world of 2d electrons is exciting modulation doping...diana mahalu m. heiblum unexpected...
TRANSCRIPT
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)
‘exchange statistics’ in the
2d world
richer than the 3d world
exchange statistics in 3d
fermions bosons
ψ ψ→ + ψ ψ→ −
both
ψ ψ→
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
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
statistics revealed
via interference of quasiparticles
however, interference of fractional charges
was never observed by us
looking for interference of fractional charges
we stumbled upon…
Itamar Sivan, Hyungkook Choi, Amir Rosenblatt Vladimir Umansky Diana Mahalu M. Heiblum
Unexpected ‘Pairing’ in the IQHE Regime in interference
our 2d world
high mobility 2DEG
in GaAs-AlGaAs
e
e
B
applying quantizing magnetic field…QHE
skipping orbits
2d electron layer
high magnetic field
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
simplistic view of LL’s near the edge
an approximation there are inter-channel interactions
current carrying edge states
edge channels
edge channels immune to back scattering
1d edge channel carries VheI
2
=
Ef +eV
Ef Ef
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
interfering edge channels
two-path Mach-Zehnder & many-path Fabry-Perot
B X
every interferometer needs a beam splitter Quantum Point Contact (QPC)
Vgate
Vsource
r λF
t
QPC 0 < t < 1
preferential backscattering of edge channels
reflected higher LLs
transmitted lower LLs
partitioned LL
Vgate
ne =1-2.5 x 1011 cm-2 B = 2-9 T T =20-30 mK
interference experiments
Fabry – Perot interferometer
S BS
optical FPI
electronic FPI
in the limit of only two-path interference
etched
quantum point contact (w/bridge)
gates
2DEG in GaAs-AlGaAs
actual realization
area modulation gate
v = 2
(2,0)
increasing B → lowers Area
small capacitance keeps # electrons constant
bare FPI – is Coulomb dominated (CD)
interfering the lowest Landau level
screening Coulomb interaction in FPI
effective screening, AB >~4 µm2 effective screening, AB all sizes
tested FPI areas……..2 – 16 µm2
FPI from CD to AB
adding screening: -- grounded ohmic contact -- top gate
our experiments:
interference of outer edge channel
v =3
𝑒𝑒2𝜋𝜋𝑖𝑖∙𝐴𝐴𝐴𝐴/(ℎ𝑒𝑒) 𝑒𝑒2𝜋𝜋𝑖𝑖∙𝐴𝐴𝐴𝐴/( ℎ2𝑒𝑒)
2.25 µm2
surprising AB interference
2.25 µm2
1 ABh / e
δ − = 1 2 ABh / e
δ − =
area ~ 2.3 µm2
periodicity in B
periodicity in B …. large 12.5µm FPI
12.5 µm2
h /e h /2e h /e
12.5 µm2
periodicity in VMG …. large 12.5µm FPI
doubled slope….e*=2e
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
independence on transmission coefficient
ℎ/2𝑒𝑒 regime
large FPI ℎ/𝑒𝑒 regime
coherence @ dephasing
preferential dephasing of channels
VC
VC
adding a ‘center QPC’
preferential dephasing of channels
VC
VC
ground
role of second channel @ vB =2
dephasing of second channel is irrelevant
h/e
VC VC VC
inner edge grounded
outer edge grounded
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
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
are 2e charges interfere ?
shot noise
hot filament
cathode anode + -
emitted electrons
noisy current in vacuum tubes
classical shot noise
Schottky, 1918
it started with - noise in vacuum tubes
Vapplied
~h/eVapplied
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 *)i(
i eI)(S 2
∝≡ ><
ν∆ν ν∆∆
(A2/Hz)
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
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
-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
shot noise, charge
quasiparticle charge e* = 2e @ h/2e regime
charge evolution @ vB ~ 3
VR
VR
forming the FPI by pinching QPCR
h/2e
QPCR
QPCR
highly non-linear transmission
charge: dephasing h/2e …… 2e → e
no interference
no interference
dephasing by grounding the second edge channel
charge drops to e
– 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 :
screening Coulomb domination
revealed inter-channel interaction
leading to unexpected pairing of electrons
presently, we do not understand the effect