* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Quantum electrodynamics wikipedia, lookup

Particle in a box wikipedia, lookup

Aharonov–Bohm effect wikipedia, lookup

Scalar field theory wikipedia, lookup

Hidden variable theory wikipedia, lookup

Quantum group wikipedia, lookup

Quantum entanglement wikipedia, lookup

Electron configuration wikipedia, lookup

Topological quantum field theory wikipedia, lookup

Nitrogen-vacancy center wikipedia, lookup

Hydrogen atom wikipedia, lookup

History of quantum field theory wikipedia, lookup

Theoretical and experimental justification for the Schrödinger equation wikipedia, lookup

Quantum state wikipedia, lookup

Canonical quantization wikipedia, lookup

Bell's theorem wikipedia, lookup

Ferromagnetism wikipedia, lookup

Spin (physics) wikipedia, lookup

Transcript

Quantum Hall effect R V1 I V2 e2 I x = N Vy h Vx = 0 Gxy L e2 =N h n.b. h/e2 = 25 kOhms Quantization of Hall resistance is incredibly precise: good to 1 part in 1010 I believe. WHY?? Robustness • Why does disorder not mess it up more? • Answer: this is due to the macroscopic separation of protected chiral edge states • Two ingredients: • Scattering at a single edge does not affect it • Scattering between edges is exponentially suppressed Single edge • Scattering at one edge is ineffective because electrons are chiral: they cannot turn around! electron just picks up a phase shift In equations: H = ~vqx ! [ [email protected] + V (x)] (x) = e i ~v Rx 0 dx0 V (x0 ) iqx x e [email protected] =✏ ✏ = ~vqx Robustness • To backscatter, electron must transit across the sample to the opposite edge. How does this happen? • Think about bulk states with disorder dk = dt high field k e ⇥B+F m m k⇡ F⇥B e F= rU (r) describes center of mass drift of small cyclotron orbits - along equipotentials Robustness • Bulk orbits m k⇡ F⇥B e Note: near edge where force is normal to boundary, this just gives the edge state Edge states: smooth potent ! Robustness! In real samples, disorder is important, and • velocity equipotential line splits the degeneracy of the bulk LL states • TransitionBUT...it from N=3 to“back-scatter” N= 2 edge states cannot from one edge to another - “protected” edge state In the center of a Hall plateau, it looks like this Electrons need to quantum tunnel from one localized state to another to cross the sample and backscatter Robustness ! ! Edge states: potentialpotential 7 Edgesmooth states: smooth ! ! ! Edge states: smooth potential 7 Edge states: smooth potential • line velocityand equipotential line isvelocity In realequipotential samples, disorder important, from N=3 to N= 2 edge Transition from N=3states to N= 2 edge states ! ! splits Transition the degeneracy of the bulk LL states ! ! 7 ! ! ! • BUT...it cannot “back-scatter” from one equipotential line equipotential line velocity velocity edge to another - “protected” edge state from N=3states to N= 2 edge states Transition fromTransition N=3 to N= 2 edge somewhere here an edge state “peels off” from boundary and crosses the bulk Robustness • Quantum picture it turns out truly delocalized states occur only at one energy...this is NOT obvious consequently, IQHE “steps” in Hall conductance are expected to be infinitely sharp at T=0, for a large sample IQHE phases • Actually, the states with different integer quantum Hall conductivity are different phases of matter at T=0: they are sharply and qualitatively distinguished from one another by σxy • This means that to pass from one IQHE state to another requires a quantum phase transition: this corresponds to the point at which the edge state delocalizes and “percolates” through the bulk • However, unlike most phases of matter, IQHE states break no symmetry • They are distinguished not by symmetry but by “topology” (actually the Hall conductivity can be related to topology...a bit of a long story) Graphene IQHE What first experiments saw... Castro Neto et al.: The electronic properties of graphene zigzag: N=200, φ/φ0 1 energy / t 0.5 0 -0.5 -1 0 1 2 3 4 energy / t 0.4 0.2 µ 0 -0.2 -0.4 2 3 4 momentum ka FIG. 21. !Color online" electrons in graphene = #0 / 701, for a finite str panels are zoom-in ima represents the chemical Zhang et al, 2005 xy FIG. 20. !Color online" Quantum Hall effect in graphene as a function of charge-carrier concentration. The peak at n = 0 shows that in high magnetic fields there appears a Landau level at zero energy where no states exist in zero field. The field draws electronic states for this level from both conduction and valence bands. The dashed lines indicate plateaus in !xy described by Eq. !111". Adapted from Novoselov, Geim, Morozov, et al., 2005. Novoselov et al, 2005 e2 = ⇥ (±2, ±6, ±10, · · · ) h holding for a graphene stripe with a zigzag !z = 1" and armchair !z = −1" edges oriented along the x direction. Fourier transforming along the x direction gives H=−t # $ei"!#/#0"n$!1+z"/2%a!† !k,n"b!!k,n" *'!k"+ = # $%!k,n n,! Equations !114" and ! that the zigzag ribbo from n = 0 to n = N − edges are obtained fr E$,k%!k,0" = − teik Relativistic vs NR LLs • We expect σxy to change by ±e2/h for each filled Landau level • Each Landau level is 4-fold degenerate • 2 (spin) * 2 (K,K’) Consistent with observations, but zero is not fixed E 0 Fermi level is “in the middle” of 0th LL in undoped graphene Relativistic vs NR LLs • Guess: when EF=0, E 10 6 there are equal numbers of holes and electrons: σxy =0 2 0 Then we get the observed sequence -2 -6 -10 Fermi level is “in the middle” of 0th LL in undoped graphene tron edge states in graphene in the Quantum Hall eﬀect regime can carry both c We show that spin splitting of the zeroth Landau level gives rise to counterpropagat pposite spin polarization. These chiral spin modes lead to a rich variety of spin curr ding on the spin flip rate. A method to control the latter locally is proposed. W n spin splitting enhanced by exchange, and obtain a spin gap of a few hundred Ke Edge state picture 2 (a) 1 1 0 0 −1 −1 −2 −3 −4 2 Energy E/ε0 3 Energy E/ε0 Wewith willlow notcarrier workdensity this out system and here, but one can show recently realized in two-dimensional varying the a gate thatcarrier levelsdensity at thewith edge range of interesting states, in particubend as shown here quantum Hall eﬀect [2, 3] (QHE). In ll-known integer QHE in silicon MOSin graphene occurs at half-integer mulThe downward bending generacy due the to spin and naturally orbit. This of half levels half-integer QHE. The unusually large explains the observed ng makes QHE in graphene observable quantization, 100 K and higher. and why the filled valence statesQHE. do In the spin eﬀects in graphene eman splitting transport in to graphene not contribute σxy is nusual set of edge states which we shall ge states. These states are reminiscent −2 0 Distance y0/ lB −2 −4 FIG. 1: (a) Graphene energy spect the splitting of n != 0 levels shown is due boundary obtained from Dirac model to lifting of the spin/valley degeneracy, which we have not talked about. condition, Eq.(5), lifts the K, K ′ dege This leads to additional integer states at numbers of edge modes lead to the hal higher fields. split graphene edge states: the blue the spin up (spin down) states. Th Topological Insulators • So the IQHE states are examples of what we now call “Topological Insulators”: states which are distinguished by “protected” edge states • Until very recently, it was thought that this physics was restricted to high magnetic fields and 2 dimensions • But it turns out there are other TIs...even in zero field and in both 2d and 3d! Topological insulators • General understanding: insulators can have gaps that are “non-trivial”: electron wavefunctions of filled bands are “wound differently” than those of ordinary insulators “unwinding” of wavefunctions requires gapless edge states Topological insulators • 2005: Kane+Mele “Quantum spin Hall effect”: 2d materials with SOC can show protected edge states in zero magnetic field • 2007: QSHE - now called 2d “Z2” topological insulator - found experimentally in HgTe/CdTe quantum wells • 2007: Z2 topological insulators predicted in 3d materials • 2008: First experiments on Bi1-xSbx start wave of 3d TIs Since then there has been explosive growth QSHE To preser ve timereversal symmetry, there *must* be counterpropagating edge states, and no net spin (magnetization) • Edge states! “topological invariant” of QSHE is Z2, unlike for IQHE where the invariant is the Hall conductance (in units of e2/h), which is an integer. L,⇓ R⇑ 2d topology is “all or nothing” You can see this by looking at the edge states like IQHE but with counter-propagating edge states for opposite spin