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Anomalous Cross Section Induced by Topological Quantum Interference. De-Hone Lin Department of Physics, NSYSU 23 December 2004. Fractional Quantum Hall States. 2-D electron system inside the GaAs/AlGaAs heterostructure - PowerPoint PPT Presentation
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Anomalous Cross Section Induced by Topological Quantum
Interference
De-Hone Lin Department of Physics, NSYSU 23 December 2004
phyiscs. mesoscopic inphenomena some ingunderstand in
useful be willand system potential general quite inappear toexpected is
casedisk hard ineffect nonlocal theuniversal, is particles charged the
onflux magnetic of influence nonlocal theSince examined. is
flux magnetic theand potential likedisk harda of process scattering The
d.establishe isflux magnetic B-A nonlocala and potential rangeshort
arbitray anfor problem scattering ldimensiona a two of theory wavePartial
phyiscs. mesoscopic in
phenomena other some ingunderstand in useful be willand system
potential general quite inappear toexpected iseffect nonlocal theuniversal,
is charged theonflux magnetic of ceinterferen quantum nonlocal theSince
effect. Hallquantum fractional thein model boson and fermion composite
explain helps chenergy whilow at flux magnetic quantized specific at the
revealed is section cross totalanomalous An d.establishe is
flux magnetic B-A nonlocala and potential rangeshort arbitrary an
for problem scattering dmensional a three of theory wavePartial
•2-D electron system inside the GaAs/AlGaAs heterostructure
•High magnetic fields (B~10T)
•Low temperatures (T~0.1K)
bosons. composite to
theming transformelectrons, thequanta toflux magnetic of
attachment theas viewedbe can attachmentVortex (b)
repulsion. (Coulomb) electron-electron
reduces electron each onto vortices three Placing(a)
1/3. filling level- Landaufractionalat attraction vortex Electron
Coulomb forces flux quantum attachment
mK 100below turesat tempera
observed is 2/5
/at
plateau Hallfractional
rdenominato-even An
2ehxy
R. willett, J.P. Eisenstein,H.L. Stormer, D.C. Tsui, A.C. Gossard, andJ.H. English,PRL, vol. 59, 1776, 1989.
1998. 875, 71, RMP,
effect Hallquantum fractional The : LectureNobel Stormer, H.L.
pairs. fermion composite of formation the
is state thisof origin for they possibilit exciting An unclear.
remains state theof origin The allowed. benot should fraction
rdenominato even an suchat state FQHEA 5/2.at FQHE
ctivity.supercondu etemperatur-high
---nsinteractio repulsivepurely from arising
pairing whichin phenomenonanother explain helpmay
arrives,it whena theory, such that possible isIt
pairing. fermion composite of theory BCSfulla not yet isit but
state, 2/5 theof ingunderstandour in forward stepimportant an is this
n,calculatio 1956 sCooper' Like1/2.not but
2/5at occurs pairing such why intoinsight some gives and
pairs,Cooper form can fermions composite ofsea a Fermi
thationdemonstrat beautifula is ncalculatio s'. Scarola
alet
N. Bonesteel, Nature, Vol. 406, 841 (2000).
field. magnetic of expulsion the toleads npenetratio whose
vortices,negative as wellas ,001.0 as little ascarry that vorticesobserved have We
. fromlly substantia differs alwaysflux that thefinding film, ctingsupercondu
a in vorticesindividualby introducedflux ofamount themeasure weHere
material. theintofarther much
survivemay effect thefilms thininbut edge, thefrom distances tresubmicromeat
negligible becomesflux of reduction thisctors,supercondubulk In .2/
quantum,flux one ansmaller th be can and edge, sample thefrom distancetheir
on depends materials ctingsupercondu in sby vortice carriedflux magnetic of
amount thesixties,early thein Ginzburg and by Bardeenout pointedfirst At
eh
Summary
Quantum interference of magnetic fluxQuantum interference of magnetic flux
Quantum interference in partial wave theory and anomalQuantum interference in partial wave theory and anomalous cross section in two dimensionsous cross section in two dimensions
Quantum interference in partial wave theory and anomalQuantum interference in partial wave theory and anomalous cross section in three dimensionsous cross section in three dimensions
Composite bonsons and fermionsComposite bonsons and fermions
Introduction
a for 0)(
and
afor 1
)(2
V
V
A charged particle
Radius
Phase shifts
Bound states therein
.)(exp)()(exp)(2
)0(21
)0(1
xxxdxA
c
iexxdxA
c
iex
xxdxA
c
iex
1
)0(1 )(exp)(
xxdxA
c
iex
2
)0(2 )(exp)(
)(x
1
2
)(x
.exp)()(
)(exp)()()(
)0(2
)0(1
)0(2
)0(1
c
iexx
xdxAc
iexxx
.2)/(by given is ceinterferen of cycle periodic The ec
1. It is non-local in the sense that it exists even when the interfering beams pass through a field free region and is associated with the entire closed curve C.
2. It is topological in the sense that the phase shift is unaffected
when is deformed within the field free region.
3. It is geometrical in the sense that the above phase factor
represents parallel transport (holonomy transformation) around
with respect to the electromagnetic connection gauge.
a for 0)(
and
afor 1
)(2
V
V
A charged particle
Radius
Phase shifts
Bound states therein
Aharonov-Bohm magnetic flux
The system is very important in understanding the quantum Hall effect, superconductivity, and the transport properties of nano structures.
Partial Wave Method for a Short Range Partial Wave Method for a Short Range Potential and an Aharonov-Bohm FluxPotential and an Aharonov-Bohm Flux
).(2
where
),(;,,
22
0
2)0(0
xVH
xxExxGi
xHE
m
imm eEGExxG )'()0()0(
2
1;',;,
In polar coordinates for the cylindrically symmetric system:
Magnetic field exists in the system, then
x
xrdrA
c
ieExxGExxG
)(exp;,;, )0(
For the Aharonov-Bohm Flux
,ˆˆ
2)(22 yx
exeygxA yx
),(43 xgB
the magnetic field
.4/ g
and the magnetic flux
)();,()(1
2 2
2
2
22
EGV
d
d
d
dE
000 //2 hnumber wit reala is cegm
Where
0)()(1
2 2
2
2
22
kRVd
d
d
dE
The corresponding radial wave equation reads
./2 with Ek
)()(sin)()(cos)( kNkkJkkR k
The general solution of a scattering particle reads
im
mk ekNkkJkkx
)()(sin)()(cos)(
x
At
iki
fxdxAc
iexkixk exp)()(expexpAsymp)(
aThe solution in exterior region
yield which)( reads section cross total theThus
.)()()(by given is section
cross aldifferenti theflux, magnetic thecarrying (fermions) bosons identicalFor
-
2
d
ff
even,
2sin16
(bosons)m
t
odd,
2sin16
(fermions)m
t
包含 AB effect 的分波散射理論所繪的短範圍位能相互作用的散射截面圖,圖一橫軸是能量的大小,縱軸是散射截面的大小,可看出低能量時散射截面產生驚人的下降現象 ;圖二橫軸是磁通的大小,可看到散射截面隨著磁通以週期性變化的神奇現象。這些效應對於納米量子傳輸系統和納米量子光電系統有許多重要的應用。
.)12(flux magnetic quantized at the 0 section cross The
flux. magnetic thecarrying bosons identicalfor sections cross Total
0 nt
.5.0for 1)(2nat quantized isflux magnetic the when0
flux. magnetic thecarrying bosons of sections cross of structures Periodic
0 kat
.5.0for 2nat quantized isflux magnetic the when0
flux. magnetic thecarrying fermions identical of sections cross Total
0 kat
.5.0for 2nat quantized isflux magnetic the when0
flux. magnetic thecarrying fermions identicalfor sections cross totalof structures Periodic
0 kat
Plane wave
),(),()(4}exp{ ~*~
~0
mlkkmlll
l
lml
YYkrjirki
mqmkkqmlmq
q
r
P
ZZkrjC
rdrAc
ierki
),(),()(
})(exp{}exp{
*~,
0
Quantum interference of magnetic flux leads to
).1~
2( and ,~ where
~
,0 liCmql lmq
The angular part is defined as
immm
q
m
mq ePl
mlqZ )(cos
2sin
2cos
)1~
(
)1~
()1(),( ),(
2
0,
00
0
function. Jacobi theis )(cos where ),( 00 mmqP
The general solution for a charged particle moving in a shortrange potential, and an Aharonov-Bohm magnetic flux is found to be
m
qmkkqml
qk ZZ
r
rur ),(),(
)()( *
~
0
At large distance, we expect it to become like
r
ikrfrdrA
c
ierkiFr
r
Pk
}exp{),(})(exp{}exp{)(
)(~ rul
2
~sin)
~sin(sin ~
lkrl
l
by given is )(for behavior asymptotic The ~ rul
ll
r
l
lkr
k
kCru ~
1~
~2
~sin
)()(
),(),(~sinsin1
~cossin
)1~
2(1
),( ,*,
~)
~(
2~
0~
~
mqkkmq
l
li
l
i
mq
ZZle
lel
kf
l
l
The Scattering amplitude is found to be
)(cossin)12(1
)(0
ll
i
q
Pelk
f l
At the quantized values of flux,the result reduces to the well-known amplitude
)0,2/( kk ),( dt
where
22
2,
42
02 sinsin)cos(sinsin21
~)(cossin)12(4 mq
mqt
Z
k
.)1
~()1(
)2/1~
()2/1(~ and ),2( 2
,0
lq
lqZmq mq
reads ),( section cross total the,0 ,2/ i.e. flux,
magnetic thelar toperpendicu directionincident theof case theIn
k dtk
22
2,
42
,02 sinsin)cos(sinsin21
~)(cossin)12(16
)( mq
evenmqt
Z
kbosons
22
2,
42
,02 sinsin)cos(sinsin21
~)(cossin)12(16
)( mq
oddmqt
Z
kfermions
),(),(),( ft ),(),(),(by given areflux magnetic thecarrying
(fermions) bosons identical for the sections cross totalThe
ft
Hard Sphere Potential
The phase shift is given byThe phase shift is given by
)(/)(tan kankaj
Accordingly, the total cross sections is given by
)()()sin(2)()(
~)()(cos)12(16
2/12/12
2/12
2/1
2,
22/1
2
02 kaJkaJkaJkaJ
ZkaJ
kmq
mqt
20 2 flux with magnetica and sphere, harda
by scattering particle chargeda for section-cross totalThe
a
.,2,1,0 ,2/)12(flux of valuesquantized
at theappear Anomalous axis.-z thealongflux magnetica and sphere,
hardfor sections-cross totalscattering theof structure periodic The
0 nn
).2( )12(flux magnetic quantized
at the )(disappearappear Anomalous flux. magnetic the
carrying bosons identical for the sections-cross totalThe
00 nn
.5.0for )12(at quantized isflux the whenzero
approaches section cross The flux. magnetic thecarrying bosons identical
of sections-cross for the noscillatio AB of stucture periodic The
0 kan
).)12((
2flux magnetic quantized at the )(disappearappear Anomalous
flux. magnetic theCarrying bosons identical for the sections-cross totalThe
0
0
n
n
.5.0for 2at quantized isflux the whenzero
approaches section cross The flux. magnetic thecarrying fermions identical
of sections-cross for the noscillatio AB of stucture periodic The
0 kan
Conclusions:
(1) The total cross section is drastically decreased in the long wave length limit and(or) sufficient short range potential. This phenomenon may ascribed to the magnetic flux induced transparency (FIT).
(2) The cross section is symmetric around magnetic flux with the oscillating period ,where n is the positive integer, and is the fundamental magnetic flux quantum .
2/0 n
0ec /2
(3) For identical “Bosons” (“Fermions”), there exists the phenomenon of FIT only for odd (even) number multiple of , and the cross section is symmetric around the odd (even) number multiple of with the oscillating period . Such effect is similar to the picture of the composite Boson (Fermion) in two dimensional fractional quantum Hall effect, and is useful in the question on pinning force in superconductor.
0
02
ctivity.supercondu etemperatur-high
---nsinteractio repulsivepurely from arising
pairing whichin phenomenonanother explain helpmay
arrives,it whena theory, such that possible isIt
pairing. fermion composite of theory BCSfulla not yet isit but
state, 2/5 theof ingunderstandour in forward stepimportant an is this
n,calculatio 1956 sCooper' Like1/2.not but
2/5at occurs pairing such why intoinsight some gives and
pairs,Cooper form can fermions composite ofsea a Fermi
thationdemonstrat beautifula is ncalculatio s'. Scarola
alet
N. Bonesteel, Nature, Vol. 406, 841 (2000).