Electron-hole duality and vortex formation in quantum dots Matti Manninen Jyväskylä Matti Koskinen Jyväskylä Stephanie Reimann Lund Yongle Yu Lund Maria.

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Presentation transcript:

Electron-hole duality and vortex formation in quantum dots Matti Manninen Jyväskylä Matti Koskinen Jyväskylä Stephanie Reimann Lund Yongle Yu Lund Maria Tureblad Lund Susanne Viefers Oslo Ben Mottelson Nordita

DOUBLE BARRIER SOURCE DRAIN QUANTUM DOT Island with ELECTRONS Tarucha et al., PRL, 1996 ”ARTIFICIAL ATOM” InGaAs (5% In) well AlGaAs (22%) barriers Tarucha et al., PRL, 1996, NTT Research Labs Trapped fermions in quasi – 2D

Electron Number N Magnetic Field B Shel l Stru ctur e Conductance Gate voltage

Reimann, Koskinen, Manninen, Mottelson, Phys. Rev. Lett. (1999)

PHASE DIAGRAM S.M. Reimann, M. Manninen, M. Koskinen and B. Mottelson, PRL 83, 3270 (1999)

Tarucha et al., PRL, 1996 Oosterkamp et al., PRL, 2001 Coulomb blockade spectra in a magnetic field T.H. Oosterkamp et al, PRL 82, 2931 (1999)

H. Saarikoski (2003) PhD Thesis, Helsinki Univ. Of Technology, electron quantum dot at high magnetic fields in CSDFT

We can do exact many-particle calculations For small systems. Can we understand vortex formation in Rotating systems?

We solve exactly the problem of 20 interacting polarized (spinless) electrons or spinless bosons in a two-dimensional harmonic potential and analyze the rotational spectrum (yrast spectrum)

Result for 20 electrons E(M) - f(M) M

Harmonic oscillator in two dimensions single particle level structure angular momentum energy

Harmonic oscillator in two dimensions single particle level structure angular momentum energy Ground state of six polarized electrons

Harmonic oscillator in two dimensions single particle level structure angular momentum energy Rotational state of six polarized electrons

Radial density

Radial density

Total density of 20 electrons: Maximum Density Droplet (MDD) Radial density

Total density of 20 electrons: Maximum Density Droplet (MDD) Radial density NEW NOTATION

Radial density: Total density of 20 electrons “Chamon-Wen edge” Surface reconstruction:

Radial density: Total density of 20 electrons “3 vortices” Vortex formation: Generator of 3 vortices

Particle-hole dualism for spinless electrons

Particle-hole dualism and vortices vortex state: A B C Same state for holes: A B ++...

~ For holes: A B A B ~ q = 7 Holes localized in a ring ~ ~

Radial density: Total density of 20 electrons “3 vortices” Vortex formation: Generator of 3 vortices

Radial density: Total density of 3 holes as “3 vortices” Vortex formation: Generator of 3 vortices particles holes

20 electrons energy angular momentum

20 electrons energy angular momentum Fit a smooth function f(m)=a+bm+cm² to the yrast line f(m)=a+bm+cm²

“cusps”

E(m)-f(m) Difference spectrum N = 20

Compare to the spectrum of holes

Yrast spectrum of 20 spinless electrons energy Angular momentum

Take part of the spectrum

E = E(m) m

Change to spectrum of holes: m(holes) = m Spectrum for 20 electrons

Spectrum for 20 particles (3 holes) Spectrum for 3 particles Spectrum for 20 electrons 3 holes

Spectrum for 20 particles (3 holes) Spectrum for 3 particles Center of mass excitations Spectrum for 20 electrons 3 holes

Spectrum for 20 particles (3 holes) Spectrum for 3 particles Center of mass excitations period of 3 oscillations

Pair-correlation for 3 particles

N = 3

Compare pair-correlation of 3 particles to that of 3 holes in 20 electron system

N = 3 N = 20 hole-hole correlation m=16m=17m=18 m=231m=230 m=229

Overlaps ?

Spectrum for 20 electrons 3 holes Overlap between the particle and hole states

Pair correlation of three holes corresponding to the three vortices particles N=20, L=232 (filling factor=0.82) holes N=3, L=18 (filling factor=1/7, q=7) Radial density:

Hole-hole correlation for N = 30, M = 555 (Coulomb interaction) Ground state First excited state Six vortices have two stable coonfigurations

How about bosons ?

Is there a single-particle explanation ?

M 3-vortex region in the mean field approach: energy

M 3-vortex region in the mean field approach: energy 230

M 3-vortex region in the mean field approach: energy

M 3-vortex region in the mean field approach: energy Cusps appear at 'compact' states

N = 20 N = 4 (holes) Difference spectrum fitted with 1 parameter 1v 2v 3v 4v

The oscillations in the spectrum can be explained with the mean field model but the pair-correlation not

Exact result Mean field result Pair correlation for 3 holes

Conclusion: Universal vortex formation Particle-hole duality Pair-correlation functions (revealed by energy spectrum)

Conclusions: Yrast spectrum shows vortix-rings as oscillations Spectrum is dominated by holes in Fermi see Bosons have similar vortex formation (what are holes for bosons?)