Exact Envelope Function Theory Band Structure of Semiconductor Heterostructure

University of Southern Denmark Mads Clausen Institute Exact Envelope Function Theory Band Structure of Semiconductor Heterostructure Daniele Barettin daniele@mci.sdu.dk

Summary Introduction k p Model Exact Envelope Function Theory (EEFT) One Band Model Applications D. Barettin EEFT p.1/9

Introduction Nanoscale semiconductor heterostructures are promising candidates for optoelectronic applications, such as devices for controlling (slow-down) the speed of light or for amplifying the light intensity (LASER). D. Barettin EEFT p.2/9

Introduction Nanoscale semiconductor heterostructures are promising candidates for optoelectronic applications, such as devices for controlling (slow-down) the speed of light or for amplifying the light intensity (LASER). Due to the heterostructure the electrons are confined within the Well/Wire/Dot material, which justifies the expression of artificial atoms. D. Barettin EEFT p.2/9

Introduction Nanoscale semiconductor heterostructures are promising candidates for optoelectronic applications, such as devices for controlling (slow-down) the speed of light or for amplifying the light intensity (LASER). Due to the heterostructure the electrons are confined within the Well/Wire/Dot material, which justifies the expression of artificial atoms. In order to model and compute optical properties we need to calculate the Band structure. D. Barettin EEFT p.2/9

Band Structure Different approaches: DFT (ab initio calculations) Tight Binding (empirical) k p theory. (empirical-continuous) D. Barettin EEFT p.3/9

Band Structure Different approaches: DFT (ab initio calculations) Tight Binding (empirical) k p theory. (empirical-continuous) k p method is analogous to the Taylor expansion of a function, highly accurate in a small region (e.g. at k = 0, the Γ point), with the difference that the band structure is frequently a non-analytic function. D. Barettin EEFT p.3/9

Exact Envelope Function Theory (1) Model of electrons in nanoscale semiconductor heterostructures Single particle Schroedinger equation D. Barettin EEFT p.4/9

Exact Envelope Function Theory (1) Model of electrons in nanoscale semiconductor heterostructures Single particle Schroedinger equation Hamiltonian H = p2 2m + V = K + V with p = i, m electron mass and V crystal potential. D. Barettin EEFT p.4/9

Exact Envelope Function Theory (1) Model of electrons in nanoscale semiconductor heterostructures Single particle Schroedinger equation Hamiltonian H = p2 2m + V = K + V with p = i, m electron mass and V crystal potential. Energies E and wave functions ψ are found by solving the eigenvalue equation Hψ = Eψ Subject to some boundary conditions, V reflects the periodicity of the material. D. Barettin EEFT p.4/9

Exact Envelope Function Theory (1) Model of electrons in nanoscale semiconductor heterostructures Single particle Schroedinger equation Hamiltonian H = p2 2m + V = K + V with p = i, m electron mass and V crystal potential. Energies E and wave functions ψ are found by solving the eigenvalue equation Hψ = Eψ Subject to some boundary conditions, V reflects the periodicity of the material. Heterostructure Only local periodicity (Different crystal structure in different materials) V unknown no exact solution. D. Barettin EEFT p.4/9

Exact Envelope Function Theory (2) EEFT simplifies the problems using local periodicity D. Barettin EEFT p.5/9

Exact Envelope Function Theory (2) EEFT simplifies the problems using local periodicity Expansion of ψ in a complete set of functions U n with a given periodicity: ψ( r) = n φ n ( r)u n ( r) D. Barettin EEFT p.5/9

Exact Envelope Function Theory (2) EEFT simplifies the problems using local periodicity Expansion of ψ in a complete set of functions U n with a given periodicity: ψ( r) = n φ n ( r)u n ( r) φ n ( r) set of unknown envelope functions solutions of infinite set of coupled differential equations: 2 2m 2 φ n ( r) i m p nl φ l ( r)+ l l Ω H nl ( r, r )φ l ( r )d 3 r = Eφ n ( r) D. Barettin EEFT p.5/9

Exact Envelope Function Theory (2) EEFT simplifies the problems using local periodicity Expansion of ψ in a complete set of functions U n with a given periodicity: ψ( r) = n φ n ( r)u n ( r) φ n ( r) set of unknown envelope functions solutions of infinite set of coupled differential equations: 2 2m 2 φ n ( r) i m p nl φ l ( r)+ l l Ω H nl ( r, r )φ l ( r )d 3 r = Eφ n ( r) To reduce to a finite set we choose a finite set of periodic functions Un and treat the rest as perturbation Multiband Model for heterostructures. D. Barettin EEFT p.5/9

Exact Envelope Function Theory (2) EEFT simplifies the problems using local periodicity Expansion of ψ in a complete set of functions U n with a given periodicity: ψ( r) = n φ n ( r)u n ( r) φ n ( r) set of unknown envelope functions solutions of infinite set of coupled differential equations: 2 2m 2 φ n ( r) i m p nl φ l ( r)+ l l Ω H nl ( r, r )φ l ( r )d 3 r = Eφ n ( r) To reduce to a finite set we choose a finite set of periodic functions Un and treat the rest as perturbation Multiband Model for heterostructures. The periodic functions Un are the periodic eigenfunctions of a homogeneous system D. Barettin EEFT p.5/9

One Band Model Restricting the EEFT to only one band we get the usual effective mass approximation: D. Barettin EEFT p.6/9

One Band Model Restricting the EEFT to only one band we get the usual effective mass approximation: H = 2 2m + V D. Barettin EEFT p.6/9

One Band Model Restricting the EEFT to only one band we get the usual effective mass approximation: H = 2 2m + V where m is the effective mass given by: 1 m = 1 m + 2 ν p x,νs S 2 m 2 E SS E νν ν Γ 15 D. Barettin EEFT p.6/9

One band model Conical QD axisymetrical(inas/alas) two-dimensional model D. Barettin EEFT p.7/9

One band model Conical QD axisymetrical(inas/alas) two-dimensional model 3 nm 9 nm D. Barettin EEFT p.7/9

One band model Conical QD axisymetrical(inas/alas) two-dimensional model 3 nm 9 nm Only one band (both in conduction and in valence band) is treated exactly: 892meV 799 mev L=1 g=4 604 mev L=0 g=2 0 417 mev 572 mev L=0 g=2 627meV D. Barettin EEFT p.7/9

Applications and further steps Derivation of dipole momentum using knowledge of the bandstructures: P nm = ψ mv P ψ nc From this optical properties can be found. (e.g. oscillator strength, susceptibility). D. Barettin EEFT p.8/9

Applications and further steps Derivation of dipole momentum using knowledge of the bandstructures: P nm = ψ mv P ψ nc From this optical properties can be found. (e.g. oscillator strength, susceptibility). Multiband model (six band, eight band) D. Barettin EEFT p.8/9

Applications and further steps Derivation of dipole momentum using knowledge of the bandstructures: P nm = ψ mv P ψ nc From this optical properties can be found. (e.g. oscillator strength, susceptibility). Multiband model (six band, eight band) Inclusion of electromechanical effects: strains, piezoelectricity. D. Barettin EEFT p.8/9

Overview Optical properties of nanostructures semiconductors: modeling of bandstructures. D. Barettin EEFT p.9/9

Overview Optical properties of nanostructures semiconductors: modeling of bandstructures. k p model: EEFT. D. Barettin EEFT p.9/9

Overview Optical properties of nanostructures semiconductors: modeling of bandstructures. k p model: EEFT. EEFT: one band model effective mass approximation. D. Barettin EEFT p.9/9

Overview Optical properties of nanostructures semiconductors: modeling of bandstructures. k p model: EEFT. EEFT: one band model effective mass approximation. One band model applied to a conical InAs/AlAs QD. D. Barettin EEFT p.9/9