Letter pubs.acs.org/NanoLett
Exciton Superposition States in CdSe Nanocrystals Measured Using Broadband Two-Dimensional Electronic Spectroscopy Daniel B. Turner, Yasser Hassan, and Gregory D. Scholes* Department of Chemistry and Centre for Quantum Information and Quantum Control, University of Toronto, 80 Saint George Street, Toronto, Ontario, M5S 3H6 Canada S Supporting Information *
ABSTRACT: Coherent superpositions among eigenstates are of interest in fields as diverse as photosynthesis and quantum computation. In this report, we used two-dimensional electronic spectroscopy (2D ES) to measure the decoherence time of a superposition of the two lowestenergy excitons in colloidal CdSe nanocrystals (cubic phase) in solution at room temperature. In the electron−hole representation, the quantum coherence is, remarkably, a twelve-particle correlation. By comparing the measured 2D ES to simulations, we also explored the effects of inhomogeneous broadening and examined the spectroscopic signatures of biexcitons. KEYWORDS: Two-dimensional spectroscopy, quantum dots, electronic coherence, electron−phonon coupling, femtosecond dynamics, nanocrystals
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advanced the understanding of the dynamics and interactions in a variety of semiconductor nanostructures, especially CdSe nanocrystals. The exciton fine structure of wurtzite CdSe nanocrystals is now relatively well understood.30,43 Vibrational modes,40,41 spin relaxation,44−47 intraband relaxation48,49 processes, and state-dependent electron−phonon coupling values have been investigated.32,33 Insights into biexciton states38,39,49−52 and many-body interactions53 have also been gained through a variety of experimental approaches. These measurements and others have served to guide and test theoretical models of the electronic states in nanocrystals.54−57 Time-resolved measurements with femtosecond time resolution have yielded numerous insights into the effects of quantum confinement on the electronic states, couplings, and dynamics. 2D ES has the potential to reveal information that is not clearly resolved by one-dimensional techniques. An important example is that 2D ES reveals couplings between states through the appearance of cross peaks.58 Here we use 2D ES to identify such couplings among exciton states. The 2D ES is similar to one-dimensional four-wave-mixing measurements like transient absorption. In transient-absorption measurements, the pump beam electric field interacts twice with the sample to photoexcite a fraction of the sample into electronic excited states. A probe arrives after a variable delay we denote τ2 to stimulate a signal field that is detected as a change in transmission of the probe. The 2D ES can be understood in a similar way, but both the spectrum of states photoexcited as well as the spectrum of states probed after the delay time τ2 are recorded and correlated. This is accomplished
uantum coherences are coherent superpositions of states with well-defined relative phases. This special characteristic of quantum-mechanical systems gives them properties quite distinct from a statistical mixture of states and has stimulated their development as resources in quantum information. Recently, there have been spectacular advances in the demonstration and study of quantum coherences and related phenomena. For example, there have been many reports of coherence, and even entanglement, of spins in diamond1,2 and semiconductors.3−8 Quantum coherence in molecules resulting from coherent, nonstatistical mixtures of vibrational states has also been studied.9,10 Other work on nanocrystals has suggested coherence to be involved in the mechanism of multiexciton generation.11,12 Coherence among molecular electronic states has generated substantial interest recently in connection with studies of quantum interference13 and energy transfer in photosynthetic proteins.14−18 Femtosecond spectroscopy has facilitated the study of coherences and correlations in semiconductor nanostructures.19−29 Here we photoexcited the two lowest-energy exciton bands of CdSe nanocrystals using femtosecond pulses to produce an electronic superposition state that we studied using twodimensional electronic spectroscopy (2D ES). We found that this exciton−exciton superposition state, formally a twelveparticle correlation in the electron−hole basis, persists for about 50 fs at ambient temperature. The electronic states of nanocrystals have been widely studied. Dynamics, such as relaxation processes, in excited states (exciton states) have been examined using ultrafast nonlinear optical experiments, typically four-wave-mixing measurements such as transient absorption,30−37 transient grating,38,39 or photon echo peak shift,40−42 which use one or more femtosecond laser pulses to first induce and then probe the underlying physical processes. Such experiments have © 2011 American Chemical Society
Received: November 9, 2011 Revised: December 15, 2011 Published: December 27, 2011 880
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typically59 by scanning τ1, the delay time between the two light−matter interactions in the pump pulse sequence, for a set value of τ2. At each τ1 point, the spectrum of the signal emitted during the emission time period, τ3, is measured, providing the emission frequency axis, ω3. Fourier transformation of the coherent oscillations during time period τ1 produces a 2D spectrum. Time period τ2 is varied parametrically, and this leads to a set of 2D spectra, one for each value of τ2, that can be transformed into a 3D spectral solid if desired.60 With a properly designed optical setup, the phase and amplitude of the emitted signal is sensitive to the phase and amplitude of the oscillations during τ1, and couplings between states can be observed as cross peaks in the 2D spectrum. We present details of the 2D ES experimental setup as well as sample synthesis and characterization in the Supporting Information. Briefly, we synthesized the CdSe nanocrystals according to literature methods.61−64 Characterization by electron microscopy and X-ray diffraction indicates that the QDs have a cubic (zinc-blende) crystal structure,47,65,66 which is advantageous because the crystal-field splitting of the more common wurtzite CdSe nanocrystals leads to an excitonic fine structure51,52 that would effectively increase the heterogeneity of the optical transitions. The linear absorption spectrum in Figure 1a shows band-edge absorption at 2.070 eV (599 nm), suggesting the nanocrystals are ∼6 nm in diameter,67 which matches the size estimated from electron microscopy. We set the central wavelength of the femtosecond pulses to be 590 nm (2.19 eV) so that peaks involving state |X2⟩ (E2 ∼ 2.17 eV) will be enhanced relative to the peaks involving the brighter state | X1⟩ (E1 ∼ 2.07 eV). For CdSe nanocrystals, theoretical models agree55,56 that the band-edge exciton |X1⟩ comprises an electron in a 1S orbital and a hole in a 1S orbital (1Se−1S3/2), and that the second-lowest energy peak is due to an exciton |X2⟩ composed of a 1S electron and a 2S hole (1Se−2S3/2); their relative energies and shaded areas indicating inhomogeneity are depicted in Figure 1b. Previous studies showed that 2D ES can report on the exciton and the biexciton fine structures51,52 in wurtzite CdSe nanocrystals. Those studies used relatively narrowband pulses and thus probed only the band-edge exciton and lowest-energy biexciton states. Here we present a 2D ES data set of cubic (zinc blende) CdSe nanocrystals in toluene using femtosecond laser pulses with bandwidths that overlap the two lowest-energy exciton bands. We observe eight peaks, four having positive amplitude (red contours) and four having negative amplitude (blue contours), in the measured 2D spectra presented in Figure 1c,d and use their locations to isolate specific states and couplings. In Figure 1d, the diagonal peaks for the |X1⟩ and |X2⟩ transitions are indicated. The cross peaks, labeled as |X1⟩⟨X2| and |X2⟩⟨X1|, show that the two exciton states are “coupled”. In other words, those electronic transitions involve common electronic orbitals. The negative-amplitude features are due to signal pathways involving the indicated biexciton states in Figure 1e, which is a spectrum simulated under the condition of no inhomogeneous broadening. We discuss all of these features below. Even well-prepared colloidal nanocrystal samples have a distribution of particle sizes that leads to broadening of spectroscopic features. In addition to polydispersity, exciton fine structure can further complicate spectra. As mentioned above, we used cubic crystal structure CdSe nanocrystals to remove this latter element of complexity from the spectra. In total, inhomogeneity can account for about half of the line
Figure 1. Spectra and states of the measurement. (a) Sample absorption spectrum (solid curve) and the spectrum of the femtosecond pulses (dashed curve). The peak absorption energies of the two exciton states of interest are indicated by the dashed vertical lines. (b) Energy levels of primary interest, where |0⟩, |X1⟩, and |X2⟩ designate the ground state, the 1Se−1S3/2 exciton state, and the 1Se− 2S3/2 exciton state, respectively. The heterogeneity for each exciton state is indicated by the shaded areas surrounding each level. (c) Total 2D ES (real part) for the indicated τ2 values. (d) Measured spectrum at τ2 of 300 fs with labels indicating major contributors for the two diagonal and two cross peaks. (e) Simulated spectrum under the condition of no inhomogeneity with the six peaks containing biexcitonic information labeled.
broadening in CdSe nanocrystal samples.40 Therefore we expect that inhomogeneity will play an important role in understanding the results. A second concept central to understanding the spectra is that a significant portion of the signal is due to excited-state absorption (ESA) pathways involving biexciton states. These pathways lead to negativeamplitude peaks that are red shifted in the emission dimension away from the exciton peaks by an amount that is often called the biexciton binding energy. These features are most easily seen in simulations of the 2D ES with small inhomogeneous line broadening, Figure 1e. The biexciton binding energy in nanocrystals is mainly the difference in exchange interactions in the biexciton state compared to the exciton state52 with a small correction due to electron correlation beyond the Hartree− 881
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Comparing the rephasing to the nonrephasing spectra shows several ways in which inhomogeneous broadening and strong ESA pathways can affect 2D spectra. Details are presented in the Supporting Information. Briefly, the peaks are better resolved and diagonally elongated in the rephasing spectra, and destructive interference between the strong negative-amplitude ESA contributions and the positive-amplitude bleach contributions leads to peaks that are centered at unexpected frequency coordinates (meaning not at the corners of a perfect square). We reproduced the essential features of the measured 2D ES using simulations based on the sum-over-states theory. One minor difference is that the peak-location changes due to destructive interference are not as large in the simulations as in the experiment. Details are presented in the Supporting Information, and simulated spectra that fully accounted for inhomogeneity are shown in Figure 3.
Fock model. We find that a value of 25 meV fits our data, consistent with previous transient-absorption experiments.49,50 Photon-echo methods15,68,69 can improve spectroscopic resolution by removing inhomogeneous line broadening. The specific time ordering of pulses required for the photon-echo effect means that the line narrowing, reduction of the antidiagonal line width, is specific to a half Fourier transform of the data over positive values of τ1. These spectra, called rephasing spectra, are displayed in Figure 2 together with the
Figure 2. Rephasing (top row), nonrephasing (middle row), and total 2D ES spectra (bottom row) in real parts (left column), imaginary parts (middle column), and magnitudes (right column) for τ2 = 120 fs. The two cross peaks indicate coupling between the two excitons. Contours are linearly spaced at 3% intervals, where normalization of all spectra are relative to the maximum of the total magnitude 2D ES spectrum. Black lines are drawn in the magnitude spectra at the |X1⟩ and |X2⟩ transition energies of 2.07 and 2.17 eV, respectively.
Figure 3. Simulated components and total 2D ES in real, imaginary, and magnitude parts. The simulations adequately reproduce the features present in the measured 2D ES.
half Fourier transform for negative τ1, called nonrephasing spectra, for τ2 = 120 fs in real, imaginary, and magnitude parts. The solid black lines in the magnitude spectra correspond to the energies of states |X1⟩ and |X2⟩ as indicated in Figure 1b. There are several noteworthy features in the spectra. The most apparent are the two diagonal and two cross peaks. While the diagonal peaks essentially follow the linear absorption spectrum, the presence of the cross peaks immediately shows that states |X1⟩ and |X2⟩ are coupled in the spectroscopic sense through a common ground state,58,70 and this supports theoretical predictions that the electrons of the two excitons reside in a common conduction state.35,51,52 The common conduction state of the electrons is the physical origin for the common ground state between two spectroscopic transitions. In other words, the cross peaks indicate Pauli blocking of the electronic transition; electronic excitation of one exciton affects the spectrum of the other exciton. The nonrephasing spectra, presented in the middle row of Figure 2, do not involve photon echos and therefore contain inhomogeneous contributions to both the diagonal and antidiagonal linewidths. The sum of the rephasing and nonrephasing spectra69,71,72 leads to the morecommon total 2D ES presented in the bottom row of Figure 2 and in Figure 1c,d.
The existence of biexciton states in nanocrystals is wellknown,49,50 and a portion of the biexcitonic fine structure has been probed previously.51,52 Our measured 2D spectra contain four visible peaks that involve biexciton states. Given the frequency coordinates, we can assign different biexciton states as the origin of the four peaks and estimate binding energies for each biexciton. The ground-state biexciton (|X1X1⟩) peak appears at τ2 = 0 fs, and the other three peaks appear after the pulse overlap period, matching a previous transientabsorption result.50 The upper-leftmost peak is due to a mixed biexciton between states |X1⟩ and |X2⟩. Since it appears directly above the ground-state biexciton, we infer that binding energies of the (|X1X2⟩) and (|X1X1⟩) biexcitons are similar, and a value of 25 meV reproduces the peaks well in simulations. The peaks on the right-hand side (indicated as |X1Xa⟩ and | X2Xa⟩, where a indicates an exciton with an energy above |X2⟩) involve biexcitons from states above the |X2⟩ state, for example a 1Se−1S1/2 or 1Pe−1P3/2 exciton state, both of which could be possible given our pulse spectrum. Since the two peaks also appear at similar emission energies, the binding energies of the biexcitons are similar. Unfortunately, we cannot extract 882
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While the existence of the cross peaks implies that the two states involve elementary excitations among one or more common orbitals, oscillations in the cross-peak amplitudes have a more consequential implication. The oscillations show that a coherent superposition between two exciton states can be created by optical excitation and that it persists for 50 fs at 298 K. In studies of another semiconductor nanostructure, GaAs quantum wells, coherent superpositions between two exciton states can last for several picoseconds, but those measurements73 were performed at a temperature of 10 K, not room temperature. Extracting electronic coherence would be difficult in transient-absorption spectroscopy due to strong contributions from phonon oscillations present in the diagonal peaks, see the Supporting Information for more details. Although not a perfect bounding value due to ESA pathways, for comparison purposes, the single-exciton coherences (the signal measured during τ1 in the 2D ES measurement) dephase on about the same time scale. It also may be the case that a significant contribution to dephasing of the coherence is due to the inhomogeneous nature of the superposition.74 What is the electronic composition of the |X2⟩⟨X1| superposition state? The band-edge X1 state is known to comprise excitation from the three p-type valence orbitals (v1) to the lowest conduction orbitals (c).52,75−78 The X2 state is similarly formed, but the electron is excited from a lower-energy set of valence orbitals (v2). The possible superposition states are formed subject to the same selection rules and total angular momentum projections as biexciton states. Therefore, proceeding in precisely the same way as described in ref 52, we can construct the possible coherences in terms of the arrangement of elementary quasiparticles in the valence and conduction orbitals. On the basis of the optical selection rules and transition dipole strengths, the dominant coherence will have total angular momentum F = 0, being a superposition of F = +1 and F = −1 excitons. The ten possible F = 0 electron−hole configurations (written in the electron−electron representation) that comprise the coherence are depicted in Figure 4b. Remarkably, the |X2⟩⟨X1| superposition involves correlations of twelve particles. In this study, we measured and simulated broadband 2D electronic spectra of cubic CdSe nanocrystals and discussed four results. First, the spectra showed that the two lowestenergy exciton states are coupled, providing unambiguous confirmation of theoretical models of nanocrystal electronic structure. Future studies with varied pulse spectra have the potential to reveal couplings involving other, higher-lying excitons. Second, we detailed how 2D ES measurements can be used to reveal insights into nanocrystal spectra that are completely obscured or dominated by inhomogeneous broadening due to nanocrystal size variation in other types of spectroscopic measurements. Third, we discussed how biexcitons contribute to the spectra by participation in ESA signals. Biexciton states in semiconductor nanocrystals have been studied extensively because they affect population inversion in quantum-dot lasers and must be considered for multiexciton generation. A key characteristic of biexcitons is the value of their binding energies. Here, a comparison between the locations of ESA peaks present in our measured 2D spectra to simulations suggested values for the binding energies of several biexciton states in CdSe nanocrystals. Multiple-quantum 2D ES measurements such as those performed on GaAs quantum wells29,79 have the potential to provide even greater insights into the biexciton states and their binding energies. Fourth, we
quantitative values for the biexciton binding energies because the features are truncated by the pulse spectra, unlike transientabsorption measurements that use a white-light probe to cover the features completely. The most striking result of this study is the observation of the time dependence of the above-diagonal cross peak as a function of τ2. The time evolution of the amplitude of the 2D ES at this cross peak position is shown in Figure 4a. The rephasing (blue)
Figure 4. (a) Extractions from the spectra through time period τ2 for the above-diagonal cross peak that shows electronic coherence. The green, blue, and red lines correspond to traces of the nonrephasing, rephasing, and total 2D ES spectra, respectively. Both the rephasing and total traces oscillate at early τ2 times. The black dashed line is a theoretical prediction for a coherence with a dephasing time of 15 fs and frequency of 25 THz. (b) The ten possible F = 0 configurations in the electron−electron representation for the twelve electrons involved in the coherent superposition state. The six singlet configurations that contribute the oscillator strength to the exciton transitions mix strongly with the four triplet configurations owing to spin−orbit coupling. Filled circles indicate spin-paired electrons in occupied orbitals; open circles indicate holes in the valence-band orbitals. The orbital angular momenta values (0, ±1) and the orbital labels (c, v1, v2) are indicated above one configuration diagram.
and total (red) 2D ES traces oscillate once before decaying to a steady-state value by τ2 = 50 fs. The experimental trace matches an exponential decay characterized by a dephasing time of 15 fs and frequency of 25 THz as depicted by the dashed line in Figure 4a. The decay profile is not a perfect fit due to the nonresonant signal present during the pulse-overlap period. The oscillation indicates a |X2⟩⟨X1| coherence during τ2, implying that the first two-field interactions created a coherent superposition between the two different exciton states.15 The coherence is due to electronic, not vibrational, coupling since the nonrephasing contribution (green curve) to the cross peak does not oscillate,18 and moreover, there are no nanocrystal phonon modes with a frequency near 25 THz (103 meV). See the Supporting Information for more details. The belowdiagonal cross peak does not show this oscillation cleanly (traces not shown), instead it rises quickly then does not change. 883
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found that femtosecond excitation of the lowest two exciton states can create an excitonic coherent superposition, formally a twelve-particle correlation, that persists for 50 fs at ambient temperatures. Future studies may show that this coherence can be used as a resource for quantum information tasks; doing so will require first varying a large set of parameters to determine the mechanisms that govern the dephasing process and then synthesizing samples that can support the coherence for much longer time scales.
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ASSOCIATED CONTENT
S Supporting Information *
Synthesis and characterization details; optical experiment description; and simulation method, parameters, and results. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected].
ACKNOWLEDGMENTS We acknowledge helpful conversations with Philip Johnson and Dr. Ryan Cooney. The United States Air Force Office of Scientific Research under Contract Number FA9550-10-1-0260 and the Natural Sciences and Engineering Research Council of Canada are acknowledged for financial support.
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REFERENCES
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Nano Letters
Letter
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