Networks of Quantum Nanodots: The Role of Disorder in Modifying

J. Phys. Chem. B 1998, 102, 7727-7734

7727

Networks of Quantum Nanodots: The Role of Disorder in Modifying Electronic and Optical Properties F. Remacle† De´ partement de Chimie, B6, UniVersite´ de Lie` ge, B 4000 Lie` ge, Belgium

C. P. Collier, G. Markovich, and J. R. Heath Department of Chemistry and Biochemistry, UniVersity of California Los Angeles, Los Angeles, California 90095

U. Banin and R. D. Levine* Department of Physical Chemistry and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew UniVersity, Jerusalem 91904, Israel ReceiVed: March 5, 1998; In Final Form: July 14, 1998

Disorder is shown to induce qualitative changes in the electronic spectrum and hence in the response of assemblies of quantum dots. Lattices of quantum dots have one unique source of disorder: the dots themselves can be prepared with a narrow distribution of properties but they are never quite identical. This is unlike a lattice of atoms or molecules. In addition, lattices of quantum dots have a configurational disorder and can also be prepared with compositional disorder. The relaxation of selection rules and the splittings of degeneracies due to symmetry breaking induced by these fluctuations can be probed by optical means. Special attention is given to the enhancement and to the variation of the second harmonic response as a function of the spacing between the dots.

1. Introduction metallic1-3

and It is becoming possible to prepare both semiconducting4,5 quantum dots6 with a narrow size distribution. Fluctuations in the properties of the individual dots can be made small enough (say, below 10%) that assembling these dots into a lattice is possible and this lattice superficially appears to be quite regular. We discuss the novel electronic properties that one might expect for such a lattice. The reason why some of these properties may be unusual is that unlike a lattice made up of atoms or molecules, a lattice of quantum dots is inherently disordered. That is, even a perfect array of real quantum dots will not have the symmetry expected from its geometry because the individual dots are not identical. The translational symmetry is inherently absent. Forbidden transitions can become thereby quite strongly allowed, particularly so when the lattice is compressed. We discuss the effect of the symmetry breaking at the level of a one-electron approximation.7 However, effects due to removal of degeneracies due to symmetry breaking are expected to survive when a more refined description is used. In addition to the effects of the fluctuation in the size (and hence in the excitation energies) of the dots, we discuss two other sources of disorder. One is the packing disorder. Because neither the core of the dots nor their organic envelope is necessarily identical, the separation between two dots, in two otherwise equivalent pairs, need not be the same. Since the quantum mechanical coupling of two dots is expected to be a sensitive function of their distance, this brings another element of disorder * Corresponding author. Fax: 972-2-6513742. E-mail: [email protected]. † Chercheur Qualifie ´ , FNRS, Belgium.

into the Hamiltonian. Finally, one can have intentional disorder, when chemically different dots are assembled together into a common lattice. The discussion is for dots where we neglect their internal structure. Particularly for semiconductor8,9 and organic materials, the vibronic structure requires a more detailed discussion. In earlier studies,10-13 such effects were examined but these studies were limited to resonance coupling made possible by near uniformity of the dots. Here we are specifically concerned with deviations from resonance due to the differing sizes of adjacent dots. (Recall that, by quantum confinement, the size determines the optical response of an individual dot.) We are not aware of already available experimental results with which we can compare our conclusions on the essential effects of disorder. The experiments of Collier et al.,2 on the optical second harmonic response of a Langmuir monolayer of organically functionalized silver quantum dots as a function of the inter particle separation, come closest. That experiment measured the zxx component of the hyperpolarizability tensor, where z is the direction perpendicular to the hexagonally packed layer. The cause of the symmetry breaking is the lack of inversion symmetry along the z axis, because the monolayer has air on one side and water on the other. This is the same kind of symmetry breaking that is so usefully employed in the second harmonic generation (SHG) spectroscopy of monolayers.14,15 Here, however, we are concerned with, say, the xxx tensor component in such an experiment. For a regular hexagonal array this component will be identically zero because the Hamiltonian has an inversion symmetry so states can be classified as even or odd. A finite second harmonic response requires that the series of transitions g f n f n′ f g be dipole allowed for the x direction, where g is the ground state and n

S1089-5647(98)01394-7 CCC: $15.00 © 1998 American Chemical Society Published on Web 09/15/1998

7728 J. Phys. Chem. B, Vol. 102, No. 40, 1998 and n′ are excited states. Since x is odd, there is no nonvanishing product 〈g|x|n〉〈n|x|n′〉〈n′|x|g〉. The point of our discussion is that this sequence of transitions becomes allowed and not just allowed but it can be strongly allowed once even quite modest disorder is possible. The inversion symmetry is broken by the fluctuations discussed above. It is effects like the breaking of inversion symmetry that we expect to survive also in a many electron description of the states. We use the minimal model that will do for the purpose of exhibiting the essential modifications due to disorder: Each quantum dot is represented as an “atom” and these “atoms” are arranged in a lattice. The Hamiltonian is represented as a matrix where the indices are the sites of the lattice. It is a one-electron Hamiltonian known in quantum chemistry as the Hu¨ckel approximation16 and as the tight binding approximation in solid state physics.17 As discussed in the review of Phillips, Anderson18 has first drawn attention to the effect of disorder in this kind of an approximation. When the “atoms” are allowed to have more than one electronic state the Hamiltonian is that of the extended Hu¨ckel approximation.7 We use the BornOppenheimer approximation so that any coupling between the electronic states caused by the nuclear dynamics is neglected. The most serious shortcoming of the model is the neglect of electron-electron interaction effects, primarily the Coulomb repulsion between two electrons which are on the same site. This can be corrected by using the Hubbard Hamiltonian19,20 and additional correction terms are introduced in the PariserParr-Pople approximation.19 These improvements go beyond the one-electron approximation but still retain the description of the dots as “atoms”. Electronic properties of individual dots and, in particular, the size effects are reasonably described by treating the dots as an atom, that is, devoid of internal structure.12,21 In addition, we here allow only one state per atom so that we are using the Hu¨ckel approximation. As discussed in section 2, this restriction can be removed by going over to an extended Hu¨ckel scheme.7 The one aspect where this “atomic” description needs an essential change is that the excited states will have a large width due to the scattering of the electron within the dot. Once we allow electronic coupling between the dots, we must also specify how the width will change when the molecular orbitals are delocalized. We use the second harmonic response22 (≡ the frequencydependent microscopic hyperpolarizability) as a convenient probe of the changes in electronic structure. It is sensitive both to the relaxation of selection rules and to the distribution of spacings (≡ the transition frequencies) in the electronic spectrum. Furthermore, quantum chemical computations of the nonlinear optical response of molecules, for example,23-27 do show that it is sensitive to changes in geometry including changes induced by the environment (e.g., varying the solvent28). In addition, we discuss the spacings of electronic states because all electronic properties are ultimately dependent on them. In concluding the Introduction we reiterate why the second harmonic response is such a useful and sensitive indicator of disorder. First of all is the absence of a response for a perfectly centrosymmetric structure. Furthermore, it is a good measure of delocalization. A simple but not unreliable way to see this is to approximate the polarizabilities by using a constant energy denominator (sometimes called the U ¨ nsold approximation). For the ordinary polarizability, this approximation shows it to scale as 〈g|x2|g〉/(average energy) and hence to be a measure of the size and to satisfy Raman-type selection rules. For the hyperpolarizability the scaling is as 〈g|x3|g〉/(average energy)2

Remacle et al. which clearly shows why the symmetry needs to be broken and why the second harmonic response is sensitive to the spread in the wave function. An indication of this sensitivity is the role of fluctuations as will be discussed below. It will be seen that the computed SHG signal rises as the wave function becomes more delocalized. What this also implies is that the second harmonic response is a collective property of the entire system and therefore it exhibits fluctuations which will not scale down when the number of coupled quantum dots in the assembly increases. 2. Hamiltonian We consider a hexagonal lattice which can have 7 or 19, 37, 61, 91, ... sites. A quantum dot, which we mimic as an atom, is placed on each site. The mean dot diameter, R, is governed by the preparation procedure and is typically a few tens of angstroms. The inherent fluctuation that we are concerned with here is due to the preparation procedure yielding a distribution (which can be narrow) in the sizes of the dots. All distances and specifically the distance D between the centers of the dots will be measured in units of 2R. D will fluctuate because of the variation in the sizes and also because of imperfect packing. As can be done experimentally,29,30 we will allow a mechanically controlled variation of the separation between the dots. For, example, in the experiments of Collier et al.,2 D/2R could be varied in the range of 1.11-1.44. The variation of D through the application of external pressure will also add to the fluctuations in the packing. The model electronic Hamiltonian provides a one-electron description (known either as the tight binding or the Hu¨ckel approximation16). This approximation is most readily stated by giving the Hamiltonian as a matrix in the site basis:

Hi,j ≡ Riδi,j βi,j

when i and j near neighbors or 0 otherwise (1)

Due to the variation in the size of the dots, the site energies (the R’s) can vary from one site to another. Because of these variations the different sites need not be in resonance. The finite width of the excited states of individual dots can compensate for some deviations from exact resonance,11 but otherwise it is the function of the near neighbor coupling to seek to bridge the mismatch in the diagonal elements of the Hamiltonian. Beyond the diagonal disorder, packing imperfections cause variations in the magnitude of the coupling (the transfer integrals β’s). In the absence of fluctuations so that all the dots are identical, the Hamiltonian (1) can be written in the compact way

H ) RI + βM

(2)

where M is the adjacency matrix,31 i.e., a matrix with unit entries when the two sites are near neighbors and zero otherwise. In this resonance case the eigenvalues of H can be written as R + βy where y is an eigenvalue of M. If one allows for the overlap of the wave functions of adjacent dots then the eigenvalue equation changes to det|H - E(I + SM)| ) 0, where S is the value of the overlap integral. The eigenvalues are changed to R + βy/(1 - Sy). One can also interpret the Hamiltonian, eq 1, as describing tunneling between adjacent quantum wells. In this view, β is the tunneling amplitude and R is the energy within the well. In the extended Hu¨ckel approximation,7 the diagonal elements of H for a given site can have several values corresponding to different excited states of the site and the off-diagonal elements

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J. Phys. Chem. B, Vol. 102, No. 40, 1998 7729

are scaled to be proportional to the mean energy of the two sites. Note that, due to the size fluctuations, each one of the several possible diagonal matrix elements can show variations. The extended Hu¨ckel approximation also provides the technical means for incorporating vibronic effects and we intend to return to this point elsewhere. The operator corresponding to the Hu¨ckel Hamiltonian can be written down using the notation of second quantization where a† and a are creation and annihilation operators.

H)

Hi,jai†aj ∑ i,j

(3)

where the matrix elements are given by (1) and the indices refer to the sites of the lattice. The site energies are taken to be all the same up to a random (uniformly distributed) fluctuation. Two scenarios can be envisaged. If, as D is varied, one probes always at the very same spot then the same set of site energies is to be used at all values of D. On the other hand, if the location of the spot varies, a new set of fluctuations is to be drawn for each computation. Once we allow fluctuations in the site energies, the diagonal part of the Hamiltonian is no longer proportional to the identity matrix, cf. eq 2, and it is no longer sufficient to diagonalize the adjacency matrix. If there is also chemical disorder, then in addition, the mean site energy is drawn to be either of one or of another value, weighted by the proportion of the two kinds of dots. The sites are coupled by a transfer integral β, which at large distances is an exponentially decreasing function of the separation D. Upon compression of the lattice it cannot be that β will increase indefinitely and we mimic its saturation at low D by

β ) (β0/2)(1 + tanh(D0 - D)/4RL) f β0 exp(-D/2RL) (4) In terms of the average separation, δ, between the surfaces of the dots, the variable D/2R is (δ + 2R)/2R, where R is the mean dot diameter. We emphasize this because the decline of β with distance is of interest also in connection with long-range electron transfer.32-39 The computations below use the value of 1/2L ) 5.5 which is consistent with the experimental results of Collier et al.27 on silver dots of mean size R ) 27 Å. In other words, the transfer integral β decreases with distance as exp(-0.2 D (in Å)). The value of 5.5 for 1/2L gave the best fit but it is at the upper end of the possible range and values as low as 3.5 are still barely consistent with the results, leading to an even longer range coupling. We emphasize that with either interpretation of β, it is the transfer amplitude. It is not the rate of transfer because we are not in the kinetic regime. The exponential falloff of the transfer integral with the distance is familiar for atomic orbitals.40 One can, however, question this dependence for finite-sized quantum dots. To estimate the distance dependence we regard the dot as a quantum well where the binding energy of the highest orbital is |E| and the particle has an effective mass m*. The radial wave function (a spherical Hankel function of imaginary argument41) can then be approximated as an exponentially decreasing function, exp(-κr). Here κ is the “wave vector” κ2 ) 2m*|E|/p2. By continuity of the wave function and its derivative, it is the same wave vector inside and outside of the well. One can therefore compute the transfer integral between two dots. As in the atomic case,40 the result is a radial integral times an angular overlap. When the two quantum wells are nonoverlapping, the radial part can be readily estimated by using the exponential

decay of the radial wave function. This approximation shows that the transfer integral decreases with distance as exp(-κD). For a realistic binding energy of say 0.1 au and an effective mass which equals the mass of an electron, κ ≈ 0.9 Å-1. For semiconductor quantum dots the effective mass m* is known to be lower,42 as low as 0.024 au for InAs, and this will significantly reduce the value of κ, i.e., the transfer has a longer range. The fluctuations in the separation D between the dots makes for fluctuations in β, that is, in the off-diagonal matrix elements of the Hamiltonian. The exponential decline with D means that magnitude of these fluctuations is proportional to β itself, β = β h (1 ( δ(D/2RL)). Hence, unless the separation is small and/or β0 is large, the role of the off-diagonal fluctuations is expected to be limited. In a one-electron approach the electronic wave function of the entire lattice is a Slater determinant16 made up from the one-electron wave functions (known as the molecular orbitals (MO’s)) which are obtained by diagonalizing the Hamiltonian H. From the structure of H the molecular orbitals are linear combination of site orbitals and to distinguish the two we label the MO’s by Greek indices. In the notation of second quantization43

aµ† )

∑j aj†cjµ

(5)

where the c’s are the matrix elements of the orthogonal transformation between the site basis and the MO basis, as determined by diagonalizing H. H ) ∑µµaµ†aµ and the ground state of the lattice is

|g〉 )

(aµ†)f |vac〉 ) ∏(∑aj†cjµ)f |vac〉 ∏ j µ µ µ

µ

(6)

In the one-electron theory the molecular (spin) orbital are either empty or fully occupied so that the number fµ of electrons in the molecular orbital µ in the ground state is either zero or one. Excited states of the lattice are generated by moving an electron from an occupied MO to an unoccupied one

|n〉 ) (1 - fµ′)fµaµ†aµ|g〉

(7)

The energies of the different electronic states of the lattice have the same form as that of the Hamiltonian, Eg ) ∑µµfµ and similarly for the excited states. The building blocks for the electronic susceptibilities are the transition frequencies and the dipole moment matrix elements between the different electronic states. These require computing the dipole matrix elements between the MO’s. Since each MO is a linear combination of site functions there will be two contributions to the dipole matrix element, one that is diagonal in the site index and one that is not. The latter is a sum over matrix elements of the electron dipole operator ()its position) between orbitals located on different sites of the lattice. This is negligible if the sites are not adjacent and, in the approximation of neglect of overlap it is negligible even for neighboring sites.16 Then the electronic transition dipole is diagonal in the site basis rˆ ) ∑iriai†ai where the circumflex denotes an operator and ri is the (vector) location of lattice site i. In the computations below we do include the overlap (i.e., nondiagonal) contributions to rˆ, an effect which is particularly important at the smaller spacings. The full Hamiltonian needs to also include an imaginary part which describes the dissipation due to the damping of the excited

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Remacle et al.

states by the internal structure of the finite-sized dots. In addition there is damping due to the medium surrounding the particles.2 When the damping is comparable to the interparticle coupling, one should diagonalize the real Hamiltonian H and the damping simultaneously.44 This is possible by a biorthogonal transformation45 but when the damping is small compared to the interparticle coupling it can be well approximated in terms of its matrix elements in the orthogonal basis which diagonalizes H and this is what we do. When the spacing is large, the coupling is not effective and this damping becomes practically diagonal in the “atomic” basis, as it should. The second harmonic generation response, i.e., the microscopic second order polarizability, at the frequency ω was computed from a 12-term sum.46 Six terms are

P2ω ijk ) -

e3

∑∑ 2 n*gm*g

2p

(

(Ω*mg

{

+

(

rigm jrjmn rkgn 1

2ω)(Ω*ng

1 (Ωmg - 2ω)(Ωng - ω)

)

+ ω)

+

+ rjgm jrkmn rign

1

)

1

+ + (Ω*mg + ω)(Ω*ng + 2ω) (Ωmg - ω)(Ωng - 2ω) 1 1 rjgm jrimn rkgn + (Ω*mg + ω)(Ω*ng - ω) (Ω*ng + ω)(Ωmg - ω) (8)

(

)}

Here jr ≡ r - 〈g|r|g〉. The indices j and k are the directions of the laser field and i is the direction in the sample. The other 47 The complex fresix terms are obtained by adding P2ω ikj . quency Ω includes the damping effects Ωmg ) p-1(Em - Eg) - i(Γm - Γg). The dimension of the second harmonic response are those implied by eq 8. In atomic units these are e3a03/(e2/ a0)2. We measure energy in units of β0, cf. eq 4, and length in units of the dots diameter 2R. Note that eq 8 does not include any contributions to the polarizability due to the dots themselves since we treat them as “atoms”. However, the width of the states is larger than what it would be for atoms because it includes the damping due to the internal structure of the dots. The second harmonic response was computed as a function of frequency by diagonalizing the Hamiltonian for each value D of the lattice spacing. Since there is a very high density of excited states, then, as long as the frequency ω is resonant (i.e., is within the band of energetically allowed transitions) the response is a very oscillatory function of frequency. The damping tends to smooth the individual resonances and thereby produces a clump structure where many resonances contribute to one clump. By increasing the width Γ it is possible to generate a frequency spectrum with very few clumps. We also computed the first-order microscopic polarizability

Rij(ω) )

e2

(

∑rign rjgn Ω p n*g

ng

1 -ω

+

1

)

Ω*ng + ω

(9)

for a given excitation frequency, ω, as a function of the lattice spacing D/2R, Rij(ω) is not subject to the same set of selection rules as the second-order microscopic polarizability (eq 8 above) and does not vanish in the absence of disorder. 3. Results and Discussion The results shown below emphasize the following points about finite assemblies:

Figure 1. The components |cj|2 of the highest occupied MO on the sites j for a lattice of 91 sites (shown as an insert) with 10% fluctuations in the dot sizes (our energy unit is β0/2, 1/2L ) 5.5, D0/2R ) 1, R ) 20, |δR| ) β0/2). At small separation (D/2R ) 1.2, bottom panel), the transfer integral β is large enough to bridge the fluctuations in the R’s and the wave function is very much delocalized over all the sites. On the other hand, for larger separation (D/2R ) 1.7), β is too small to induce efficient coupling and the wave function is essentially completely localized on a single site.

(i) For an assembly of identical dots, the electronic states will be delocalized for any strength of the coupling between the dots. This can also be the case in the presence of fluctuations in the energies of the dots, if the coupling is strong enough. Technically, this requires that the coupling compensates for the fluctuations, δR, in the site energies or, when this is relevant, even for the differences in the site energies due to changes in the composition of the dots. However, in order to do so the transfer integral β must be large enough, where “enough” means that 4β h g |δR|. In other words, if β0 is large and/or if the packing is close enough, the coupling can smooth out the local fluctuations. The factor of about 4 is that for a hexagonal lattice but it is of that order for all planar arrays. (ii) Since β varies with distance, then at large lattice spacings the dots will be de facto uncoupled. At large separations the electronic wave functions will therefore be localized. This will always be the case because of the inhomogeneity in the preparation of the dots. Additional sources of disorder will only reinforce the localization of the electronic states. Figure 1 shows the components (the c’s of eq 5) of the highest occupied MO on the different sites at a large and a small value of the lattice spacing D. A plot of the site lattice of 91 dots used in the computations is shown as an insert. We reiterate that the barrier to delocalization of the wave function are the fluctuations of the site energies (the R’s). If the site energies are the same, the wave function is completely delocalized due to the coupling. This is because the eigenvectors of the adjacency matrix are delocalized and remain delocalized even if there are fluctuations in the packing (i.e., in the transfer integral β). (iii) A regular lattice, whether hexagonal or otherwise, will have symmetry elements which will lead to degenerate molecular orbitals. We emphasize that these are degeneracies which remain in the presence of the coupling between the sites, as long as the symmetry is maintained. Only the symmetry breaking fluctuations can split these degeneracies. This will give rise to atypically small electronic spacings between those electronic states which would be degenerate in the absence of the fluctuations. Figure 2 and its legend elaborate on this point. There are two kinds of splittings, the small and the atypically very small. These have to do with the breaking of two different kinds of degeneracies. The small spacings are due to the fluctuation in the site energies. Figure 2 (top) shows the transition frequencies as a function of the separation between

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Figure 3. Electronic transition frequencies between excited states and the ground state, logarithmic scale (ωng ) ωn - ωg, cf. eq 7), for a 91-site lattice, vs D/2R, same parameters as in Figure 1. Computed in the presence of fluctuations only in the transfer integral β(|δ(D/2R)| ) 0.1β0). Note (cf. Figure 2, bottom) how the splitting of the degeneracy due to all the R’s being the same is now much smaller because the long-range fluctuations in β are exponentially small. As discussed in Figure 2, there are states of the regular lattice which are strictly degenerate. These are split by the fluctuations and, as seen, they are even smaller when only fluctuations in transfer signals are allowed. Figure 2. Electronic transition frequencies between excited states and the ground state (ωng ) ωn - ωg, cf. eq 7, in units of β0, see text for more details), for a 91-site lattice vs D/2R (same parameters as in Figure 1). In the absence of fluctuations in the R’s (upper panel), the eigenstates are the eigenstates of the adjacency matrix M. The frequency spacings between states are therefore proportional to β and at the larger separations they decrease exponentially with D/2R. This can be seen from the logarithmic scale used in the insert. This logarithmic plot also serves to indicate the numerical accuracy. (The very smallest transitions shown, which are