Quantum Confinement Theory of Auger-assisted Biexciton

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C: Physical Processes in Nanomaterials and Nanostructures

Quantum Confinement Theory of Auger-Assisted Biexciton Recombination Dynamics in Type-I and Quasi Type-II Quantum Dots Alexey L Kaledin, Degui Kong, Kaifeng Wu, Tianquan Lian, and Djamaladdin G Musaev J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b04874 • Publication Date (Web): 22 Jul 2018 Downloaded from http://pubs.acs.org on July 25, 2018

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The Journal of Physical Chemistry

Quantum

Confinement

Theory

of

Auger-assisted

Biexciton

Recombination Dynamics in Type-I and quasi Type-II Quantum Dots Alexey L. Kaledin,1,* Degui Kong2, Kaifeng Wu3, Tianquan Lian1, Djamaladdin G. Musaev1,* 1

Cherry L. Emerson Center for Scientific Computation and Department of Chemistry,

Emory University, Atlanta, GA 30322 2

College of Electronic Engineering, Heilongjiang University, Harbin 150080, China

3

State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical

Physics, Chinese Academy of Sciences, 457 Zhongshan Road, Dalian 116023, China Abstract Use of quantum confinement as a tool to control biexciton recombination in quantum dots is investigated theoretically for a series of quasi-Type-II and Type-I CdSe/CdS core/shell spherical quantum dots. Recent experimental measurements show that in such nanostructures, the CdS shell may act as a type of an efficient retarder for biexciton recombination in the quasi-Type-II regime, but not as efficient in the Type-I regime, and that this phenomenon is achieved by a strong charge separation in the former. These findings are interpreted on the basis of Auger-assisted biexciton decay using a quantum confinement theory. We perform single-band effective mass calculations simulating the quasi-Type-II regime by a CdSe 2.4 nm core + CdS shell, and the Type-I by a CdSe 3.8 nm core + CdS shell. The calculations reveal a tightly confined hole for both types of regimes. The key difference occurs in the behavior of the electron, whose wavefunction is progressively delocalized into the growing shell region in the quasi-Type-II regime but remains partially confined in the core in the Type-I regime despite the increasing shell thickness. This behavior yields the calculated Auger transition amplitudes, and consequently the lifetimes, which closely correlate with the measured ones. The calculations expose further details of biexciton Auger recombination dynamics, such as the dominance of the hot electron channel over the hot hole channel, and the fact that the cold-to-hot electron (and cold-to-hot hole) transitions are qualitatively correctly described by the single-band effective mass approximation, giving this primitive theory a useful validation for treating high-energy excitonic transitions in Type-II/Type-I core/shell quantum dot structures.

*[email protected], [email protected]

ACS Paragon Plus Environment

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I. Introduction The phenomenon of quantum confinement has long been used to model the spectral and dynamical behaviors of semiconductor nanoparticles (quantum dots) as a function of particle size.1,2 As is well-known, quantum dots possess a variety of attractive optoelectronic properties, such as size-dependent photo absorption and emission, long exciton lifetimes, multiple exciton generation, efficient control of charge carriers spatial distribution over a size range, and many other size-tunable properties. 3 , 4 Of special interest in the past few decades have been Auger effects, which play an important role in charge carrier processes in quantum dots, including electron/hole intraband cooling, biexicton recombination, interfacial electron transfer to molecular acceptors, single quantum dot photoluminescence intermittency (the so-called “blinking”), and others.5,6,7 Among these, multiexciton generation (MEG) by the absorption of one high-energy photon in colloidal quantum dots, especially those of core/shell composition, has received particular attention recently due to its potentials for dramatically enhancing the efficiency of quantum dot based solar cells and solar-to-chemical reactions.8,9 However, ultrafast Auger recombination of multiexcitons (on the picosecond timescale) would erase all the efficiency gains due to MEG if carrier extraction rates could not compete with Auger recombination, highlighting the importance of understanding and suppressing Auger decay for many applications of quantum dots. Since quantum confinement involves, by definition, controllable size effects on nanoparticle’s wavefunction, computational modeling requires the use of rather simplified approaches.10,11 Particularly challenging for high-level, atomistic theories, are the core/shell type quantum dots, where in addition to the size (the number of atoms) there is the difficulty of treating material interface.12 Thus, one of the more widely used, practical wavefunction-based theories is in the family of the effective mass approximation (EMA),1, 13 which includes the standard single band EMA as well as progressively more sophisticated multi-band EMA theories, so called k*p.14 The single band EMA has been shown to work well for describing quantum dots ionization potentials, electron affinities, exciton binding in their lowest excited states,2 radiative transitions near band edges and also these properties for more complicated core/shell quantum dots.10,15,16,17,18 In other words, even in the single band limit, the EMA is a


2

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reasonable approximation for describing excitons with the energy comparable to bulk band gap Eg. However, the single parabolic band may not be a good approximation at much higher energies,19 including the cases involving multiple excitons. This problem has been addressed in a number of recent studies of Auger-mediated biexciton and trion decay lifetimes. In colloidal CdTe/CdSe core/shell dots, the authors treat the 1S electron and hole states at a 2-band EMA level of theory while using uniform waves for high energy excited states on the assumption of a fully delocalized excited state wavefunction.20 In two other studies, specifically on one-dimensional CdSe quantum dots, the authors use plane waves to describe (quasi)continuum excited state.21,22 In the present work, we take the single band EMA approach to biexciton decay a step further and calculate the excited electron (hole) state explicitly. We propose that both the 1S states and the excited states with the exciton energies of the order of ~2Eg, in both conduction and valence bands, when calculated explicitly with the single-band EMA while simultaneously taking into account all the existing electrostatic interactions, can provide a qualitatively correct representation of the particles wavefunctions in core/shell quantum dots and therefore the dynamics of biexciton recombination. We describe EMA calculations for a series of size-varying core/shell CdSe/CdS quantum dots designed in two confinement regimes, quasi-Type-II (small CdSe core) and Type-I (large CdSe core), to reveal key differences between the two regimes in the variation of the CdS shell thickness. We model biexciton decay on the basis of an Augerassisted recombination theory, applied to bound-to-bound transitions, using Fermi’s golden rule. Note that the radiative decay time of biexcitons in these quantum dots is on the tens to hundreds of nanoseconds timescale, and hence is neglected in our analysis of biexciton decay. Our results are analyzed and compared with the recently published experimental measurements of the same systems. 23 As will be shown below, the calculated data closely correlate with the measured ones and also prove that the key difference between the biexciton Auger recombination rates in quasi-Type-II and Type-I quantum dots indeed lies in the different behaviors of electron delocalization in these two regimes, as described by Lian et al in their experimental work.23 In light of the two competing explanations for Auger rate suppression in core/shell dots, namely, the envelope wavefunction overlap argument23 and the interface alloying argument,21,22 we


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note that the experimental measurements interpreted in the present work were made in the regime where CdSe/CdS alloying at interface is negligible.23 Therefore, our computational approach using the overlap argument is justifiable at the level of the single band EMA, which is capable of such description. In addition, our calculations expose further details of biexciton Auger recombination in these core/shell quantum dots such as the shell-thickness dependent Auger rates of the negative and positive trion channels, some of which have been reported in recent experiments.24 Thus, our work not only gives the primitive EMA theory a useful validation for treating high-energy excitonic transitions in heterostructured quantum dots, but also could stimulate future experimental studies into the Auger decay of the detailed trion channels.

Scheme 1. A schematic representation of Auger-assisted biexciton decay as a twoparticle three-orbital process. In biexciton (a), electrons e1 and e2 occupy the lowest energy CB orbital 1Se; holes h1 and h2 are in the VB orbital Sh. e1-h1 recombination promotes e2 into a high energy CB orbital NSe (horizontal bars) while the hole h2 remains in Sh culminating in the hot electron exciton (b). Similarly, in biexciton (c) the h1-e1 pair recombine, and hole h2 is excited into the orbital MSh (horizontal bars) deep in the VB leading to the final state, hot hole exciton (d). The total energy, the orbital angular momentum and the spin are conserved in these processes. Several hot electron/hole Swave excitons (black horizontal bars in (a) and (c)) are possible as final states for the target energy level (red horizontal bar). The allowable range is defined by the width of the Lorentzian (red curve), Γ; see equation 2. II. Theoretical approach to biexciton recombination rate


4

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Auger-assisted biexciton (BX) decay is approximated by a three-particle, threeorbital process,20,25 comprised of a pair of trion decay processes, illustrated in Scheme 1. As the two electrons occupy the lowest orbital 1Se in the conduction band and the holes occupy the corresponding orbital 1Sh in the valence band, an electron hole pair, say e1-h1, recombines. From this initial configuration, the other electron, e2, is promoted to an excited orbital NSe, mediated by e1-e2 Coulomb repulsion and conserving the energy and the angular momentum (orbital and spin). This process is referred to as the negative trion decay which leads to a hot electron exciton. Alternatively, the hole h2 may be promoted deeper into the valence band, MSh, while also conserving the energy (and also the angular momentum), mediated by h1-h2 repulsion. This is referred to as the positive trion decay, yielding a hot hole exciton. The total rate of the biexciton decay is a sum of the rates of the two trion processes, negative and positive, respectively,

kBX = 2kX- + 2kX+ ,

(1)

where X- and X+ are the negative and positive trions, respectively. The factor of 2 reflects the BX degeneracy. The Auger lifetime is then defined as τ = 1 kBX . Application of the time-dependent perturbation theory to each of the two rates in (1) yields the well-known Fermi golden rule formula.14,26 In the present work, we use an equivalent expression but with a Lorentzian width function, with a width parameter Γ, to represent an allowable transition range of the final bound states,25

(2a)

(2b)


5


with the summations running over the final states in the conduction band, for X-, and valence band for X+ trions. The operator is a two-particle Coulomb repulsion in a nonuniform dielectric medium. Since the spin is conserved, it is sufficient to write the initial and final state two-particle wavefunctions using only the spatial orbital parts. The initial and final 2-electron singlet states, written as Slater determinants, of the negative trion are, respectively,

i(−) = φS (1)φ S (2)

(3a)

1 ( ψS (1)φ N (2) + ψS (2)φ N (1) ) . 2

(3b)

and

f N(−) =

And the initial and final 2-hole states, of the positive trion are, respectively,

i(+) = ψ S (1)ψS (2)

(4a)

and

fM(+) =

1 ( φS (1)ψ M (2) + φS (2)ψ M (1) 2

)

(4b)

where the numbers in the parentheses of the r.h.s. are particle’s indices. In (3b), φN is an excited electron orbital in the CB, and in (4b), ψM is an excited hole orbital in the VB.

Absorption peak (eV)

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2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 0

EMA (2.4 nm) exp. (2.4 nm) EMA (3.8 nm) exp. (3.8 nm)

0.5

1

1.5

2

2.5

3

CdS shell thickness (nm)

Figure 1. The calculated, at the EMA level of theory, and the measured23 absorption peaks for the two types of quantum dots, with the 2.4 nm CdSe core (black line and back dots) and 3.8 nm CdSe core (read line and red dots) as functions of CdS shell thickness.


6

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The orbital for periodic cell structure, i.e. crystal structure of bulk material, is a product of the local periodic and global periodic parts. 27 The true wavefunctions of a finite size crystal (Eqs 3 and 4) can also be approximated as such a product of the local (periodic Bloch function) and the envelope parts.20 In such a separable approximation the one-particle orbitals are

φS (r) = uCB (r)1Se (r),

(5a)

ψ S (r) = uVB (r)1Sh (r) ,

(5b)

φ N (r) = uCB (r)NSe (r) ,

(5c)

ψ M (r) = uVB (r)MSh (r) ,

(5d)

where uCB and uVB are the Bloch functions with the periodicity of the direct lattice in the bulk, respectively in the conduction and valence bands, and 1Se, 1Sh, NSe and MSh are the corresponding spherically symmetric envelope parts. The Bloch functions describe the local character within a unit cell, which here we assume does not change on the 1Se → NSe and 1Sh ← MSh electron transitions, while the envelope functions describe the spatial extent and depend sensitively on the size and shape of the dot. Substituting (5) into (3) and (4) the local and envelope parts are separated, leading to the following expressions for the matrix elements appearing in the numerators of (2),

Kif(−)N = 2F ∫ dr2 dr1

1Se (r2 )NSe (r2 )1Se (r1 )1Sh (r1 ) ε (r1, r2 ) r1 − r2

(6a)

Kif(+)M = 2F ∫ dr2 dr1

1Sh (r2 )MSh (r2 )1Se (r1 )1Sh (r1 ) , ε (r1, r2 ) r1 − r2

(6b)

and

where the common scale factor is the overlap between the Bloch functions,

F = uCB uVB


(6c)

7


Page 8 of 25

arising due to Cd(5s) ← Se(4p) and Cd(5s) ← S(3p) bulk transitions. In a 2-band

(

approximation,20 we may write uCB = 1+ c12

) (φ + c φ ) and u −1 2

s

1 p

VB

= (1+ c22 )

−1 2

(c φ

2 s

+ φp ) ,

in an orthogonal s-p basis, where the mixing coefficients are solutions to a parameterized secular equation. However, the exact value of F is not critical if one is interested in Auger rate dependence on dot size, e.g. the CdS shell thickness, since the atomic transitions are localized and likely to remain independent of overall size. The remaining task is to compute the double integrals in (6). Thus, in what follows we set F = 1 and analyze the Auger rate as a function of CdS shell thickness for the two different CdSe core diameters. As in previous works,28,29,30 we simplify the dielectric function as the average of the two positions, i.e. ε (r1, r2 ) = (ε (r1 ) + ε (r2 )) 2 , properly capturing direct interactions at any two positions within same dielectric environment, and those involving positions on different sides of an interface.31 This approximation has been shown to be quite accurate for quasiType-II CdSe/CdS exciton binding energies.29

III. Effective mass treatment of CdSe/CdS exciton bands In the present work, the envelope orbitals of CdSe/CdS quantum dots are calculated within the effective mass approximation.1,27 The material parameters for bulk wurtzite CdSe and CdS used in the present EMA calculations are those used by us previously29 and are also given in Table 1. We note the presence of a light electron and a heavy hole for both CdSe and CdS. In conjunction with a low CdS conduction band edge, only 0.22 eV above CdSe CB edge, the light electron is prone to delocalization into the QD shell. The much heavier hole has a higher escape barrier into the CdS VB, about 0.6 eV, which is expected to cause 1Sh localization in the CdSe core region. The CdSe/CdS conduction and valence band offsets used in the calculations are consistent with the experimental bulk values.32 Table 1. Material parameters used in the EMA calculations: m0 is the electron mass; ε0 is the vacuum permittivity; Ve and Vh are conduction and valence band edges respectively. CdSe CdS chloroform me* / m0

0.13

0.21

1


8

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mh* / m0

0.45

0.80

1

ε / ε0

10

8.9

4.7

Ve / eV

-4.0

-3.78

0

Vh / eV

-5.7

-6.29

-8.4

Calculation of orbitals requires numerically solving the coupled SchrödingerPoisson equations for a single exciton in the initial state:28,29,30 for the electron in 1Se orbital,

, 2

∇ ⋅ ε r ∇Φe (r) = 4π 1Se (r) .

(7a) (7b)

and for the hole in 1Sh orbital,

, 2

∇ ⋅ ε r ∇Φh (r) = −4π 1Sh (r) ,

(8a) (8b)

and corresponding equations for all final state excitons (1Sh,NSe) and (MSh,1Se) for the various N and M values of the excited S-orbitals. The curly brackets in 7a and 8a contain, respectively, the effective mass kinetic energy terms, the band potentials for electron and hole, the electrostatic source potentials due to hole and electron, and the self-induced electron and hole potentials with the proper factor of 1/2.29,33 The Poisson equations are solved subject to the condition that the inner and outer potentials match at the surface boundary.28,30 We note that the outer potential in the surrounding environment (presently an organic solvent), at a point sufficiently far from the QD, is given by −1 εoutside ∫ d r′|1Se (r′) |2 r′ − r , for the electron, with the corresponding expression for the

hole, where εoutside is the dielectric constant of the medium outside the quantum dot,


9


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presently chloroform (cf Table 1). The surface, defined by all points with the dot center 2 at which the inner and outer potentials are as the origin satisfying x 2 + y 2 + z 2 = Rmax

matched, coincides with the surface beyond which the wavefunction is made to vanish, which in the present study is a distance ∆R from the outer surface of a quantum dot, i.e. Rmax=Rcore+L+∆R, where Rcore is the radius of the CdSe core, L is the CdS shell thickness, and ∆R = 1 nm. This sets the necessary boundary conditions for the differential equations 7b, 8b.

Table 2. DVR grid parameters for the two types of the quantum dots: CdSe core diameters 2.4 nm and 3.8 nm, with the variable shell thickness L2.4 and L3.8, respectively, in nm. Rmax is the sphere radius in nm beyond which the wavefunction is made to vanish; grid denotes the ∆x=∆y=∆z spacing in bohr, and size is the resulting DVR matrix dimension. L2.4 Rmax grid size L3.8 Rmax grid size 0

2.2

4.0

4729

0

2.9

4.0

10827

0.2

2.4

4.0

6043

0.8

3.7

5.0

11459

0.7

2.9

4.0

10827

1.1

4.0

5.0

14411

1.35

3.55

5.0

10131

1.5

4.4

6.0

11152

1.65

3.85

5.0

12965

1.7

4.6

6.0

12712

2.95

5.15

6.0

17904

The differential operators, wavefunctions and electrostatic potentials are recovered using a contracted, infinite order, discrete variable representation (DVR) method described by us in detail previously.28 We construct a uniform isotropic Cartesian grid with ∆x = ∆y = ∆z and retain the points within the cutoff radius Rmax (see Table 2 for details). The above conditions define the coupled matrix equations: (i) the eigenvalue equations (7a,8a) for the wavefunctions, which are solved using Davidson’s subspace expansion,34 and (ii) linear equations (7b,8b) for the electrostatic potentials, which are solved using a generic linear conjugate gradient algorithm.


10

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Table 3A. Summary of CdSe/CdS core/shell spherical dots for core diameter 2.4 nm and shell thickness L, nm. The initial state (1Sh,1Se) biexciton with energy Ei and final state (MSh,NSe) exciton with energy Ef are reported in eV, in the order of increasing |Ei - Ef|. The integer pairs in parentheses next to each energy are the respective indices of excited S orbitals (M,N). The Auger matrix element Kif (equation 6) is given below each final state energy in meV. L Ei Ef (MSh,NSe) Kif 0

0.2

0.7

1.35

1.65

2.95

5.323

4.912

4.553

4.401

4.372

4.330

5.314(3,1)

6.025(1,2)

3.830(2,1)

2.0

19.2

23.2

4.837(3,1)

5.129(1,2)

6.279(1,3)

3.438(2,1)

1.3

20.6

2.9

21.0

4.669(4,1)

3.671(1,2)

3.579(3,1)

5.952(1,3)

0.3

20.7

6.2

5.3

4.294(1,3)

4.292(5,1)

4.917(6,1)

3.540(4,1)

3.075(3,1)

5.995(1,4)

6.1

0.7

0.1

3.7

9.6

2.6

4.579(6,1)

3.865(5,1)

3.818(1,3)

5.309(1,4)

0.2

1.5

6.4

3.3

4.171(8,1)

4.562(1,4)

4.652(9,1)

3.745(7,1)

3.611(1,3)

0.5

2.0

0.2

1.3

4.0

The induced potentials Φeind , Φhind are derived from the source potentials Φe, Φh by integrating the induced surface polarization density31 over all interfacial (CdSe/solvent, CdSe/CdS and CdS/solvent) surfaces I,


,

(9a)

.

(9b)

11


Page 12 of 25

I are By the convention used in deriving equations (9) the dielectric constants εinI and ε out in measured on either side of the interface along the surface norm nˆout pointing from inside

to outside; the potential gradient is evaluated just inside the interfacial surface. We have also used the fact that the electric field

, crossing interface I along the normal

direction, has a discontinuity:

. In the trivial case of a uniform ε, the

induced potentials are identically zero. It can be shown that for a grid contracted down from an infinite expansion, as is done in the present work, the electric potential (the solution of the Poisson equation) has a simple analytic representation,28,35

(10) where Φijk is the DVR solution of equations (7b, 8b) as the value of the electrostatic potential at grid point ijk;

, etc, and the sum runs over the entire grid.

The Cartesian gradient of equation (10), which is to be substituted into equations (9), readily follows at any point on the grid.

Table 3B. Summary of CdSe/CdS core/shell spherical dots for core diameter 3.8 nm and shell thickness L (in nm). For definition of other parameters and the presented values see the caption of Table 3A. L Ei Ef (MSh,NSe) Kif 0

0.8

1.1

1.5

1.7

4.359

4.026

3.986

3.956

3.945

4.809(4,1)

3.733(1,2)

3.623(3,1)

6.100(1,3)

0.5

19.0

0.1

0.9

3.989(1,3)

4.169(5,1)

3.386(4,1)

2.856(1,2)

3.0

0.1

0.5

16.2

3.726(5,1)

3.623(1,3)

4.441(6,1)

5.197(1,4)

0.4

5.4

< 0.1

0.7

3.942(6,1)

4.401(1,4)

4.507(7,1)

3.375(5,1)

3.299(1,3)

3.023(4,1)

0.2

1.7

0.1

0.5

6.6

0.2

4.097(1,4)

3.726(6,1)

4.318(7,1)

3.266(5,1)

3.162(1,3)

2.937(4,1)

2.4

0.3

0.0

0.4

7.0

0.4


12

Page 13 of 25

With these definitions, the coupled equation systems 7-9 are solved to selfconsistency. Typically, convergence of the wavefunction to 10-7 is reached after a few iterations. For the special case of an excited state calculation, the Hamiltonians in 7a, 8a are transformed to yield the sought excited state to be the ground state, i.e. by applying a shift to the Hamiltonian matrix (the DVR matrix representation of the quantity in the curly brackets in equations (7a) and (8a)), and squaring the result: H ⇒ (H − λ I)2 . The transformed matrix shares the same eigenvectors as the original, while scanning over a sufficiently wide range of λ yields the desired excited states.

-1

kX- (ps )

10

CdSe core diameter 2.4 nm CdSe core diameter 3.8 nm

8 6 4 2 0

1.2 -1

kX+ (ps )

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


0.8 0.4 0 0

1

2

3


Figure 2. Biexciton decay rates, for the small and large CdSe core, decomposed into two trion decay channels: the negative trion X- and the positive trion X+. The rates are calculated using equations 2a and 2b, respectively. The lifetime parameter Γ is 0.8 eV.

IV. Results and discussion The results of the EMA calculations, and the Auger matrix elements, are given in Tables 3A and 3B. Additionally, Figure 1 compares the measured and calculated absorption energies, reported here as the 1Sh → 1Se excitations. Two confinement regimes, with a small 2.4 nm CdSe core diameter, corresponding to a quasi-Type-II quantum dot, and a large 3.8 nm CdSe core diameter, corresponding to a Type-I quantum dot are considered.


13


The number of layers of CdS, with the corresponding shell thickness L, matches that used in the experiment. The agreement with experiment is overall quite close, despite the noticeable overestimation (~0.2 eV) of the EMA excitation energy for the bare core CdSe(2.4 nm) dot. It is conceivable that in strong quantum confinement regimes the energy is overestimated by the EMA model. 36 Therefore, we expect a better relative description of the energies and wavefunctions of the dots in the weak confinement regime, i.e. with increasing shell thickness, which is the focal point of the present work. As single exciton energies, the 1Sh → 1Se transitions, when multiplied by 2 are treated as biexciton energies, or the initial energies Ei entering equations (2). With this definition, the biexciton energies range from ~5.3 to ~4.3 eV in the small core dots and from ~4.4 to ~3.9 eV in the large core dots. In these ranges we search for all excited S-wave excitons (above and below the target energy, see Scheme 1), separately in the conduction band as (1Sh,NSe) single excitons, and in the valence band as (MSh,1Se) single excitons. We note that at least three available S-wave excitons were found for the quasi-Type-II dots, and four for the Type-I dots, with the number increasing by 1-2 with the growing shell thickness, as seen in Tables 3A,3B.

τexp (ps)

400

CdSe core diameter 2.4 nm CdSe core diameter 3.8 nm

300 200 100 0 400

ατEMA (ps)

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

Page 14 of 25

300 200 100 0 0

1

2

3

CdS shell thickness (nm) Figure 3. A direct comparison of the measured23 biexciton Auger lifetimes (τexp) and those calculated with the EMA approximation (τEMA). The calculated values are scaled by a constant (α=78.0) chosen so that ατEMA=τexp for the CdSe(2.4 nm)/CdS(2.95 nm) dots.


14

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Using the final states Ef as the basis for the negative and positive trion decay channels, we calculate the corresponding transition matrix elements (6) from which one can derive the rate constants (2). The choice of the Lorentzian width parameter Γ, entering the rate equation, should be dictated by the S-wave level spacing, yet, since the energy levels do not correspond to true electronic eigenstates, it was necessary to gradually increase it from a near-resonance regime (Γ ~ 0) to an off-resonance regime (Γ ~ 1 eV) until the rate curves converged. A common choice for both the negative and positive trion channels was Γ = 0.8 eV, a value comparable to an average level spacing for all the states reported in Tables 3. We use the same Γ for all calculations to properly reflect the change in the number of available S-states with shell thickness. These results are presented in Figure 2. Immediately, we observe that the X- channel, or the hot electron channel, is about an order of magnitude faster than its hole counterpart in the quasi Type-II dots, with the exception of the bare core 2.4 nm CdSe dot, and about two orders of magnitude faster in the Type-I dots. The Auger matrix elements and the energies, reported alongside each other in Tables 3A and 3B, suggest that in the first case, the preference for the electron excitation channel is due to the available S-wave in the conduction band lying in close proximity of the target energy. While in the second case, the preference for the hot electron channel is dictated by the Auger matrix elements. As we will analyze below in more detail, this is due to the generally stronger wavefunction mismatch in the valence band and a much smaller resulting Auger matrix element. Secondly, we note the abrupt jump in the hot electron channel going from a bare core CdSe(2.4 nm) to a one-layer CdSe(2.4)/CdS(0.2 nm) structure. This is explained by the opening of a near resonant channel (1Sh,2Se) and the fact that it has a strong Auger transition element (cf. Table 3A) due to a favorable overlap with the 1Se orbital. Further growth of CdS shell layers has a common effect on both rates, causing them to monotonically decrease. It is important to stress, however, that this preference for the hot electron channel may arise from the structure of CdSe/CdS band alignment, as well as from escape barrier heights (see Table 1). That is, in the present EMA model, the electron, given also its lighter mass, is bound in a deeper, less confining potential than the hole; and thus the final NSe orbitals are less


15


oscillatory than their MSh hole counterparts, resulting in the larger Auger matrix elements seen in Tables 3A,3B. Combining both X- and X+ channels provides the total lifetimes, which are plotted in Figure 3, along with the experimental measurements. At this stage we recall the Bloch function overlap element, Eq. 6c, which has been factored out of the Auger transition amplitudes due to its being assumed independent of quantum dot size. Since its magnitude is not known, we scale the calculated lifetimes by a constant chosen so that ατEMA = τexp for CdSe(2.4 nm)/CdS(2.95 nm), yielding α = 78.0. It is, of course, possible to relate Eq. 6c to α by simple substitution, yielding an average value over the dot size of = 0.11, which is comparable to the ones calculated for CdTe quantum dots.20 More importantly, however, a visual inspection of Figure 3 shows an excellent correlation between theory and experiment. The lifetime curve of the quasi Type-II dot increases gradually until about 1.3 nm shell thickness to ~100 ps before making a steeper jump to reach ~350 ps at approximately 3 nm shell thickness. The Type-I curve, on the other hand, increases up to ~200 ps near 1 nm shell thickness and hits a plateau going further to the

probability in 2.4 nm CdSe core

thicker shells.

probability in 3.8 nm CdSe core

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

Page 16 of 25

1

0.4

1 1Se 0.8 N1 S e 0.6 N2 S e square wave 0.4

0.2

0.2

0.8 0.6

0

0 1

0.5

1

1.5

2

2.5

3

0

0 1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

1 0.5 1.5 CdS shell thickness (nm)

2

0

0

1Sh M 1 Sh M 2 Sh

2.5

3

1 0.5 1.5 CdS shell thickness (nm)

2

0.5

1

1.5

2

Figure 4. Localization of electron (left) and hole (right) orbitals, both in the initial (black curves) and final (red and green) exciton states, in the CdSe core as a function of CdS shell thickness. For conciseness, two most important final states are shown. The blue


16

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curve, shown here for reference (see text), is an analytic function derived for a normalized square wave final state defined as const. inside the dot and 0 outside. We attempt to interpret the lifetime curves on the basis of quantum confinement in the two different regimes: the small and the large core. To this end, we analyze the probability of finding an electron and a hole within the CdSe core, both in the initial and the final exciton states, as a function of shell thickness. Here, we find it useful to delineate a lower bound of core-localization probability by using a volume argument, i.e. the core-to-shell volume ratio must always decrease and eventually approach zero for a fixed core and an expanding shell system. The simplest orbital type fulfilling this requirement is a normalized square wave: constant everywhere inside the core+shell space and zero outside, i.e. a maximally delocalized wave. This analysis is presented in Figure 4, where the “square wave” core-localization probability is shown by a blue line. In both confinement regimes, the 1Sh orbital is fully localized within the core, yet the 1Se orbital behaves differently: in the quasi Type-II regime it escapes into the growing shell region while in Type-I regime it tends to stay in the core, even for the thick shells. This essentially implies that the initial state is a charge-separated biexciton in quasi-Type-II regime, i.e. a [CdSe]2+/[CdS]2- configuration, and a non-charge-separated biexciton in the Type-I regime, approximately a [CdSe]0.5+/[CdS]0.5- configuration at 1 nm shell thickness. A further revelation can be made by a more detailed examination the behavior of the excited NSe orbitals (two of several are shown in Figure 4, for conciseness), which mimics that of the corresponding 1Se orbitals, in their respective regimes. The excited electron S-waves in the quasi-Type-II regime almost completely escape the core, while the Type-I excited electron S-waves prefer to stay within the core. The excited hole S-waves, on the other hand, penetrate out into the shell regardless of the confinement regime. Recalling the integrands of equations 6, three orbitals must simultaneously have a good spatial match for the integral to be significant, i.e. 1Sh, 1Se, NSe for the X- channel, and 1Sh, 1Se, MSh for the X+ channel. Using the spatial localization analysis given above, we may conclude that as CdS increases: (A) in quasiType-II regime both products 1Sh1SeMSh and 1Sh1SeNSe become smaller, implying both X+ and X- channels become effectively closed to biexciton recombination, and thus the total Auger lifetime becomes longer; and (B) in the Type-I regime the product 1Sh1SeMSh


17


becomes smaller, while the product 1Sh1SeNSe becomes smaller only until ~1 nm and then acquires a steady value beyond the 1 nm mark, implying that the X+ channel becomes closed while the X- channel remains open, thus flattening the total Auger lifetime curve. This qualitative derivation of biexciton recombination dependence on shell thickness based on spatial localization is in agreement with the full calculation of

Exciton binding energy (eV)

Figure 3.

(A)

-0.08

(1Sh,1Se) (1Sh,NSe) (MSh,1Se)

-0.12 -0.16 0

Exciton binding energy(eV)

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

Page 18 of 25

0.5

1

1.5

2

2.5

3

-0.06

(B) -0.08 -0.10 -0.12 0

0.5

1

1.5

2


Figure 5. Exciton binding energies We-h in the two confinement regimes, (A) quasi TypeII (2.4 nm core), and (B) Type-I (3.8 nm core). The black curves are the single exciton binding energies in the initial (1Sh,1Se) state. The red curve is exciton binding in the hot electron final state, and green curve is exciton binding in the hot hole final state. As a part of wavefunction localization analysis, we calculate two additional, highly revealing properties, the exciton binding energy We-h , and “exciton radius” change ∆reh upon Auger transition to the conduction or valence band, in each of the two confinement regimes. Exciton binding is calculated as a sum of three component interactions,29

We−h = W(e−h) + We−(e) + Wh−(h ) ,

(11)

where W(e-h) is the energy of the electron in the field of the hole (or equivalently the hole in the field of the electron), shown below for the (1Sh,1Se) exciton,


18

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W(e−h) = − ∫ drΦh (r) 1Se (r) = 2

∫ drΦ (r) 1S (r) e

2

h

,

(12a)

and We-(e) and Wh-(h) are electron’s and hole’s self-energy due to interface polarization, respectively,

We−(e) = − Wh−(h) =

1 2

1 2 drΦind ∫ e (r) 1Se (r) 2

∫ drΦ

ind h

(12b)

2

(r) 1Sh (r) .

(12c)

These results are shown in Figure 5. The exciton radius change in calculated as the difference in the expectation values of the electron-hole instantaneous position in two quantum states,

∆reh(−) =

∫ dr

h

1Sh (rh )

2

∫ dr ( NS (r ) e

e

2

e

− 1Se (re )

2

) r −r e

h

(13a)

for the X- trion decay channel, or the hot electron channel, and

∆reh(+) =

∫ dr 1S (r ) ∫ dr ( MS (r ) 2

e

e

e

h

h

h

2

− 1Sh (rh )

2

) r −r e

h

(13b)

for the X+ trion decay channel, or the hot hole channel. These results are shown in Figure 6. As can be seen in Figure 5, there is noticeable difference in the binding energy behavior in the two confinement regimes as a function of CdS shell. In the quasi-Type-II regime, the exciton binding energy monotonically decreases with shell thickness in the initial and final states. This supports a dissociating exciton proposition. In the Type-I regime, the binding energies in the initial and final states slow their decrease rate around the 1nm shell thickness, supporting the proposition of a bound, i.e. non-dissociating exciton.


19


∆reh (nm)

1.5

hot electron hot hole

(A) 1.0 0.5 0.0 0 1.5

∆reh (nm)

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

Page 20 of 25

1

0.5

1.5

2

2.5

3

(B) 1.0 0.5 0.0 0

0.5

1

1.5

2


Figure 6. The change in the exciton radius reh occurring upon the Auger transition from the initial (ground) state to the final state (hot electron/hole) as a function of shell thickness, presented for the two regimes: quasi-Type-II (A), and type-I (B).

Exciton radius change behavior closely correlates with the binding energy. There is a well-pronounced, linear growth in exciton radius, as a function of CdS shell thickness, in the quasi-Type-II regime in both decay channels (Figure 6A). This is a strong indication that Auger transitions in this regime are weak because they require energetically unfavorable charge separation. In the type-I regime, only the hot hole channel marks a very modest increase in the exciton radius. The exciton radius change in the hot electron channel, however, experiences a leveling-off just before the 1nm CdS shell thickness, a common feature in all of the analysis presented here. This flattening of the red curve at 1 nm CdS (Figure 6B), which reflects the static Auger decay rate, is another computationally derived evidence in support of the experimentally observed flattening Auger lifetime in Type-I regime.

V. Conclusions We have described calculations of biexciton recombination lifetimes in two types of quantum dots: quasi-Type-II and Type-I CdSe/CdS spherical core/shell dots, with respective core diameters of 2.4 and 3.8 nm, and CdS shell thickness in the 0-3 nm range. Analysis of the results and comparison with the available experimental data23 show that the EMA calculations are able to capture the essential qualitative features of biexciton


20

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decay in the two confinement regimes as functions of the shell thickness. The calculations support the presence of a tightly confined hole for both types of regimes. The key difference occurs in the behavior of the electron, whose wavefunction is progressively delocalized into the growing shell region in the quasi-Type-II regime but remains partially confined in the core in the Type-I regime despite the increasing shell thickness. This leads to the monotonically increasing, with shell thickness, lifetime in the quasi-Type-II regime, and a flattening lifetime curve in the Type-I regime for CdS shell thicker than ~1 nm. In summary, we recall that the chosen level of theory used in the present calculations is the single-band effective mass approximation, i.e. generally the most primitive theoretical treatment of quantum confinement effects in nanoparticles. Secondly, we calculate explicitly both the low energy exciton states (with EX = Eg) and the highenergy exciton states (EX = 2Eg), required for computing the Auger matrix elements. Based on the above we make the following conclusions, (i) with the properly chosen bulk parameters (effective masses, band edge energies, dielectric constants and ligand potentials), the single-band EMA is a qualitatively reliable method for treating quantum confinement effects in Type-I,II core/shell quantum dots in strong and weak confinement regimes, and (ii) the theory recovers crucial details of Auger-mediated biexciton recombination, in the said quantum dot types, by virtue of a correct description of the high-energy exited states, in the conduction and valence bands. The latter point, which is proven by a detailed wavefunction analysis, is quite encouraging for large-scale applications which are beyond the grasp of atomistic-level theories, examples including charge-separation effects in nanorods, nanoplatelets, nanodisks, as well as the more sophisticated constructions such as bulbs and dumbbell nanostructures.

Acknowledgements This work was funded by the U.S. Department of Energy, Office of Basic Energy Sciences, Solar Photochemistry Program (DE-FG02-07ER-15906). We also gratefully acknowledge NSF MRI-R2 grant (CHE-0958205) and the use of the resources of the Cherry Emerson Center for Scientific Computation.


21


Page 22 of 25

Supporting Information. Tabulated orbital energy levels of the final state excitons in Type-I and quasi Type-II quantum dots.

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TOC Graphic


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Quantum Confinement Theory of Auger-assisted Biexciton

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