world of 2d electrons is exciting modulation doping...diana mahalu m. heiblum unexpected...
TRANSCRIPT
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world of 2d electrons is exciting modulation doping
– high purity enables ‘ballistic transport’
– easy electrostatic control gates close to surface
– unique excitations (e.g., fractional statistics)
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‘exchange statistics’ in the
2d world
richer than the 3d world
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exchange statistics in 3d
fermions bosons
ψ ψ→ + ψ ψ→ −
both
ψ ψ→
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exchange statistics in 2d → abelian (Laughlin qp’s)
anyons
ψ ie θψ→ψ 2ie θψ→
ψ 2 4i ie eθ θψ→
2 4 4 2i i i ie e e eθ θ θ θ=
2ie θ
4ie θ
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exchange statistics in 2d → non-abelian
degenerate ground state
i ii
a aψ ψ ψ= = ⋅∑
aψ ψ= ⋅ ( )a ψ→ ⋅
U
exchange → unitary
ψ1 2ψ→U U
1 2 2 1U U U U≠
non-abelian anyons 1U
2U
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statistics revealed
via interference of quasiparticles
however, interference of fractional charges
was never observed by us
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looking for interference of fractional charges
we stumbled upon…
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Itamar Sivan, Hyungkook Choi, Amir Rosenblatt Vladimir Umansky Diana Mahalu M. Heiblum
Unexpected ‘Pairing’ in the IQHE Regime in interference
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our 2d world
high mobility 2DEG
in GaAs-AlGaAs
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e
e
B
applying quantizing magnetic field…QHE
skipping orbits
2d electron layer
high magnetic field
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ν = number of filled LL = number of electrons per flux quantum
φ0=h/e
EF cω
energy
*c meB
=ω
Ne
e
EF
Ne
ν =2
)( 21+= nE cn ω
2 =
Bl eB
in the bulk
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simplistic view of LL’s near the edge
an approximation there are inter-channel interactions
current carrying edge states
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edge channels
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edge channels immune to back scattering
1d edge channel carries VheI
2
=
Ef +eV
Ef Ef
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interference with edge channels
• incompressible bulk; current carried by edge channels
• edge channels encloses a definite area minimizes phase averaging, high visibility fringes
• electrons directed along definite paths flexible design of interferometers
• no back-scattering
insensitive to impurities
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interfering edge channels
two-path Mach-Zehnder & many-path Fabry-Perot
B X
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every interferometer needs a beam splitter Quantum Point Contact (QPC)
Vgate
Vsource
r λF
t
QPC 0 < t < 1
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preferential backscattering of edge channels
reflected higher LLs
transmitted lower LLs
partitioned LL
Vgate
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ne =1-2.5 x 1011 cm-2 B = 2-9 T T =20-30 mK
interference experiments
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Fabry – Perot interferometer
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S BS
optical FPI
electronic FPI
in the limit of only two-path interference
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etched
quantum point contact (w/bridge)
gates
2DEG in GaAs-AlGaAs
actual realization
area modulation gate
v = 2
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(2,0)
increasing B → lowers Area
small capacitance keeps # electrons constant
bare FPI – is Coulomb dominated (CD)
interfering the lowest Landau level
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screening Coulomb interaction in FPI
effective screening, AB >~4 µm2 effective screening, AB all sizes
tested FPI areas……..2 – 16 µm2
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FPI from CD to AB
adding screening: -- grounded ohmic contact -- top gate
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our experiments:
interference of outer edge channel
v =3
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𝑒𝑒2𝜋𝜋𝑖𝑖∙𝐴𝐴𝐴𝐴/(ℎ𝑒𝑒) 𝑒𝑒2𝜋𝜋𝑖𝑖∙𝐴𝐴𝐴𝐴/( ℎ2𝑒𝑒)
2.25 µm2
surprising AB interference
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2.25 µm2
1 ABh / e
δ − = 1 2 ABh / e
δ − =
area ~ 2.3 µm2
periodicity in B
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periodicity in B …. large 12.5µm FPI
12.5 µm2
h /e h /2e h /e
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12.5 µm2
periodicity in VMG …. large 12.5µm FPI
doubled slope….e*=2e
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can it be related to preferred even windings ?
unlikely two windings we measured h /2e ~ 50%
AB transmission…
𝐴𝐴 2 = 𝐴𝐴0 + 𝐴𝐴1 cos 2𝜋𝜋 ∙ 𝐴𝐴𝐴𝐴/𝜙𝜙0 + 𝐴𝐴2 cos 2𝜋𝜋 ∙ 2𝐴𝐴𝐴𝐴/𝜙𝜙0 + ⋯
h /e
h /2e
measured h/e…
h /e
h /2e
ideally…
4 x 4µm2
w/ dephasing
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independence on transmission coefficient
ℎ/2𝑒𝑒 regime
large FPI ℎ/𝑒𝑒 regime
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coherence @ dephasing
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preferential dephasing of channels
VC
VC
adding a ‘center QPC’
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preferential dephasing of channels
VC
VC
ground
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role of second channel @ vB =2
dephasing of second channel is irrelevant
h/e
VC VC VC
inner edge grounded
outer edge grounded
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dephasing inner channel fully dephases the outer channel
role of second channel @ vB =3
h/ 2e
VC VC VC
‘two-channel entanglement’
second edge grounded third edge grounded
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summarizing interference :
h /e & h /2e independent of FPI pinching
h /2e not due to preferred even windings
appearance of h /2e depends on ff (not on B or ne)
h /2e only when outer channel interferes
no inter-channel tunneling
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are 2e charges interfere ?
shot noise
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hot filament
cathode anode + -
emitted electrons
noisy current in vacuum tubes
classical shot noise
Schottky, 1918
it started with - noise in vacuum tubes
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Vapplied
~h/eVapplied
zero temperature ordered electrons are noiseless !
shot noise =0 …. full Fermi sea (non-partitioned electrons)
Khlus, 1987 Lesovik, 1989
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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 *)i(
i eI)(S 2
∝≡ ><
ν∆ν ν∆∆
(A2/Hz)
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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
f0 ,∆f0 C<<
50 Ω ; 300 K
1 GΩ
warm preamp
voltage gain = 1000
coax
averaging time,τ
noise <i2> DC current
VDC
cryostat
‘home made’
kHzRCπ
f,MHzLCπ
f 30≈2
1=Δ4→2≈
21
= 00
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shot noise in partitioning QPC
0
2
4
6
0 1 2 3
curr
ent n
oise
, S
i (1
0 -28 A
2 /Hz)
current, I (nA)
T = 57mK t = 0.37
−
−+=
eVTk
TkeVcoth)t(eITgk)(S B
BBi
22
1240
e
Reznikov et al. PRL 1995
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-20 -10 0 10 20
0
1
2
3
4
5
6
ISD
(nA)
SI(0
) (×1
0-27 A
2 Hz-1
)
3
0 128 254-276 -135V
SD (µV)
Cooper pairs
e*=e
e*=2e
shot noise in a superconductor
Das et al., Nature Comm. 2012
0 20 400
1
2
3
4T=9mK
e/3
Shot
Noi
se, S
(10-3
0 A2 /Hz)
Back Scattered Current, IB (pA)
Chung et. al. PRL 2003
fractional charge
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shot noise, charge
quasiparticle charge e* = 2e @ h/2e regime
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charge evolution @ vB ~ 3
VR
VR
forming the FPI by pinching QPCR
h/2e
QPCR
QPCR
highly non-linear transmission
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charge: dephasing h/2e …… 2e → e
no interference
no interference
dephasing by grounding the second edge channel
charge drops to e
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– a screened FPI …. AB interference
– AB periodicity vB ~1 - 2.5 …… 𝜙𝜙𝑜𝑜 = ℎ𝑒𝑒
• quasi-particles charge: 𝑒𝑒∗~𝑒𝑒
– AB periodicity: vB ~3 - 4.5 … . .𝜙𝜙𝑜𝑜∗ = ℎ2𝑒𝑒
• quasi-particles charge: 𝑒𝑒∗~2𝑒𝑒
– two edge-channels entanglement
summarizing :
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screening Coulomb domination
revealed inter-channel interaction
leading to unexpected pairing of electrons
presently, we do not understand the effect