the world of 2d electrons is exciting: enabling ‘ballistic high mobility transport’ modulation...
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the world of 2d electrons is exciting:
• enabling ‘ballistic high mobility transport’ modulation doping
• easy electrostatic control close to surface
• unique excitations (quasiparticles)
exchange statistics in the
2d world
richer than the 3d world
exchange statistics in 3d
fermionsbosons
both
fermionsbosons
1
1BE
kT
fe
1
1FD
kT
fe
consequences…
exchange statistics in 2d abelian (Laughlin qp’s)
anyons
ie 2ie
2 4i ie e
2 4 4 2i i i ie e e e
2ie
4ie
exchange statistics in 2d non-abelian
degenerate ground
statei i
i
a a
a a
U
exchange unitary
1 2 UU
1 2 2 1UU U U
non-abelian anyons1U
2U
• 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
Edge States in QHE Regime …. their nature & use
my 2d world
high mobility 2DEG
GaAs-AlGaAs
B
VH Vxx
I
L
WX
sxy en
BR
Lorentzelectric field
B
Rxx
Rxy
classical Hall effect
xx = xx =0
quantizing the Hall effect
choice of gauges for
H
A
m=0
harmonic oscillatorEn
n n
2
Bl eBdegeneracy=B /0
0
useful in interference experiments
m = 1, 2, 3
C
eB
mBn m r lL , , ,
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
EFc
energy
*cm
eB
Ne
e
EF
Ne
=2
)( 21 nE cn
2
Bl eB
summary
what started it… in a Si MOSFET
ns
Rxx= 0
RH = ( e 2/h )-1
B = const.
= number of electrons per flux quantum
RH=(e 2/3h)-1
continued with… in a GaAs MODFET
= number of electrons per flux quantum
plateaus in xy and minima in xx energy gap
gauge invariance E (0) = E (0)
xy= e 2/h
xx=0
Magnetic Field, B (Tesla)
/1m
eB*c EF
Ne c
localizeddoes not contribute to current
delocalizedcontributes to current
e
/h weak disorder
disordered 2DEG bulk picture of QHE
plateau
gauge invariance, =1/3…
1
2o
r t
trrJπtrIq Δ)(2=Δ)(= =e / 3
φφr εσrJ =)(
h
er
2
3
1
ohmic contact
J (r )
0
r
state of the art QHE states
few basics of
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
vd =/B
truncated harmonic oscillator
simplistic view
B >>0
B=0
incompressible
compressible
density increasing reachingmaximum density of LL
EF
electrons move from LL n+1 to ngaining 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 Vh
eI
2
Ef +eV
Ef
Ef
hot spot
He4 bubble
He4 bubble
ballistic, but with energy
dissipation
is the simplistic view of non-interacting1d edge channels correct ?
inter-channel interaction
intra-channel interaction
inter-channel…
=2
gQ= 2 e2/h
e2/he2/h
e/2
e/2
e/2
-e/2
neutral charge
slow ‘neutral’ mode fast ‘charge’ mode
LL1
LL2
non interacting Landau levels
e injecting electrons >0
=0
with interactions new basis
Berg et al., PRL (2009)
electronic beam splitterQuantum Point Contact (QPC)
Vgate
Vsource
r
lF
t
QPC0 < t < 1
preferential backscattering of edge channels
reflected higher LLs
transmitted lower LLs
partitioned LL
Vgate
how to test ? injecting and detecting
LL1
LL2
e
inducing noise in LL1 (hot mode) via QPC
e/2
e/2
e/2
-e/2
neutral charge
projecting via QPC to LL2
(cold mode)
v
v
injecting electrons into LL1 looking for fluctuations in LL2
with no net current
current fluctuations
shot noise
hotfilament
cathode anode+-
emitted electrons
noisy current in vacuum tubes
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
IDC
0
current
time
IDC
Si (0)=2 e I spectral density (A2/Hz)
Vapplied
~h/eVapplied
zero temperature ordered electrons are noiseless !
shot noise =0 …. full Fermi sea (non-partitioned electrons)
Khlus, 1987Lesovik, 1989
shot noise - single channel
t <<1poissonianS =2eISchottky formula
incoming transmitted
binomialS =2eI (1-t )
t
Khlus, 1987Lesovik, 1989
spectral density of current fluctuations*)i(
i eI)(S 2
(A2/Hz)
i
ii00Ti )t1(tgVe2)0(S
conductance and shot noise in QPC
con
du
ctan
ce (
g0)
gate voltage, Vg
3
2
1
noise
current responsiblefor noise = V g0 ti
It1=1
t2=1
2DEG
QPC
Vg
S t (1-t )
shot noise in QPC- experimental results -
0
2
4
6
0 1 2 3
curr
ent
nois
e,
S (1
0 -2
8 A2 /H
z)
current, I (nA)
T =57 mKt =0.37
I
eV
Tk2
Tk2
eVcoth)t1(eI2Tgk4)0(S B
BBi
experimental considerations
2DEG : ns=1.1x1011 cm-2 ; m=4 x106 cm2/Vs
04=)0( TGkS Bi
shot noise signal Si(0)=2e*I=10 -29 A2/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*~100-200 mK at f0 = 1 - 4 MHz (above 1/f noise knee)
difficulties in measurements
QPC resistance R ~ 100 k W
coax capacitance C ~ 60 pF
fmax = 1/(2 p RC) ~ 30 kHz
\ 1/f noise is large
coaxial cable
60cm 60pF
QPC( d i)2
dV=R di
cooledpreamp
(hot, 4.2 K)(cold, 50 mK)
spectrumanalyzer
R
experimental setup
* frequency above 1/f noise corner of preamplifier;* capacitance compensated by resonant circuit;
QPC
cooledpreamp
L RC
calibration signal
spectrumanalyzer
f0 ,f0
C<<
50 ; 300 K
1 G
warmpreamp
voltage gain = 1000
coax
averaging time,
noise <i2<DC current
VDC
cryostat
‘home made’
kHzRCπ
f,MHzLCπ
f 30≈2
1=Δ4→2≈
2
1= 00
injecting and detecting
LL1
LL2
e
inducing noise in LL1 (hot mode) via QPC
e/2
e/2
e/2
-e/2
neutral charge
projecting via QPC to LL2
(‘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 LL1 is induced n LL2
• stochastic partitioning by QPC adds low-frequency spectrum
partitioning also the ‘cold channel’
G
G
G G
T
Inoue et al, PRL (2014)
T2 dependence
current of ‘cold mode’
T1=0.5
no net current
T1 dependence~T2(1-T2)
intra-channel…
EfEfEf DDD 21 1
cold channel
hot channel
VD1
warm
cha
nnel
VD2
QD
QD – narrow BPF
EfEfEf DDD 21 1
energy distribution (QD)
=1
=0
VμVV DD 36=21
-
5.0
36
VVD
energy equilibration with distance
equilibration due to e - e interactions
simplistic view of integer edge channels fails
interactions are dominant
we make use of it…
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