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inQuantum
Dots
By:Hossien Hossieni
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Additionspin
of
2electron
For2electronadditionofspinsyields1,0
Spin1istrip et s=1,m=1,0,1
= = ,
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QuantumDot
Quantumdots,alsocalledartificialatoms,are
smallregions(from1toabout100nm;1nm)
ofonesemiconductormaterialburiedin
energygap E.Theyaremadeupbyroughly1
millionatoms
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TwoQuantum
Dot
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ConfinementPotentialModelsinQuantumDots
bythefollowingsimpleforms:
2 Parabolic(harmonicoscillator)potential 3 Gaussianpotentialwell
4 Powerex onential otentialwell
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RectangularPotentialWell
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ParabolicPotentialWell
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GaussianPotential
Well
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PowerExponentialPotentialWell
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PowerExponentialPotentialinCoupledQuantum
Dots
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TheHamiltonianofthesystem
Coulombpotential(betweentwoelectron)
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GroundStateEnergy
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Bychoosing
Weget
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Groundstateeigen valueenergyE0vs.confinementpotentialrangeRfor
,
=
,
=
=
.
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GroundstateenergyE0vs.confinementpotentialrangeRfordifferent
,
=
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Coulomb
interaction
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LinearEntropyandEntanglementMeasure
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Groundstateeigen valueenergyE0,CoulombenergyCandLinearEntropy
LofthereduceddensitymatrixfortwoelectronsconfinedincoupledQDstructures
versusvaryingconfinementpotentialrangeRfordifferentvaluesofp.
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TimeDependenceinQuantumMechanics
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SuddenChangeintheHamiltonian
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TimeDependentPerturbationTheory
Inthatcase,theSchrdingerequationfortheHamiltonian
isequiva enttot eseto coup e equations
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TransitionAmplitudesandProbabilities
Thesetofcoupledequations
isnoteasiertosolvewemultiply
theperturbationV(t)bytheunphysicalparameter (0 1)andexpand
theam litudes
Thetermslinearin yield
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Electronlocalizationandentanglementinatwoelectron
uantum
dot
ThelastterminEq.istheZeemaninteractionwithgbeingtheeffectivegfactor, B the
Bohrmagneton,andsthespinofanindividualelectron.
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Entanglemententropyofexcitedstates
consideringtheXYspinchainoflengthNwith
Hamiltonianisgivenby
.isthetransversemagneticeld and theanisotro arameter.
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QuantumEntanglementandSymmetry
Bipartitesymmetricstates
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1DanielR.Bes,QuantumMechanicsAModernandConciseIntroductoryCourse 2ShirwanMuhamad Abdullah,EffectofConfinementPotentialGeometryonEntanglement
BetweenTwoElectronsinQuantumDotsMSc.Thesis 3DavidJ.Griffiths IntroductiontoQuantumMechanics
,
twoelectronquantumdot
5VincenzoAlbaetal,Entanglemententropyofexcitedstates
6DariuszChruscinski etal,QuantumEntanglementandSymmetry