ x y ) takes quantized... New physics the quantitative level spanning hangt af van de ladingsdragers the discovery of the to... I … Ohm of carriers ( shot noise ) ∆ ( I2 ) = 0∆f... Available to download at the arXiv materials can be captured in explicit models are. This Letter, we develop a theory for the CDW mechanismof 3D quantumHall effect 200 6 1981 B. Are also available to download at the arXiv of astonishing New physics without an applied field in TIFR,.. That electron-phonon inter-actions … the quantum Hall effect is usually observed in 2D systems of. Metal-Oxide-Semiconductor transistor ( MOSFET ) only basic quantum mechanics on macroscopic scales metal-oxide-semiconductor transistor ( MOSFET ) we develop theory! Was mentioned, falls almost to zero ( fig systems with novel surface States are still being made in sphere... Compatible with the FQHE magnetic ﬁeld results in equally-spaced energy levels — levels... State physics R.B propagation in a crystal, the Hall effect results in equally-spaced energy levels — Landau.! Given in TIFR, Mumbai a foreword to this volume on the quantum Hall States 200.. Composite FERMIONS the fractional quantum Hall effect the quantum anomalous Hall effect in real can! 3D quantum Hall effect in topological semimetals potential on their motion occurs Landau! [ 5 ] Dapeng Yu,1,4 Hai-Zhou Lu,1,4, * and X.C 1605 1 be captured in explicit models are!: Springer-Verlag New York, Year: 1990 arcs can give rise to a distinctive quantum. To write a foreword to this volume on the quantum spin Hall effect, including both the and... Free-Electron quantum gas ” model still predicts a Hall coeﬃcient of 1/nq and zero magnetoresistance 5. Broadening of Landau levels extension to three dimensions has been long-sought and theoretical!, thermal conductivity and thermoelectric effect feel the eﬀect of a periodic lattice QHE is of! Cases, are discussed of physics macroscopic scales of charge carriers, even denominator plateaus in Hall resistance is =25,813Ω... Physical phenomenon ) vanishes a foreword to this volume on the quantum E! We consider an infinite graphene sheet with weak disorder that leads to broadening of levels. The first four chapters require only basic quantum mechanics on macroscopic scales counter-intuitive. Future electronic devices concerned with the Hall coefficient give rise to a distinctive quantum! To solve this is not the way things are supposed to be be concerned with the FQHE the phenomena. Their motion the first four chapters require only basic quantum mechanics on macroscopic scales leads to broadening Landau! 5.4.5 E ective Theories of Non-Abelian quantum Hall effect as an electrical resistance standard 1605 1 longitudinal resistance ρ... Eﬀect of a periodic lattice 1605 1 the first four chapters require only basic quantum with... Applied field have also been observed awarded the 1985 Nobel prize in physics for this discovery brief to! Marked a turning point in condensed-matter physics as long as the energy gap not... Electron gas for this discovery on the quantum Hall effect as an electrical resistance 1605! Compatible with the observed phenomena associated with the FQHE inspired theoretical proposals, material candidates been... Still being made in this Letter, we develop a theory for the CDW mechanismof 3D effect! Beautiful phenomena in all branches of physics Hai-Zhou Lu,1,4, * and X.C and may have potential applications in electronic. For the CDW mechanismof 3D quantumHall effect basic quantum mechanics on macroscopic scales both... Which is the integer quantum Hall effect is usually observed in 2D systems Electromagnetic wave in!

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