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A Hybrid Quantum Mechanical Approach: Intimate Details of Electron Transfer Between Type I CdSe/ZnS Quantum Dots and an Anthraquinone Molecule Alexey Leonid Kaledin, Tianquan Lian, Craig L Hill, and Djamaladdin G Musaev J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp511935z • Publication Date (Web): 21 Jan 2015 Downloaded from http://pubs.acs.org on February 8, 2015

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The Journal of Physical Chemistry

A Hybrid Quantum Mechanical Approach: Intimate Details of Electron Transfer Between Type I CdSe/ZnS Quantum Dots and an Anthraquinone Molecule Alexey L. Kaledin,*,§ Tianquan Lian*,† Craig L. Hill,† and Djamaladdin G. Musaev*,§ §Cherry

L. Emerson Center for Scientific Computation, Emory University, 1515 Dickey Dr.,

Atlanta, GA 30322, United States †Department

of Chemistry, Emory University, 1515 Dickey Dr. Atlanta, GA 30322 United States

ABSTRACT: We report a hybrid computational approach to calculate electron transfer between a Type I CdSe/ZnS core-shell quantum dot (QD) with a varying shell thickness and the functionalized anthraquinone (AQ) molecule. This novel approach combines the traditional electron/hole confinement theory in the effective mass approximation for QD and molecular orbital theory for AQ molecule. In present study, the QD’s electron and hole envelope wavefunctions are solutions of the effective-mass Schrödinger equation and the AQ wavefunction is obtained at the density functional level. Electron transfer rate calculations are based on Marcus’s theory with the coupling strength computed according to an one-electron orbital perturbation model. We show that in a heptane solution the LUMO of AQ and the 1Se electron orbital of QD are involved in the charge separation (CS) process. The charge recombination (CR) process, on the other hand, occurs from the singly occupied molecular orbital of AQ radical (which corresponds to the LUMO in AQ) to a trapped hole state of QD within the band gap. The calculations support previously reported interpretations of the role of ZnS shell as a hindrance in the charge separation and recombination process.

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1. INTRODUCTION Quantum confined semiconductor nanostructures have emerged as a new class of light harvesting and charge separation materials for solar energy conversion.1,2,3,4,5,6,7,8 Significant progress has been made in the preparation of many nano-hetero-structures consisting of two or more materials and of various shapes (spheres, rods, tetrapods).2,3,9,10 The ability: (a) to choose component materials with desirable bulk band alignment, (b) to control the confinement energy of each component through its dimension, and (c) to arrange the spatial distribution through shapes of these materials, has led to exciting opportunities for designing hetero-structures (or artificial molecules) with desirable properties for light absorption and charge separation. For example, some of us have demonstrated that compared to core only CdSe quantum dots, type II (CdTe/CdSe) and quasi-type II (CdSe/CdS) core/shell QDs can maintain ultrafast electron transfer to acceptors and slow down charge recombination by extending the electron density to the shell and localizing the hole in the core.11,12 Dramatically different quantum efficiency for photoreduction of adsorbates and photodriven H2 formation was also demonstrated using heterostructure of different shapes, sizes and band alignments.11 Despite the advances in material synthesis and their promising performance in solar energy conversion, our ability to model the electron and hole transfer processes between these quantum confined nano-hetero-structures and adsorbed acceptors is still lacking, hindering further design and improvement of nano-materials. Although formal theoretical framework for computing charge transfer has been well established and applied successfully for inter- and intramolecular donor-acceptor systems, 13 the extension of this method to charge transfer at nanocrystal/molecule interface remains challenging because of the difficulty in treating realistic quantum dots at the ab initio theory level. Previous studies have utilized smaller nanoclusters as a model for the larger QDs to enable DFT level studies14 or employed empirical pseudopotential methods15 to model QDs of sizes comparable to experiment. Despite these successes in limited systems, more general and faster methodologies are still needed. In this paper, we report a new hybrid method for calculating charge transfer dynamics of QD-molecule complexes. This hybrid method overcomes the challenge of computing exact orbitals for QDs by realizing that the coupling strength for electron (hole) transfer from QDs to acceptors should depend on the electron (hole) density that extends outside the QDs. Therefore, a method that can accurately compute this part of the QD wave function is sufficient for modeling

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charge transfer. It has been well established that for many quantum-confined semiconductor nanocrystals, the effects of quantum confinement on electronic structure and optical property can be well described by the effective mass approximation (EMA).16,17,18 This method treats the electrons and holes as independent particles confined in a potential well and can readily calculate their confinement energy and envelope wavefunction for nanocrystals of any shape. A recent theoretical study has confirmed that the wavefunction outside the QD can be also well represented by the envelope functions with proper treatment of the boundary conditions at the QD surface without explicit calculation of the Bloch functions of the QD.19 Admittedly, the EMA wavefunction suffers from an ambiguous behavior at the vacuum interface due to the band discontinuity;20 however, recent studies reported by some of us11,21 provide encouraging results on the utility of the EMA envelope wavefunction at the QD/vacuum interfaces in treating interfacial electron transfer. Thus, in this work, our hybrid approach treats (i) the QD wavefunctions at the envelope function level in the EMA, (ii) the molecular acceptor at the DFT level, and (iii) the charge transfer between the QD and acceptor within the well-established perturbation theory framework.13 To demonstrate the validity of this method, we apply it to the study of electron and charge transfer processes in CdSe/ZnS QD-Anthraquinon (AQ) complexes, for which these charge transfer rates have been experimentally determined.11

Figure 1. Schematic illustration of the (COOH)2-AQ molecule attached to a spherical core/shell QD (drawn not to scale) at the distance Δr between the terminal oxygens of carboxylic linkers of the AQ and the QD’s outer layer. The red inner sphere is CdSe core of radius R0 (13.7 Å in the present work); the blue outer sphere is the ZnS shell of thickness D.

2. DESCRIPTION OF THE METHODS USED. 2.1 Brief description of the electron/hole confinement theory in the effective mass approximation (EMA) for QDs.

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To describe electron (hole) states in the conduction (valence) band of the QD we use the envelope wavefunction approach in the effective mass approximation. The computational procedure employed in the present work yields exact envelope wave functions, within a given numerical accuracy, and was described extensively in our earlier publication.21 Below we only provide a brief overview of the method. Within this approach, the envelope wavefunctions ψ e and ψ h for the electron and hole, respectively, are the solutions of the corresponding Schrödinger equation

€

€

+ 2  % 1  ( . -− ∇ ⋅ ' * ∇ * +Ve (r)0ψ e (r) = ΔEeψ e (r) , & me (r) ) , 2 /

€

+ 2  % 1  ( . -− ∇ ⋅ ' * ∇ * +Vh (r)0ψ h (r) = ΔEhψ h (r) . & mh (r) ) , 2 /

(1a)

(1a)

In the above, me* and€mh* are the position dependent electron and hole effective masses. We note that this particular form of the kinetic energy operator in Eqs. 1a ensures the continuity of probability current at core/shell and shell/vacuum interfaces.22 Ve and Vh are the conduction and valence band levels of the bulk materials (relative to the vacuum) and ΔEe and ΔEh are the corresponding confinement energies. The electron-hole Coulomb interaction is computed using the converged ground state wavefunctions and the position dependent dielectric constants (see Table 1), and the electron and hole energy levels are adjusted by the amount of the interaction. The total energy of the electron(hole) can be calculated from electron (hole) confinement energy and bulk CB (VB) band edge of CdSe (taken from literature values). Such a perturbative approach is known to yield accurate results for the spherical Type I QDs.11 We represent both equations on a 3D uniform Cartesian grid, where each grid point is a discrete “basis function”. The discrete variable representation, or DVR,23 results in a very simple expression for the matrix elements of the Hamiltonian, many of which are identically zero and, more importantly, bypasses multidimensional integral evaluation. Additional improvement, i.e. convergence toward the exact result, can be achieved by taking the limit of an infinitely large grid followed by grid contraction, as discussed previously in our recent work21 and by others.24,25 Briefly, here we use a uniform ΔX = ΔY = ΔZ = 2 bohr grid laid initially on [-800, 800] bohr in

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all three dimensions. The matrix representations of Eqs 1a and 1b were computed using the full grid but consisting of only those points satisfying the condition ( x2 + y2 + z2 )1/2 < Rc, with the cutoff radius of Rc = 50 bohr, resulting in a total of 65117 contracted grid points. By this definition the wavefunction vanishes on the surface x2 + y2 + z2 = Rc2. After solving Eqs. 1 for the ground state 1Se and 1Sh the electron (hole) wavefunction are expressed using the grid as

ψ ( x, y,z) =

1 ΔXΔYΔZ

∑ψ

DVR ijk

ijk

& y−yj & x − xi ) sinc( π + sinc( π ' ΔX * ' ΔY

) & z − zk ) + + sinc( π ' ΔZ * *

(2)

DVR €where ψ ijk is the value of the eigenfunction at grid point (xi, yj, zk), ΔX=xi+1-xi, and

sinc(x)=sin(x)/x. Convergence of the calculation with respect to grid contraction scheme was € done for Rc = 60, 70 and 80 a0 with the largest error in the eigenvalue being ~1 meV. The

parameters me*, mh*, εr, Ve and Vh required for solving equations 1a and 1b are taken from the experiments (see Table 1). Table 1. The material parameters used in the quantum dot calculations: effective masses me*, mh* of electron and hole, respectively, dielectric constant εr, and conduction and valence band edges Ve and Vh, respectively. me *

mh *

εr

Ve (eV)

Vh (eV)

CdSea

0.13

0.45

10

-3.4

-5.7

ZnSa

0.25

1.3

8.9

-3.1

-6.6

heptane

1

1

1.9b

-1.0a

-8.4a

a: these values are taken from Ref. 11 b: this value is taken from the Gaussian 09 program26 revision D01 used in the calculations. 2.2 Calculations of anthraquinone A key assumption we make in this work is that orbitals and geometry of fragments, i.e. QD and molecule, are not significantly perturbed upon molecule’s adsorption on QD’s surface. This is justified by the negligibly small shift of QD and adsorbate absorption spectra upon the

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formation of QD-adsorbate complexes, including the CdSe/ZnS QD-anthraquinone complexes studied in this work.11,27,28,29,30,31 This assumption allows us to solve the wavefunctions of the standalone QD and the isolated molecule at their global minimum geometry separately and use perturbation theory to calculate the coupling. It is also important to note that there are examples of strong QD-adsorbate coupling,32,33 for which our perturbation theory based method would not be applicable and more advanced computational approaches would be needed.34,35 Figure 1 illustrates a functionalized anthraquinone molecule at the interface of a spherical Type I CdSe/ZnS quantum dot. We use the terminal oxygens of the carboxylic linkers of the AQ to define the distance of the molecule to the QD’s surface. For electronic structure of AQ, functionalized with two COOH groups, we use the hybrid B3LYP density functional36,37,38 with the 6-31++G(d,p) basis sets. The inclusion of diffuse functions was found to be important for a proper description of the unoccupied orbitals and the electron affinity. After optimizing the geometry, the orbitals were saved on a uniform 40 a0 x 40 a0 x 40 a0 grid. This grid extends ~10 a0 and ~8.3 a0 beyond the left and right edges of (COOH)2AQ along the X axis, respectively (see Figure 1). Systematic decrease of the grid spacing produced a satisfactory orbital normalization value with Δx = Δy = Δz = 0.25 a0 and 161 grid points per Cartesian coordinate. These calculations were performed with Gaussian 09.26 Table 2. Frontier molecular orbital and their energies of (COOH)2-AQ (in C2-symmetry and a heptane solution) calculated at the PCM//B3LYP/6-31++G(d,p) level of theory. MO

irrep

EMO (eV)

LUMO+1

a2

-2.474

LUMOa

a1

-3.629b

HOMO

a2

-7.690

HOMO-1

a1

-7.837

a: LUMO of AQ and SOMO of AQ- (see text for more details). b: In the calculations we use the experimental value of -3.8 eV reported in Ref. 11 The structure of the (COOH)2-AQ complex (or dyad) can be seen in Figure 1. The molecule has a rotational symmetry about the X-axis, and the corresponding point group is C2. Molecular dimensions are 10.5 Å along X, 6.1 Å along Y and 1.9 Å along Z. The longest end-to

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end distance is thus more than twice shorter than the smallest QD’s diameter (The Cartesian coordinates are provided in the SI). Frontier molecular orbitals are summarized in Table 2. Given the valence and conduction band edges of the QD core (see Table 1), only the LUMO orbital is energetically accessible for accepting an electron from the QD and then donating it back. The other orbitals can only be spectators in the charge transfer processes. The LUMO is a totally symmetric antibonding π orbital consisting predominantly of the C(p) and O(p) orbitals of AQ. A small amount of its density derives from the carboxylic group’s π* orbital which constitutes the majority of the overlap with the 1Se orbital of the QD. Invoking Koopmans theorem, LUMO’s energy of -3.629 eV may be interpreted as the negative molecular electron affinity (EA), which is in very good agreement with the experimental value of 3.8 eV.11 Calculations of the electron affinity (EA) based on the total energy difference gives 2.57 eV, significantly underestimating the experimental value. It is thus reasonable to treat the LUMO as a state describing AQ with all the other electrons forming a non-participating “core”. In the following analysis the LUMO being unoccupied and singly occupied shall correspond to AQ and AQ-, respectively, with its energy independent of the occupation number. Additionally, to be consistent with the experiment we will use the measured value for the LUMO energy in the electron transfer calculations described below, i.e. -3.8 eV.

Figure 2. The LUMO of the (COOH)2-AQ molecule.

To estimate the distance, Δr, between QD

surface and molecule, as well as an optimal coordination motif of (COOH)2-AQ to the QD surface we performed plane-wave density functional calculations of several binding geometries by utilizing VASP program package.39 The QD’s surface in the vicinity of contact with the much smaller molecule was treated as periodic bulk surface. We used the standard wurtzite (hexagonal) cell with a = b = 3.81 (3.723) and c=6.23 (7.015) Å for ZnS (CdSe), respectively. A supercell with dimensions 5x3x3 was used to model a surface slab, with the slab thickness (along

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b-axis) of 3 ZnS (CdSe) layers, resulting in Zn90S90 (Cd90Se90) composition. Each surface slab filled with a 5b vacuum space along the b-axis was then optimized using the PW9140 functional with a plane wave basis, with a 300 eV energy cutoff, Γ-point integration and using projectoraugmented pseudopotentials.41 The (COOH)2-AQ molecule was then placed on the slab in several orientations, e.g. vertical end-on, side-on and parallel, and the structures were fully optimized. We found the lowest energy configuration to be the vertical end-on approach, as shown in Figure 3.

Figure 3. Optimized geometries of (a) CdSe-(COOH)2-AQ, and (b) ZnS-(COOH)2-AQ models corresponding to core-AQ and core/shell-AQ real systems. Both the core and shell are modeled as 5x3x3 wurtzite (100/010) surface slabs, resulting in three CdSe/ZnS layers of 11.17/11.43 Å thickness. The OCd bond is 2.56 Å, and the two O-Zn bonds are 2.19 and 2.20 Å. The shortest distance between a carboxylic oxygen and the plane defined by the contact CdSe (or ZnS) layer of surface atoms, outlined by the boxes, was found to be 2.3 Å for both surfaces.

From the calculated geometry we determined the Δr parameter (see Figure 1) as the shortest distance between a terminating carbonyl oxygen of COOH linker and the plane defined by the outer CdSe (or ZnS) layer consisting of 30 Cd(or Zn) and 30 Se(or S) atoms. The choice of the layer of atoms to define the plane of contact is somewhat arbitrary, however we believe that the outer layer is the best representation of the spherical surface of the real QDs; these atoms

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can be seen in Figure 3 contained in rectangular boxes. The planes are least-squares fits to the atoms Cartesian positions. In both cases, Δr was found to be ~2.3 Å.

3. RESULTS AND DISCUSSION 3.1 Type I CdSe/ZnS core/shell quantum dot. Following the experimental setup,11 here we use: (i) the CdSe core radius R0=25.89 a0 (13.7 Å), (ii) three ZnS shell layers of 5 a0 (2.646 Å) each, which corresponds roughly to a single layer of ZnS, and (iii) the effective masses, dielectric constants, and valence and conduction band edges given in Table 1. All parameters used in the calculations are given in Table 1. The Schrödinger equation (Eqs. 1a and 1b) was then solved for the four cases, with ZnS shell thickness: D = 0 (i.e. bare core), 5, 10, 15 a0. Table 3a lists the calculated ground state 1S energies of electron and hole, exciton energy, and particle density at interface. Table 3a. Electron and hole Coulomb adjusted 1S energy levels (Ee and Eh, respectively), QD exciton energy Eexc and probability density ρ at the interface with environment (heptane) in

CdSe/ZnS quantum dot with various ZnS shell thickness. All energies are in eV. €

ρe / a0-1

ρh / a0-1

D (a0)

Ee

Eh

QD Eexc

0

-3.582

-5.858

2.425

4.86*10-2 1.35*10-2

5

-3.573

-5.795 €

2.356

1.75*10-2 8.74*10-4

10

-3.571

-5.793

2.351

1.41*10-2 8.93*10-5

15

-3.572

-5.795

2.350

6.12*10-3 4.91*10-6

As previously shown,11,21 the role of the ZnS shell in the Type I CdSe/ZnS quantum dot is to lower the exciton energy and reduce electron/hole density at the surface of the QD as the shell thickness grows. The surface densities summarized in Table 3a are consistent with these observations. To provide further support, we replaced ZnS shell by the CdSe shell. The results of these model calculations are shown in Table 3b. It is necessary to note that due to the smaller

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confinement energies the wavefunctions are more diffuse, as is evidenced by the higher densities at the interface compared to the CdSe/ZnS core-shell structures. This is especially true for the hole where the density is quite flat with shell thickness (i.e. same order of magnitude). Below, we will demonstrate that this property greatly facilitates electron-hole recombination while barely impacting charge separation.

Table 3b. Electron and hole Coulomb adjusted 1S energy levels (Ee and Eh, respectively), QD exciton energy Eexc and probability density ρ at the interface with environment (heptane) in the

spherical CdSe core QDs of radius R. All energies are in eV. €

ρe / a0-1

ρh / a0-1

R (a0)

Ee

Eh

QD Eexc

30.89

-3.677

-5.801

2.251

2.84*10-2 6.80*10-3

35.89

-3.742

-5.766 €

2.135

3.46*10-2 7.01*10-3

40.89

-3.790

-5.742

2.051

2.17*10-2 3.89*10-3

3.2 Interfacial electron transfer Based on the assumption that QD-molecule interaction is weak and non-covalent in nature, we utilize a simple frontier orbital overlap picture and calculate the rate of electron transfer using the Marcus’s theory42

kET (D) =

2 & λ + ΔG(D) 2 ) ( )+ 2 π H12 (D) exp(−  4 πλkBT 4 λkBT (' +*

(3)

where the quantum mechanical coupling element H12 and the reaction free energy change (or € driving force) ΔG are dependent on the shell thickness. The term λ represents the energy change due to structural deformation in the course of the reaction. As argued previously, the reorganization energy in QDs is assumed to be negligible, and the main contribution to λ comes from the molecule that is independent of shell thickness. For the charge separation process, λCS is

10


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defined as the energy difference between AQ-(R1) and AQ-(R2), where R1 and R2 are the equilibrium geometries of AQ and AQ-, respectively. For the charge recombination process, λCR is calculated as the energy difference between AQ(R2) and AQ(R1). Calculations at the B3LYP/6-31++G(d,p) level of theory yielded λCS = λCR = 0.21 eV. To derive the coupling strength H12, we follow a one-electron orbital perturbation model advocated for processes involving a single electron transfer.13 Here, we consider two weakly interacting systems, the QD and the molecule, and define a combined single-particle Hamiltonian using spectral resolution in terms of the eigenstates, Hˆ ≡

∑E

k

(4a)

χk χk

k

where the summation includes both€ QD’s as well as molecule’s single-particle orbitals χi and energies Ei. (We note that the doubly occupied molecular orbitals, as well as the valence band electrons in QDs, are excluded from the summation.) The diagonal matrix elements of the Hamiltonian are

H ii = Ei +

∑E

k

χi χk

2

,

(4b)

k≠i

and its off-diagonal elements are €

H ij = (Ei + E j ) χ i χ j +

∑E

k

χi χk χk χ j .

(4c)

k≠i, j

Taking the limit of € infinitely separated fragments, the Hamiltonian matrix becomes diagonal with the non-zero elements corresponding to electron energies in non-interacting QD and the molecule. In the sense of a perturbation theory of weakly interacting fragments, the higher order terms in the equations 4b and 4c can be ignored. For instance, the overlap of an electron orbital of the QD with a molecular orbital of molecule is very small, typically on the order of 10-4. This guarantees that the second term in Eqs. 4b and 4c is much smaller than the first term and is of the

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order of 10-8, i.e. practically zero, leading to much simpler expressions for the matrix elements, namely

and

H ii ≈ Ei

(5a)

H ij ≈ (Ei + E j ) χ i χ j .

(5b)

€

Although independent of the€hole, implicit electron-hole Coulomb interaction is included in the orbital energies.

Figure 4. Schematic illustration of the charge separation (CS) and charge recombination (CR) processes between the CdSe/ZnS core/shell QD and anthraquinone AQ. The conduction (CB) and valence (VB) bands of QD are shown along with the hole (open circle) and electrons (filled circles). The dashed arrow indicates the hole’s drift from the VB to the trap region (shaded area). The trapped hole level is estimated to be ~1.3 eV above the valence band edge (for more details, see main text).

A) Charge Separation As seen in Figure 4, the driving force for the charge separation (CS), i.e. an electron transferring from 1Se conduction band level of QD to the LUMO of AQ, is ΔG12 = ELUMO - Ee, and the quantum coupling strength of this process is H12CS = (ELUMO + Ee), as given by Eq. 5b. As seen in Table 4a, where the calculated quantities for the CdSe/ZnS - AQ dyad are given, charge separation in the CdSe/ZnS - AQ dyad is an exergonic process with the exergonicity (i.e. ΔGCS) slightly increasing as the ZnS shell thickens (D) passing from normal to

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inverted Marcus region. The dependence of the driving force on shell thickness appears to be insignificant compared to the rate of change of the coupling strength (i.e. H12CS), and the latter determines the rate constant’s (i.e. kCS) decay with increasing D. From the plot shown in Figure 5, the slope of the fit ln(kCS) = -βCSD is ~0.358 Å-1, which is just within the error margins of the experimental measurement11 of 0.35±0.03 Å-1. More importantly, the absolute value of the rate constant is off by no more than a factor of 3 from the experimental measurements for the bare core CdSe and the three ZnS shell layers (see Supplementary Information for more details). It is important to note that in the present calculations we used only the envelope part of of the full wavefunction of the QD, which appears to be sufficient for calculating the orbital overlap with the adsorbate. Also worth noting is the relationship of the so-called evanescent part of the full wavefunction19 to the vacuum region of the EMA envelope. As mentioned in Introduction, the EMA wavefunction is not well-defined outside the semiconductor, and its validity has been questioned previously.20 It is thus encouraging to see a close correlation between the experimentally measured rate constant and the QD/molecule orbital overlap at the EMA level of theory. Furthermore, a recent combined experimental-theoretical study, based on the EMA, of the driving force dependence of electron transfer from QDs to adsorbates has revealed an Auger assisted electron transfer model, in which the excess energy in electron transfer is used to excite the hole in the inverted regime.43 The same study has also shown that the effect of this pathway is minor in the Marcus normal regime, which is consistent with the finding of our study. We believe that this result can be used as a benchmark of the method in further calculations. Table 4a. The calculated parameters, such as an orbital overlap S12, coupling strength H12, reaction driving force ΔG12 and the reaction rate k, for the charge separation (CS) and charge recombination (CR) processes in the CdSe/ZnS – AQ dyad as a function of ZnS shell thickness D. The calculated reorganization energy is λ=0.21 eV. The rates are given for 300K. D /a0

S12CS

H12CS

/meV

ΔG CS

kCS /s-1

S12CR

/eV

H12CR

/meV

ΔG CR

kCR /s-1

/eV

€0

-4 6.44*10 €

€ 4.75

-0.218 €

11 8.23*10 €

-4 4.10*10 €

€ 3.35

-0.56 €

1.45*109

5

4.72*10-4

3.48

-0.227

4.44*1011

1.40*10-4

1.14

-0.56

1.70*108

10

2.32*10-4

1.71

-0.229

1.04*1011

2.96*10-5

0.24

-0.56

7.58*106

13


Tim Lian Formatte Superscr

Tim Lian Deleted:


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

15

1.80*10-4

1.32

-0.228

6.27*1010

5.85*10-6

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0.05

-0.56

2.98*105

B) Charge Recombination Based on the arguments given in Section 2.2, charge transfer properties of AQ- can be accurately described by the singly occupied frontier orbital (SOMO, which is the LUMO in the charge separation process) and treating the other electrons as a non-participating “core” electrons, a consequence of Koopmans theorem, used by others13 in similar applications. The charge recombination in the CdSe/ZnS-AQ dyad, i.e. electron transfer from SOMO of AQ- to the valence band of QD is a highly exergonic process, by at least ΔG = -1.9 eV [i.e. -5.7 eV (the VB of the QD) less the experimental value of -3.8 eV (the SOMO/LUMO of AQ-/AQ) see Tables 1 and 2]. And since the reorganization energy is much smaller (λCR = 0.21 eV, see above) than |ΔG|, the process occurs in a deeply inverted Marcus regime. An estimate of the room temperature charge recombination rate constant for the bare core QD, with ESOMO = -3.8 eV, Eh = -5.421 eV (cf. Table 3a) and S=4.1*10-4 (cf. Table 4a) yields a value of the order of 10-29 s-1, too small to be considered viable. Thus, it is reasonable to assume that the hole becomes trapped upon QD oxidation and occupies an energy level deep inside the band gap, energetically closer to the SOMO level of AQ-. Such traps are possible due to surface defects. 44 The electron then recombines with a trapped hole whose energy level is much close to the SOMO of AQ-. This scheme is illustrated in Figure 4 where the trap region is emphasized by the shaded box. By varying the trapped hole energy level and assuming the same hole transfer coupling strength as 1Sh hole, we were able to match the calculated rate for the bare core QD with the experimentally measured one. This approach placed the hole energy at -4.36 eV, or 1.34 eV above the valence band edge. For the calculations of QDs with a ZnS shell we assumed the trapped hole energy level to be the same as in the CdSe core, i.e. independent of shell thickness. These calculations are summarized in Table 4a for the core/shell structures. The decay of charge recombination rate constant (kCR) with increasing shell thickness is fully determined by the coupling strength (i.e. H12CR) since the driving force for the charge recombination (ΔGCR) is constant. From the plot shown in Figure 5, the slope of the fit ln(kCR) = -βCRD is ~1.08 Å-1. Compared with the experimentally determined11 slope of 0.91±0.14 Å-1, the calculated one is just

14


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outside the error margins and is in agreement with the βCR > βCS relationship. We note that because the hole transfer coupling strengths (i.e. H12CR) are calculated for valence band holes, the absolute values of the calculated rates and hole energy levels are likely to be less reliable. However, similar -βCR, reflecting the relative change of the hole transfer rate as a function of the shell thickness, can be expected for trapped holes because it is determined, mainly, by the shell material.

Table 4b. Electron transfer reaction parameters for charge separation (CS) and charge recombination (CR) processes as functions of CdSe shell radius R (core radius R0 = 25.89 a0) in CdSe/CdSe QDs: orbital overlap S12, coupling strength H12, reaction driving force ΔG12 and the rate constant k. The reorganization energy is calculated to be λ = 0.21 eV. The R-R0=0 is equivalent to the D=0 in Table 4a. The rates are given for 300K. R-R0

S12CS

/a0

H12CS

ΔG12CS

/meV

/eV

kCS /s-1

S12CR

H12CR

ΔG12CR

/meV

/eV

kCR /s-1

€5

-4 5.81*10 €

4.34 €

-0.123 €

11 4.86*10 €

-4 3.33*10 €

2.72 €

-0.56 €

9.56*108

10

3.35*10-4

2.53

-0.058

0.81*1011

1.88*10-4

1.53

-0.56

3.04*108

15

3.27*10-4

2.48

-0.010

0.36*1011

1.69*10-4

1.38

-0.56

2.47*108

C) The effect of the ZnS shell on electron transfer rate The present calculations, as well as previous computational and experimental explorations,11,21 suggest that the role of ZnS shell in Type I CdSe/ZnS core-shell quantum dot is to effectively block charge recombination by drastically slowing its rate relative to charge separation. It is understood that the effect of an outer shell, surrounding the bare core CdSe dot, derives from two primary components: (i) the spatial extent beyond the core, and (ii) the mass and tunneling barrier relative to the core. Here we separate the two components by carrying out a calculation with the shell made of the same material as the core, i.e. CdSe. In other words, we eliminate the effect of the mass and potential gradients, and consider only the effect of extending the core radius.

15



28 24 kCS (CdSe/ZnS)

-1

ln( k / s )

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Page 16 of 24

kCR (CdSe/ZnS)

20

kCS (CdSe/CdSe) kCR (CdSe/CdSe)

16 12 0

2 6 8 10 12 4 shell thickness D ( Å )

14

Figure 5. The calculated rate constants for charge separation kCS and charge recombination kCR, in CdSe/ZnS – AQ and CdSe/CdSe - AQ dyads, as functions of the ZnS shell thickness D (solid points and lines) or the extended core radius R (in this scheme the R-R0 values, where R0 = 12.7 Å is a radius of CdSe core, were used, see dashed points and lines). The straight lines are the corresponding fits to the data with slope values of 0.358 / 1.08 Å-1 for CdSe/ZnS QDs and 0.424 / 0.244 Å-1 for CdSe/CdSe QDs, as CS / CR, respectively.

The results are summarized in Table 4b and Figure 5. Remarkably, both charge separation and recombination rate constants of the extended core (i.e. CdSe/CdSe) QDs decay exponentially with similar exponents comparable to that of kCS of CdSe/ZnS core/shell QDs. In other words, filling the shell space with ZnS significantly affects only the charge recombination rate constant reducing it several orders of magnitude over the ~8 Å range, while barely changing the charge separation constant over the same range (see also Table 4a). We investigated the origin of this disparity by analyzing the wavefunctions and by comparing QDs with mixed shell properties, separately emphasizing the effect of the potential and the effective mass. Choosing a QD with a three-layer shell thickness (D=15 bohr) for analysis, two additional calculations were done using the following parameters for the shell material: (i) Ve/h(ZnS), me/h*(CdSe), ε(CdSe);

16


Page 17 of 24

and (ii) Ve/h(CdSe), me/h*(ZnS), ε(CdSe). These calculations are summarized in Figure 6 where they are respectively labeled CdSe/ZnS-(V) and CdSe/ZnS-(m*).

-3

CdSe/CdSe CdSe/ZnS-(V) CdSe/ZnS-(m*) CdSe/ZnS

1Se

3×10

-3

2×10

-3

1×10

0 20

25

30

35

40

45

50

25

30

35

40

45

50

-3

3×10

1Sh

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-3

2×10

-3

1×10

0 20

distance from QD center / bohr Figure 6. Electron and hole wavefunctions, 1Se and 1Sh, respectively, of four core/shell QDs with core radius of R0 = 25.89 bohr and shell thickness D = 15 bohr. In all these cases the core is CdSe while the shell is filled with: CdSe (black curve), CdSe with the ZnS potentials (red curve), CdSe with the ZnS effective masses (green curve), and ZnS (blue curve). The vertical lines mark the positions of the core/shell and shell/heptane interfaces.

Comparing the black and blue curves in Figure 6 (CdSe/CdSe and CdSe/ZnS core/shell ODs, respectively) we note that the effect of ZnS as shell material on the hole is much greater than on the electron. While the electron’s amplitude at the outer interface drops by about a factor of two, the hole’s amplitude is suppressed by a factor of ~30. In other words, the electron is relatively insensitive to ZnS’s coating of the QD core. Examining the mixed shell wavefunctions reveals that relative to CdSe the higher hole barrier of ZnS (red curve) has noticeably more

17



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impact on the hole’s amplitude than the heavier hole mass in ZnS (green curve). The effect of the mass gradient appears to be local, as manifested by the inflection point at the core/shell interface. 4. CONCLUDING REMARKS High level calculations of interfacial electron transfer processes involving bulk materials and single molecules, using the first principles-based theories, have traditionally been a challenging task. In this paper we describe an application of a simple hybrid quantum mechanical method to quantum dot–molecule interactions with the ensuing electron/hole separation and recombination processes. The main assumptions we make in this work are that the quantum dot and the molecule do not significantly perturb each other’s orbitals and that the electron transfer is caused by orbital mixing between the external part of the QD wavefunction and the LUMO of the molecule. To describe electron and hole wavefunctions in the quantum dot we use the envelope wavefunction method in the effective mass approximation. The molecule is treated at a density functional level of theory. The orbital energies and the corresponding wavefunctions form the basis for estimating the driving force of the reaction and the quantum mechanical coupling strength. We then apply Marcus’s electron transfer theory to calculate transfer rate constants. The calculations of electron transfer rate between CdSe/ZnS core/shell quantum dots interacting with a functionalized anthraquinone support the earlier experimental observations. 11 Namely, charge separation occurs on a significantly faster time scale than charge recombination. And in contrast to the charge separation process, charge recombination is greatly impeded by the shell of ZnS around the CdSe core. The origin of this disparity is mainly due to the higher potential step for the hole created by ZnS relative to CdSe: 0.9 eV for the hole compared with 0.45 eV for the electron. As a result, the hole 1S wavefunction decays much faster at the shell/molecule interface. Additionally, we affirm the earlier speculation about the existence of a trapped hole following charge separation. Finally, grid based methods have been successfully implemented for solving the multiple exciton states in QDs.45 Therefore, the hybrid DVR method described in this work can also be extended to multi electron/hole transfer processes and to systems involving two or more molecular adsorbates, opening the possibility to better interpret more complex experiments and help design more efficient light harvesting devices.

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AUTHOR INFORMATION Corresponding Authors * E-mail: [email protected], [email protected], [email protected] ACKNOWLEDGMENTS The authors acknowledge U.S. Department of Energy, Office of Basic Energy Sciences, Solar Photochemistry Program (DE-FG02-07ER-15906) for support. We also gratefully acknowledge NSF MRI-R2 grant (CHE-0958205) and the use of the resources of the Cherry Emerson Center for Scientific Computation. SUPPORTING INFORMATION (1) Charge separation and recombination rates, and (2) optimized geometry of anthraquinone on CdSe. This material is available free of charge via the Internet at http://pubs.acs.org.

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45

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Page 24 of 24

Table of Contents Use only: Alexey L. Kaledin,* Tianquan Lian,* Craig L. Hill, and Djamaladdin G. Musaev*, kCS$

,$

ZnS$

LUMO+1$ 1Se"

AQ$

CS$

trapped'hole'

CdSe D$ $ +$

CR$

LUMO$(AQ)$ SOMO$(AQ,)$

(AQ,)$ 1Sh"

kCR$

24

HOMO$


A Hybrid Quantum Mechanical Approach: Intimate ... - ACS Publications

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