Numerical simulation of electronic properties of quantum dotsTherefore, a quantum dot can be thought...

Numerical simulation of electronic properties ofquantum dots

Heinrich [email protected]

Hamburg University of TechnologyInstitute of Mathematics

TUHH Heinrich Voss Quantum dots Isfahan, July, 2016 1 / 69

Quantum dots

Semiconductor nanostructures have attracted tremendous interest in the pastfew years because of their special physical properties and their potential forapplications in micro– and optoelectronic devices.

In such nanostructures, the free carriers are confined to a small region ofspace by potential barriers, and if the size of this region is less than theelectron wavelength, the electronic states become quantized at discreteenergy levels.

The ultimate limit of low dimensional structures is the quantum dot, in whichthe carriers are confined in all three directions, thus reducing the degrees offreedom to zero.Therefore, a quantum dot can be thought of as an artificial atom.

Quantum dots

Electronic structure of quantum dots

Electron Spectroscopy Group, Fritz-Haber-Institute, BerlinTUHH Heinrich Voss Quantum dots Isfahan, July, 2016 3 / 69

Outline

1 Quantum Dots

2 Nonlinear minmax characterization

3 Full Approximation Method

4 Iterative projection methodsNonlinear Arnoldi methodJacobi–Davidson method

5 Numerical example

6 Quantum dot on wetting layer

Quantum Dots

Outline

1 Quantum Dots

5 Numerical example

Quantum Dots

ProblemDetermine relevant energy states (i.e. eigenvalues) and corresponding wavefunctions (i.e. eigenfunctions) of a three-dimensional quantum dot embeddedin a matrix.

Quantum Dots

Problem ct.

Governing equation: Schrödinger equation

−∇ ·(

2m(x ,E)∇Φ

)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm

where ~ is the reduced Planck constant, m(x ,E) is the electron effectivemass, and V (x) is the confinement potential.

m and V are discontinous across the heterojunction.

Boundary and interface conditions

Φ = 0 on outer boundary of matrix Ωm

BenDaniel–Duke condition1

∣∣∣∣∂Ωm

∣∣∣∣∂Ωq

on interface

Quantum Dots

Problem ct.

−∇ ·(

2m(x ,E)∇Φ

)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm

∣∣∣∣∂Ωm

∣∣∣∣∂Ωq

on interface

Quantum Dots

Problem ct.

−∇ ·(

2m(x ,E)∇Φ

)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm

∣∣∣∣∂Ωm

∣∣∣∣∂Ωq

on interface

Quantum Dots

Electron effective mass

The electron effective mass m = mj , j ∈ q,m depends an the energy levelE .

Most simulations in the literature consider constant electron effective masses(e.g. mq = 0.023m0 and mm = 0.067m0 for an InAs quantum dot and theGaAs matrix, respectively, where m0 is the free electron mass).

Numerical examples demonstrate that the electronic behavior is substantiallydifferent: For the pyramidal quantum dot in our examples there are only 3confined electron states for the linear model whereas the nonlinear modelexhibits 7 confined states (i.e. energy levels which are smaller than theconfinement potential).The ground state of the nonlinear model is about 7% and the first exited stateabout 18% smaller than for the linear approximation.

Quantum Dots

Numerical examples demonstrate that the electronic behavior is substantiallydifferent: For the pyramidal quantum dot in our examples there are only 3confined electron states for the linear model whereas the nonlinear modelexhibits 7 confined states (i.e. energy levels which are smaller than theconfinement potential).

The ground state of the nonlinear model is about 7% and the first exited stateabout 18% smaller than for the linear approximation.

Quantum Dots

The dependence of m(x ,E) on E can be derived from the eight-band k · panalysis and effective mass theory. Projecting the 8× 8 Hamiltonian onto theconduction band results in the single Hamiltonian eigenvalue problem with

m(x ,E) =

mq(E), x ∈ Ωqmm(E), x ∈ Ωm

1mj (E)

E + gj − Vj+

1E + gj − Vj + δj

), j ∈ m,q

where mj is the electron effective mass, Vj the confinement potential, Pj themomentum, gj the main energy gap, and δj the spin-orbit splitting in the j thregion.

Other types of effective mass (taking into account the effect of strain, e.g.)appear in the literature. They are all rational functions of E where 1/m(x ,E) ismonotonically decreasing with respect to E , and that’s all we need.

Quantum Dots

The dependence of m(x ,E) on E can be derived from the eight-band k · panalysis and effective mass theory. Projecting the 8× 8 Hamiltonian onto theconduction band results in the single Hamiltonian eigenvalue problem with

m(x ,E) =

mq(E), x ∈ Ωqmm(E), x ∈ Ωm

1mj (E)

E + gj − Vj+

1E + gj − Vj + δj

), j ∈ m,q

where mj is the electron effective mass, Vj the confinement potential, Pj themomentum, gj the main energy gap, and δj the spin-orbit splitting in the j thregion.

Other types of effective mass (taking into account the effect of strain, e.g.)appear in the literature. They are all rational functions of E where 1/m(x ,E) ismonotonically decreasing with respect to E , and that’s all we need.

Quantum Dots

Variational form

Find E ∈ R and Φ ∈ H10 (Ω), Φ 6= 0, Ω := Ωq ∪ Ωm, such that

a(Φ,Ψ; E) :=~2

∫Ωq

1mq(x ,E)

∇Φ · ∇Ψ dx +~2

∫Ωm

1mm(x ,E)

∇Φ · ∇Ψ dx

∫Ωq

Vq(x)ΦΨ dx +

∫Ωm

Vm(x)ΦΨ dx

= E∫Ω

ΦΨ dx =: Eb(Φ,Ψ) for every Ψ ∈ H10 (Ω)

Quantum Dots

Properties

a(·, ·,E) bilinear, symmetric, bounded, H10 (Ω)–elliptic for E ≥ 0

b(·, ·) bilinear, positive definite, bounded, completely continuous

By the Lax–Milgram lemma the variational eigenproblem is equivalent to

T (E)Φ = 0

whereT (E) : H1

0 (Ω)→ H10 (Ω), E ≥ 0,

is a family of bounded operators such that . . .

Quantum Dots

Properties ct.

Forf (E ; Φ) := 〈T (E)Φ,Φ〉 = Eb(Φ,Φ)− a(Φ,Φ; E)

it holdsf (0; Φ) < 0 < lim

E→∞f (E ; Φ) =∞ for every Φ 6= 0

∂∂E f (E ; Φ) > 0 for every Φ 6= 0 and E ≥ 0

For fixed E ≥ 0 the eigenvalues of the linear eigenproblem T (E)Φ = µΦsatisfy a maxmin characterization.

For every Φ 6= 0 the real equation f (E ; Φ) = 0 has unique solution p(Φ) whichis called Rayleigh functional at Φ. The eigenvalues of the quantum dotproblem can be characterized as minmax values of p(Hadeler (1968), V. & Werner(1982), V. (2009))

Quantum Dots

Properties ct.

Forf (E ; Φ) := 〈T (E)Φ,Φ〉 = Eb(Φ,Φ)− a(Φ,Φ; E)

it holdsf (0; Φ) < 0 < lim

E→∞f (E ; Φ) =∞ for every Φ 6= 0

∂∂E f (E ; Φ) > 0 for every Φ 6= 0 and E ≥ 0

For fixed E ≥ 0 the eigenvalues of the linear eigenproblem T (E)Φ = µΦsatisfy a maxmin characterization.

For every Φ 6= 0 the real equation f (E ; Φ) = 0 has unique solution p(Φ) whichis called Rayleigh functional at Φ. The eigenvalues of the quantum dotproblem can be characterized as minmax values of p(Hadeler (1968), V. & Werner(1982), V. (2009))

Nonlinear minmax characterization

Outline

1 Quantum Dots

5 Numerical example

Minmax characterization

The Schrödinger equation modelling the quantum dot with effective masshas a countable set of eigenvalues

0 < E1 ≤ E2 ≤ E3 ≤ . . .

the only cluster point of which is∞.

Ej = mindim V =j

maxΦ∈V ,Φ6=0

p(Φ) (∗).

E is the j smallest eigenvalue if and only if µ = 0 is the j largesteigenvalue of the linear eigenproblem

T (E)Φ = µΦ

The minimum in (*) is attained for the invariant subspace of T (Ej )corresponding to its j largest eigenvalues.

0 < E1 ≤ E2 ≤ E3 ≤ . . .

Ej = mindim V =j

maxΦ∈V ,Φ6=0

p(Φ) (∗).

T (E)Φ = µΦ

0 < E1 ≤ E2 ≤ E3 ≤ . . .

Ej = mindim V =j

maxΦ∈V ,Φ6=0

p(Φ) (∗).

T (E)Φ = µΦ

0 < E1 ≤ E2 ≤ E3 ≤ . . .

Ej = mindim V =j

maxΦ∈V ,Φ6=0

p(Φ) (∗).

T (E)Φ = µΦ

Discretization

(by FEM or FVM, e.g.) yields a nonlinear matrix eigenvalue problem

S(E)x := EMx − 1mq(E)

Aqx − 1mm(E)

Amx − Bx = 0

where M, Aq , Am and B are typically large and sparse.

If M is positive definite, and Aq , Am and B are positive semi–definite, and

1mq(0)

mm(0)Am + B

is positive definite, then the minmax characterization holds for S(E)x = 0.

If S(·) is obtained by orthogonal projection to some finite dimensionalsubspace V of H1

0 (Ω), then the eigenvalues of S(·) are upper bounds of thecorresponding eigenvalues of T (·).

Discretization

Aqx − 1mm(E)

Amx − Bx = 0

1mq(0)

mm(0)Am + B

Discretization

Aqx − 1mm(E)

Amx − Bx = 0

1mq(0)

mm(0)Am + B

Full Approximation Method

Outline

1 Quantum Dots

5 Numerical example

1: Start with initial energy level E0k

2: determine the effective masses mq(E0k ) and mm(E0

k ) for the quantum dotand the matrix

3: for n = 1,2, . . . until convergence do4: Determine the k :th smallest eigenvalue En

k and correspondingeigenfunction Φk of the linear eigenvalue problem

−∇ ·

2mj (En−1k )

)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm. (1)

with Ben Daniel–Duke condition5: Update the effective masses mj (En

k ), j ∈ m,q6: end for

Y. Li, O. Voskoboynikov, C.P. Lee, and S.M. Sze:Electron energy level calculations for cylindrical narrow gap semiconductorquantum dot. Comput.Phys.Comm. 141 (2001) 66

1: Start with initial energy level E0k

2: determine the effective masses mq(E0k ) and mm(E0

k ) for the quantum dotand the matrix

3: for n = 1,2, . . . until convergence do4: Determine the k :th smallest eigenvalue En

k and correspondingeigenfunction Φk of the linear eigenvalue problem

−∇ ·

2mj (En−1k )

)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm. (1)

with Ben Daniel–Duke condition5: Update the effective masses mj (En

k ), j ∈ m,q6: end for

Y. Li, O. Voskoboynikov, C.P. Lee, and S.M. Sze:Electron energy level calculations for cylindrical narrow gap semiconductorquantum dot. Comput.Phys.Comm. 141 (2001) 66

To analyze the iterative method of the last section we consider the lineareigenproblem

−∇ ·(

2mj (µ)∇Φ

)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm.

with interface condition and homogeneous Dirichlet condition on the outerboundary of Ωm, which depends on a nonnegative parameter µ.

The Rayleigh quotient

R(Φ;µ) :=a(Φ,Φ;µ)

b(Φ,Φ)

is monotonically decreasing with respect to µ ≥ 0 for every fixed Φ ∈ H10 (Ω),

Φ 6= 0.

Hence, it follows from the minmax characterization of the eigenvalues of thelinear eigenvalue problem that for every fixed k the mapping µ 7→ Ek (µ) ismonotonically decreasing with respect to µ.

−∇ ·(

2mj (µ)∇Φ

)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm.

b(Φ,Φ)

Φ 6= 0.

−∇ ·(

2mj (µ)∇Φ

)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm.

b(Φ,Φ)

Φ 6= 0.

Convergence behaviour

Assume that cj := gj − Vj > 0 for j ∈ m,q, and let Ek and λk (µ) be the k :thsmallest eigenvalue of the nonlinear Schrödinger equation and the parameterdependent equation, both with Ben Daniel–Duke condition on the interfaceΩq ∩ Ωm and homogeneous boundary condition on the outer boundary of Ωm.

Let E0 ≥ 0 be any initial value, and set En := λk (En−1) for n ∈ N. Then itholds

E0 < E2 ≤ · · · ≤ E2n−2 ≤ E2n ≤ Ek ≤ E2n+1 ≤ E2n−1 ≤ . . .E3 ≤ E1, (2)

limn→∞

En = Ek , (3)

and the convergence is linear, i.e. there exists a constant C, 0 < C < 1 andN ∈ N such that

|En − Ek | ≤ C|En−1 − Ek | for every n ∈ N, n ≥ N. (4)

Convergence behaviour

Assume that cj := gj − Vj > 0 for j ∈ m,q, and let Ek and λk (µ) be the k :thsmallest eigenvalue of the nonlinear Schrödinger equation and the parameterdependent equation, both with Ben Daniel–Duke condition on the interfaceΩq ∩ Ωm and homogeneous boundary condition on the outer boundary of Ωm.

Let E0 ≥ 0 be any initial value, and set En := λk (En−1) for n ∈ N. Then itholds

E0 < E2 ≤ · · · ≤ E2n−2 ≤ E2n ≤ Ek ≤ E2n+1 ≤ E2n−1 ≤ . . .E3 ≤ E1, (2)

limn→∞

En = Ek , (3)

and the convergence is linear, i.e. there exists a constant C, 0 < C < 1 andN ∈ N such that

|En − Ek | ≤ C|En−1 − Ek | for every n ∈ N, n ≥ N. (4)

Modified FAM

1: Start with initial energy level E0

2: Determine the effective masses mq(E0) and mm(E0) for the quantum dotand the matrix

3: for n = 1,2, . . . until convergence do4: Determine the k :th smallest eigenvalue En and corresponding

eigenfunction Φk of the linear eigenvalue problem

−∇ ·(

2mj (En−1)∇Φ

)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm. (5)

with Ben Daniel–Duke condition5: Determine the Rayleigh functional En =: p(Φk ) at Φk6: Update the effective masses mj (En), j ∈ m,q7: end for

Convergence of MFAM

The Modified Full Approximation Method (MFAM) is nothing else but the fixedpoint iteration En+1 = h(En) := p(Φk (En−1)), and since Φ(Ek ) is a stationaryelement of p it follows that h′(Ek ) = 0.

Hence, the Modified Full Approximation Method converges quadratically toEk , i.e. there exist some constant C > 0 such that

|En − Ek | ≤ C|En−1 − Ek |2 for every n ∈ N. (6)

Convergence of MFAM

The Modified Full Approximation Method (MFAM) is nothing else but the fixedpoint iteration En+1 = h(En) := p(Φk (En−1)), and since Φ(Ek ) is a stationaryelement of p it follows that h′(Ek ) = 0.

Hence, the Modified Full Approximation Method converges quadratically toEk , i.e. there exist some constant C > 0 such that

|En − Ek | ≤ C|En−1 − Ek |2 for every n ∈ N. (6)

Example 1: Conical QD

Consider a conical quantum dot with radius R0 = 10 nm and height z0 = 10nm, which is embedded in cylindrical matrix with Radius R1 = 40 nm andheight z1 = 30 nm.

Since the system is rotationally symmetric, the wave function can be written(in cylindrical coordinates) as Φ(x) = φ(r , z) exp(i`ϕ), where` = 0,±1,±2, . . . is the electron orbital quantum number.

Then the Schrödinger equation obtains the form

− ~2

2mj (E)

(1r∂

∂r(r∂

∂r) +

∂z2 −`2

)φ(r , z) + Vjφ(r , z) = Eφ(r , z), (7)

with (r , z) ∈ (0,40)× (0,30), and the Ben Daniel–Duke interface condition.

The boundary conditions are φ(r ,0) = φ(r ,30) = 0, 0 ≤ r ≤ 40, andφ(40, z) = 0, 0 ≤ z ≤ 30, and for continuity reasons ∂

∂r φ(0, z) = 0 for ` = 0and φ(0, z) = 0 for ` ≥ 1.

− ~2

2mj (E)

(1r∂

∂r(r∂

∂r) +

∂z2 −`2

)φ(r , z) + Vjφ(r , z) = Eφ(r , z), (7)

∂r φ(0, z) = 0 for ` = 0and φ(0, z) = 0 for ` ≥ 1.

− ~2

2mj (E)

(1r∂

∂r(r∂

∂r) +

∂z2 −`2

)φ(r , z) + Vjφ(r , z) = Eφ(r , z), (7)

∂r φ(0, z) = 0 for ` = 0and φ(0, z) = 0 for ` ≥ 1.

− ~2

2mj (E)

(1r∂

∂r(r∂

∂r) +

∂z2 −`2

)φ(r , z) + Vjφ(r , z) = Eφ(r , z), (7)

∂r φ(0, z) = 0 for ` = 0and φ(0, z) = 0 for ` ≥ 1.

Discretization

Discretization using FEMLAB by quadratic elements on a triangular net.dimension =6720 for ` = 0 and 6653 for ` ≥ 1.

Convergence history

Convergence

global ` k eigenvalue FAM MFAMCPU iterations CPU iterations

1 0 1 0.254585 6.76 10 2.97 42 1 1 0.384162 10.56 12 3.84 43 0 2 0.467239 13.91 13 4.77 44 2 1 0.503847 12.83 14 4.23 45 0 3 0.561319 19.28 14 6.32 46 1 2 0.598963 20.30 14 6.59 47 3 1 0.617759 16.05 14 4.91 48 0 4 0.688563 25.29 12 8.40 4

Iterative projection methods

Outline

1 Quantum Dots

5 Numerical example

For linear sparse eigenproblems

S(λ) = λB − A

very efficient methods are iterative projection methods (Lanczos, Arnoldi,Jacobi–Davidson method, e.g.), where approximations to the wantedeigenvalues and eigenvectors are obtained from projections of theeigenproblem to subspaces of small dimension which are expanded in thecourse of the algorithm.

Essentially two types of methods are in use:

methods which project the problem to a sequence of Krylov spaceslike Lanczos or Arnoldi, andmethods which aim at a specific eigenpair likethe Jacobi–Davidson method.

For linear sparse eigenproblems

S(λ) = λB − A

very efficient methods are iterative projection methods (Lanczos, Arnoldi,Jacobi–Davidson method, e.g.), where approximations to the wantedeigenvalues and eigenvectors are obtained from projections of theeigenproblem to subspaces of small dimension which are expanded in thecourse of the algorithm.

Essentially two types of methods are in use:

methods which project the problem to a sequence of Krylov spaceslike Lanczos or Arnoldi, andmethods which aim at a specific eigenpair likethe Jacobi–Davidson method.

Generalizations to nonlinear sparse eigenproblems

nonlinear rational Krylov: Ruhe (2000,2005)Regula falsi for the nonlinear problem and Arnoldi for the linear problemare knit together to form a sequence of subspaces Vk ∈ Cn andcorresponding Hessenberg matrices Hk which approximate the projectionof S(σ)−1S(λk ) on Vk for some shift σ close to the wanted eigenvalues.

Can be interpreted as projection method for S(σ)−1S(λ) (Jarlebring, V.(2005)) .

Arnoldi method: V. (2003, 2004); quadratic probl.: Meerbergen (2001)

Jacobi-Davidson for polynomial problems: Sleijpen, Boten, Fokkema, vander Vorst (1996), Hwang, Lin, Wang, Wang (2003,2004)for general problems: Betcke, V. (2004),

Expanding the subspace

Given subspace V ⊂ Rn . Expand V by a direction such that the expandedspace has a high approximation potential for the next wanted eigenvector.

Let θ be an eigenvalue of the projected problem

V T S(λ)Vy = 0

and x = Vy corresponding Ritz vector, then inverse iteration yieldssuitable candidate

v := S(θ)−1S′(θ)x

BUT: In each step have to solve large linear system with varying matrix

V T S(λ)Vy = 0

v := S(θ)−1S′(θ)x

V T S(λ)Vy = 0

v := S(θ)−1S′(θ)x

Iterative projection methods Nonlinear Arnoldi method

Outline

1 Quantum Dots

5 Numerical example

Way out: Residual inverse It.

1: start with an approximation x1 to an eigenvector of S(λ)x = 02: for ` = 1,2, . . . until convergence do3: solve xH

` S(µ`+1)x` = 0 for µ`+14: compute the residual r` = S(µ`+1)x`5: solve S(σ)d` = r`6: set x`+1 = x` − d`, x`+1 = x`+1/‖x`+1‖7: end for

Theorem (Neumaier 1985)

Let S(λ) be twice continously differentiable. Assume that λ is a simpleeigenvalue of S(λ)x = 0, and let x be a corresponding eigenvectornormalized by ‖x‖ = 1. Then the residual inverse iteration converges for all σsufficently close to λ, and it holds

‖x`+1 − x‖‖x` − x‖

= O(|σ − λ|), and |λ`+1 − λ| = O(‖x` − x‖2).

Way out: Residual inverse It.

1: start with an approximation x1 to an eigenvector of S(λ)x = 02: for ` = 1,2, . . . until convergence do3: solve xH

` S(µ`+1)x` = 0 for µ`+14: compute the residual r` = S(µ`+1)x`5: solve S(σ)d` = r`6: set x`+1 = x` − d`, x`+1 = x`+1/‖x`+1‖7: end for

Theorem (Neumaier 1985)

Let S(λ) be twice continously differentiable. Assume that λ is a simpleeigenvalue of S(λ)x = 0, and let x be a corresponding eigenvectornormalized by ‖x‖ = 1. Then the residual inverse iteration converges for all σsufficently close to λ, and it holds

‖x`+1 − x‖‖x` − x‖

= O(|σ − λ|), and |λ`+1 − λ| = O(‖x` − x‖2).

Arnoldi method

If θ is an eigenvalue of the projected problem V HS(λ)Vy = 0 and x = Vy is acorresponding Ritz vector, then expand V by new direction

v = x − S(σ)−1S(θ)x

In projection methods the new direction is orthonormalized against theprevious ansatz vectors. Since the Ritz vector x is contained in span V weexpand V by

v = S(σ)−1S(θ)x .

For the linear problem S(λ) = λI − A this is exactly the Cayley transformationor shifted-and-inverted Arnoldi method.Therefore the resulting iterative projection method is called nonlinear Arnoldimethod although no Krylov space is constructed and no Arnoldi recursionholds

Arnoldi method

v = x − S(σ)−1S(θ)x

v = S(σ)−1S(θ)x .

Arnoldi method

v = x − S(σ)−1S(θ)x

v = S(σ)−1S(θ)x .

Nonlinear Arnoldi Method

1: start with initial basis V , V HV = I; set k = m = 12: determine preconditioner M ≈ S(σ)−1, σ close to first wanted eigenvalue3: while m ≤ number of wanted eigenvalues do4: solve V HS(µ)Vy = 0 for (µ, y) and set u = Vy , rk = S(µ)u5: if ‖rk‖/‖u‖ < ε then6: Accept eigenpair λm = µ, xm = u,7: if m == number of wanted eigenvalues then STOP end if8: m = m + 19: if ‖rk−1‖/‖rk‖ > tol then

10: choose new pole σ, determine preconditioner M ≈ S(σ)−1

11: end if12: restart if necessary13: Choose approximations µ and u to next eigenvalue and eigenvector14: determine r = S(µ)u and set k = 015: end if16: v = Mr , k = k + 117: v = v − VV Hv ,v = v/‖v‖, V = [V , v ] and reorthogonalize if necessary18: end while

Comments

Comments 1:

Here preinformation on eigenvectors can be introduced into the algorithm(approximate eigenvectors of contigous problems in reanalysis, e.g.)

If no information on eigenvectors is at hand, and if we are interested ineigenvalues close to the parameter σ ∈ D:

choose initial vector at random, and execute a few Arnoldi steps for the lineareigenproblem S(σ)u = θu or S(σ)u = θS′(σ)u, and choose the eigenvectorcorresponding to the smallest eigenvalue in modulus or a small number ofSchur vectors as initial basis of the search space.

Starting with a random vector without this preprocessing usually will yield avalue µ in step 3 which is far away from σ and will avert convergence.

Comments 1:

Comments

Comments 4:

Since the dimension of the projected problem is small it can be solved by anydense solver like— methods based on characteristic function det S(λ) = 0— inverse iteration— residual inverse iteration— successive linear problems

Notice that symmetry properties that the original problem may have areinherited by the projected problem, and one can take advantage of theseproperties when solving the projected problem (safeguarded iteration,structure preserving methods).

Comments 4:

Since the dimension of the projected problem is small it can be solved by anydense solver like— methods based on characteristic function det S(λ) = 0— inverse iteration— residual inverse iteration— successive linear problems

Notice that symmetry properties that the original problem may have areinherited by the projected problem, and one can take advantage of theseproperties when solving the projected problem (safeguarded iteration,structure preserving methods).

Comments

Comments 9 – 11:

Residual inverse iteration converges (at least) linearly where the contractionrate satisfies

O(|σ − λ|).

Therefore, the preconditioner is updated if the convergence (measured by thereduction of the residual norm in the final step before convergence) hasbecome too large.

Comments

Comments 12:

As the subspaces expand in the course of the algorithm the increasingstorage or the computational cost for solving the projected eigenvalueproblems may make it necessary to restart the algorithm and purge some ofthe basis vectors.

Since a restart destroys information on the eigenvectors and particularly onthe one the method is just aiming at, we restart only if an eigenvector has justconverged.

Resonable search spaces after restart are— the space spanned by the already converged eigenvectors (or a space

slightly larger)— an invariant space of S(σ) or eigenspace of S(σ)y = µS′(σ)y

corresponding to small eigenvalues in modulus.

Comments 12:

Restarts

0 20 40 60 80 100 1200

iteration

CPU time

nonlinear eigensolver

Restarts

0 20 40 60 80 100 1200

iteration

CPU time

LU update

nonlin. solver

restart

Locking of converged eigenvectors

A major problem with iterative projection methods for nonlinear eigenproblemswhen approximating more than one eigenvalue is to inhibit the method fromconverging to an eigenpair which was detected already previously.

linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitean problems one can often take advantage of a variationalcharacterization of eigenvalues

linear problems: (incomplete) Schur decomposition

quadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitean problems one can often take advantage of a variationalcharacterization of eigenvalues

linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)

cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitean problems one can often take advantage of a variationalcharacterization of eigenvalues

linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problem

Approach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitean problems one can often take advantage of a variationalcharacterization of eigenvalues

linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determined

for Hermitean problems one can often take advantage of a variationalcharacterization of eigenvalues

linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitean problems one can often take advantage of a variationalcharacterization of eigenvalues

Safeguarded iteration

The minimum inλj = min

dim V =jmax

v∈V , v 6=0p(v)

is attained by the invariant subspace of S(λj ) corresponding to the j largesteigenvalues, and the maximum by every eigenvector corresponding to theeigenvalue 0. This suggests

1: Start with an approximation µ1 to the `-th eigenvalue of S(λ)x = 02: for k = 1,2, . . . until convergence do3: determine eigenvector u corresponding to the `-largest eigenvalue of

S(µk )4: evaluate µk+1 = p(u)5: end for

The minimum inλj = min

dim V =jmax

v∈V , v 6=0p(v)

is attained by the invariant subspace of S(λj ) corresponding to the j largesteigenvalues, and the maximum by every eigenvector corresponding to theeigenvalue 0. This suggests

1: Start with an approximation µ1 to the `-th eigenvalue of S(λ)x = 02: for k = 1,2, . . . until convergence do3: determine eigenvector u corresponding to the `-largest eigenvalue of

S(µk )4: evaluate µk+1 = p(u)5: end for

Convergence of safeguarded iteration

For ` = 1 the safeguarded iteration converges globally to λ1

If λ` is a simple eigenvalue then the (local) convergence is quadraticIf S′(λ) is positive definite and xk in Step 3 is replaced by an eigenvectorof

S(σk )x = µS′(σk )x

corresponding to the `-th largest eigenvalue, then the convergence iseven cubic.A variant exists which is globally convergent also for higher eigenvalues.

If λ` is a simple eigenvalue then the (local) convergence is quadratic

If S′(λ) is positive definite and xk in Step 3 is replaced by an eigenvectorof

corresponding to the `-th largest eigenvalue, then the convergence iseven cubic.

A variant exists which is globally convergent also for higher eigenvalues.

Iterative projection methods Jacobi–Davidson method

Outline

1 Quantum Dots

5 Numerical example

Alternative expansion of subspaceInverse iteration x = S′(θ)−1S(θ)x

For truly large problems v is not availabe, but has to be replaced by an inexactsolution v + e of S(θ)v = S′(θ)x .

Alternative expansion of subspaceInverse iteration x = S′(θ)−1S(θ)x

For truly large problems v is not availabe, but has to be replaced by an inexactsolution v + e of S(θ)v = S′(θ)x .

Alternative expandsion of subspace

Since v is used to expand the search space and the next iterate is obtainedfrom the projection to the expanded space, v can be replaced by (an inexactversion of) any linear combination x + αS(θ)−1S′(θ)x , α ∈ R.

Jacobi–Davidson expansion

It can be shown (V. 2007) that the expansion of the search space by aninexact realization of t := x + αv is most robust with respect to smallperturbations, if α is chosen such that x and x + αv are orthogonal.

Hence,

t = x − ‖x‖2

xT S(θ)−1S′(θ)xS(θ)−1S′(θ)x ,

and this is easily seen to be the solution of the correction equation

(I − S′(θ)xxT

xT S′(θ)x)S(θ)(I − xxT )t = −S(θ)x , t ⊥ x .

For linear eigenproblems the corresponding iterative projection method wasintroduced by Sleijpen and van der Vorst (1996) and is calledJacobi–Davidson method.Polynomial eigenproblems: Sleijpen, Boten, Fokkema, van der Vorst (1996),general nonlinear eigenproblems: Betcke, V. (2004)

Hence,

t = x − ‖x‖2

(I − S′(θ)xxT

Hence,

t = x − ‖x‖2

(I − S′(θ)xxT

For linear eigenproblems the corresponding iterative projection method wasintroduced by Sleijpen and van der Vorst (1996) and is calledJacobi–Davidson method.

Polynomial eigenproblems: Sleijpen, Boten, Fokkema, van der Vorst (1996),general nonlinear eigenproblems: Betcke, V. (2004)

Hence,

t = x − ‖x‖2

(I − S′(θ)xxT

Nonlinear Jacobi-Davidson

1: Start with orthonormal basis V ; set m = 12: determine preconditioner M ≈ S(σ)−1; σ close to first wanted eigenvalue3: while m ≤ number of wanted eigenvalues do4: compute eigenpair (µ, y) of projected problem V T S(λ)Vy = 0.5: determine Ritz vector u = Vy , ‖u‖ = 1,and residual r = S(µ)u6: if ‖r‖ < ε then7: accept approximate eigenpair λm = µ, xm = u; increase m← m + 18: reduce search space V if necessary9: choose new preconditioner M ≈ S(µ) if indicated

10: choose approximation (λm, u) to next eigenpair, and compute r = S(λm)u11: end if12: solve approximately correction equation(

I − S′(µ)uuH

uHS′(µ)u

)S(µ)

(I − uuH

)t = −r , t ⊥ u

13: t = t − VV T t ,v = t/‖t‖, reorthogonalize if necessary14: expand search space V = [V , v ]15: update projected problem16: end while

Solving correction equation

The correction equation is solved approximately by a few steps of an iterativesolver (GMRES or BiCGStab).

The operator S(σ) is restricted to map the subspace x⊥ into itself. Hence, ifM ≈ S(σ) is a preconditioner of S(σ), σ ≈ µ, then a preconditioner for aniterativ solver of the correction equation should be modified correspondingly to

M := (I − S′(µ)xxH

xHS(µ)x)M(I − xxH

Taking into account the projectors in the preconditioner, i.e. using M instead ofM, raises the cost of the preconditioned Krylov solver only slightly (cf.Sleijpen, van der Vorst). Only one additional linear solve with system matrix Mis required.

M := (I − S′(µ)xxH

Variant of Hwang, Lin, Wang, Wang (2004)

Solve (I − S′(µ)xxH

xHS′(µ)x

)S(µ)

(I − xxH

)t = −r , t ⊥ u

approximately by computing

t = M−1r + αM−1S′(µ)u with α :=xHM−1r

xHM−1S′(µ)x

where M is a preconditioner of S(µ).

Method combines preconditioned Arnoldi method (M−1r ) and simplifiedinverse iteration (M−1S′(µ)x)

Variant of Hwang, Lin, Wang, Wang (2004)

Solve (I − S′(µ)xxH

xHS′(µ)x

)S(µ)

(I − xxH

)t = −r , t ⊥ u

approximately by computing

t = M−1r + αM−1S′(µ)u with α :=xHM−1r

xHM−1S′(µ)x

where M is a preconditioner of S(µ).

Method combines preconditioned Arnoldi method (M−1r ) and simplifiedinverse iteration (M−1S′(µ)x)

Numerical example

Outline

1 Quantum Dots

5 Numerical example

Numerical example

pyramidal quantum dot: baselength: 12.4 nm, height: 6.2 nmcubic matrix: 24.8× 24.8× 18.6 nm3

Parameters (Hwang, Lin, Wang, Wang 2004)P1 = 0.8503, g1 = 0.42, δ1 = 0.48, V1 = 0.0P2 = 0.8878, g2 = 1.52, δ2 = 0.34, V1 = 0.7

Discretization by FEM or FVM yields rational eigenproblem

S(λ)x = λMx − 1m1(λ)

A1x − 1m2(λ)

A2x − Bx = 0

where S(λ) is symmetric and satisfies conditions of minmax characterizationfor λ ≥ 0

All timings for MATLAB 7.0.4 on AMD Opteron Processor 248× 860_64 with2.2 GHz and 4 GB RAM

Numerical example

A1x − 1m2(λ)

A2x − Bx = 0

Numerical example

A1x − 1m2(λ)

A2x − Bx = 0

Numerical example

A1x − 1m2(λ)

A2x − Bx = 0

Numerical example

Numerical example ct.

FVM: Hwang, Lin, Wang, Wang (2004)

dim λ1 λ2/3 λ4 λ5 CPU time2′475 0.41195 0.58350 0.67945 0.70478 0.68 s

22′103 0.40166 0.57668 0.68418 0.69922 8.06 s186′543 0.39878 0.57477 0.68516 0.69767 150.92 s

1′532′255 0.39804 0.57427 0.68539 0.69727 4017.67 s12′419′775 0.39785 0.57415 overnight

FEM: Cubic Lagrangian elements on tetrahedal grid

dimension: 96’640 ((DofQD,Dofmat,Dofinterf) = (43′615,43′897,9′128)

dim λ1 λ2 λ3 λ4 λ5 CPU time96′640 0.39779 0.57411 0.57411 0.68547 0.69714Arnoldi 44 it. 29 it. 29 it. 24 it. 21 it. 188.8 s

JD 15 it. 9 it. 1 it. 7 it. 7 it. 204.4 sHLWW 45 it. 49 it. 5 it. 24 it. 21 it. 226.7 s

Numerical example

Numerical example ct.

FVM: Hwang, Lin, Wang, Wang (2004)

dim λ1 λ2/3 λ4 λ5 CPU time2′475 0.41195 0.58350 0.67945 0.70478 0.68 s

22′103 0.40166 0.57668 0.68418 0.69922 8.06 s186′543 0.39878 0.57477 0.68516 0.69767 150.92 s

1′532′255 0.39804 0.57427 0.68539 0.69727 4017.67 s12′419′775 0.39785 0.57415 overnight

FEM: Cubic Lagrangian elements on tetrahedal grid

dimension: 96’640 ((DofQD,Dofmat,Dofinterf) = (43′615,43′897,9′128)

dim λ1 λ2 λ3 λ4 λ5 CPU time96′640 0.39779 0.57411 0.57411 0.68547 0.69714Arnoldi 44 it. 29 it. 29 it. 24 it. 21 it. 188.8 s

JD 15 it. 9 it. 1 it. 7 it. 7 it. 204.4 sHLWW 45 it. 49 it. 5 it. 24 it. 21 it. 226.7 s

Numerical example

Convergence history

Numerical example

Preconditioning

incomplete LU with cut-off threshold τ

τ JD Arnoldi HLWW precond.0.1 261.4 1084.1 1212.4 3.4

0.01 132.7 117.1 155.7 71.70.001 118.9 61.2 96.0 246.6

0.0001 155.6 46.6 71.1 665.6

Sparse approximate inverse

τ JD Arnoldi HLWW precond.0.4 968.6 3105.7 4073.4 314.00.3 819.9 2027.6 2744.7 641.20.2 718.1 1517.5 2157.2 1557.00.1 694.9 1461.5 2124.4 1560.9

Numerical example

Preconditioning

incomplete LU with cut-off threshold τ

τ JD Arnoldi HLWW precond.0.1 261.4 1084.1 1212.4 3.4

0.01 132.7 117.1 155.7 71.70.001 118.9 61.2 96.0 246.6

0.0001 155.6 46.6 71.1 665.6

Sparse approximate inverse

τ JD Arnoldi HLWW precond.0.4 968.6 3105.7 4073.4 314.00.3 819.9 2027.6 2744.7 641.20.2 718.1 1517.5 2157.2 1557.00.1 694.9 1461.5 2124.4 1560.9

Quantum dot on wetting layer

Outline

1 Quantum Dots

5 Numerical example

Stranski-Krastanow growth

Quantum dots can be produced today by the Stranski–Krastanov processwhich uses the relief of the elastic energy when two materials with a largelattice mismatch form an epitaxial structure.

The deposited layer initially grows as a thin two dimensional (2D) wettinglayer. As the deposited layer exceeds a critical thickness, the growth modeswitches from 2D to 3D leading to the formation of self-assembled quantumdots on top of the wetting layer.

Stranski–Krastanov

Actually one has to consider arrays of quantum dots, and even more generallyarrays of quantum dot/wetting layers.

Quantum dot on wetting layerAgain, we have to solve the Schrödinger equation, but the domain of the QDhas to be expanded the QD plus wetting layer.

1. Wave function

2. Wave function

4. Wave function

Generalization

Numerical simulation of electronic properties of coupled quantum dots onwetting layers require the same techniques. The Rayleigh functional isdefined on the entire space. (M.M. Betcke & V. 2008)

Incorporating spin orbit splitting yields an additional term in the Schrödingerequation, such that the Rayleigh functional p is only defined on a subspace.However, the range of p is big enough that the relevant energy levels arecaptured, and iterative projection methods apply (M.M. Betcke & V. 2007)

Incorporating the influence of a magnetic field yields a more complicatedmagnetic one-band effective Hamiltonian, but still the relevant energy levelsallow for a minmax characterization (M.M. Betcke & V. 2012).

Generalization

Conclusions

Electron energy levels of semiconductors and corresponding wavefunctions satisfy a rational eigenvalue problem.

For pure quantum dots we demonstrated that they can be determinedefficiently by iterative projection methods combined with safeguardediteration.The same approach applies to quantum dot/wetting layer structures andto arrays of such structures. Spin orbit interaction and magnetic effectscan be included (joint work with Marta Betcke)

Conclusions

Electron energy levels of semiconductors and corresponding wavefunctions satisfy a rational eigenvalue problem.For pure quantum dots we demonstrated that they can be determinedefficiently by iterative projection methods combined with safeguardediteration.

The same approach applies to quantum dot/wetting layer structures andto arrays of such structures. Spin orbit interaction and magnetic effectscan be included (joint work with Marta Betcke)

Conclusions

Electron energy levels of semiconductors and corresponding wavefunctions satisfy a rational eigenvalue problem.For pure quantum dots we demonstrated that they can be determinedefficiently by iterative projection methods combined with safeguardediteration.The same approach applies to quantum dot/wetting layer structures andto arrays of such structures. Spin orbit interaction and magnetic effectscan be included (joint work with Marta Betcke)

References

M.M. Betcke & H. Voss: Stationary Schrödinger equations governing electronicstates of quantum dots in the presence of spin-orbit splitting. Appl. Math. 52, 267– 284 (2007)

M.M. Betcke & H. Voss: Numerical simulation of electronic properties of coupledquantum dots on wetting layers. Nanotechnology 19, 165204 (2008)

M.M. Betcke & H. Voss: Analysis and efficient solution of stationary Schrödingerequation governing electronic states of quantum dots and rings in magnetic field.Commun.Comput.Phys. 12, 1591 – 1617 (2012)

T. Betcke & H. Voss. A Jacobi–Davidson–type projection method for nonlineareigenvalue problems. Future Generation Comput. Syst. 20, 363-372 (2004)

K.P. Hadeler: Variationsprinzipien bei nichtlinearen Eigenwertaufgaben. Arch.Ration. Mech. Anal. 30, 297-307 (1968)

T.-M. Hwang, W.-W. Lin, W.-C. Wang & W. Wang: Numerical simulation of threedimensional quantum dot. J. Comput. Phys. 196, 208-232 (2004)

E. Jarlebring & H. Voss: Rational Krylov for nonlinear eigenproblems, an iterativeprojection method. Appl. Math. 50, 543-554 (2005)

K. Meerbergen: Locking and restarting quadratic eigenvalue solvers. SIAM J. Sci.Comput. 22, 1814-1839 (2001)

References ct.

A. Neumaier: Residual inverse iteration for the nonlinear eigenvalue problem.SIAM J. Numer. Anal. 22, 914-923 (1985)A. Ruhe: Rational Krylov for large nonlinear eigenproblems. pp. 357 – 363 in J.Dongarra, K. Madsen & J. Wasniewski (eds.): Applied Parallel Computing. Stateof the Art in Scientific Computing. Lect, Notes Comput. Sc. 3732, Springer 2006G.L. Sleijpen, G.L. Booten, D.R. Fokkema & H.A. van der Vorst: Jacobi-Davidsontype methods for generalized eigenproblems and polynomial eigenproblems. BITNumer. Math. 36, 595-633 (1996)G.L. Sleijpen & H.A. van der Vorst: A Jacobi-Davidson iteration method for lineareigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 401-425 (1996)H. Voss: A maxmin principle for nonlinear eigenvalue problems with application toa rational spectral problem in fluid–solid vibration. Appl. Math. 48, year = 2003,607 – 622 (2003)H. Voss: An Arnoldi method for nonlinear symmetric eigenvalue problems. OnlineProceedings SIAM Conf. on Applied Linear Algebra, Williamsburg 2003H. Voss: An Arnoldi method for nonlinear eigenvalue problems. BIT Numer. Math.44, 387-401 (2004)H. Voss: Electron energy level calculation for quantum dots. Comput. Phys.Comm. 174, 441-446 2006

References ct.

H. Voss: Iterative projection methods for computing relevant energy states of aquantum dot. J. Comput. Phys. 217, 824-833 (2006)

H. Voss: A minmax principle for nonlinear eigenproblems depending continuouslyon the eigenparameter. Numer. Lin. Algebra Appl. 16, 899-913 (2009)

H. Voss & B. Werner: A minimax principle for nonlinear eigenvalue problems withapplications to nonoverdamped systems. Math. Meth. Appl. Sci. 4, 415-424(1982)

Numerical simulation of electronic properties of quantum dotsTherefore, a quantum dot can be thought...

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