Date Published: August 15, 2017
Author(s): Galan Moody, Steven T. Cundiff.
Multi-dimensional coherent spectroscopy (MDCS) has become an extremely versatile and sensitive technique for elucidating the structure, composition, and dynamics of condensed matter, atomic, and molecular systems. The appeal of MDCS lies in its ability to resolve both individual-emitter and ensemble-averaged dynamics of optically created excitations in disordered systems. When applied to semiconductors, MDCS enables unambiguous separation of homogeneous and inhomogeneous contributions to the optical linewidth, pinpoints the nature of coupling between resonances, and reveals signatures of many-body interactions. In this review, we discuss the implementation of MDCS to measure the nonlinear optical response of excitonic transitions in semiconductor nanostructures. Capabilities of the technique are illustrated with recent experimental studies that advance our understanding of optical decoherence and dissipation, energy transfer, and many-body phenomena in quantum dots and quantum wells, semiconductor microcavities, layered semiconductors, and photovoltaic materials.
A central challenge in condensed-matter, atomic, and molecular science is to understand the roles that numerous physical processes responsible for decoherence and relaxation play in the complexity of light-matter interaction. Condensed matter comprises ~1023 particles cm−3 that interact through the infinite-range Coulomb force, which is responsible for the formation of excitons (bound electron-hole pairs) and residual interactions between them. In semiconductors, the optical response near the fundamental bandgap is profoundly influenced by Coulomb correlations and other many-body effects associated with excitons, in marked contrast to atomic systems [1,2]. Substantial physical insight into the implications of many-body phenomena on exciton dynamics has been acquired through time-resolved spectroscopy [3,4]. An effective technique must have the dynamic range to probe the optical response across femtosecond to nanosecond timescales and the sensitivity to resolve disorder due to stochastic fluctuations in structural and material composition ubiquitous to semiconductor nanostructures. One-dimensional (1D) optical spectroscopies have the requisite timing resolution, but a 1D spectrum provides an ensemble-averaged response that can be challenging to interpret, especially for heterogeneous systems . In semiconductors, random disorder manifests as a distribution of transition frequencies that inhomogeneously broadens the optical lines. Resolving the intrinsic homogeneous linewidth of each individual emitter in an ensemble is generally impossible with 1D spectroscopy. Although 1D spectra may contain signatures of couplings between transitions, insight into their origin and dynamics is usually obscured by multiple overlapping contributions in the optical response.
The Fourier-transform methods of MDCS were originally developed at radio frequencies for nuclear magnetic resonance spectroscopy of spins . Over the last two decades, much work has been devoted to implementing these concepts in the optical regime. In the infrared, MDCS provides a detailed view of vibronic coherences that elucidates the structure and dynamics of molecules [25–27]. At visible optical frequencies, MDCS has becoming an increasingly useful tool for probing electronic transitions, with significant contributions in identifying long-lived coherences in light-harvesting photosynthetic complexes [28–33], electronic-vibrational coupling in nitrogen-vacancy centers , collective resonances in atomic vapors [35,36], and electronic structure in colloidal nanocrystals [37–40]. The limits of MDCS have recently been extended to the extreme ultraviolet spectral region to explore inner-valence transitions in complex molecules [41,42]. In this article, recent experimental developments in the application of MDCS to study excitonic dynamics, interactions, and transport in semiconductor nanostructures are reviewed.
Semiconductor QDs are an exemplary system to demonstrate the capabilities of 2DCS for dissecting disordered ensembles. The stochastic nature of the epitaxial growth process leads to an inhomogeneous distribution of QD sizes that translates into ~50–100 meV dispersion of the exciton transition energies for self-assembled InAs/GaAs QDs and ~1–5 meV for ‘natural’ GaAs/AlGaAs QDs. The in-plane QD confinement potential tends to be asymmetric with principle axes along the  ≡ V and
[11¯0]≡H crystal directions due to several sources, including strain, shape, and piezoelectricity [50,87–89] (see Figure 7(a)). When considering the electron-hole exchange interaction, anisotropy mixes the electronic transitions, resulting in two bright exciton states |H〉 and |V〉 that are separated in energy by the so-called fine-structure splitting.
Semiconducting compounds in which the covalent bonds within the crystal plane are significantly stronger than the weak bonds between planes comprise the family of layered semiconductors. Examples including InSe, GaSe, Bi2Se3, and van der Waals materials. Excitons in single monolayers exhibit interesting physics in two dimensions, which can give rise to a host of new electronic and optical phenomena. For this review, we consider narrow GaAs QWs within this group, since they represent a canonical example of exciton physics in a quasi-2D system.
By extending 2DCS into three dimensions, the different quantum pathways can be further separated to reveal hidden details of the mechanisms contributing to a system’s nonlinear optical response. 3DCS techniques have been developed in the infrared to examine vibrational and structural relaxation dynamics of molecules [131–133] and extended to the visible regime to resolve high-frequency vibronic coherences , to fully distinguish all quantum pathways in an atomic vapor up to χ(3) , and to separate pathways involving weakly bound mixed-biexciton states in a GaAs QW .
The resonant interaction between semiconductor excitons and an optical field inside a microcavity can strongly modify the light-matter interaction. In the strong coupling regime, the exciton and cavity modes form new admixed eigenstates known as upper (UP) and lower (LP) exciton-polaritons that exhibit an anti-crossing in their dispersion curves (see Figure 12(a)). Exciton-polariton states can be created by embedding a QD or a QW within a semiconductor microcavity . A typical group III–V microcavity system consists of a single or few layers of InxGa1–xAs QWs sandwiched between GaAs/AlAs distributed Bragg reflectors (DBRs) acting as the cavity end mirrors. Due to their mixed light-matter character, exciton-polaritons have extremely low effective mass that facilitates Bose-Einstein condensation  and superfluidity  in the solid-state. Because they retain an excitonic component, exciton-polaritons are strongly influenced by the many-body interactions inherent to semiconductor nanostructures. Transient FWM experiments reveal that two-exciton Coulomb correlations clearly play a role in exciton-polariton scattering and decoherence [143–146].
An enticing alternative approach to the conventional optical heterodyne-detection techniques of MDCS is to instead use a sequence of four collinear pulses to drive a fourth-order population in the system. The resulting excited-state populations can be detected either optically as fluorescence, which is the basis for 2D fluorescence spectroscopy previously demonstrated on a rubidium vapor , or through photocurrent in a semiconductor photoconductive device . Isolation of the fourth-order population from other signals is achieved by utilizing acousto-optic phase modulation of the excitation pulses discussed in Section 2.4.2 in combination with phase-synchronous detection using a lock-in amplifier. Mechanical fluctuations in the setup can be partially compensated for using an auxiliary continuous-wave laser running through the same optics as the excitation laser pulses, which eliminates the need for active phase stabilization schemes. This approach offers several advantages compared to conventional MDCS methods, including simultaneous detection of the rephasing, non-rephasing, and two-quantum signals. Due to the collinear geometry, diffraction-limited spatial resolution is possible.
Multi-dimensional coherent spectroscopy is a powerful tool for probing the structure, composition, and dynamics of condensed matter, atoms, and molecules. The field continues to evolve with new developments in the implementation of the technique and the materials that are accessible. The frequency range of two-dimensional coherent spectroscopy covers terahertz to XUV frequencies. Rapid advances in table-top high-harmonic and X-ray sources have extended this range to enable XUV and X-ray optical spectroscopy of super-excited states and coupled electron-nuclear dynamics in molecules [149,150]. When applied to nanostructures, multi-dimensional XUV/X-ray spectroscopy may improve our understanding of the relation between local atomic structure and the physical properties of low-dimensional semiconductors on a sub-nanometer level and attosecond timescale . These studies may lead to improvements in the realization of nanostructures with optimized size control, tailored structure and material composition, and increased homogeneity and synthesis reproducibility.