NANO LETTERS
Nanocrystal Shape and the Mechanism of Exciton Spin Relaxation
2006 Vol. 6, No. 8 1765-1771
Gregory D. Scholes,* Jeongho Kim, Cathy Y. Wong, Vanessa M. Huxter, P. Sreekumari Nair, Karolina P. Fritz, and Sandeep Kumar Department of Chemistry, 80 St. George Street, Institute for Optical Sciences, and Centre for Quantum Information and Quantum Control, UniVersity of Toronto, Toronto, Ontario M5S 3H6 Canada Received June 20, 2006
ABSTRACT The rate of exciton spin relaxation (flips) between the bright exciton states (F ) ±1) of CdSe nanocrystals is reported as a function of shape, for dots and nanorods. The spin relaxation is measured using an ultrafast transient grating method with a crossed linearly polarization sequence. It is found that the spin relaxation rate depends on the radius, not length, of the nanocrystals. That observation is explained by deriving an expression for the electronic coupling matrix element that mixes the bright exciton states.
In recent years, the size- and shape-tunable properties of semiconductor nanocrystals (quantum dots) have garnered much attention.1-4 To understand these size-tunable properties, an analogy is drawn with the quantum levels of a particle-in-a-box.5 Nanocrystals are sufficiently large that the precise details of chemical bonds blur into an average picture of how the electron density occupies volume, and the electronic states conform to the three-dimensional particlein-a-box model, leading to the notion of quantum confinement. The spectroscopy of nanocrystals is thereby linked to size and shape, and that has been utilized in, for example, wavelength tuning of lasers, light-emitting diodes, or other optical technologies.6-9 It has been shown that the band gap10 and excitation energies11-13 of excitons in CdSe nanocrystals depend strongly on the radius a and weakly on the length L. However, with the increasing sophistication by which the shape of colloidal nanocrystals can be controlled,3,14-17 it is of interest to discover further implications of shape for spectroscopy. Here, we report measurements of exciton spin relaxation rates as a function of the shape of CdSe nanocrystals. Spin degrees of freedom have received interest recently as a consequence of the potential of spintronics and quantum information processing.18 Rather than examining electron or hole spin relaxation, we investigate here exciton spin relaxation. A consequence of rapid exciton spin relaxation is that quantum information stored optically in exciton states of quantum dots would be randomized. One example of such a phenomenon is relaxation of the bright quantum dot exciton, which has total angular momentum F ) (1, to the * To whom correspondence should be addressed. E-mail: gscholes@ chem.utoronto.ca. 10.1021/nl061414e CCC: $33.50 Published on Web 07/22/2006
© 2006 American Chemical Society
corresponding dark exciton with F ) (2.19 That relaxation is caused by spin-orbit coupling. In the present work, we report the measurement of the rate of spin flips between F ) (1 nanocrystal exciton states with the aim of elucidating the mechanism allowing this spin flip. The experimental approach we use is specific in targeting just these dynamics.20 The samples we have studied are CdSe wurtzite nanorods and dots that span mean radii of 1.3 to 3.1 nm, lengths 1.3 to 14.5 nm, and aspect ratios of ∼1 to 3.5. We find the decisive shape parameter for interactions that flip spin between the bright excitons to be the plane normal to the crystal c-axis. Surprisingly, the nanocrystal length does not influence the spin flip rate at all. To elucidate the shape dependence of the exciton spin flip process, we describe a simple “spectroscopic” model for nanocrystal exciton states and use that prescription to suggest the mechanism promoting the direct exciton spin flip. The exciton fine structure21-23 of nanocrystals is an analogous spectral quantity to the singlet-triplet manifold of molecules.24 However, it is challenging to identify the exciton fine structure states precisely in the inhomogeneously broadened nanocrystal absorption spectrum.23,25 Fluorescence line narrowing spectroscopy and photoluminescence studies of single nanocrystals have successfully revealed the brightdark splitting (e.g., between the F ) 1 and F ) 2 states) for a range of nanocrystalline quantum dots.23,26-32 That splitting is principally determined by electron-hole exchange interactions that are several millielectronvolts in magnitude. In the present work, we investigate exciton spin flips between the F ) (1 bright exciton states. The observation of these dynamics exposes small electronic interactions that are not considered in the usual Hamiltonian used to predict
the exciton fine structure of nanocrystals. We will suggest the origin of these interactions below, where we identify a short-range three-center, two-electron integral that is capable of mixing the F ) (1 bright exciton states, thus causing exciton spin flips. Exciton spin flips are known to be mediated by long-range exchange interactions,33-36 but those interactions are only nonzero for nanocrystal shapes very different than the samples we have studied. For example, to obtain a nonzero interaction coupling the F ) (1 states, the nanocrystal would need to have an elliptical cross-section in the plane normal to the c-axis, which is inconsistent with the structure of colloidal CdSe nanocrystals. Alternative mechanisms include a two-step process via the dark exciton states that is mediated by spin-orbit coupling.37,38 Such an effect is expected to be diminished as the nanocrystal size decreases as a consequence of the concomitant increase in bright-dark fine structure splitting due to the exchange interaction. We observe an increase in the spin relaxation rate with decreasing nanocrystal size, which we attribute to a small, direct coupling between the F ) (1 states. The exciton fine structure cannot be revealed easily by frequency domain experiments because their location is obscured by substantial inhomogeneous line broadening. To obtain evidence for the even weaker interaction between the exciton states labeled F ) (1, we measure exciton spin relaxation. To do that we have developed an ultrafast laser experiment that, rather than being sensitive to exciton population dynamics as they recover to the ground state, keeps track of the history of an exciton state,20,39,40 thereby providing a means to tell if an F ) (1 exciton state has flipped its spin (i.e., its total angular momentum has changed sign). The cross-polarized 3-TG (third-order transient grating) experimental method we have employed thus measures spin relaxation among quantum dot F ) (1 exciton states for an isotropic ensemble of nanocrystal orientations.20,40 Note that there is some confusion among some workers regarding the process of photoexcitation of randomly oriented colloidal quantum dots using circularly polarized light, as well as the kinds of information measured by time-resolved Faraday rotation compared to our cross-polarized transient grating experiment. These issues, though subsidiary to the present work, are carefully addressed in ref 20. Resonant third-order transient grating (3-TG) spectroscopy is typically used to investigate excited-state relaxation dynamics.41,42 Two laser pulses with wave vectors k1 and k2 are crossed at a small angle (2.2° in our case) in the sample. The resultant pair of interactions with the sample initiates photoexcitation. It is often useful to think about a population or polarization grating that is formed across the sample due to interference of these beams.42 The probe pulse, with wave vector k3, arrives at a delay time tp later, inducing the signal field to radiate in the phase-matched direction ks ) -k1 + k2 + k3. We detect induced third-order polarization P(3)(0,tp,t), integrated over its evolution with time t by optical heterodyne detection. The measured signal is IHET(tp) ∝
∫0∞ dt Re[E/LO(tp,∆φ)‚〈P(3)(0,tp,t)〉]
(1)
where ∆φ is the phase delay between the local oscillator 1766
field E/LO and the probe field Es that we tune by rotating a cover slip. The angle brackets indicate that a rotational average is taken to account for the random orientation of each nanocrystal in the ensemble relative to the laboratory frame. An important facet of the 3-TG experiment is the ability to control independently the polarization of each of the three incident fields and select a polarization component of the signal field. We make use of these degrees of freedom to measure spin relaxation. As we have described and demonstrated,20 the experiment can be applied to materials with the zinc-blend-type selection rules (including wurtzite and rock salt semiconductors). For typical molecules, there is no difference between 3-TG measurements with the various polarization sequences (apart from signal intensity). To summarize how the experiment works, we expand the rotationally averaged induced polarization (the signal field) for the cases of all-vertically polarized pulses and analyzer, VVVV, and the two cross-polarized experiments, VHVH and VHHV 1 〈PVVVV(3)(0,tp,t)〉 ) P(3)(0,tp,t) 5
(2a)
〈PVHVH(3)(0,tp,t)〉 ) P1(3)(0,tp,t) + P2(3)(0,tp,t) + P3(3) (0,tp,t) + P4(3)(0,tp,t) + P0(3)(0,tp,t) )
2 2 +1 (3) P(3)(0,tp,t) n P (0,tp,t) - n-1 15 C 15 F
+
2 2 -1 (3) P(3)(0,tp,t) + n P (0,tp,t) - n+1 15 C 15 F cn0P(3)(0,tp,t) (2b)
〈PVHHV(3)(0,tp,t)〉 ) P1(3)(0,tp,t) + P2(3)(0,tp,t) + P3(3) (0,tp,t) + P4(3)(0,tp,t) + P0(3)(0,tp,t) )-
2 2 +1 (3) P(3)(0,tp,t) n P (0,tp,t) + n-1 15 C 15 F
2 -1 (3) 2 n P (0,tp,t) + n+1 P(3)(0,tp,t) + 15 C 15 F cn0P(3)(0,tp,t) (2c)
where the fractions and their associated signs are obtained from the rotational averaging procedure.39 The populations of excitons that were photoexcited to, or probed in, the F ) 0 bright state are labeled n0. This population contributes a -1 small offset to the VHVH and VHHV transients. n+1 C ) nC are the populations of the F ) +1 and F ) -1 excitons, respectively, at any time tp after photoexcitation that were initially photoexcited to that same state. In other words, these are exciton populations that have not flipped their spins (or that have flipped an even number of times). The populations indicated by the subscript F are excitons that have flipped their spins. In other words, n+1 F is the population of F ) +1 excitons that were originally photoexcited to the F ) -1 +1 state. The initial condition is that n-1 F ) nF ) 0. Clearly, it is the interference of polarizationssnot their decaysthat leads to a decay component in the measured total signal Nano Lett., Vol. 6, No. 8, 2006
Figure 1. Introduction to the 3-TG experiment for nanocrystals. Top: a schematic of the nanorod ensemble where black nanorods are in an excited state and silver nanorods are in the ground state. For the VVVV 3-TG experiment, the amplitude of the radiated polarization, illustrated on the right-hand side, is proportional to the number of nanorods in any excited state. The decay of the signal intensity with time delay tp therefore mirrors the exciton recombination dynamics. Bottom: here, the photoexcited nanorods are color-coded. Black is an F ) 0 exciton, red is an F ) +1 exciton, and blue is an F ) -1 exciton. Consistent with the diagram, photoexcitation of an orientationally isotropic ensemble leads to equal numbers of the F ) (1 excitons, no matter what excitation polarization is used. For the VHVH (or VHHV) experiments, we need to consider four processes occurring after photoexcitation of the nanorods. (1) If a nanorod is excited to an F ) 0 exciton, then it will radiate a polarization as in the case above. (2) The excited nanorods return to the ground state, as above, thus decaying the overall signal intensity. (3) The exciton spin states of these nanorods have not changed since photoexcitation. They radiate the (3) polarizations P(3) 1 + P3 when interrogated by the probe beam. The radiated polarizations plotted in the figure are calculated by solving the equations of motion for the nonlinear optical process, as described in ref 20, but note the transition frequency is set to be small simply for the illustration. (4) In contrast to the previous case, the exciton spin states of these nanorods have flipped during the delay time tp. They (3) (3) (3) radiate the polarizations P(3) 2 + P4 . Notice how these fields are phase shifted by π from P1 + P3 . That phase shift is the origin of the (3) (3) (3) signal decay; as the population of flipped excitons increases, the extent of destructive interference between P(3) 1 + P3 and P2 + P4 grows.
intensity as a direct result of exciton spin flips. That is further explained in Figure 1. We emphasize that the experiment measures bright exciton spin relaxation only (i.e., F ) +1 T -1). Other kinds of spin relaxation that can be detected, for instance, in timeresolved Faraday rotation,43 such as spin relaxation of dark excitons or electrons, do not contribute to our data. All our experimental results were obtained at room temperature (294 K) in the absence of an external magnetic field. To summarize: when all three beams and the analyzer are vertically polarized in the laboratory frame, signified as VVVV, the experiment measures exciton recombination dynamics. On the other hand, when the polarizations are cross-linearly polarized, VHVH or VHHV (H ) horizontal), then the signal decays according to the number of excitons that have flipped their spin since they were optically excited. Our experimental setup is described elsewhere.20 Each sample, a dilute solution in toluene, is photoexcited on resonance with the first absorption band using ultrashort, approximately transform-limited laser pulses. Experiments were carried out at a power sufficiently low to avoid biexciton formation and, more importantly, artifacts from thermal gratings. The optical heterodyne detected 3-TG signal was measured using a silicon photodiode and a lock-in Nano Lett., Vol. 6, No. 8, 2006
amplifier. To ensure that only the signal and local oscillator (LO) fields were incident on the detector, all unwanted beams and scattered light were blocked by a mask after the sample. A third polarizer was mounted after the mask to set the signal polarization. Depending on the laser center frequency, pulse durations of 30 to 40 fs were obtained from autocorrelation measurements at the sample position. To prevent any sample degradation and thermal grating contributions to the signal, the pulse energy was kept at less than 5 nJ/pulse. Both pump and probe beams were attenuated until the early-time signal shape was independent of pulse energy. To ensure that the samples were not photodegrading, the absorption spectrum of each sample was measured before and after the 3-TG scans. To extract the exciton spin relaxation dynamics from the measured VHVH (or VHHV) 3-TG signal, the experimental data need to be carefully fitted since the signal contains exciton population relaxation dynamics as well as spin relaxation dynamics. If the population and spin relaxation occur on very different time scales, then the data can be simply fitted by a linear combination of exponentials to retrieve the spin relaxation rate. However, when exciton recombination dynamics are complicated by fast surface trapping effects, it is more accurate to use the information 1767
about the exciton recombination obtained from the independently measured VVVV signal in conjunction with an analysis of the VHVH and VHHV data. In that case, the cross-polarized 3-TG data are fitted by the function IVHVH(tp) ) [A1 exp(-2kstp) + A2] IVVVV(tp)
(3)
where IVVVV(tp) is the decay profile of the VVVV signal as a function of pump-probe delay time, tp, and ks is the rate of exciton spin relaxation. The initial sub-50 fs ultrafast decay was not included in the fitting procedure since it could contain a coherent spike or other nonlinear effect occurring when the pump and probe pulses are overlapped, which is beyond our ability to analyze. When we compare the spin relaxation rates determined from each fitting procedure, nearly identical values are obtained confirming that an accurate value for the spin relaxation rate is retrieved. The reader will notice that we focus only on the F ) (1 fine structure states, even though we photoexcited all the bright fine structure states (e.g., the linearly polarized F ) 0 state). We do that to clarify the presentation, but account has been taken of all states in realistic modeling of the data to demonstrate that our fitting procedure extracts the F ) +1 T F ) -1 spin relaxation and is unaffected by other relaxation pathways, for example, F ) +1 to F ) +2. The polydispersity of nanocrystals in the ensemble may also pose a concern. In these experiments, we achieve a degree of photoselection determined by the convolution of the laser pulse spectrum with the inhomogeneously broadened absorption band. The photoselection is greatly enhanced about the carrier frequency of the laser pulse because of the cubic intensity dependence of this nonlinear process. Nonetheless, we are still measuring a distribution of spin relaxation rates. The average spin relaxation rate of that distribution is obtained by an exponential fit to the decay curve, as has been described by Ware and co-workers.44 CdSe nanorods were prepared by a thermolysis reaction between Cd and Se precursor compounds in the presence of a structure-directing agent. Size and shape tuning was achieved by control of growth kinetics as described in the Supporting Information. Samples were characterized by powder X-ray diffraction, absorption and photoluminescence spectroscopy, as well as careful transmission electron microscopy (TEM) analysis to determine size and shape distributions. The radius of each rod is smaller than the exciton radius in the bulk material. A TEM image, absorption, photoluminescence, and the imaginary (absorptive) contribution to heterodyne-detected 3-TG data for one of the nanocrystal samples are shown in Figure 2. 3-TG data are shown for VVVV, VHVH, and VHHV polarization sequences (V ) vertical and H ) horizontal polarization in the laboratory frame). The VVVV data decay according to exciton recombination dynamics. In contrast, decay of the VHVH signal as a function of the pump-probe delay time tp, mirrored as a rise of the VHHV signal, depends sensitively on the sample. Analysis of these data obtains the exciton spin relaxation dynamics. Those dynamics, caused by exciton spin flip transitions between F 1768
Figure 2. Representative data for a CdSe nanorod sample (a ) 2.2 nm, L ) 7.4 nm). The upper panel shows the absorption and photoluminescence spectra (toluene) and a TEM image. The lower panel displays the imaginary component of the heterodyne-detected 3-TG data for the VVVV, VHVH, and VHHV polarization sequences (V ) vertical and H ) horizontal). The oscillations evident in all the signals are quantum beats assigned to the LO phonon.
) (1 states, are found to follow single-exponential kinetics. Plots of the spin relaxation rate ks vs structural parameters including the volume, surface area, aspect ratio, and length show no evidence for a systematic correlation (see Supporting Information). A conspicuous dependence of ks on sample radius a, however is apparent. Our results for CdSe nanocrystals that span mean radii of 1.3 to 3.1 nm, lengths 1.3 to 14.5 nm, and aspect ratios of ∼1 to 3.5 are summarized in Figure 3. A power-law fit through these data adduces the relationship ks ∝ 1/a4. A linear fit to a plot of ks vs 1/a4 extrapolates to 1/ks ≈ 20 ps at infinite radius. That does not correspond to the spin relaxation time for bulk CdSe but may indicate an approximate limiting value for a large nanocrystal. To examine the cause of a direct spin flip further, let us assume a Fermi golden rule rate expression for the spin flip rate ks ks )
2π |Ms|2Ff p
(4)
where Ff is the density of final states, largely determined by homogeneous line broadening, and Ms is the matrix element that promotes the exciton spin flip. In the present work, we consider just the direct spin flip between the F ) +1 and F ) -1 states, which we believe dominates the spin relaxation for small nanocrystals. According to our experimental observations, the interaction matrix element coupling the F ) +1 and F ) -1 exciton states, Ms, scales as 1/a2. This result suggests that it is quantum confinement in a plane normal to the c-axis that decides the interaction between F Nano Lett., Vol. 6, No. 8, 2006
Table 1. Antisymmetrized Product Functions that Described Zeroth-order Excitations of a Nanocrystal in Terms of the Valence and Conduction Orbital Bloch Sums
Figure 3. Spin relaxation rate ks vs CdSe quantum dot and nanorod radius, a. Open circles are the quantum dot data points and black squares are the nanorod data points, plotted on a log-log scale. Horizontal error bars indicate the standard deviation of the nanorod diameters obtained from TEM analysis, though the ultrafast measurement recovers a spin relaxation rate representative of a significantly more narrow distribution of nanorod sizes about the mean. Vertical error bars are within the symbol dimensions. The solid line is a power-law fit to the entire data set that finds ks ∝ a-4. The open rectangle shapes show, to scale, the mean shape projection of each nanorod sample and are located near the error bar of the corresponding data point.
) +1 and F ) -1 excitons in CdSe nanocrystals rather than the volume or even an effective radius such as (a2L)1/3. Remarkably, ks is not at all influenced by the length of a nanocrystal but is strongly affected by variation in the radius. To understand why that is the case, we must first consider the origin of this matrix element. We begin by sketching a “spectroscopic” description of the exciton states in terms of the valence and conduction Bloch functions that facilitates the identification of the electronic coupling matrix element connecting the F ) (1 bright exciton states. Although atomistic methods are quantitatively superior,45 useful physical insights have been provided by describing quantum dot exciton states according to the effective mass approximation.21 In such models, each wave function is separated into the product of a Bloch function and an envelope function. The envelope function is important in determining the overall size-scaling of twoelectron integrals, while integrals involving the Bloch function are factored out to represent the microscopic origin of a matrix element. For our purposes, such a prescription is appealing because we can identify electronic interactions between states without having to perform lengthy numerical calculations. Moreover, we are not restricted by the currently accepted set of approximations used to admit electronelectron (electron-hole) interactions into the Hamiltonian. Recall that the integral over space that is carried out to evaluate an electronic coupling matrix element yields a product of a term quantifying confinement and a two-electron integral. In the following development, the envelope functions will not be explicitly given since we are interested first in identifying the two-electron integral. Useful background can be found in the work of Slater and of Lo¨wdin.46,47 We will describe the valence band Bloch functions, as usual, in terms of the functions a ) |2-1/2)(X + iY)R〉, b ) |ZR〉, c ) |2-1/2(X - iY)R〉 and similar functions aj, bh, cj for electrons Nano Lett., Vol. 6, No. 8, 2006
F
S
MS
configuration
-2 +2 -1 -1 -1 +1 +1 +1 0 0 0 0
1 1 1 1 0 1 1 0 0 1 1 1
-1 +1 -1 0 0 +1 0 0 0 0 +1 -1
ψ1 ) |sja j bb h ccj| ψ2 ) |aa j bb h cs| ψ3 ) |aa j sjb h ccj| ψ4 ) (|sa j bb h ccj| - |asjbb h ccj|)/x 2 ψ5 ) (|sa j bb h ccj| + |asjbb h ccj|)/x 2 ψ6 ) |aa j bsccj| ψ7 ) (|aa j bb h scj| - |aa j bb h csj|)/x 2 ψ8 ) (|aa j bb h scj| + |aa j bb h csj|)/x 2 ψ9 ) (|aa j sb h ccj| + |aa j bsjccj|)/x 2 ψ10 ) (|aa j sb h ccj| - |aa j bsjccj|)/x 2 ψ11 ) |asbb h ccj| ψ12 ) |aa j bb h sjcj|
of β spin. These functions have the symmetry properties of px, py, and pz orbitals with respect to the symmetry operations of the crystal and refer to primitive cells not atoms. The related conduction band functions are s ) |SR〉 and js ) |Sβ〉. Bloch sums are taken of these basic functions to provide spin-orbitals that are then used to construct properly antisymmetrized product functions ψi, given in Table 1. These ψi values serve as basis configurations to obtain the nanocrystal exciton states. For clarity of presentation, we do not explicitly write the Bloch sums here. If we take such a representation literally, it is equivalent to writing only the on-site two-electron integrals in the developments below. We consider the case only of k ) 0. The Hamiltonian from k‚p theory describing the energies of the functions aj, bh, cj, js, a, b, c, and s and their mixing by spin-orbital coupling ∆ is well-known.48 We take this together with a crystal field splitting ∆xf and the relevant exchange integrals Kmn ) (nm|nm), where (ik|jl) ≡ 〈i(1)j(2)|1/r12|k(1)l(2)〉, to obtain the Hamiltonian
[
HA 0 H) 0 0
0 HB 0 0
0 0 HC 0
0 0 0 HD
]
(5)
where HA )
[
- ∆/3 - K0 0 - ∆/3 - K0 0
[ [
∆xf - K0 ∆/3 - ∆/3 - K0 - ∆/3 HB ) ∆/3 - ∆/3 - ∆/3 - K1 ∆xf - K0 - ∆/3 - ∆/3 - K0 ∆/3 HC ) - ∆/3 - K3 - ∆/3 ∆/3
[
- K2 0 HD ) ∆/3 - ∆/3
0 ∆xf - K0 ∆/3 ∆/3
∆/3 ∆/3 ∆/3 - K0 0
]
] ]
- ∆/3 ∆/3 0 ∆/3 - K0
(6a)
(6b)
]
(6c)
(6d)
1769
with K0 ) Kas + Kbs + Kcs, K1 ) -Kas + Kbs + Kcs, K2 ) Kas - Kbs + Kcs, and K3 ) Kas + Kbs - Kcs. That allows us to evaluate the wave functions and energies relative to the band gap for the exciton states of the fine structure by solving the associated secular equations. Those results can be related to the representation reported by Efros et al.21 For such a comparison, it should be noted that each K ) 3η. Using the representative parameters ∆ ) 415 meV, ∆xf ) 25 meV, and Ki ) 12 meV, we obtain Ψ-1 ) -0.38ψ3 + 0.83ψ4 + 0.40ψ5
(7a)
Ψ+1 ) 0.38ψ6 + 0.83ψ7 - 0.40ψ8
(7b)
Note that we have set the relative energies of each valence band configuration, a, b, and so forth, to be equal. When that is not the case, for example, for nanorods,49 the coefficients of ψ3 and ψ6 will be influenced. While that has implications for the calculated eigenvalues, it is a diversion from the present aim: to determine how these states identified in eq 7 are coupled. Now we can examine the origin of the matrix element that promotes exciton spin flips between the F ) +1 and F ) -1 exciton states Ms ) 〈Ψ+1|H′|Ψ-1〉
(8)
A well-known interaction from molecular spectroscopy can be discerned from inspection of eq 8 after expanding Ψ+1 and Ψ-1 using eq 7, that is, H′36 ) 〈ψ6|H′|ψ3〉, which gives rise to the E term in zero field splitting of the Ms ) (1 triplet states.50-52 Physically, this is a spin-spin interaction involving the aligned triplet spin-oriented electrons. That observation provides an intuition for the origin of the splitting, however, such effects are typically very small. Moreover, owing to the cylindrical symmetry of the CdSe primitive cell, H′36 should be zero. Consider the interaction between the singlet components of each wave function, H′58 ) 〈ψ8|H′|ψ5〉. That matrix element can be shown to be H′58 ≈ 2(sa|cs). Such threecenter integrals are always ignored in electronic structure calculations of solids. The integral represents the interaction between the spinless transition density for exciting an electron from a to s (Psa) with that for the de-excitation s to c (Pcs).53-55 Such integrals can be considered semiquantitatively using the dipole approximation, so if the functions a, c, and s were really atomic orbitals, then it is easy to see that the integral would vanish (the dipole transition moments are perpendicular in that case). However, for binary semiconductors such as CdSe, we defined these basis functions in terms of the primitive cell, which contains one anion and one cation. Qualitatively, the excitation involves promotion of an electron from Se2- to Cd2+.56 Hence, each transition density acquires a z-component that provides the interaction to couple Ψ+1 and Ψ-1 via the three-center, two-electron integral from H′58. This coupling is therefore nonzero for a polar semiconductor like CdSe. Our observation that ks ∝ 1/a4 can now be understood by thinking about how the valence band might be delocalized 1770
in the nanocrystal, which in turn will indicate how quantum confinement influences the scaling of the two-electron integral (sa|cs) that mixes the F ) (1 fine structure states. According to our experimental results, spectral splitting caused by this interaction is magnified by confinement in the plane defined exclusively by the nanocrystal radius. This now makes sense because a and c are delocalized in the plane defined by the nanocrystal radius, while only b (which does not enter into the integral in H′58) spans the length of the nanocrystal. Drawing analogy to the particle-in-a-box model, the spin relaxation is governed exclusively by the shape of the box in the x-y plane. Thus, exciton spin relaxation among the lowest bright exciton states in CdSe nanocrystals depends only on the width of the nanocrystal, according to the interaction H′58 scaling as 1/a2 once we include the normalization condition and assume that long-range contributions to the integral are small. The fact that H′58 contains an integral not included in standard computational approaches for calculating the exciton states of solids explains why an F ) (1 splitting has not been reported in papers describing the electronic states of nanocrystals. In nanoscale materials, the details of chemical structure and bonding are averaged into a characteristic shape, which in turn influences the quantum details of nanocrystal spectroscopy. We found that the rate of exciton spin flips between the CdSe nanocrystal bright exciton fine structure states is controlled by the nanocrystal radius and is not influenced by the length. We concluded, on the basis of a theoretical description of the spectroscopy, that this observed shape dependence of the interaction between the F ) +1 and F ) -1 states is a consequence of the form of a threecenter, two-electron integral that governs the interaction. The spectroscopic technique we have employed may therefore allow the measurement of the effective exciton width in a nanocrystal of more complex shape by considering the results plotted in Figure 3 as a calibration curve for spin flip rate vs exciton lateral radius. Such information is useful to inspire a deeper understanding of quantum dot materials and facilitate the design of shape-tunable properties. Acknowledgment. The Natural Sciences and Engineering Research Council of Canada and the Alfred P. Sloan Foundation are gratefully acknowledged for support of this research. Supporting Information Available: Further plots showing the spin flip rates as a function of nanocrystal structure, details of the nanorod synthesis, and details of the structural characterization. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Alivisatos, A. P. J. Phys. Chem. B 1996, 100, 13226-13239. (2) Bawendi, M. G.; Steigerwald, M. L.; Brus, L. E. Annu. ReV. Phys. Chem. 1990, 41, 477. (3) Burda, C.; Chen, X. B.; Narayanan R.; El-Sayed, M. A. Chem. ReV. 2005, 105, 1025-1102. (4) Weller, H. Angew. Chem., Int. Ed. Engl. 1993, 32, 41-53. (5) Ekimov, A. I.; Hache, F.; Schanne-Klein, M. C.; Ricard, D.; Flytzanis, C.; Kudryavtsev, I. A.; Yazeva, T. V.; Rodina, A. V.; Efros, A. L. J. Opt. Soc. Am. B 1993, 10, 100.
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