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Background Model Results Summary and outlook

Understanding the spectra of few electrons confined in a quasi-one-dimensional nanostructure

Tokuei Sako1 Geerd HF Diercksen2

1Nihon University, College of Science and Technology

Funabashi, Chiba, JAPAN

2Max-Planck-Institut für Astrophysik

Garching, GERMANY

October 9, 2008

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Introduction

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Introduction

Introduction

Semiconductor technology permits to engineer quantum systems consisting of a small number of electrons confined in nanostructures on semiconductor surfaces. Because of their finite size these quantum systems have a discrete energy-level structure that follows Hund’s rules known for atoms. Therefore they are referred to as artificial atoms or quantum dots.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Introduction

Introduction

One of the most significant difference between quantum dots and atoms is that the electronic properties of quantum dots can be controlled by the size of the dots, namely, the strength of confinement. For practical applications of quantum dots as future quantum devices the relation between the form of the confining potential, the resultant energy spectra, and the dynamics of the electrons needs to be well established.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Introduction

Introduction

The energy spectra and wave functions of few electrons have been studied

confined in a quasi one-dimensional Gaussian potential of different strength and unharmonicity, by using the quantum chemical configuration-interaction method, and by employing a reduced Cartesian anisotropic Gaussian basis set.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Introduction

Introduction

The model has been chosen for the following reasons:

The quasi one-dimensional Gaussian potential allows the wave functions of two electrons to be visualized in a two-dimensional plane and permits a detailed analysis of the correlation of the confined electrons. The Gaussian potential is approximated in the low energy region by a harmonic-oscillator potential used typically for modelling semiconductor quantum dots. The anharmonicity of the Gaussian potential permits to study the breakdown of the generalized Kohn theorem.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Hamiltonian

The Hamiltonian operator adopted in the present study is given by H =

2

• i=1
• −1

2∇2 i

• +

2

• i=1

w(ri) + 1 |r1 − r2|, (1)

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Confining potential

The one-electron confining potential is chosen as w(r) = 1

2ω2

x2 + y2 − Dexp

• − ω2

z

2D z2

• .

(2)

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Anisotropic harmonic oscillator potential

Anisotropic harmonic oscillator potential: w(ri) = 1 2

• ω2

xx2 i + ω2 yy2 i + ω2 zz2 i

• Tokuei Sako, Geerd HF Diercksen

Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Anisotropic harmonic oscillator eigenfunctions

Anisotropic harmonic oscillator eigenfunctions: χ

ω

• ν (

r) = N

ω

• ν Hνx(x)Hνy(y)Hνz(z) exp
• −1

2(ωxx2 + ωyy2 + ωzz2)

• .

N

ω

• ν : normalization constant

Hνx(x), etc.: Hermite polynomial

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Basis set

A set of properly chosen Cartesian anisotropic Gaussian-type orbitals (c-aniGTOs) is the most convenient choice to correctly approximate the wavefunction of electrons confined by an anisotropic harmonic oscillator potential.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Cartesian anisotropic Gaussian-type orbitals

Cartesian anisotropic Gaussian-type orbital centered at (bx, by, bz): χ

a, ζ ani(

r; b) = xax

bx yay by zaz bz exp(−ζxx2 bx − ζyy2 by − ζzz2 bz),

(3) xbx = (x − bx), etc.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Cartesian anisotropic Gaussian-type orbitals

Following the quantum chemical convention the orbitals are classified as s-type, p-type, . . . for a = ax + ay + az = 0, 1, . . ., respectively. The (bx, by, bz) parameters are chosen in general to coincide with the center of the confining potential, i.e. the

• rigin of the Cartesian coordinate system.

The optimal orbital exponents (ζx, ζy, ζz) are chosen to be (ωx/2, ωy/2, ωz/2).

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Gaussian potential

• 15
• 10
• 5

• 40
• 30
• 20
• 10

Figure: One-dimensional attractive Gaussian potential with small (figure (a)) and large (figure (b)) anharmonicity.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Anharmonicity

A Gaussian potential can be approximated close to the minimum by a harmonic-oscillator potential with ω defined by: ω =

• 2Dγ

(4) ω: force constant of the harmonic oszillator γ: exponent of the Gaussian potential D: depth of the Gaussian potential The anharmonicity of a Gaussian potential may be characterized by: α = ω/D (5)

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Gaussian potential

• 15
• 10
• 5

• 40
• 30
• 20
• 10

Figure: One-dimensional attractive Gaussian potential with small (figure (a)) and large (figure (b)) anharmonicity.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Potential Basis sets

Basis set

The number of basis functions is increased stepwise by adding a new function with an additional nodal line to the previous basis set. The maximum deviation of the energy levels covered by the present study was shown to be smaller than 2×10−4 for the results obtained by using basis sets with 13 and 14 functions, respectively, including functions with 12 and 13 nodal lines, respectively. Therefore in this study the basis set of 13 functions has been used.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Energy spectrum

Figure: Energy spectrum of two electrons confined by a quasi-one-dimensional Gaussian potential with different strength of confinement ω represented as relative energies from the ground state. The anharmonicity parameter α of the Gaussian potential is 0.05 for all cases.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Classification of confinement regimes

The chosen ωz values of 5.0, 1.0, and 0.1 correspond, respectively, to three regimes of the strength of confinement, namely, strong, medium, and weak. The classification of the three regimes is defined by the relative importance of the one-electron energy E1 compared to the two-electron energy E2.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Classification of confinement regimes

The one-electron energy is scaled by ωz as E1 ∼ ωz since the eigenenergy of the one-dimensional harmonic

• scillator is given by ωz(n + 1

2) where n denotes the

harmonic-oscillator quantum number.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Classification of confinement regimes

The two-electron energy E2 can be estimated by considering the size of the system: The characteristic length lz of the system along the z direction is related to ωz by lz ∼ 1/√ωz. Therefore, the two electron energy is scaled by ωz as E2 ∼ √ωz since it is related to lz by E2 ∼ 1/lz.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Classification of confinement regimes

The one-electron energy E1 dominates the two-electron energy E2 for ωz ≫ 1.0 (large ωz), The one-electron energy E1 becomes similar to E2 for ωz ∼ 1.0 (medium ωz) The one-electron energy E1 becomes dominated by E2 for ωz ≪ 1.0 (small ωz).

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Strong confinement

The zeroth order Hamiltonian for the system can be approximated by a sum of the three harmonic-oscillator Hamiltonians as H0 =

3

• i=1
• −1

2

∂ ∂zi 2 + 1

2ω2 zz2 i

• ,

(6) where the x and y degrees of freedom are neglected and the Gaussian potential along the z direction is approximated by a harmonic oscillator with the frequency ωz.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Strong confinement

The energy of the Hamiltonian is written in terms of vp as En1,n2,n3 = ωz

• vp + 3

2

• ,

(7) with vp = n1 + n2 + n3, (8) where n1, n2, and n3 represent the harmonic-oscillator quantum numbers for the electron 1, 2, and 3, respectively.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Energy levels and wavefunctions

• 10

• 10

• 3

• 3

Figure: The correspondence of low-lying energy levels and the wave functions of two electrons confined in a quasi-one-dimensional Gaussian potential with the strength of anharmonicity used previously.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Notation

The assignment 2S+1[ns, na] given to the wave functions of ω = 1.0 has been made by the spin multiplicity 2S + 1 and by a pair

• f numbers ns and na counting the number of nodal lines along

the symmetric coordinate zs and antisymmetric coordinate za, respectively, that are defined by zs =

1 √ 2[z1 + z2],

(9) za =

1 √ 2[z1 − z2],

(10) which coincides with the 45-degree diagonal line and the other diagonal line in the density plots, respectively.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Potential function

In order to understand the origin of the appearance of the new nodal line in the singlet wave functions for small ω, the sum of the one- and two-electron potential functions in the Hamiltonian

• f Eq. (1) projected onto the z1-z2 plane,

V(z1, z2) =

2

• i=1

D

• 1 − exp
• − ω2

z

2D zi

• +

1 |z1 − z2|, (11) has been calculated.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Potential surfaces

• 10
• 5

• 10
• 5

• 4
• 2

• 4
• 2

• 2
• 1

• 2
• 1

Figure: Two-dimensional contour plot of the sum of the Gaussian and electron repulsion potentials for ωz = 5.0 (a), 1.0 (b), and 0.1 (c). The diagonal line separating the contours into two regions represents the potential wall of the electron repulsion potential.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Estimated energy increase

An energy increase owing to the potential wall may be estimated by an integration of the density ρ(z1, z2) multiplied by the electron-electron interaction over the region around the potential wall denoted by ∆Ewall =

• Ω

ρ(z1, z2) 1 |z1 − z2|dz1dz2, (12) where Ω denotes a domain of integration specifying a vicinity of the potential wall.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Square density plot

• 1.5

• 1.5

• 3

• 3

Figure: The square desity plot

• f wave functions for the three

states in the polyad manifold of vp = 2 of two electrons confined in a quasi-one-dimensional Gaussian potential with ωz = 5.0 and 1.0 with the strength of anharmonicity used previously.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Coordinate transformation

By transforming the coordinates from (z1, z2) to (zs, za) and introducing a harmonic approximation to the Gaussian potential in Eq. (11) the zeroth-order Hamiltonian for the upper (+) and lower (−) bound regions seperated by the potential wall in figure (c) can be approximated as H±

ωz=0.1 = hs + h± a .

(13)

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Coordinate transformation

The operators hs and h±

a are defined, respectively, by

hs = −1 2 ∂2 ∂z2

s

+ 1 2ω2

zz2 s ,

(14) h±

a = −1

2 ∂2 ∂z2

a

+ 1 2ω2

zz2 a,

(15) where the domain of the antisymmetric coordinate za in

• Eq. (15) is defined as za < 0 for h+

a and za > 0 for h− a .

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Eigenvalues

Consequently, the eigenvalues for the zeroth order Hamiltonian (13) is calculated as E±(vs, va) = ωz [vs + va + 1] , (16) where vs = 0, 1, 2, · · · and va = 1, 3, 5, · · · . It is natural that the E+ and E− have the same energy spectrum.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Eigenvalues

Table: The zeroth-order energy levels E± for ωz = 0.1 defined by

• Eq. (16).

vs va v∗

p 1

E±/ωz 1 1 2 1 1 2 3 3 3 4 2 1 3 4 1 3 4 5 3 1 4 5

1The extended polyad quantum number. Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Exitation energy spectrum

Figure: Excitation energy spectrum of two electrons confined in a quasi-one-dimensional Gassian potential with different strength

• f confinement ω. The

anharmonicity of the Gaussian potential α is 0.125 for all cases.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Wavefunction square density

• 10

• 10

Figure: The square density plot

• f wavefunctions for the

low-lying states of two electrons confined in a quasi-one-dimensional Gaussian potential with (D, ω) = (0.8, 0.1).

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Wavefunction square density

• 10

• 10

• 10

• 10

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Wavefunction square density

• 10

• 10

• 10

• 10

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Wavefunction square density

• 10

• 10

• 10

• 10

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Wavefunction square density: 1[1, 4] and 3[1, 4]

• 10

• 10

• 10

• 10

Figure: The square-density plot

• f wavefunctions and their sum

and difference for the local-mode pair 1[1, 4] and

3[1, 4].

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Wavefunction square density: 1[4, 1] and 3[4, 1]

• 10

• 10

• 10

• 10

Figure: The square-density plot

• f wavefunctions and their sum

and difference for the local-mode pair 1[4, 1] and

3[4, 1].

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Potential surface

• 10
• 5

• 10
• 5

Figure: Two-dimensional plot of the sum of the Gaussian and electron-repulsion potentials with (α, ω) = (0.125, 0.1).

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Energy spectrum

Figure: Energy spectrum of three electrons confined by a quasi-one-dimensional Gaussian potential with different strength of confinement, ωz, represented as relative energies from the ground state. The doublet and quartet levels are colored by green and red,

• respectively. The vertical axis of

each of the three energy diagrams is scaled by ωz.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Cartesian coordinate system

• Figure: Three-dimensional

Cartesian coordinate system for displaying the three-electron wave functions (upper figure). The square-density plot of the three-electron wave function for the lowest quartet 4[0, 3, 0] state is displayed As an example (lower figure).

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Internal plane

• 1

• 1

Figure: Cross section with respect to the internal plane of the square-density plot of the wave function of the lowest quartet 4[0, 3, 0] state

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Internal plane

The internal plane will serve to make a correspondence between the density distribution of the wave function and the underlying correlated motion of the three electrons. It is defined by the equation, z1 + z2 + z3 = 0. (17) numbers for the electron 1, 2, and 3, respectively.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Correlated coordinates

The independent-electron coordinates (z1, z2, z3) are transformed into correlated coordinates (za, zb, zc) by the unitary transformation, za =

1 √ 3[z1 + z2 + z3],

zb =

1 √ 6[2z1 − z2 − z3],

zc =

1 √ 2[z2 − z3].

(18)

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Correlated coordinates

The two red lines represent the projection of the z2 and z3 axes onto the internal plane. These two lines are obtained from Eq. (18) by putting za = 0 and solving the simultaneous equations for z2 and z3 as z2 =

1 √ 2

• zc −

1 √ 3zb

• ,

z3 = − 1

√ 2

• zc +

1 √ 3zb

• (19)

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Wavefunction square density

(a) 4[1,3,0] (b) 4[0,3,1]

Figure: The square-density plot

• f the three-electron wave

function for the second lowest quartet state (a) and the third lowest quartet state (b) with their assignments 4[1, 3, 0] and

4[0, 3, 1], respectively.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Energy spectrum

Figure: Energy spectrum of three electrons confined by a quasi-one-dimensional Gaussian potential with different strength of confinement ω. The anharmonicity parameter α of the Gaussian potential is 0.1 for all cases.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

Wavefunction

4[4,3,0] 4[2,3,1]

harmonic anharmonic

4[1,6,0] 4[0,3,2]

Figure: The square-density plot

• f the three-electron wave

functions for the four quartet states in the polyad manifold of v∗

p = 7 for α = 0.0 (harmonic)

and 0.1 (anharmonic). The red and blue surfaces represent the positive and negative phase of the wave functions, respectively.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Summary Outlook Downloads Acknowledgement

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Summary Outlook Downloads Acknowledgement

Summary

The energy spectra and wave functions of two electrons have been studied

confined in a quasi one-dimensional Gaussian potential of different strength and unharmonicity, by using the quantum chemical configuration-interaction method, and by employing a reduced Cartesian anisotropic Gaussian basis set.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Summary Outlook Downloads Acknowledgement

Summary

The electrons have been studied in

a nearly harmonic and a strongly unharmonic

Gaussian potential for three regimes of confinement strength:

strong (ωz = 5.0), medium (ωz = 1.0) and weak (ωz = 0.1).

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Summary Outlook Downloads Acknowledgement

Summary - nearly harmonic Gaussian potential

For strong confinement the energy spectrum shows a regular band structure with a band-gap close to ωz. The energy levels of each band are well localized and are characterzied by the polyad quantum number vp. 2-electron case: The number of energy levels belonging to each band is vp + 1.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

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Summary - nearly harmonic Gaussian potential

For medium confinement the energy spectrum shows also a band structure characterized by the polyad quantum number vp. The splitting of the energy levels belonging to the same vp manifold is so large that adjacent polyad manifolds get close to each other. As the strength of confinement becomes even weaker energy levels belonging to different vp manifolds start to

• verlap with each other.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

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Summary - nearly harmonic Gaussian potential

2-electron case: For weak confinement the triplet energy levels having the set of nodal lines (ns, na) become nearly degenerate with the singlet energy levels that for strong and medium confinement have the set of nodal lines (ns, na − 1). The energy spectrum of the weak confinement has a band structure with a band-gap of about ωz as observed for strong confinement but each band is characterized by the extended polyad quantum number v∗

p.

The number of levels belonging to the v∗

p manifold is equal

to:

v∗

p + 1 for odd v∗ p and

v∗

p for even v∗ p .

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

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Summary

3-electron case For weak confinement the energy spectrum shows a band structure as observed for strong confinement but each band is characterized by the extended polyad quantum number v∗

p.

The energy levels belonging to each v∗

p manifold form

nearly degenerate triplets each of which consists of two doublet states and one quartet state. The nodal pattern of the wave functions belonging to each nearly degenerate triplet are almost identical to each other except for their phases.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

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Summary - nearly harmonic Gaussian potential

2-electron case: The density of the singlet wave functions along the symmetric coordinate decreases as ωz decreases. It becomes negligibly small for ωz = 0.1. The singlet wave functions at ωz = 0.1 have the same number of nodal lines as their counterpart triplet wave functions of the degenerate pairs using the extended assignment. Their nodal pattern become almost identical to each other except for their phases.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

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Summary - nearly harmonic Gaussian potential

2-electron case: The sum V of the one- and two-electron potentials has been projected onto the z1-z2 plane. The contour plot of the two-dimensional potential V(z1, z2) shows that for decreasing ωz the decreasing density along the symmetric coordinate in the singlet wave functions is caused by the increasingly stronger potential wall of the electron-electron interaction along the this coordinate.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

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Summary - strongly unharmonic Gaussian potential

2-electron case: For strong and medium confinement the energy spectra look quite similar to those of the nearly harmonic case. For weak confinement of ωz = 0.1 the spectrum shows an irregular level structure in the high energy region above ∆E ≥ 0.3. The nodal lines of the wave functions in this high energy region get curved increasingly strongly as the energy increases.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

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Summary - strongly unharmonic Gaussian potential

2-electron case: By taking the sum and the difference of the singlet and triplet wave functions of the degenerate pairs bent nodal lines result:

• ne pair of lines passing through the valleys of the potential

V(z1, z2) and

• ne pair passing along the hillside.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

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Summary - strongly unharmonic Gaussian potential

2-electron case: The wave functions having nodal lines

through the valleys correspond to the classical motion of two electrons performing an elastic collision and along the hillside correspond to a classical motion in which

• ne electron follows the movement of the other electron

after a quarter cycle.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

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Summary

3-electron case The origin of the tripling of energy levels and the similarity

• f the wave functions for different spin states has been

rationalized by using the projection of one- and two-electron potentials onto the internal plane.

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Summary Outlook Downloads Acknowledgement

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Summary Outlook Downloads Acknowledgement

Outlook

Anharmonic oscillator potentials, chaos Gaussian potentials, double quantum dots, surfaces Intense laser fields Strong magnetic fields

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Summary Outlook Downloads Acknowledgement

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Summary Outlook Downloads Acknowledgement

Downloads

The lecture and relevant papers may be downloaded from: URL: http://www.mpa-garching.mpg.de/mol_physics

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Summary Outlook Downloads Acknowledgement

Outline

1

Background Introduction

2

Model Potential Basis sets

3

Results 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential

4

Summary and outlook Summary Outlook Downloads Acknowledgement

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure

Background Model Results Summary and outlook Summary Outlook Downloads Acknowledgement

Acknowledgement: Institutions

Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure