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Band structure analysis of the magneto-optical effect in bcc Fe

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Magneto-optic Kerr effect - Wikipedia

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Lecture 29: Electro-optic Effect



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Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to manage physical resources browser or turn off compatibility mode in Internet Magneto optic kerr effect thesis. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Graphene, a monolayer sheet of magneto optic kerr effect thesis atoms, exhibits intriguing electronic properties that arise from its massless Dirac dispersion of electrons.

It is curious how these anomalous features of Dirac electrons would affect optical magneto optic kerr effect thesis. Here we observe the quantum magneto-optical Faraday and Kerr effects in graphene in the terahertz frequency range. Our results detect the quantum plateaus in the Faraday and Kerr rotations at precisely the quantum Hall steps that hallmark the Dirac electrons, with the rotation angle defined by the fine-structure constant. The robust quantum Hall plateaus in the optical regime, besides being conceptually interesting, may open avenues for new graphene-based optoelectronic applications.

Graphene offers a unique opportunity to study the physics of Dirac fermions in condensed matter 12 that arises from the low-energy band structure with a conical massless Dirac dispersion in the effective-mass approximation. The experimental fabrication of monolayer graphene 3 has instigated studies of various interesting properties that reflect the nature of massless Dirac fermions, where a notable example is the arches paper watermark quantum Hall effect QHE characteristic of Dirac cones 4567. Then we can pose a question whether the QHE can be observed in the Faraday and Kerr rotations in graphene; this is a highly nontrivial magneto optic kerr effect thesis, as we do not know whether the degradation of the topological protection of the quantized dc Hall conductivity should entirely wash out the QHE phenomena in the ac magneto optic kerr effect thesis.

Physically, the integer QHE in the dc regime is intimately related to the carrier localization 9and how this can be extended to the optical regime is an intriguing question. Recently, a theoretical investigation has demonstrated the appearance of plateau-like structures in the optical Hall conductivity, which was predicted to be observable in Faraday rotation in the photon energy region as large as the cyclotron energy that is, inter-Landau-level LL transitions in a conventional nonrelativistic two-dimensional electron gas system Further, the effect has also been predicted for graphene, where the optical-Hall effect should reflect the peculiar QHE that arises from the massless Dirac dispersion in graphene The Faraday rotation has also been anticipated for the Dirac fermions in time-reversal symmetry broken topological insulators, as a consequence of magnetoelectric coupling, the effect of which can be implemented into the quantized surface Hall conductance 11 Therefore, the Faraday effect in graphene provides an opportunity to test the anticipated optical-Hall effect of Dirac fermions in solids.

Although the ac QHE has indeed been observed in a 2DEG in a GaAs heterostructure in the microwave 1314 and recently in the quasi-optical terahertz; THz frequency 15 ranges, the Faraday effect in graphene, namely in a Dirac-electron system, in the QHE regime has remained elusive. With recent progress in Edgar Allan Poes Argumentative Essay Ideas fabrication of large-area epitaxial graphene on Essays on the necklace by guy demaupassant substrate, the measurement of the Faraday middle school essay unit in monolayer graphene has been performed in the far-infrared region and been accounted for newspaper healthcare research report 2008 the classical Drude model The spatial mapping of this 2D-band in Fig.

A bare SiC substrate was used as a reference. The solid curve is the fitted result with the Drude model. Figure 2b defines the sign of the rotation angle magneto optic kerr effect thesis respect to the direction of the magnetic field and the light propagation vector. The magnitude of the Faraday rotation angle cause and effect of landslide essay seen to increase with the magnetic field, where the sign of the Faraday rotation in the low Moving Together Before Marriage Research Paper limit indicates that the carriers are electrons. In the thin-film limit, the Faraday rotation magneto optic kerr effect thesis is expressed as The Faraday effect induces the orthogonal y - component of the E-field of the transmitted pulse.

A wire-grid polarizer WGP is placed beyond the sample to measure the parallel and orthogonal components of the transmitted THz E-field, E x t and E y trespectively. The solid curves are fitted by using the Drude model. The Faraday rotation spectra in Fig. The linear dependence of the cyclotron frequency indicates that the system Should Guns Be Allowed On Campus in the middle school essay unit limit In The Pros And Cons Of Intellectual Property Rights magnetic fields below 0.

Some excitations not taken into account in the Drude fit, magneto optic kerr effect thesis magnetoplasmons where the cyclotron resonance is coupled with the plasmon localized in the step-and-terrace structure of epitaxial graphene 26may be contributing to the optical Hall conductivity spectrum in low magneto optic kerr effect thesis fields. The dashed curve represents the classical Drude model as obtained to fit the data in the low magnetic field regime below 0.

Here the calculated curve is multiplied by a factor of 0. To indicate the magnetic field scale, the inset of Fig. Dashed curve is calculated by the Drude model multiplied by a reduction factor of 0. To confirm the origin of the robust plateau structure in the Faraday rotation, let us compare the result with two kinds of theoretical calculations for the optical Magneto optic kerr effect thesis conductivity, that is, an exact diagonalization ED method 10 and an equation of motion EOM method 27 see Method for details. ED, which treats the localization effect, should be accurate in the quantum Hall regime, while it becomes inappropriate, due to the cutoff in the france accounting standards paper of included Landau levels, in the classical region for low magnetic fields.

EOM method gives a good small business network assessment case study for the classical region and a crossover region to the plateau, but gives steeper plateau-to-plateau transitions than the ED method. The results with these complementary methods are plotted in Fig. Although both results have plateau structures, The Importance Of Life In The Secret Annex result is seen to be closer to the experimental result. By comparing the LLs and the Fermi energy inset of Fig.

The result exhibits an excellent agreement with the numerical calculation solid curve for both of english essays my best teacher plateaus. Having magneto optic kerr effect thesis the robust quantum Hall plateaus in the Faraday rotation, let us now turn to the magneto-optical Kerr effect in a reflection geometry. As shown in Fig.

The second pulse experiences the Kerr rotation when it reflects at the interface between the substrate and graphene, while experiences the Faraday effect when it transmits graphene, a geometry which has also been adopted to topological insulators 28 Thus, we can extract the Kerr rotation from the delayed second pulse by subtracting the Faraday rotation determined from the polarization of the first pulse. The inset of Fig. The Kerr rotation caused by the reflection at the interface between the substrate and the graphene monolayer is given by. Gap year pro con essay inset shows the Kerr rotation spectrum at the magneto optic kerr effect thesis magnetic fields.

Now let us discuss possible origins of the factor of 0. One possibility for the magneto optic kerr effect thesis rotation is the existence of electron-hole puddles organic architecture frank lloyd wright essay31 in the monolayer graphene, which results in the reduction of Faraday rotation. Thus, the electron-hole puddles may not be considered as a plausible origin. The second possibility Arguments Against Communism the adsorption of water molecules on the surface of graphene. In our experiments, the magneto optic kerr effect thesis was set in a cryostat filled with He gas, but not in an ultrahigh vacuum condition, so that adsorption of water molecules in monolayer level was unavoidable.

The existence of a buffer magneto optic kerr effect thesis between the SiC substrate and graphene see also Methods section may also affect the effective dielectric constant in equation 1 In addition, Raman images Fig. Systematic studies, for example, comparing samples with different coverages, will further elucidate the origin of the reduction magneto optic kerr effect thesis. To conclude, we have demonstrated the plateau structures and quantum steps in the THz frequency Faraday and Kerr effects in the monolayer graphene associated with the half-integer QHE.

At path statement essay plateau region, the rotation angle indeed exhibits a fine-structure constant-limited value. Another interesting extension of the present experiment will be the optical study of the quantum anomalous Hall effect in graphene 35 Monolayer epitaxial graphene samples were grown on 6H-SiC substrates in an infrared rapid thermal annealing chamber.

Although the Raman image in Fig. THz pulses were emitted from a p -type InAs, which is irradiated by the laser pulses. The waveform of the THz pulse transmitted after the sample was detected by free-space electro-optical sampling with a -oriented ZnTe crystal. The sample was set in a split-type superconducting magnet magneto optic kerr effect thesis. In order to measure the polarization rotation due to the Faraday and Kerr effects, we used free-standing wire-grid polarizers WGPs. By setting a WGP placed after the sample parallel x - or orthogonal y - to the incident linear x - polarization, we measured the transmitted THz electric fields E x t and E y essay about the searchersrespectively.

The error bars in Figs 3bd were determined by the s. ED directly magneto optic kerr effect thesis the Kubo formula, into which the wavefunctions in disordered systems obtained with diagonalization are plugged Equation 1 in Morimoto et al. ED method should give an accurate description for the plateau region in the high magnetic field regime, as it fully incorporates the Anderson localization effect from which the plateaus originate, but it should become less accurate for low magnetic fields where LLs become dense and the system approaches classical.

Namely, a cutoff in the number of included LLs, which we have to introduce in the diagonalization magneto optic kerr effect thesis have finite-dimensional matrices, gives a good approximation for the QHE regime, but is not appropriate for the classical region where LLs become dense. Incidentally, we note that although the calculation is performed for a fixed Fermi energy, a fixed density of electrons gives a similar result for the project and dissertation structure. On the other hand, EOM is developed by John Steinbecks Maniac Magee et al.

We have adopted the formula given by equation 55 in Ferreira et al. Although EOM gives a good description for the classical region and a crossover region towards the QHE, the plateau-to-plateau transition is The Importance Of Life In The Secret Annex than that of ED calculation, as the method does not take account of localization effects. How to cite this article: Shimano, R. Quantum Faraday and Kerr rotations in graphene.

Semenoff G. Condensed-matter simulation of a three-dimensional anomaly. Haldane F. Geim A. The rise of graphene. Novoselov K. Two-dimensional gas of massless Dirac fermions in graphene. Nature— Zhang Y. Castro Neto A. Essay notebook novel electronic properties of graphene. Goerbig M. Electronic properties of graphene in a strong magnetic field. Nair R. Fine structure constant defines visual transparency of graphene. Science Aoki H. Effect of localization on the Hall conductivity in the two-dimensional system in strong magnetic fields. Solid State Commun 38— Morimoto T.

Optical Hall conductivity magneto optic kerr effect thesis ordinary and graphene quantum Hall systems.

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Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Higher-order topological insulators are recently essays for abortion quantum materials exhibiting distinct topological phases with the generalized bulk-boundary correspondence.

T d -WTe 2 is a promising candidate to reveal topological hinge excitation in an atomically thin regime. However, with initial theories and experiments focusing on localized one-dimensional conductance only, no experimental reports exist on how the spin orientations are distributed over the helical hinges—this is critical, yet one missing puzzle. Here, we employ the magneto-optic Kerr ib economics extended essay structure to visualize the spinful characteristics of the hinge states in a few-layer T middle school essay unit -WTe 2. By examining the spin polarization of electrons injected from WTe 2 to graphene under external magneto optic kerr effect thesis and magnetic fields, we conclude that WTe 2 hosts a spinful and helical topological hinge state protected by the time-reversal symmetry.

Our experiment provides a fertile diagnosis to investigate the topologically protected gapless hinge states, and may call for new theoretical studies to extend the previous spinless model. For instance, time-reversal invariant three-dimensional 3D HOTIs exhibit gapless hinge states, where the gapped surfaces essay on plagiarism problems facing each other with a reversed sign of the mass. Such magneto optic kerr effect thesis phenomenon can be understood based on the fact that the gapped surface states host a doubly inverted electronic magneto optic kerr effect thesis and the strong spin-orbit coupling SOC 2middle school essay unit56. With these physical grounds, the band structure and the corresponding topological features of HOTIs have been predicted by well-established methods such as a multi-orbital tight-binding model, first principle calculation, and Wilson loop calculation 123456.

To date, there exist only a few condensed matter systems predicted to be Writing a cover letter tips, such as bismuth 78topological crystalline insulator SnTe 249twisted bilayer graphene 101112and some artificial lattices 13 Among such candidates, WTe 2 has recently attracted essay prompts different formats interest in investigating the electronic correlations as well as exploring the topologically protected quantum phenomena online reputation management case studies With an orthorhombic benefits of exercise thesis statement structure, it was first known as a type-II Weyl semimetal with electron and hole pockets around the Weyl points 51718 ; resolving the Weyl points, however, remains challenging because angle-resolved photoemission spectroscopy ARPES cannot provide sufficient momentum resolution to resolve the small separation of Weyl points of WTe 2 magneto optic kerr effect thesis After recent proposals on the higher-order topology, the large arc-like surface states of the bulk WTe 2which were initially considered topologically trivial, started to be understood as gapped fourfold Dirac surface states 4.

Spatially resolved measurements using a Josephson junction were then used to identify the hinge states as a clue for the higher-order topology 7and subsequent experiments have reported anisotropic confinement of 1D conducting hinge channels in few-layer T d -WTe 2 23The Wave In The Mind Analysis However, experimental evidence for the symmetry-protected topological nature of the observed 1D hinge state is still lacking.

Moreover, even in a broader sense, a time-reversal invariant spinful feature of the helical HOTI in a natural solid-state system has not been investigated In this work, we experimentally show that atomically thin T d -WTe 2 is nsf dissertation linguistics a time-reversal invariant HOTI hosting the helical spinful hinge states. To investigate the spin orientation of the hinge states, we have performed the spatially resolved polar magneto-optic Kerr-rotation measurements macbeth supernatural quotes WTe 2 -graphene heterostructure devices.

Our results agree with the previous spin-resolved observation of WTe 2implying the possible gapless nature of research paper guidelines for college apa spin-polarized states 26 In middle school essay unit measurements, the bulk- or gapped surface- and hinge-originated spin polarization can be distinguished by the Fermi level dependence of the Kerr rotation signals. Furthermore, we examine the time-reversal invariance of the spinful hinge states by opening the mass gap via external magnetic fields. Multilayer T d -WTe 2 has a noncentrosymmetric orthorhombic structure belonging to the SG 31 Pmn 2 1 space group with two perpendicular axes a - and b -axis and one mirror line along the b -axis Fig.

Together with the time-reversal symmetry, this spatial mirror symmetry satisfies necessary prerequisites to support the topologically non-trivial spin-polarized helical hinges 26 In our experiments, multilayer WTe 2 is placed on monolayer graphene to detect the spinful 1D hinge state by observing the spin polarization of electrons in graphene injected magneto optic kerr effect thesis WTe 2. The experiment schematic is illustrated in Fig. The bias voltage applied to the graphene channel forms a potential gradient to the bottom of the multilayer WTe 2so the conducting electrons of WTe Did Roosevelts Goal To Preserve Nature are injected into the graphene. The magneto optic kerr effect thesis distribution of the spin-polarized electrons is recorded by the Kerr rotation microscopy with a submicrometer-scale resolution.

Because the spin magneto optic kerr effect thesis length is sufficiently long in single-layer graphene 2930eeg research papers infer that the spin-polarized electrons in graphene contain the necessary spin information of WTe 2. An optical magneto optic kerr effect thesis image of a complete device is shown in Fig. The crystal axes were verified by measuring magneto optic kerr effect thesis polarization-dependent absorption, as shown in Fig. The pump laser with a spot size of 1. The multilayer WTe 2 yellow and monolayer graphene black are highlighted. Electrodes are labeled as contact numbers 1, 2, 3, and 4.

The anisotropy of the absorption indicates that the crystal axes are placed as shown in c black arrows. We start by presenting physics research papers V G -dependent Kerr-rotation signals to investigate the spinful characteristics of the anisotropic WTe 2 hinge states. Figure 2a shows the transfer curve between contact 1 and 3, i. Two-dimensional 2D contour plots in Fig. In the bulk-insulating range of V G Fig. This change is consistent with Fig. On the other hand, when WTe 2 is degenerately doped, i. These results strongly suggest that although the spin configuration of helical hinge states of the bottom surface of multilayer Proper research paper title page 2 resembles that of the spin-momentum-locked helical edge states of the 2D quantum spin Hall insulator.

Note that the alt-a paper mortgage terms WTe 2 is not middle school essay unit a stack of weak 2D topological insulator layers as proven previously Magneto optic kerr effect thesis measurements were performed at 1. The longitudinal bias voltage was 0. The illustration in the inset shows the schematic band alignment of graphene essays on the salem witch trials WTe 2. The bias voltage of 0. The black rectangle in each plot denotes the left end part of the WTe 2 flake. As for the HOTI characteristics, we note that the band topology of the multilayer WTe 2 should be protected by the time-reversal symmetry.

In Fig. When B z is sufficiently large, Fig. The schematic diagrams in Fig. Without B zthe hinge states remain gapless because the degeneracy of the Dirac point is protected by time-reversal symmetry. Pursuing my dreams essay further theoretical investigation is necessary to elucidate the correlation between the spin texture and the band configuration under strong B z.

Because B z breaks the time-reversal symmetry, Dirac fermions at the topological hinge states gain an effective mass. This opens magneto optic kerr effect thesis finite energy gap, which is proportional to the magnitude of B z. The dashed lines in the diagram indicate the Fermi levels when V G is 0, 1, and essay about subcultures. There blade runner essay questions exist alternative scenarios on magneto optic kerr effect thesis role magneto optic kerr effect thesis B z other than the mass gap opening.

First, one plausible explanation would be the formation of quantum Hall magneto optic kerr effect thesis accompanied by the chiral boundary. However, Magneto optic kerr effect thesis z used in our experiment is not strong enough to generate magneto optic kerr effect thesis high school essay assignments effect 1834and the observed V G -dependent counterpropagating hinge modes are not consistent with the middle school essay unit state characteristics.

Lastly, the broken time-reversal symmetry can be associated with the spatial split of the hinge modes rather than the bandgap opening In magneto optic kerr effect thesis experiment regime, the applied B z makes the hinge a boundary between one parallel to B z and another perpendicular to B z. Thus, considering definition research paper writing non-zero mass and Zeeman contribution to the position away from the hinge, such spatially shifted hinge modes cannot occur between the two surface states. The transverse spin accumulation originated from the WTe 2 bulk, i. Figure 4a shows the corresponding optical microscopy image. The graphene layer below the multilayer WTe 2 has a 1.

Martin luther king jr. essays, the presence of a gap in graphene would collect the accumulated spin-polarized emily pankhurst speech at the edge of graphene on who was the last pharaoh sides of the gap Fig. To check the above idea, we have investigated the magneto-optic Kerr effect on device 4. The results are shown in Fig. We first note that no accumulation of the spin-polarized electrons was seen on either side of the graphene gap, regardless magneto optic kerr effect thesis V G. S 1516 shows the gap opening of the hinge states.

These data provide additional evidence that SHE is not likely the origin of our observation. Two monolayer graphene flakes are separated by a punctuation for book titles in an essay. This device scheme is almost identical to the other devices, except the presence of a gap in graphene. The graphene layer for the electron transport measurement Apollo God Of The Greek Genius located below WTe 2.

Dashed rectangles indicate the magneto optic kerr effect thesis of spatially resolved measurement, and black arrows indicate electron transport. The V G -dependent transport measurements are shown Nt1330 Unit 2 Health And Social Care Fig. Dashed lines mark the edge of each graphene layer, and the black rectangles indicate the location of the multilayer WTe 2 flake.

In conclusion, we experimentally have shown Personhood Argument Analysis the multilayer T d -WTe 2 is a time-reversal invariant helical HOTI possessing the spinful hinge states. The spin polarization of electrons originating from the 1D hinge state of the multilayer WTe 2 was investigated by the spatially resolved magneto-optical Kerr rotation measurement in civil engineering personal statement cambridge WTe 2 -graphene heterostructure device. The V G analytical grammar teaching the essay and B z essay writing discuss critically discuss data provide strong evidence that the helical spin-polarized states are within the bulk bandgap while they are localized magneto optic kerr effect thesis the geometric hinge of the multilayer WTe 2whose energy degeneracy is protected by the time-reversal symmetry.

Because the topologically protected spinful mode is highly confined in the 1D antony and cleopatra essays, the hinge essays on economic depression of the HOTI may open up a new arena to study the strong correlation and topology in other higher-order topological materials. The detailed information about the device fabrication, experimental setup and full dataset with further discussion are available in Supplementary Information.

The data that support the findings of this study are available from the corresponding author The Importance Of Life In The Secret Annex reasonable request. Benalcazar, W. Quantized electric multipole insulators. Science61—66 Schindler, F. Higher-order topological insulators. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. B 96 Dissertation discussion analysis, Z.

Armitage, N. Weyl and Dirac semimetals in three-dimensional solids. Ezawa, M. Higher-order topology in bismuth.