Si CoreâShell Nanocrystal: An

Article pubs.acs.org/JPCC

Laser-Driven Hole Trapping in a Ge/Si Core−Shell Nanocrystal: An Atomistic Configuration Interaction Perspective G. Hermann and J. C. Tremblay* Institut für Chemie und Biochemie, Freie Universität Berlin, Takustraße 3, 14195 Berlin, Germany ABSTRACT: This contribution reports on the first example of laser-driven charge carrier confinement in a solid state quantum dot investigated using a fully atomistic, correlated many-electron ansatz. Specifically, a Ge/Si model nanocrystal is designed to retain the main structural characteristics and excitonic properties of experimentally observed self-assembled pyramidal heterostructures. State-selective laser excitations yielding hole confinement in the Ge nanostructure are simulated using the reduced density matrix variant of the time-dependent configuration interaction method (ρ-TDCI). The degree of carrier localization in the quantum dot is determined by analyzing the correlated one-electron densities from configuration interaction electronic states at a singles level. Additionally, dissipation and pure dephasing are included to treat the coupling of the local core−shell structure with the vibrations of the surrounding silicon matrix. For this purpose, a new microscopic model for these nonadiabatic coupling rates is derived. The results reveal that, despite the presence of dissipation, charge carriers can be efficiently confined by localized optical excitations in the model Ge/Si quantum dot to create long-lived, large permanent dipoles in the nanocrystal.

■

with different computational methods.1 The theoretical studies of Ge/Si heterostructures in the form of solid-state materials or large clusters are chiefly treated with scalable approximate techniques such as the effective mass approximation,21 the pseudopotential method,22 and the k·p perturbation formalism.23 For the freestanding Ge/Si nanocrystals, small molecular cluster models are constructed and described with more accurate methods including the density functional theory (DFT) within the local density approximation (LDA)24 and the generalized gradient approximation (GGA),25 the timedependent DFT (TD-DFT),26 and the tight-binding method.27 Since we are interested in the microscopic structure of Ge/Si quantum dots, we advocate a freestanding model of a Ge/Si quantum dot retaining its principal structural features. An important extension to the existing theoretical studies is the inclusion of electronic correlation, which can be described using wave-function-based ab initio methods. Molecular electronic structure calculations at this methodological level scale steeply with the size of the system. Consequently, a size-reduced Ge/Si model quantum dot with a core−shell type structure is constructed, which is composed of a small Ge impurity surrounded by a few Si shells mimicking the Si bulk. Recently, de Oliveira et al. studied small Si/Ge and Ge/Si core/shell nanocrystals employing a configuration interaction formalism28,29 to calculate their exciton structure.30 It was found that this method is an appropriate choice to reproduce the crucial optical features of our Ge/Si heterostructure.

INTRODUCTION In the last decades, the fundamental physics of semiconductor nanocrystals (quantum dots) has been extensively investigated due to their unique properties, which are eminently different compared to their bulk counterparts.1−3 One essential characteristic is the substantial tunability of their material properties by varying the size, morphology, or chemical composition of these zero-dimensional structures. In particular, their natural tendency to confine charge carriers in all three spatial dimensions qualifies them for advanced applications such as light emitting/absorbing devices. Consequently, much research effort has been devoted to the design, fabrication, and characterization of semiconductor heterostructures.4 Especially nanostructures composed of self-assembled Ge quantum dots embedded in a Si host matrix are promising candidates of this material class. These type-II semiconductor heterostructures stand out by their large valence-band offset (∼0.7 eV), which engenders an electron hole localization in the Ge regions and an electron confinement in the Si layers.5,6 Despite the difficulties due to the lattice mismatch between Si and Ge, multiple fabrication methods7−10 have been developed to enable the control of the composition, size, and growth of these Ge/Si systems. Meanwhile, considerable progress has been achieved in the detection and analysis of the spatial quantum confinement of charge carriers and optical transitions in Ge/Si heterostructures11−15 for potential applications such as light-emitting diodes,16,17 solar cells,18,19 or quantum computers.20 From a structural point of view, two theoretical model classes of semiconductor quantum dots can be distinguished embedded and freestanding quantum dotswhich are tackled © XXXX American Chemical Society

Received: August 14, 2015 Revised: October 21, 2015

A

DOI: 10.1021/acs.jpcc.5b08606 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

Hamiltonian in a basis of singly excited configuration state CIS functions to obtain the eigenstates ΨCIS λ and eigenvalues Eλ of the system within the variation principle. The λth CIS wave function consists of a linear combination of the Hartree−Fock ground state Slater determinant Ψ0HF and singly excited configuration state functions Ψra, which for a closed-shell system reads

Consequently, we follow a similar approach to compute the excitonic properties of a Ge/Si model quantum dot using standard configuration interaction methodology at a single excitation level (CIS). We propose a timely extension to the existing theoretical studies in the form of time-resolved excitation dynamics to achieve a more precise description of the transient electron dynamics. The electron dynamics is studied using the reduced density matrix variant of the timedependent configuration interaction method (ρ-TDCI), which is designed to study the time evolution of a many-electron density coupled to arbitrary environments.31−33 This allows seamless incorporation of the effect of energy and phase relaxation on the electron dynamics due to the contact with the system vibrations. A perturbative microscopic model for the nonadiabatic coupling to the internal vibrations and the external phonons of the silicon matrix is derived to determine the first-order rates associated with energy relaxation and pure dephasing in the electronic density propagation. The objective of this paper is the investigation of charge carrier confinement during the laser-controlled excitation of selected states in a model Ge/Si quantum dot. For this purpose, various definitions of atomic charges are used to evaluate the one-electron densities of the correlated CIS eigenstates. From this static analysis, we can identify interlevel transition series that exhibit a high propensity for the spatial localization of holes in the quantum dot and electrons below, leading to the creation of a large permanent dipole in the nanocrystal with potential application for information storage. The focus of this work is to present new methodologies that allow studying laserdriven many-electron dynamics in semiconductor nanostructures using an atomistic configuration interaction ansatz, while including the effects of lattice vibrations. The toy system is chosen to retain the characteristics of larger, experimentally relevant nanostructures. This paper is organized as follows. In the Theory section, the ρ-TDCI method is summarized and novel rate expressions for the treatment of dissipation induced by nonadiabatic coupling are derived. Subsequently, the properties of the Ge/Si model quantum dot are presented along with the methodologies to specify its electronic structure. In Results and Discussion, the results of the quantum chemical calculations and the laserdriven electron dynamics are presented and discussed before the most significant results are summarized in the Conclusion.

N /2

|ΨCIS λ ⟩

1 Ĥ el = − 2

NA

∑ ∇i 2 − ∑ ∑ i=1

i=1 A=1

ZA + riA

N

N

∑∑ i=1 j>i

1 rij

Dar(λ)|Ψ ra⟩

∑ ∑

(2)

r(λ) where {D(λ) 0 ,Da } are expansion coefficients. The indices L and M refer to the lowest occupied and highest unoccupied orbitals included in the generation of the singly excited configuration state function basis, respectively. These singly excited configuration state functions are spin-adapted linear combinations of Slater determinants formed by moving an electron from an occupied molecular orbital a to an unoccupied orbital r from the reference state ΨHF 0 . Equation of Motion and Field. The laser-induced manyelectron dynamics in the presence of energy and phase relaxation is treated within the reduced density matrix formalism. The time propagation of the reduced density operator ρ̂(t) is carried out by solving the Liouville−von Neumann equation

∂ρ ̂(t ) = −ι[Ĥ , ρ (̂ t )] + 3 Dρ (̂ t ) ∂t

(3)

In the present context, the reduced density operator is represented in terms of CIS eigenfunctions ρ ̂( t ) =

CIS ∑ ρλν |ΨCIS λ ⟩⟨Ψ ν |

(4)

λν

with ρλν as time-dependent expansion coefficients. The dissipative Liouvillian 3 D describes the dissipation of energy and phase due to the contact with an environment. The choice of restricting the CI expansion to singly excited configuration state functions is physically justified to represent the dynamics of all one-electron processes such as the ones investigated here. In order to simulate laser-driven excitations, the total timedependent electronic Hamiltonian Ĥ (t) is first defined as the sum of the time-independent electronic Hamiltonian Ĥ el and the semiclassical interaction between the molecular system and the laser field F(t)

THEORY ρ-TDCI. Configuration Interaction Method. In the present work, we describe the laser-driven many-electron dynamics in a dissipative environment using the time-dependent configuration interaction method34,35 in its reduced density matrix form (ρ-TDCI).31−33 First, the relevant time-independent electronic structure method is briefly established, namely configuration interaction singles (CIS).36 To start with, the field-free electronic Hamiltonian for a system with N electrons and NA nuclei is written in the clamped nuclei approximation as (ℏ = 4πϵ0 = me = e = 1) N

+

M

a = L r = N /2 + 1

■

N

=

D0(λ)|Ψ0HF⟩

̂ t) Ĥ = Ĥ el − μF(

(5) N

where μ̂ = −∑Ni ri + ∑A A ZARA is the molecular dipole operator. The electric field is expressed as a series of individual pulses with a sin2 envelope F (t ) =

∑ fl(t ) cos(ωl(t − tp,l)) l

⎧ ⎛ π (t − t ) ⎞ l ⎟ ⎪ ⎪ f 0, l sin 2⎜⎜ ⎟ if tl < t < tl + t p , l fl(t ) = ⎨ t ⎠ ⎝ p,l ⎪ ⎪ ⎩0 otherwise

(1)

where rij is the distance between electrons i and j, riA labels the distance between the ith electron and the Ath nucleus, and ZA designates the charge of nucleus A. The basic idea of the CIS approach relies on the diagonalization of the N-electron

(6)

where tp,l (tl) indicates the duration (starting time) of the lth pulse with carrier frequency ωl. An initial guess for the pulse amplitudes is obtained from the π-pulse condition. Under the rotating wave approximation, π-pulses lead to a complete B


Article


1. Energy Relaxation Rates. Approximating the vibronic wave functions, {|n,λ⟩, |n′,ν⟩}, as factorizable allows rewriting the energy relaxation rates for a given mode q as

population inversion in an idealized isolated two-level system. Accordingly, the amplitude of the lth pulse has the form f 0, l =

2π t p , l|μλν |

(7)

1 (q) Γλ → ν

with μλν as the transition dipole moment between the electronic states λ and ν. A more detailed description of the ρ-TDCI methodology and its equations of motion is presented elsewhere.31 Treatment of Dissipation. A convenient model for the description of dissipation is the Lindblad semigroup formalism,37 which ensures semipositivity of the reduced density operator and, thus, enables a probabilistic interpretation of its diagonal elements as population of the CIS eigenstates. The dissipative Liouvillian from eq 3 in Lindblad representation reads 3 Dρ (̂ t ) = −

1 2

†

k

1 (q) CIS Γλ → ν |ΨCIS λ ⟩⟨Ψ ν |

2π ℏ

(8)

nn ′

2

∂ ⟨n| |n′⟩ ∂q

(9)

∂ ⟨λ| |ν⟩ ∂q

Mqωq

=

0

2ℏ Mqωq

=

(14)

2ℏ

|⟨n|aq|n′⟩ − ⟨n|aq†|n′⟩|2 (15)

0

|n′δnn ′− 1 − (n′ + 1)δnn ′+ 1|2

For the relaxation process, only the first term on the righthand side contributes: Eλ > Eν → En′ > En, where the transition frequency is defined as positive ωn′n > 0. It follows that 1 (q) Γλ → ν

∂2 |n′, ν⟩ ρn (1 − ρn ′) ∂q2

(10)

2 2π ℏ3 ⎛ Mqωq0 ⎞ ∂ ⎜ ⎟ ≅ ⎜ ⎟ ⟨λ| |ν⟩ ∂q Mq2 ⎝ 2ℏ ⎠

≅

∑ (n + 1) n

π ℏ2ωq

0

Mq

∂ ⟨λ| |ν⟩ ∂q

(16)

2

∑ (n + 1) n

ρn (1 − ρn + 1) δ(ΔEλν − ℏωq ) 0

δ(En , λ − En ′ , ν)

To represent the delta distribution, we choose a normalized Lorentzian function

where {n,n′} and {λ,ν} are respectively vibrational and electronic states, Tq is the nuclear kinetic energy operator associated with coordinate q of mass Mq, and ρn is a thermal distribution function. The first order rate expression can be expressed as follows

δ(ΔEλν − ℏωq ) = 0

γq/2 1 π (ΔEλν − ℏωq )2 + (γq/2)2 0

q) 2 (q) Γ(λq→) ν = 1Γλ(→ ν + Γλ δλν

(11)

=

in which the first term on the right-hand side is related to energy relaxation appearing in eq 9. The separation reveals an elastic contribution to second order leading to pure dephasing, for which the Lindblad operator takes the following simplified form Ĉ k →

2 (q) CIS Γλ |ΨCIS λ ⟩⟨Ψ λ |

(13)

2

ρn (1 − ρn + 1) δ(ΔEλν − ℏωnn + 1) 2

⟨n , λ|

nn ′

2

Substituting the definition in the expression for the energy relaxation rates yields

nn ′

∑

∑

∂ ⟨n| |n′⟩ ∂q

⎛ Mqωq ⎞1/2 ∂ 0 ⎟⎟ (aq − aq†) = ⎜⎜ ∂q ⎝ 2ℏ ⎠

∑ |⟨n , λ|Tq|n′, ν⟩|2 ρn (1 − ρn′)

π ℏ3 2Mq 2

nn ′

∂ ∂ |n′⟩⟨λ| |ν⟩ ρn (1 − ρn ′) ∂q ∂q

In the second quantization, the gradient operator takes the form

δ(En , λ − En ′ , ν) =

⟨n|

ρn (1 − ρn ′) δ(ΔEλν − ℏωnn ′)

The associated dissipation rates 1Γλ→ν define the state-to-state lifetimes τλ→ν = (1Γλ→ν)−1. For electronic systems in the rigid nuclei environments, transitions leading to energy relaxation are induced by nonadiabatic coupling to small harmonic nuclear displacements. The rates from the different channels are treated additively, and the total rate in eq 9 is given by 1Γλ→ν = ∑q Γ(q) λ→ν. In the weak coupling regime, each single nonadiabatic contribution to the electron dynamics can be evaluated using first order time-dependent perturbation theory Γ(λq→) ν =

∑

2π ℏ3 ≅ Mq 2

where Ĉ k is a Lindblad operator representing the kth dissipation channel. To investigate the effect of energy relaxation, raising/ lowering operators as Lindblad operators are used Ĉ k →

2

2π ℏ3 Mq 2

δ(Eλ − Eν − ℏωnn ′)

†

∑ ([Ĉkρ (̂ t ), Ĉk ] + [Ĉk , ρ (̂ t )Ĉk ])

≅

2 π ℏωq

ΔEλν =ℏωq0

(17)

0

where the width is set to the vibrational transition energy, γg = ℏωq0. Employing the notation ⟨λ|

(12)

Closed expressions for both rates will be derived in the following sections.

∂ |ν ⟩ = ∂q

∂ ∂q

λν

eq 16 becomes C


Article

The Journal of Physical Chemistry C 2 ∂ ∂q

2

1 (q) Γλ → ν

≅

2π ℏ ωq

0

Mq

λν

π ℏωq

2 (q) Γλ → ν

≅

⟨n⟩(T )

≅

∂ ∂q

0

⟨n⟩(T )

≅

λν

2ℏ |⟨Hel(q)⟩λν |2 ⟨n⟩(T ) 2 Mq |ΔEλν|

2ℏ |⟨Hel(q)⟩|2 ⟨n⟩(T ) Mq |ΔEλν|2

(19)

∂2 ∂q2

1 2ℏ = |⟨Hel(q)⟩|2 τq Mq(ℏωq )2 0

2τq

≅

ℏωq

2 0

ΔEλν

2 (q) Γλ → λ

≅

2

δλν λλ

∂ ∂ |ν⟩⟨ν| |λ⟩ ∂q ∂q ∂Hel ∂H |ν⟩⟨ν| ∂qel |λ⟩ ∂q

(25)

ΔEλνΔEνλ

ℏ2 |⟨Hel(q)⟩|4 | ∑ ΔEλν−2|2 Mq 2ωq ν 0

(21)

1 ≅ 4ωq τq 2 0

0

⎛ ℏωq ⎞2 ∑ ⎜⎜ 0 ⎟⎟ ΔEλν ⎠ ν ⎝

2

(26)

where the last line is obtained by using the definition of the average coupling strength, eq 22. The pure dephasing rate does not contribute to energy relaxation and is seen to scale with the fourth power of the inverse energy differences in the system. Note that, contrary to the relaxation rates, the pure dephasing rates are independent of temperature. Here again, the contribution from different phonons is treated as additive. Due to their common origin, both relaxation and dephasing rates are scaled using the same phonon frequencies and lifetimes. As pure dephasing exhibits an inverse quartic energy dependence, these second-order contributions are expected to have only a minor impact on the dynamics. Model System. Germanium/Silicon Quantum Dot. A prerequisite to the current dynamical study is the construction of a size-reduced model for a Ge/Si quantum dot (QD) which mainly retains the main characteristics of these systems, in particular the local electronic structure and the charge confinement. Since we advocate employing wave-functionbased ab initio methods in the form of configuration interaction method, the model system is limited to a rather small number of atoms. Experimentally studied Ge/Si quantum dots are generally large self-assembled Ge islands grown on a Si(001) surface and embedded in a Si host matrix. Despite the high lattice mismatch and miscibility between Ge and Si, welldefined Ge/Si quantum dots are observed with various growth techniques.11,15 In previous theoretical studies, a microscopic core−shell type system proved to be an appropriate model

(22)

⟨n⟩(T ) 0

(24)

Again, using the average nonadiabatic coupling strength and inserting the result in eq 24 yields

0

τq

∑ ν

Substituting in eq 20 yields 1 (q) Γλ → ν

0

ν

λλ

=

2

0

∂2 ∂q2

∑ ⟨λ|

=

(20)

This in turns allows for a unique definition of the average nonadiabatic coupling strength

Mqℏωq

π ℏ3 ∂2 ⟨λ| 2 |ν⟩ δ(ΔEλν) ∑ ρn (1 − ρn ) 2 2Mq ∂q n

⟨λ|

For a reference energy equal to that of a phonon at the Γ-point, ΔEλν = ℏωq0, the transition rate at 0 K (⟨n⟩(T=0) = 1) can be set equal to the inverse reference lifetime for a given mode

|⟨Hel(q)⟩|2 =

∂2 |ν⟩ ρn (1 − ρn ′) ∂q2

where the continuous limit of ∑n ρn(1 − ρn) → ∫ dn ρn(1 − ρn) = 1 is taken for simplicity, as well as the same definition for the delta function, eq 17, with ΔEλν = 0. The second derivative with respect to the nuclear coordinate can be resolved by inserting the complete basis of electronic eigenstates and using the Hellmann−Feynman theorem

The thermal distribution factor is defined as ⟨n⟩(T) = ∑n (n + 1)ρn(1 − ρn+1) . Replacing the nonadiabatic coupling strength (q) by its average value, ⟨H(q) el ⟩λν ≃ ⟨Hel ⟩, allows simplification to

0

nn ′

ℏ2 ≅ Mq 2ωq

∂H

−⟨λ| ∂qel |ν⟩ −⟨Hel(q)⟩λν ∂ ⟨λ| |ν⟩ = = ∂q ΔEλν ΔEλν

≅

⟨n|n′⟩⟨λ|

2

(18)

where the last equality is obtained by virtue of the Hellmann− Feynman theorem

1 (q) Γλ → ν

2

∑

δ(Eλ − Eν − ℏωnn ′)

2

2ℏ ≅ Mq

π ℏ3 2Mq 2

(23)

where the rate is seen to have a simple inverse dependence on the transition energy squared. Interestingly, the same energy dependence is found in the three-dimensional Fermi liquid theory of Bauer and Aeschlimann.38 The rates in eq 23 can be understood as the Markovian closure of the electron−phonon coupling mechanism, while short-time memory effects of the phonon bath are neglected. In the present work, the phonons reference lifetime τq0 and frequency ωq0 are extracted from theoretical calculations based on the ab initio pseudopotentialtotal-energy approach by Chang et al.39 Since both optical and acoustic phonons have been experimentally observed in selforganized Ge quantum dot superlattices,40 contributions from the transverse acoustic, transverse optic, and longitudinal optic and acoustic modes are taken into account in the present work. 2. Pure Dephasing Rates. The nonadiabatic contribution of mode q on the pure dephasing rates can be derived from the second term in eq 11. Starting from eq 10, one finds D


Article


Figure 1. Isosurface plots of the one-electron difference densities between (a) ground state |0⟩ and eighth excited state |8⟩, (b) ground state |0⟩ and sixth excited state |6⟩, and (c) sixth excited state |6⟩ and eighth excited state |8⟩. The atomic species Si, Ge, and H are colored black, purple, and gray, respectively. The program Amira50 was used for the data visualization. The isocontour values for the difference densities are 0.001 a0−3 (yellow) and −0.001 a0−3 (blue).

Voronoi deformation density (VDD) method.48 These two schemes are particularly advantageous due to their rapid convergence with respect to the basis set size.47 The atomic charge in the Hirshfeld population analysis takes the form

structure to capture the main features of both the structure and excitonic properties.24,27,30 The general construction principle for the model quantum dot used in this work is the embedding of a small Ge core in a small Si host matrix which mimics the properties of the silicon bulk. For the construction of the core−shell model, the diamond-structured Si bulk with a Si−Si distance of 2.352 Å41 serves as a starting point. Frequently occurring morphologies of self-assembled Ge islands on a Si (001) surface are multifaceted dome clusters and pyramidal structures.7 Accordingly, the smallest conceivable covalently bound Ge cluster with a pyramidal shape in the diamond structure, Ge7, is defined as the QD core. The germanium core is then deposited on the Si(001) surface at the expected bulk lattice position of the Si atoms before a silicon wetting layer is added to cover the Ge7 pyramid. The final Si host matrix (47 atoms) is designed in such a way that two shells surround the Ge cluster; i.e., the nearest and the second nearest neighbors to each Ge atom correspond to the first and second silicon shells. The outer Si atoms are saturated with hydrogen atoms at the appropriate tetrahedral positions to saturate all dangling bonds. As an initial guess, a Si−H distance equal to that in SiH4 (1.480 Å) is chosen.42 The final model quantum dot has a C2v symmetry, consistent with experiment, and is depicted in Figure 1. Structure optimization is performed at the restricted Hartree−Fock (RHF) level of theory by freezing the outer silicon shell in order to imitate the strain of the Si bulk structure on the quantum dot. To keep the electronic calculations tractable, effective core potentials, ECP10MWB (Si) and ECP28MWB (Ge), are used with their corresponding doubleζ basis sets.43 Since configuration interaction calculations are sensitive to the diffuse nature of the atomic basis, the bases for Si and Ge are extended with a d-function.44 The hydrogen atoms are treated using a cc-pVDZ basis set.45 All constrained structure optimizations are performed with TURBOMOLE.46 Charge Analysis. To identify a possible hole-trapping state, the one-electron density associated with each electronic state is reconstructed by integrating the CIS wave functions and projected on a regular spatial grid. The N electron CIS wave function ΨCIS is related to the one-electron density ρ(r) as λ follows ρ(r) = N

CIS ∫ dr2 ··· drN ΨCIS λ (r , ..., rN )Ψ λ (r , ..., rN )

8 Hirshfeld = ZA − A

∫

ρA (r) ρpro (r)

ρ(r) dr (28)

where ρpro is a promolecular density, which is defined as the sum over the ground-state electron densities ρA for the isolated atom A NA

ρpro (r) =

∑ ρA (r)

(29)

A

In the Hirshfeld scheme, a one-electron density ρ(r) is partitioned according to a weighting function ωHirshfeld = A ρA(r)/ρpro(r), as can be seen from eq 28. In the VDD method, the space is divided into atomic domains defined as Voronoi polyhedrons. These so-called Voronoi cells are the regions of space closer to the observed atom than to any other. Following this geometric partitioning scheme, the VDD atomic charge 8VDD of atom A is defined as A 8VDD =− A

∫Voronoi cell of A ρdef (r) dr

(30)

where ρdef(r) = ρ(r) − ρpro(r) refers to the deformation density. For both schemes, each atomic component of the promolecular charge density must first be computed and projected on the grid used for the representation of the oneelectron density. Additionally, a modified scheme with the following definition for the atomic charge is used 8VCD = ZA − A

∫Voronoi cell of A ρ(r) dr

(31)

This scheme is very time-saving compared to the two preceding methods, since there is no need for the calculation of the promolecular density. For the sake of simplicity, we designate it the Voronoi cell density method (VCD). All one-electron densities and partial charges are computed using orbkit,49 which is a Python toolbox for postprocessing electronic structure data. The programs Amira50 and matplotlib51 are used for the data visualization.

(27)

■

with r, r2, ..., rN as sets of coordinates for the electrons 1, 2, ..., N. The charge analysis of the one-electron densities is then investigated employing various atomic charge determination schemes, including the Hirshfeld population analysis47 and the

RESULTS AND DISCUSSION Static Considerations. The lowest-lying 250 eigenstates of the CIS Hamiltonian to be used as a dynamical basis are

E


Article

The Journal of Physical Chemistry C calculated for the Ge/Si model quantum dot optimized at the Hartree−Fock level of theory (cf. Figure 1), in accordance with the description in the previous section. In the present work, the Hartree−Fock and all subsequent CIS calculations are performed with the GAMESS-US program package.52 For analysis purposes, the one-electron densities and pairwise difference densities for the ground state and the first 10 excited CIS electronic eigenstates are projected on a 251 × 251 × 251 equidistant Cartesian grid. Further, the partial charges for the Ge atoms in the cluster are determined by means of three different atomic population schemes: explicitly, the Hirshfeld, the VDD, and the VCD methods (cf. section Charge Analysis). From the static charge density difference, it is possible to identify series of state-to-state excitations where a charge carrier is becoming confined in the vicinity of the Ge impurity. The partial charge differences within the Ge7 fragment are defined as the sum of atomic contributions from all germanium atoms in the Ge/Si cluster. For all excitations from the ground state, an increase of the partial charge of the Ge nanoparticle is observed. This implies the localization of a hole in the Ge fragment as a consequence of the excitation from the ground state. The highest degree of hole localization inside the Ge impurity is seen for the transition from the ground state to the eighth excited state (cf. Table 1). About a fifth of an electron is transferred from the Ge atoms to the surrounding Si shells independently of the partial charge method used.

Figure 2. Schematic picture of three selected CIS eigenstates, |0⟩, |6⟩, and |8⟩, of the Ge/Si model quantum dot. The nonzero transition dipole moments μλν,q and the lifetimes τλ→ν are assigned to the respective state-to-state transitions. The arrow marked with an “A” symbolizes the direct transition from the ground state |0⟩ to the eighth excited state |8⟩. The arrows labeled with B1 and B2 correspond to the excitations from the ground state |0⟩ to the sixth excited state |6⟩ and from the sixth excited state |6⟩ to the eighth excited state |8⟩, respectively. The state-to-state transition energies are ΔE06 = 5.64 eV, ΔE68 = 106.80 meV, and ΔE08 = 5.75 eV.

into the Si matrix, although in a very different manner. It can thus be anticipated that the electron dynamics will have a drastically different transient behavior upon excitation using the one and two step mechanism. Selective Laser-Driven Excitation. As a first step, pulse sequences are optimized to promote the two selected transition pathways: the direct transition from the ground state |0⟩ to the eighth excited state |8⟩, and the indirect excitation via the sixth excited state |6⟩. For a high population yield in the final state |8⟩, sequential optimization of the laser pulse parametersthe lengths, the amplitudes, and the frequencies of the respective pulsesis performed. In the case of the indirect path |0⟩ → |6⟩ → |8⟩, the time delay between the two sin2 pulses is also optimized. In order to keep computational costs tractable, only 25 CIS eigenstates are used in the dynamical simulation for the laser field optimization. As a general trend, it can be observed that these transitions all remain close to the π-pulse conditions, with a somewhat lower intensity due to nonlinear field effects. All pulse frequencies are almost at the resonant transition, within less than 0.1% of the reference frequency. The pulses are found to be relatively long and, consequently, very little dynamical broadening is observed. For the final dynamical simulations, all 250 electronic CIS states are included to ensure proper description of the system polarizability and provide a better description of the coupling of the system with the laser field. The time evolution of selected state populations for the two investigated transition series is depicted in Figures 3a and 4a, along with the simulations with 25 CIS eigenstates used for pulse optimization purposes. For the direct excitation pathway |0⟩ → |8⟩, a nearly complete population inversion to the target excited state |8⟩ (total transfer = ∼98%) is obtained in the simulation with 25 CIS eigenstates. The intensity of the laser pulse induces dynamical broadening of the energy levels, which marginally decreases the efficiency of the population transfer due to competing excitation channels. In contrast, the propagation with 250 CIS eigenstates exhibits reduced transfer yield and state selectivity. Due to the improved description of the system polarizability by using a larger basis for the variational representation of the coupling to the electric field, the interaction of the system with the laser pulse is modified by nonlinear effects. This behavior can be seen in Figure 3b, where

Table 1. Partial Charge Differences between Selected CIS Electronic Eigenstates for the Ge7 Fragment in the Ge/Si Model Nanocrystala λ→ν

Hirshfeld

VDD/VCD

energy (eV)

0→8 0→6 6→8

0.194 0.118 0.076

0.213 0.120 0.093

5.75 5.64 0.11

a

The Hirshfeld, VDD, and VCD atomic population methods have been employed. The last column reports the transition energies.

The difference density between the ground and eighth excited states shown in Figure 1a confirms the spatial trapping of a hole inside the Ge fragment (yellow isocontour). This observation holds for all excitations from the ground state considered, but they were omitted for clarity. For the transition of interest, the electron appears to be displaced from the germanium fragment to below the pyramid. An advantage of this particular charge distribution is the induction of a large dipole moment along the z-axis of the quantum dot, which could be used for information storage as a part of a larger device. Another auspicious transition sequence is found as the indirect excitation via the sixth excited state. Arguments in favor of this transition path are the relatively high transition dipole moments involved and the well-separated transition energies, which are depicted schematically in Figure 2. Figures 1b,c depict the difference density for the indirect mechanism, confirming the stepwise charge carrier confinement. The figures reveal that both the transition from the ground state to the sixth excited state and the transition from the sixth excited state to the eighth excited state exhibit a spatial localization of the electron hole in the region of the Ge7 cluster. This is in agreement with the charge density differences reported in Table 1. From Table 1, it can be recognized that for both excitations, |0⟩ → |6⟩ and |6⟩ → |8⟩, a similar amount of electron density from the Ge atoms in the model quantum dot is transferred F


Article


z are solely due to the creation of a wave packet, with main components from the ground and eighth excited states. In the case of the transition path |0⟩ → |6⟩ → |8⟩, two ypolarized laser pulses are employed, with the result that a sharp dipole switching of the z-component is observed. Both dynamical simulations with 25 and 250 CIS eigenstates lead to an adequate target state selectivity (respectively 96 and 92%). This is a strong indication that lateral excitation, i.e., using pulses polarized along the x and/or y axes, is less sensitive to polarizability and higher-order field effects. The response of the system is seen solely in the y-component of dipole expectation value, as shown in Figure 4b. The x-component remains completely inactive during the simulation, and the zcomponent smoothly switches from a modest initial value (∼ −1 D) to a large final value (∼ −3 D). Nevertheless, dynamic broadening effects are seen to impact population transfer, as the two-pulse sequence does not quantitatively reach the target. Small amplitude Rabi oscillations due to the creation of an electronic wave packet can be seen at times longer than 7 ps. The smooth behavior of the dipole moment along the z axis renders this excitation pathway promising for dynamic dipole switching applications such as information storage and quantum computing. Varying the time delay between the two pulses reveals that the optimal population transition is accomplished with a STIRAP mechanism: the two tuned pulses are overlapping and the first pulse promoting the transition |0⟩ → |6⟩ starts after the second pulse with a delay of 720 fs (cf. Figure 4b). Conceptually speaking, a dressed state is initially created as a coherent superposition of states |6⟩ and |8⟩, and the dressed state is then directly accessed by an excitation from the ground state |0⟩.53 This type of excitation mechanism was already found to help alleviate the effects of relaxation in dissipative systems.54 For all excitations from the ground state, the effects of dissipation and pure dephasing are negligible. This is due to the comparatively high gap between the ground state and the first excited state (ΔE01 = 5.34 eV). Recall that the rates scale inversely with the square of the excitation energies (cf. eqs 23 and 26), with a Markovian closure defined by a typical phonon vibrational quantum. In the present work, the influence from the longitudinal optic and acoustic (LOA: ω = 19.9 meV, τ = 50.0 ps), transverse acoustic (TA: ω = 17.1 meV, τ = 100.0 ps), and transverse optic (TO: ω = 52.8 meV, τ = 4.8 ps) modes are considered explicitly. The theoretical frequencies and lifetimes are obtained from the literature.39 The three components are treated additively, so that the total energy relaxation and pure dephasing rates are respectively given by

Figure 3. Laser-driven electron dynamics in the Ge/Si quantum dot with coupling to a dissipative environment for the direct transition, |0⟩ → |8⟩. (a) Population evolution for a propagation with 25 CIS eigenstates (dashed line) and with 250 CIS eigenstates (solid line). The direct excitation, |0⟩ → |8⟩, is promoted using a z-polarized tp,08 = 3090 fs sin2 pulse with amplitude f08,z = 4.55 MV/cm (cf. eq 7) and frequency ω08 = 5.75 eV. (b) Time evolution of the electric field (upper panel) and of the dipole moment components (three lower panels); 250 CIS eigenstates are incorporated in the simulation.

Figure 4. Laser-driven electron dynamics in the presence of energy and phase relaxation for the interlevel transition series, |0⟩ → |6⟩ → | 8⟩. (a) Population evolution with 25 CIS eigenstates (dashed line) and with 250 CIS eigenstates (solid line) incorporated in the simulation. The first excitation, |0⟩ → |6⟩, was promoted using a y-polarized tp,08 = 330 fs sin2 pulse with amplitude f06,z = 5.32 MV/cm and frequency ω06 = 5.64 eV. The second transition, |6⟩ → |8⟩, was driven by a ypolarized tp,08 = 7435 fs sin2 pulse with amplitude f68,z = 1.00 MV/cm and frequency ω68 = 106.75 meV. (b) Time evolution of the electric field (upper panel) and of the dipole moment components (three lower panels); 250 CIS eigenstates are taken into account for the propagation.

the time evolution of the laser field and the components of the dipole moment are reported. It is noteworthy that the effect of dephasing due to nonadiabatic coupling is more strongly felt in the larger dynamical basis. To understand the highly nonlinear behavior of the laser− molecule interaction, it is instructive to look at the expectation of the dipole moment as the reaction proceeds. In particular, since no states at an energy about twice that of the |0⟩ → |8⟩ transition are included in the dynamical basis, multiphoton effects can be safely ruled out. Both the x- and y-components of the dipole moment are hardly affected by the interaction with the z-polarized pulse, which is in stark contrast with the zcomponent. During the |0⟩ → |8⟩ transition, an electron becomes confined below and a hole inside the Ge7 nanocrystal (cf. Figure 1a), which strengthens the dipole moment along the z-direction. Due to the increasingly strong interaction with the z-polarized electric field, a fast and broad oscillation of the dipole moment expectation value along z is observed. During the population transfer, the dynamical stark shift of the target states is felt much more strongly than by the ground states, which means that the energy difference between ground and excited states changes importantly in time. At the end of the pulse, residual oscillations of the dipole expectation value along

(TA) (TO) Γλ → ν = Γ(LOA) λ→ν + Γ λ→ν + Γ λ→ν

γλ = γλ(LOA) + γλ(TA) + γλ(TO)

(32)

It comes as no surprise that excitations in the 5 eV regime feel only weakly the effect of both relaxation and dephasing, with a typical relaxation lifetime of τ01 = 1/Γ01 = 1.91 μs. Note that the pure dephasing time, T1 = 1/γ1, is about 2 orders of magnitude longer and its effect could be neglected to a good approximation. On the contrary, for the final excitation |6⟩ → |8⟩ with a transition energy (ΔE68 = 106.8 meV) comparable to the phonon frequencies, energy relaxation and pure dephasing have a slight effect on the electron dynamics. This can be seen first and foremost from the small decay of the G


Article

The Journal of Physical Chemistry C |8⟩ population in the last picoseconds of the simulation. Further, the incomplete population transfer from state |6⟩ → | 8⟩ finds its origin in both dynamical broadening and dephasing.31 The latter has two components stemming from energy dissipation and from pure dephasing. Despite the imperfections of both excitation pathways, they can be considered to provide robust population transfer to an electron−hole pair state in the presence of energy and phase relaxation. The target state |8⟩ is predominantly populated and, thus, a hole is spatially localized in the area of the Ge7 cluster, with the residual electron density displaced below the pyramid base to generate a large permanent dipole. A high stability of the final state can be inferred from its long lifetime, with the shortest state-to-state relaxation lifetime of τ6→8 = 0.76 ns.

Finally, energy relaxation and pure dephasing induced by nonadiabatic coupling are seen to marginally affect the population yields in the target state. It can thus be anticipated that optically induced charge carrier confinement in Ge/Si quantum dots of small to moderate size will be similarly robust to the effects of energy and phase relaxation induced by nonadiabatic couplings.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +49-30-83858150. Fax: +49-30-838-452051. Notes

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The authors declare no competing financial interest.

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CONCLUSION In the present work, the laser-controlled many-electron dynamics in a Ge/Si quantum dot in the presence of energy relaxation and pure dephasing was simulated. For this purpose, a size reduced core−shell type model of a Ge/Si quantum dot was constructed possessing the main characteristics and morphology of experimental self-assembled Ge quantum dots in a Si host matrix. The electronic structure of the system was determined using wave-function-based ab initio methods, namely Hartree−Fock, and it reveals similar attributes in comparison to previous studies on Ge/Si nanocrystals. Investigation of the system excited electronic states and associated properties was performed using the configuration interaction (CI) method at the singles level. The electron dynamics was performed using the reduced density matrix variant of the time-dependent configuration interaction method (ρ-TDCI). A perturbative model for nonadiabatic coupling of localized electronic states with small amplitude harmonic vibrations was introduced, and its effect on the electron dynamics was treated using the Lindblad semigroup formalism. Ensuing dynamical simulations aimed at the confinement of hole density in the dome-shaped Ge7 cluster. As an indicator for the degree of charge carrier trapping, the one-electron density difference between various CIS eigenstates was calculated. Two selected transition pathways, direct and indirect, were promoted using series of sin2 pulses, for which a sequential parameter optimization in a small basis of CIS eigenstates was performed. The dynamical electronic simulation with a variationally better CIS eigenstates basis provided an improved description of the system polarizability and other nonlinear field−molecule effects, leading to a reduction of the transfer yield and state selectivity for the direct excitation mechanism. This strong nonlinear dynamical effect is due to the dipole moment enhancement in the z-direction upon excitation, since an electron and a hole are dynamically getting confined respectively under and inside the Ge7 cluster. Since the indirect transition path involves y-polarized laser pulses, it was found to be very robust to the effects of polarization. Note that the toy problem chosen here retains the structural characteristics of larger germanium quantum dots in silicon matrixes and the electron/hole localization observed in our nanocrystal is further in qualitative agreement with previous work on similar structures. We thus expect the trends observed in our dynamical studies to hold for larger, experimentally relevant quantum dots. On the other hand, the degree of charge localization and the magnitude of the created dipole should be taken with care.

ACKNOWLEDGMENTS The authors acknowledge funding from the Deutsche Forschungsgemeinschaft through Project TR1109/2-1, the Freie Universität Berlin (ZEDAT) for computer time, and Hans-Christian Hege for providing the ZIBAmira visualization program.

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REFERENCES

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