the world of 2d electrons is exciting: enabling ‘ballistic high mobility transport’ modulation doping enabling ‘ballistic high mobility transport’ modulation doping easy electrostatic control close to surface easy electrostatic control close to surface unique excitations (quasiparticles) unique excitations (quasiparticles)

exchange statistics in the 2d world richer than the 3d world

exchange statistics in 3d fermions bosons both

fermionsbosons consequences…

exchange statistics in 2d  abelian (Laughlin qp’s) anyons

exchange statistics in 2d  non-abelian degenerate ground state exchange  unitary non-abelian anyons

basics of edge channels in IQHE doing physics with integer edge channels studies of edge transport in FQHE regime deviations from the ‘accepted’ picture Moty Heiblum QHE Regime Edge States in QHE Regime …. their nature & use

my 2d world high mobility 2DEG GaAs-AlGaAs

B V H V xx I L W X Lorentz electric field B R xx R xy classical Hall effect  xx =  xx =0

classical electron trajectories B most convenient picture…

quantizing the Hall effect

choice of gauges for H

m=0 harmonic oscillator EnEn n n degeneracy=B /  0 00 useful in interference experiments m = 1, 2, 3

useful in understanding edge modes do not carry energy / current harmonic oscillator

most convenient (another) gauge resembling classical orbits

= number of filled LL = number of electrons per flux quantum  0 =h/e EFEF energy NeNe e EFEF NeNe =2 summary

what started it… in a Si MOSFET nsns R xx = 0 R H = ( e 2 /h ) -1 B = const.

= number of electrons per flux quantum

R H =(e 2 /3h) -1 continued with… in a GaAs MODFET

= number of electrons per flux quantum

energy gap plateaus in  xy and minima in  xx  energy gap E (  0 ) = E (0) gauge invariance  E (  0 ) = E (0)  xy = e 2 /h  xx =0 Magnetic Field, B (Tesla)

EFEF NeNe localized does not contribute to current delocalized contributes to current e weak disorder disordered 2DEG  bulk picture of QHE plateau

gauge invariance, =1/3… =e / 3 ohmic contact J (r ) 00 r

state of the art QHE states

few basics of IQHE IQHE edge channels

e e B near the edge skipping orbits 2d layer

approaching the edge … approaching the edge  energy curves up finite drift velocity v d =  /B truncated harmonic oscillator

simplistic view B >>0 B=0

incompressible compressible density increasing reaching maximum density of LL EFEF electrons move from LL n + 1 to n gaining cyclotron (Zeeman) energy charge imbalance charging energy competing with cyclotron energy LL

proposed edge structure compressible strips separated by incompressible strips where does current flow? in the insulating strips? in the conducting strips? arguments continue until today… local measurements of current distributions still lacking

edge channels immune to back scattering 1d edge channel carries E f +eV EfEf EfEf

hot spot He4 bubble ballistic, but with energy dissipation

is the simplistic view of non-interacting 1d edge channels correct ?  inter-channel interaction  intra-channel interaction

inter-channel… = 2 g Q = 2 e 2 /h e2/he2/h e2/he2/h e/2 -e/2 neutral charge slow ‘neutral’ mode fast ‘charge’ mode LL 1 LL 2 non interacting  Landau levels e injecting electrons  >0  =0 with interactions  new basis Berg et al., PRL (2009)

electronic beam splitter Quantum Point Contact (QPC) V gate V source r F t QPC 0 < t < 1

preferential backscattering of edge channels reflected higher LLs transmitted lower LLs partitioned LL V gate

how to test ? injecting and detecting LL 1 LL 2 e inducing noise in LL 1 (hot mode) via QPC e/2 -e/2 neutral charge projecting via QPC to LL 2 (cold mode) v v injecting electrons into LL 1 looking for fluctuations in LL 2 with no net current

current fluctuations shot noise

hot filament cathode anode + - emitted electrons noisy current in vacuum tubes shot noise classical shot noise Schottky, 1918 it started with - noise in vacuum tubes

classical shot noise  large number of impinging electrons  very small escape probability time 0 current time S i (0)=2 e I spectral density (A 2 /Hz)

V applied ~h/eV applied zero temperature ordered electrons are noiseless ! shot noise =0 …. full Fermi sea ( non-partitioned electrons) Khlus, 1987 Lesovik, 1989

shot noise - single channel t <<1 poissonian S =2eI Schottky formula incoming transmitted binomial S =2eI (1-t ) t Khlus, 1987 Lesovik, 1989 spectral density of current fluctuations (A 2 /Hz)

conductance and shot noise in QPC conductance (g 0 ) gate voltage, V g noise current responsible for noise = V g 0 t i I t 1 =1 t 2 =1 2DEG QPC VgVg S  t (1-t )

shot noise in QPC - experimental results current noise, S ( A 2 /Hz) current, I (nA) T =57 mK t =0.37 I

experimental considerations 2DEG : n s =1.1x10 11 cm -2 ;  =4 x10 6 cm 2 /Vs shot noise signal S i (0)=2e * I= A 2 /Hz...…………… T*~ 40 mK Johnson noise……………………………………………. T ~ (10-30) mK noise in ‘warm electronics’……….....……………………T*~ 3.5 K “home made” (MODFET) cryogenic preamplifier (T= 4.2 K) T*~ mK at f 0 = MHz (above 1/f noise knee)

difficulties in measurements  QPC resistance R ~ 100 k    coax capacitance C ~ 60 pF f max = 1/(2  RC) ~ 30 kHz  1/f noise is large coaxial cable 60cm 60pF QPC  i) 2  V=R  i cooled preamp (hot, 4.2 K)(cold, 50 mK) spectrum analyzer R

experimental setup * frequency above 1/f noise corner of preamplifier; * capacitance compensated by resonant circuit; QPC cooled preamp L R C calibration signal spectrum analyzer f 0,  f 0 C<< 50  ; 300 K 1 G  warm preamp voltage gain = 1000 coax averaging time,  noise DC current V DC cryostat ‘home made’

injecting and detecting LL 1 LL 2 e inducing noise in LL 1 (hot mode) via QPC e/2 -e/2 neutral charge projecting via QPC to LL 2 (‘cold ‘ channel ) open QPC …………………………... no net current no low frequency noise partition QPC ………………………no net current low frequency noise semi-classically: mutual capacitance between the two edge modes high frequency (~GHz) noise in hot LL 1 is induced n LL 2 stochastic partitioning by QPC adds low-frequency spectrum

partitioning also the ‘cold channel’ G G GG T Inoue et al, PRL (2014)

T 2 dependence current of ‘cold mode’ T 1 =0.5 no net current T 1 dependence ~T 2 (1-T 2 )

intra-channel…

cold channel hot channel V D1 warm channel V D2  QD QD – narrow BPF

energy distribution (QD)  =1  =0 -

energy equilibration with distance e - e equilibration due to e - e interactions

simplistic view of integer edge channels fails interactions are dominant we make use of it…