Background Model Results Summary and outlook Understanding the spectra of few electrons confined in a quasi-one-dimensional nanostructure Tokuei Sako 1 Geerd HF Diercksen 2 1 Nihon University, College of Science and Technology Funabashi, Chiba, JAPAN 2 Max-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 Background 1 Introduction Model 2 Potential Basis sets Results 3 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential Summary and outlook 4 Summary Outlook Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Introduction Results Summary and outlook Outline Background 1 Introduction Model 2 Potential Basis sets Results 3 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential Summary and outlook 4 Summary Outlook Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Introduction Results Summary and outlook 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 Introduction Results Summary and outlook 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 Introduction Results Summary and outlook 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 Introduction Results Summary and outlook 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 Potential Results Basis sets Summary and outlook Outline Background 1 Introduction Model 2 Potential Basis sets Results 3 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential Summary and outlook 4 Summary Outlook Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Potential Results Basis sets Summary and outlook Hamiltonian The Hamiltonian operator adopted in the present study is given by 2 2 1 � � � 2 ∇ 2 � − 1 H = + w ( r i ) + | r 1 − r 2 | , (1) i i = 1 i = 1 Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Potential Results Basis sets Summary and outlook Confining potential The one-electron confining potential is chosen as − ω 2 � � 2 ω 2 � x 2 + y 2 � w ( r ) = 1 2 D z 2 z − D exp . (2) Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Potential Results Basis sets Summary and outlook Outline Background 1 Introduction Model 2 Potential Basis sets Results 3 2e case: nearly harmonic potential 2e case: anharmonic potential 3e case: nearly harmonic potential 3e case: anharmonic potential Summary and outlook 4 Summary Outlook Downloads Acknowledgement Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Potential Results Basis sets Summary and outlook Anisotropic harmonic oscillator potential Anisotropic harmonic oscillator potential: w ( r i ) = 1 � � ω 2 x x 2 i + ω 2 y y 2 i + ω 2 z z 2 i 2 Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Potential Results Basis sets Summary and outlook Anisotropic harmonic oscillator eigenfunctions Anisotropic harmonic oscillator eigenfunctions: � − 1 � 2 ( ω x x 2 + ω y y 2 + ω z z 2 ) χ � ω r ) = N � ω ν ( � ν H ν x ( x ) H ν y ( y ) H ν z ( z ) exp . � � N � ω ν : normalization constant � H ν x ( x ) , etc.: Hermite polynomial Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Potential Results Basis sets Summary and outlook 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 Potential Results Basis sets Summary and outlook Cartesian anisotropic Gaussian-type orbitals Cartesian anisotropic Gaussian-type orbital centered at ( b x , b y , b z ) : a ,� χ � r ; � b x y a y ζ b ) = x a x b y z a z b z exp ( − ζ x x 2 b x − ζ y y 2 b y − ζ z z 2 ani ( � b z ) , (3) x b x = ( x − b x ) , etc. Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Potential Results Basis sets Summary and outlook Cartesian anisotropic Gaussian-type orbitals Following the quantum chemical convention the orbitals are classified as s -type, p -type, . . . for a = a x + a y + a z = 0, 1, . . . , respectively. The ( b x , b y , b z ) parameters are chosen in general to coincide with the center of the confining potential, i.e. the origin of the Cartesian coordinate system. The optimal orbital exponents ( ζ x , ζ y , ζ z ) are chosen to be ( ω x / 2 , ω y / 2 , ω z / 2 ) . Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Potential Results Basis sets Summary and outlook Gaussian potential ( a ) 0.10 0.08 Figure: One-dimensional 0.06 v = 5 attractive Gaussian potential E /a .u. v = 4 0.04 with small (figure (a)) and large v = 3 v = 2 (figure (b)) anharmonicity. 0.02 v = 1 D = 1.0 = 0.01 v = 0 0.00 -40 -30 -20 -10 0 10 20 30 40 x /a .u. (b) 1.0 0.8 0.6 v = 5 E /a .u. v = 4 0.4 v = 3 v = 2 0.2 D = 1.0 v = 1 = 0.1 v = 0 0.0 -15 -10 -5 0 5 10 15 x /a .u. Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
Background Model Potential Results Basis sets Summary and outlook Anharmonicity A Gaussian potential can be approximated close to the minimum by a harmonic-oscillator potential with ω defined by: � ω = 2 D γ (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 Potential Results Basis sets Summary and outlook Gaussian potential ( a ) 0.10 0.08 Figure: One-dimensional 0.06 v = 5 attractive Gaussian potential E /a .u. v = 4 0.04 with small (figure (a)) and large v = 3 v = 2 (figure (b)) anharmonicity. 0.02 v = 1 D = 1.0 = 0.01 v = 0 0.00 -40 -30 -20 -10 0 10 20 30 40 x /a .u. (b) 1.0 0.8 0.6 v = 5 E /a .u. v = 4 0.4 v = 3 v = 2 0.2 D = 1.0 v = 1 = 0.1 v = 0 0.0 -15 -10 -5 0 5 10 15 x /a .u. Tokuei Sako, Geerd HF Diercksen Few electrons confined in a one-dimensional-nanostructure
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