Directed Two-Photon Absorption in CdSe Nanoplatelets Revealed by

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Directed Two-Photon Absorption in CdSe Nanoplatelets Revealed by k‑Space Spectroscopy Jan Heckmann,† Riccardo Scott,† Anatol V. Prudnikau,‡ Artsiom Antanovich,‡ Nina Owschimikow,† Mikhail Artemyev,‡ Juan I. Climente,§ Ulrike Woggon,† Nicolai B. Grosse,† and Alexander W. Achtstein*,† †

Institute of Optics and Atomic Physics, Technical University of Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany Research Institute for Physical Chemical Problems of Belarusian State University, 220006, Minsk, Belarus § Departament de Quı ́mica Fı ́sica i Analı ́tica, Universitat Jaume I, E-12080, Castelló de la Plana, Spain ‡

S Supporting Information *

ABSTRACT: We show that two-photon absorption (TPA) is highly anisotropic in CdSe nanoplatelets, thus promoting them as a new class of directional two-photon absorbers with large cross sections. Comparing two-dimensional k-space spectroscopic measurements of the one-photon and two-photon excitation of an oriented monolayer of platelets, it is revealed that TPA into the continuum is a directional phenomenon. This is in contrast to one-photon absorption. The observed directional TPA is shown to be related to fundamental band anisotropies of zincblende CdSe and the ultrastrong anisotropic confinement. We recover the internal transition dipole distribution and find that this directionality arises from the intrinsic directionality of the underlying Bloch and envelope functions of the states involved. We note that the photoemission from the CdSe platelets is highly anisotropic following either one- or two-photon excitation. Given the directionality and high TPA cross-section of these platelets, they may, for example, find employment as efficient logic AND elements in integrated photonic devices, or directional photon converters. KEYWORDS: 2D k-space spectroscopy, 2D semiconductors, CdSe nanoplatelets, angle-dependent two-photon absorption, transition dipole distribution, bright plane

T

for both (degenerate) two-photon excitation and subsequent photoluminescence (PL) emission. In our 2D k-space spectroscopy setup, we vary the excitation k-vector (wavevector) and measure the k-space resolved emission. We analyze the internal distribution of transition dipole moments (TDM) involved in TP absorption to determine their degree of in-plane orientation. This allows one to reconstruct the angle dependent (highly anisotropic) TPA cross section of the oriented NPLs. The TPA results in this work complement our recent investigation19 of k-space resolved one-photon absorption and emission of these nanoplatelets. The linear absorption in the continuum at 3 eV (413 nm) is found to be isotropic, whereas the heavy-hole exciton ground state emission is highly directed because it stems from a bright plane of TDMs that are oriented in the platelet plane−in-plane (IP) dipoles. The strong confinement lifts the degeneracy of the heavy hole (hh) and light hole (lh) in zinc blende systems, subsequently leading to the loss of isotropy.

wo-photon absorber nanocrystals are rapidly gaining interest for their diverse applicability to nonlinear gain media,1 power-limiting, photodynamic two-photon (TP) cancer therapy,2 and biolabeling.3 In addition, confocal twophoton imaging, which combines high spatial resolution and deep tissue penetration, greatly benefits in vivo cell and animal imaging.4,5 Further applications have been demonstrated in microfabrication, lithography, polymerization, data storage, and spectroscopy.6−8 Thus, efficient TP absorbers with extremely high TPA cross sections per particle or per particle volume such as CdSe nanoplatelets are highly desirable.9 Compared to other semiconductor nanoparticles like CdSe or CdS dots and rods10−12 or typical organic dyes with TPA cross sections up to ∼103−105GM (where 1 GM = 1 Göppert Mayer = 10−50 cm4 s photon−1),13,14 it has been found that CdSe nanoplatelets have much larger TPA cross sections of up to 5 × 107 GM.9 However, a less investigated aspect is the anisotropy of TPA in II−VI semiconductors and lowdimensional systems of those materials.15−18 In this work, we study the k-space resolved two-photon absorption in oriented CdSe nanoplatelets to gain a microscopic understanding of the underlying electronic properties. We use a set of in-plane oriented CdSe nanoplatelet samples © 2017 American Chemical Society

Received: July 17, 2017 Revised: September 7, 2017 Published: September 12, 2017 6321

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Figure 1. (a) Linear absorption spectrum of the studied 4.5 monolayer CdSe nanoplatelets. The spectral positions of two- and one-photon excitation are marked with red and blue arrows, respectively. The heavy-hole photoluminescence (PL) is also shown. Inset: TEM image of the oriented NPLs. (b) Excitation and emission scheme. An immersion oil objective is used to excite the oriented nanoplatelets. By projecting the back aperture of the objective onto a highly sensitive CCD camera, the wave-vector dependent (i.e., its angular) emission pattern can be observed. The k-scale is normalized to the wave-vector in air (k0 = ω/c). Total internal reflection occurs for wave-vectors larger than the wave-vector in the surrounding air, (k/k0 > 1). The variation of the excitation wave-vector in the back aperture enables 2D k-space spectroscopic measurements. (c) Typical emission pattern for oriented nanoplatelets. The kx- and ky-cuts can be referred to as p- and s-polarized emissions, respectively.

Differences between linear and TP absorption are discussed in terms of their respective TDM distribution and the underlying Bloch functions of the hh, lh, and split-off (so) bands. The observed strongly directional TP absorption is shown to be related predominantly to transitions involving either hh−cb or cb−cb contributions, which have an anisotropic transition dipole distribution. We also show how the confinement-induced anisotropies stem from the envelope functions for the intraband transitions and from the Bloch part of the wave functions for the interband transitions. Hence, combining two-photon and one-photon 2D k-space spectroscopy is a powerful tool for analyzing the electronic nature of nonlinear (and linear) optical properties and their anisotropy. The investigated zinc blende (ZB) CdSe nanoplatelets, covered with oleic acid ligands, were deposited on 170 μm thick fused silica substrates by a Langmuir technique. This produces a monolayer of flat-lying nanoplatelets which are oriented on the substrate (x−y-plane);19,20 see inset in Figure 1a. The sample can thus be modeled as a three-layer system consisting of the glass substrate, an effective medium of a monolayer of nanoplatelets and their ligands, and air (Figure 1b). The CdSe NPLs have a thickness of 4.5 monolayers (Lz = 1.37 nm). The lateral size is determined by TEM-analysis as 19.6 × 9.6 nm2 (see inset in Figure 1a), yielding an aspect ratio AR = Lz / LxLy = 0.10. For TP excitation we use a

information about the samples and the experimental configuration can be found in the SI. The wavevector composition of the emission can be characterized in greater detail by imaging the back aperture of the objective onto a sensitive CCD camera (Figure1c). To determine the distribution of transition dipoles involved in the emission processes, we introduce a polarizer into the detection path. The cuts in Figure 1c can then be related to pure in-plane (ky cut, s-polarized light) and a superposition of in-plane (IP) and out-of-plane (OP) dipole emission (kx cut, p-polarized light). We concentrate on the p-polarized emission since it contains information on IP as well as OP emission. The radiation of the excitation laser is also p-polarized (electric field and the wavevector coincide in the plane of incidence) to excite IP as well as OP dipoles as a function of the incoming wavevector kinx .19 We model the k-vector dependent platelet emission and excitation by extending a local density of states formalism,21,22 described in the following. Fermi’s Golden Rule states that the radiative rate of an emitter is proportional to the product of the Einstein coefficient A (proportional to the TDM |μ2|) and the normalized density of (photon) states ρ̃. For modeling the radiation of IP and OP transition dipoles in our platelet monolayer on SiO2, we use a normalized effective optical density of states ρ̃IP,OP(ω) = ρIP,OP(ω)/ρ0(ω) accounting for the alteration of the radiative rate in our heterogeneous system with respect to free space.19,22,23 The density of electromagnetic modes in the emitter medium is defined as ρ0(ω) = ω2(εμ)3/2/π. The radiative rate is then given by

Ti:sapphire laser (fwhm 150 fs, 75.4 MHz, 0.8 kW/cm2 CWequivalent excitation density) providing pulsed radiation at 830 nm wavelength. For linear excitation (at a wavelength of 415 nm), a beta barium borate (BBO) crystal is used for frequency doubling. The k-space analysis is performed using a high numeric aperture (1.49) immersion oil objective shown in Figure 1b. Excitation and detection are symmetric in the confocal setup. Via Fourier transform the beam position in the back aperture of the objective relates to the in-plane wave-vector component and hence the incident angle of a light ray. The excitation characteristics are obtained by translating the laser beam parallel with respect to the optical axis of the objective. Detailed

Γr(ω) = A(ω)ρ ̃(ω) = ρ(ω)

πω |μ(ω)|2 ε 3ℏ

with ε = εrϵ0.24 The total radiative decay rate and the TDM are both decomposed into IP and OP components using the related Einstein coefficients AX (ω) = ρ0 (ω)

πω Nr,X|μ X (ω)|2 3ℏε

(1)

The index X = IP, OP denotes in-plane or out-of-plane, respectively. Nr,X|μX(ω)|2 are the IP and OP projections of the 6322

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Nano Letters dipole strengths with respect to the principle axes of the dipole ellipsoid. Nr,X are the relative weights, depending on its eccentricity. The density of photon states ρ̃(ω) relates the IP and OP transition dipoles to the radiative rates. For ppolarization ρ̃(ω) can also be decomposed into IP and OP components. Due to the Lorentz reciprocity,25 we can treat the absorption and emission process equivalently in this formalism and relate them to ρ̃(ω) in the absorbers/emitters environment. We can model the (p-polarized) k-vector dependent onephoton absorption/emission characteristics as the superposition of IP and OP absorption/emission rates: p S1PA (ω , kx) = C(1)[ρIP ̃ p (ω , kx)AIP(ω)

+ ρOP ̃p (ω , kx)A OP(ω)]

(2)

A two-photon absorption process can be interpreted in this picture via concatenating two one-photon transitions. As discussed later, the (weighted) summation of the respective concatenated one-photon transition dipole orientations gives the overall orientation of the TPA process with effective (2) (2) Einstein coefficients AIP and AOP . To account for the 26 nonlinearity in modeling the TPA process, the calculated intensities from eq 2 have to be squared:

Figure 2. Calculated k-space spectra. (a) Calculated two-photon excitation at a wavelength of 830 nm of a 100% IP and an isotropic (67% IP and 33% OP) dipole distribution, calculated from eq 3. (b) Same for one-photon excitation at 415 nm, using eq 2. The curves are normalized to their area.

medium of the nanoplatelet monolayer and ligands have different dispersion relations at 830 and 415 nm; see also Table S1 in the SI. Our experimental results of in-plane k-vector dependent TP and linear excitation and the related emission patterns are shown in Figure 3. Experimental 2D k-space spectra are shown as false color plots in Figure 3a and b (left). They are obtained by plotting the cut in x-direction of the back aperture image (see Figure 1c) for every excitation angle kinx . Vertical cross sections in these 2D k-space maps are shown on the right in Figure 3a and b. A proxy for the excitation wavevector (kinx ) dependent absorption, they can be modeled with eq 3 for two-photon absorption, panel a and eq 2 for onephoton absorption, panel b. The contributions of IP and OP transition dipoles involved in TPA and 1PA processes are obtained by fitting the experimental results with a least-squares routine. IP contributions are given in Figure 3a and b. For the degenerate TPA process at 830 nm into the continuum, the model delivers an 85% IP to 15% OP ratio of the transition dipoles. This reflects a considerable net orientation of the continuum TPA transition. In contrast, the one-photon continuum absorption at a wavelength of 415 nm is nearly isotropic with a ratio of 70% IP and 30% OP transition dipoles. We estimate the uncertainty of the OP to IP relation determination to ±5%. The emission at 512 nm after two- and one-photon (horizontal cross sections in 2D k-space maps) is compared in Figure 3c. We observe a strong decrease of emitted intensity at kout x /k0 = 1. As discussed in Figure 2, only OP dipoles can radiate here to the far-field leading to the characteristic minimum. In fact the fit using eq 2 reveals a highly anisotropic transition dipole distribution with 95% IP and only 5% OP dipoles, which can be considered within our above-mentioned error margin as IP only emission. This finding results in strongly directed emission in z-direction, perpendicular to the nanoplatelet plane.19 The emission pattern is not altered by the nature of the excitation. In both cases, recombination luminescence originates from the lowest hh exciton as the continuum generated hot excitation cools down to the band edge. The quadratic pump-power dependence of the TP induced photoluminescence proves two-photon absorption related emission and is shown in Figure S3 of the SI.

p (2) STPA (ω , kx) = C(2)[ρIP (ω) ̃ p (ω , kx)AIP (2) + ρOP (ω)]2 ̃p (ω , kx)A OP

(3)

C(1) (in eq 2) and C(2) (in eq 3) are proportionality constants related to the setup sensitivity for one- and two-photon excitation, respectively. All other parameters in this model are defined by the experimental setup or literature values (listed in the SI) except for the ratio of IP and OP oriented transition dipoles. Figure 2a shows calculated k-space absorption spectra for two-photon excitation at 830 nm using eq 3. The curves follow the two-photon induced PL signal as a function of the incident k-vector kinx /k0. The cases for 100% IP (dashed line) and an isotropic (solid line) transition dipole distribution are shown. Due to two in-plane axes (x and y) a ratio of 67% IP to 33% OP dipoles represents an isotropic distribution. Figure 2b shows the calculated k-space absorption spectra for one-photon excitation at 415 nm. The emission characteristics (not shown here) at 512 nm for the same TDM distributions differ only slightly from the curves in Figure 2b due to differing material dispersion effects at different energies. The maxima in the pure IP dipole absorption patterns (for TPA and 1PA) correspond to modes beyond the angle of total internal reflection (TIR) of the glass to air interface (|kx|/k0 = 1). They are observable only with our objective due to the index matching immersion optics. A perfectly in-plane (x−y plane) oriented transition dipole has no electric field component in the z-direction and is not expected to interact with the light field around the TIR angle. On the other hand pure OP dipoles can only interact with the light field in a small region of k-vectors (angles) around the TIR. This leads to a shallower minimum at |kx|/k0 = 1 for an isotropic distribution of transition dipoles. A further crucial characteristic of k-space spectra is the height difference between the maximum and the local maximum (at |kx|/k0 = 0, see Figure 2). It also increases with increasing OP contributions. The differences between the calculated two-photon and onephoton k-space excitation curves are related to the quadratic nature of eq 3. Furthermore, the glass substrate and effective 6323

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Figure 3. Comparison of k-resolved two- and one-photon excitation of oriented nanoplatelets and the related emission. (a) 2D k-space spectrum (false color plot) for two-photon excitation at 830 nm is obtained by plotting the measured p-polarized emission I(kout x ) (horizontal) for every excitation wave vector (kinx , vertical). The emission at kout x /k0 = 1.25 (vertical cross section, dashed line) and fit is shown on the right. The calculated 2D k-space spectrum (model) is in excellent agreement with experimental data. (b) The same for one-photon excitation at 415 nm. (c) The (heavyhole) emission at 512 nm I(kout x ) (taken from horizontal cross sections in panels a and b) is identical after two- and one-photon excitation. In-plane (IP) transition dipole contributions are given for TP (a) and 1P (b) excitation and emission (c).

electronic states as Ψ ≈ f u, where f is the envelope function which varies slowly over the entire plateletand u a periodic (Bloch) function defined within each unit cell.28 f can be calculated using effective mass Hamiltonians, but for current purposes we only need to determine the symmetry, which ultimately determines the optical selection rules. Thus, for states in band j (j = cb, hh, lh, so), we write the wave function as Ψj = f jνj, nj uj, where nj denotes the n-th state with the envelope function point symmetry νj. A rectangular nanoplatelet belongs to the D2h point group. By inspecting the nodes in fνj,nj, one can easily ascertain the symmetry (irreducible representation) corresponding to each state. For instance, in Figure 4a, we schematically plot the two lowest envelope states. The (nodeless) ground state has νj = Ag (totally symmetric), while the first excited state (one node on the x−z plane) has νj = B2u. As for the Bloch functions uj, nanoplatelets have a ZB crystal structure, and their point group (in the absence of spin−orbit interaction) is Td. The lowest conduction (highest valence) band is found to have A1(T2) symmetry, using Mulliken notation. Because the spherical harmonics of atomic orbitals {s} ({px , py , pz}) form bases of such irreducible representations, it is customary to use them to represent conduction (valence) band |uj⟩ states. Once spin−orbit interaction relevant in CdSeis considered, the corresponding group is Td*. The conduction band evolves into E1/2 and |ucb⟩ is found to preserve s-orbital symmetry. In turn, the valence band splits into hh and lh subbands (G symmetry) plus the so subband (E5/2 symmetry); see Figure 4b. It follows that |uhh⟩ has mixed px and py symmetry, while |ulh⟩ and |uso⟩ have mixed px, py, and pz character.28 Here x and y are in-plane directions, while z is

The 95% IP emission transition dipole fraction also confirms the nearly perfect parallel orientation of the platelet film, seen in Figure S1 of the SI. A considerable amount of platelets tilted with respect to the SiO2 substrate would result in a higher contribution of OP transition dipoles. In such a case the high IP fraction of 95% could not be measured. Using these resulting IP and OP contributions for excitation and emission, we can calculate 2D k-space maps as a function of the excitation and emission wave-vector. Shown as the righthand false color plots (model) in Figure 3a and b, they demonstrate the very good coincidence of our IP-OP transition dipole model with our experimental results both for linear and two-photon excitation. Summarizing, we observe highly anisotropic directed emission from mainly IP dipoles (95%) after both one- and two-photon absorption. In zinc blende CdSe quantum wells, hh exciton transition dipoles are in-plane oriented. As discussed later (see also SI),15,27 the hh Bloch function has components in the x−y plane only, and the conduction band Bloch function is isotropic. In photoluminescence experiments at room temperature, electron−hole recombination takes place at the lowest hh excitons. This transition occurs only with in-plane polarized light and accounts for the directed emission. The transition dipole distributions for one- and two-photon continuum absorption, however, differ. TPA shows a high degree of orientation, 85% IP transition dipoles, considerably higher than the isotropic (67% IP, 33% OP) distribution for linear excitation. To understand the different dipole distributions, we need to focus on the optical selection rules involved in one- and twophoton absorption processes. For convenience of the analysis, we use Bloch’s theorem to factorize the wave function of 6324

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Figure 4. (a) Schematic of a rectangular nanoplatelet (point group D2h) and its lowest-energy envelope functions fν. (b) Schematic of ZB band diagram (double group T*d ) indicating the symmetry and atomic orbital basis set for each band. Plots of periodic Bloch functions ucb and uhh (real part) are shown. (c) Examples of complete wave functions, Ψ, obtained as the product of envelope and Bloch parts.

nanoplatelet emission in Figure 3c, after both off-resonant linear or two-photon excitation. By contrast, cb−lh and cb−so band transitions have finite dipole projection along z, so that ⟨ucb|μ|ulh⟩(⟨ucb|μ|uso⟩) is finite in all space directions (see also SI, section 8, Table S2). At linear (2.98 eV) excitation in the continuum, hh−cb, lh−cb, and so−cb transitions can occur; see Figure 5a. The band related Bloch functions form a complete orthonormal basis. This leads to an isotropic (one-photon) absorption dipole distribution in the continuum,19 in-line with the measured 67% to 33% IP to OP ratio, Figure 3b. With the one-photon absorption understood, we now discuss the two-photon absorption in Figure 5b. The rate of a TPA process is given by second-order Fermi’s golden rule:

the [001] direction (platelet thickness direction). To illustrate this point we schematically plot the ucb and uhh functions in Figure 4b. We recall that the complete wave functions Ψj are obtained as the product of envelope and Bloch parts. Figure 4c shows a few examples. In the above approximation, the transition probability between states Ψa and Ψb is related to transition dipole moments of the form: ⟨a p ⃗ b⟩ = ⟨fa ⟨ua p ⃗ ub⟩ fb ⟩ = ⟨fa fb ⟩⟨ua p ⃗ ub⟩ + ⟨fa p ⃗ fb ⟩⟨ua ub⟩

(4)

The first summand is finite for interband transitions (a ≠ b) and the second one for intraband ones (a = b). We first focus on one photon absorption processes, which is a prerequisite to understand the origin of the anistropic TPA in our CdSe nanoplatelets. The probability of absorption is proportional to the transition dipole moment |μcb−j|2 = |⟨f cb νcb,ncb| f jνj,nj⟩⟨ucb|μ|uj⟩|2, with j = hh, lh, so. The envelope integral provides selection rules via the symmetry of the envelope function ν, since only transitions fulfilling δνcb,νj will be allowed. The unit cell integral (Bloch part) determines the orientation of the absorbed/emitted light and therefore its directionality; see SI section 8. The integral ⟨ucb|μ|uhh⟩ is nonzero only for (x,y)-polarized light, and emission takes place predominantly orthogonal to the NPL surface. Therefore, a hh exciton Ψ = ΨcbΨhh forms an inplane electronic dipole. Identified as the bright plane, it is observed as a 95% IP transition dipole contribution for

2

WTPA ~ ∑ i,f

∑ m

⟨f | e ⃗ ·p ⃗ |m⟩⟨m| e ⃗ ·p ⃗ |i⟩ δ(Ef − Ei − 2hν) Em − Ei − hν (5)

where |j⟩ is the wave function and Ej the energy in the initial (j = i), the intermediate (j = m), and the final (j = f) state. Here, e ⃗ is the light polarization vector. In our degenerate two-photon excitation, it is identical for both photons. p⃗ is the momentum operator. The initial states (i) of the TPA transition are in the valence band (hh-band, lh-band, and so-band), while the final states ( f) are in the conduction band, represented by the outer summation over i and f in eq 5. Intermediate states for twophoton transitions are virtual, nonresonant, states, which are short-lived and hence energetically not well-defined. The transition probability of such processes is proportional to the sum of perturbation terms involving all possible intermediate 6325

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Figure 5. One- and two-photon transitions. Schematic of the conduction band electron and valence band hole energy levels in a rectangular (D2h) NPL with ZB crystal structure. The point symmetry of a few near-band-edge envelope functions are shown on the left. The Bloch states (|uj⟩) are given color-coded at the bottom. (a) In continuum absorption (1PA), hh, lh, and so holes are equally involved, resulting in isotropic absorption. (b) Two-photon excitation from an initial state |i⟩ to a final state |f⟩ takes place via a virtual nonresonant state. The efficiency of this process is proportional to the sum of perturbational terms involving all possible intermediate states |m⟩. Transitions with Em close to (Ef − Ei)/2 and thus close to the bandgap energy have the highest probability (most relevant paths). These contributions can further be analyzed in terms of their optical orientation (IP and OP), which directly relates to the polarization of the excitation. It should be noted that the displayed paths are only examples of all of the contributing linear combinations.

states |m⟩ (summation over m in eq 5). It can thus be decomposed into a linear combination of transitions between real states. Figure 5b shows some exemplary combined pathways of these transitions. The optical selection rules for these transitions are given by the matrix elements in the numerator of eq 5. However, the interpretation of the involved processes is further complicated because, in comparison to single-photon absorption, not only interband transitions among hh, lh, so, and the cb take place. There are also pathways with one intraband and one interband transition connecting the initial and final state (Figure 5). Hence the related transition dipole moments in eq 5 and their matrix elements may involve not only the first summand of eq 4, but also the second one. The applicable polarization selection rules for the transition come either from the Bloch functions (first summand, for interband transitions) or from the envelope functions (last summand, for intraband transitions). A detailed discussion of possible interband and intraband transitions and their optical orientationwhich corresponds to the polarization of the optical excitationcan be found in the SI section 8. As discussed before, the luminescent emission takes place through recombination of the hh exciton. This interband transition is related to an orientation of the TDM inside the platelet-plane (IP dipole) which causes the strong anisotropy of the emission pattern.19 However, the decomposition of the TPA excitation comprises a multiplicity of transitions with different transition dipole orientations. Their overall orientation is thus given by the sum of allowed transitions with their TDM orientation weighted with their respective relative probabilities. It can be seen from the denominator in eq 5, that the probability for transitions is maximized for intermediate states near the laser photon energy, where Em ≈ (Ef − Ei)/2. Therefore, the transition probabilities are highest for state energy differences near the laser energy (1.5 eV at 830 nm), where the intermediate state |m⟩ is near the band edge. Because of the strong confinement of NPLs along the [001] direction, holes near the top of the valence band have almost exclusively hh character.29−31 Therefore, the highest probability transitions

involve hh−cb excitations, as shown in Figure 5b. The corresponding transition dipoles are optically in-plane (IP) oriented, leading to the observed strong anisotropy in the twophoton absorption. Hence the intrinsic orientation of the transition dipoles can be largely attributed to the Bloch functions of the bands involved in the interband transitions. TPA paths arising from lh states, which are 0.15−0.2 eV away from the valence band top, are less favored but also significant. In this case the interband transition selection rule allows OP dipoles. However, the second step of the TPA path is a cb−cb intraband transition, for which OP oriented dipoles are forbidden by envelope point symmetry (see SI), as illustrated in Figure 5b. Some TPA processes with two interband transitions, involve OP oriented dipoles are still possible, but energetically unfavored; see an example on the right side of Figure 5b. The observed TP absorption anisotropy at 830 nm reflects the high, yet not exclusive, probability of transitions with optically in-plane orientation that are involved in TPA excitation of oriented nanoplatelets. On the basis of the considerations above, we can reason that depending on the laser energy this anisotropy changes. For the minimal energy for a two-photon process, corresponding to half of the bandgap, a maximal anisotropy of the TPA process can be expected, while with increasing laser energy the distribution of possible transitions will tend to the isotropic limit observed for the 1PA process. Furthermore, we can use the occurrence of IP and OP transition dipoles obtained in our analysis to reconstruct the two-photon absorption pattern of CdSe nanoplatelets in an isotropic medium−with and without dielectric contrast to CdSe. The two-photon induced fluorescence signal ITPL can be related to the TDM distribution and the local field factors (the dielectric contrast to the surrounding). By expressing the electric field amplitude of the exciting photon field in polar coordinates, we can obtain ITPL as a function of the incident angle θin. Since ITP−PL ∝ σ(2)I2 and I2 constant in angle, ITP−PL and σ(2) have identical angular distribution functions. This 6326

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Nano Letters yields an analytical expression of the angular dependence of the TPA cross section as a function of IP and OP contributions and local field factors. The TPA cross sections of CdSe nanoplatelets of different sizes have been reported in ref 9. There the nanoplatelets were measured in a randomly orientated ensemble in solution, that is, with dielectric contrast to the surrounding (oleic acid). From the volume scaling given in ref 9, we can approximate the angularly averaged TPA cross section ⟨σ(2) OA⟩Ω of the NPL size used here to be ∼3 × 105 GM. This corresponds to the integral of the angular distribution σ(2) OA(θin)/Ω over 4π. The index OA denotes the surrounding ligands (oleic acid). With this absolute value we can calculate the TPA cross section and its angular distribution in a medium without dielectric contrast to CdSe, that is, the electronic contribution to the TP absorption pattern. To the best of our knowledge this is the first time the TPA cross section is quantified as a function of incidence and the excitation solid angle. For the sake of clarity Figure 5 shows σ(θ,ϕ) for fixed azimuth angle ϕ, as σ(θ,ϕ) is rotationally symmetric in ϕ. A detailed discussion of the angular dependence of the TPA cross section is found in section 6 of the SI. Figure 6 shows the incidence angle θin dependence of σ(2)(θin)/Ω for an in-plane oriented ensemble of our platelets in (2) an isotropic medium. σ(2) OA and σCdSe denote, respectively, the distributions of the TPA cross section of platelets with and without dielectric contrast to CdSe, that is, in oleic acid (OA) and in a medium with the same dielectric constant as CdSe. A strong change of the characteristic is observed. An oriented and controllable TPA can be of real practical interest, starting from basic issues like the achievable efficiency in practical implementations of TPA. Usually excitation is performed using the whole numerical aperture (NA) of an objective. The k-space distribution of the irradiance Ir(k) in the focus is Gaussian due to the underlying Fourier transformation properties of the objective. The width of the Gaussian depends on the objective’s NA and determines the range of accessible kvectors. The k-resolved two-photon excitation spectracalculated in Figure 2a and measured in Figure 3arepresent the twophoton induced PL for a constant excitation probed in constant steps of dk. Thus, they can be understood as response functions R(k) of the three layer system (glass/NPLs with ligands/air) for all accessible k-vectors up to NA = 1.49. The two-photon induced PL is then determined by the overlap integral of R(k) and I2r (k) over the whole NA of the objective used for excitation. Compared to an isotropic two-photon absorber, higher IP contributions lead to a higher response in the range within the angle of total internal reflection |k/k0| < 1, that is, for small angles; see also Figure 2a. Consequently for a small NA more TP transitions can be excited in a directional TP absorber compared to an isotropic TP absorber of the same total TPA cross section. In fact, for numerical apertures from 0.2 to ∼0.75 our NPL monolayer with 85% IP transition dipoles is 40% more efficient than the isotropic case. Even at NA = 1.49, this directionality yields 25% more efficient TPA. A detailed discussion is found in the SI section 7. The phase-matching condition between the fundamental and frequency doubled beam in autocorrelators based on second harmonic generation (SHG) limits their spectral bandwidth. A typically used BBO crystal, for instance, has acceptance angles of about 0.2° for an efficient conversion of 800 nm light.32 Autocorrelators based on TPA and two-photon induced

Figure 6. Two-photon absorption cross section σ(2) per solid angle Ω. (a) Excitation geometry. (b) TPA cross sections as a function of θin for oriented NPLs (4.5 monolayer 19.6 × 9.6 nm2) embedded in oleic acid (blue) and in an isotropic medium with no dielectric contrast to CdSe (green). (c) Close-up of the region indicated in panel b: The reduction of TP absorption for large angles clearly shows the further directionality induced by the anisotropic shape of the platelets and the dielectric contrast to oleic acid.

photoluminescence are not subject to phase-matching conditions.33 The conversion efficiency per interaction length of two-photon induced PL using a monolayer of oriented nanoplatelets and that of SHG34 is in the same order of magnitude. The employment of oriented nanoplatelets in TPA based autocorrelators would benefit greatly from their record high TPA cross sections, spectrally broad TPA spectra,9 and from the directed emission of the two-photon induced PL for further signal processing. Additionally, as seen in Figure 6b, they have a much greater acceptance angle, for example σ(2)/Ω decreases to half its value at an incident angle of θin ≈ 30°. An enhanced conversion efficiency is desirable for many photonic applications, like in the realization of logic AND elements. As we have not only directed two photon absorption but also highly directed luminescence emission with 95% IP dipole orientation, the platelets may be used as a directional photon converter, for example, in integrated optics as a photon redirector. Conclusion. We have shown, that in contrast to linear absorption, continuum two-photon absorption in oriented CdSe nanoplatelets is highly directed, making them a new class of directional two photon absorbers with high cross sections and additionally directed emission. We demonstrated that the described directional TPA is related to basic band anisotropies of zinc blende CdSe and the ultrastrong anisotropic confinement in nanoplatelets. It was demonstrated that this high directionality originates from dominating hh−cb interband and cb−cb intraband transitions, which have a high directionality due to the underlying Bloch and envelope function selection rules. Hence the used 2D k-space spectroscopy is shown to be a powerful tool to investigate the electronic nature of linear and nonlinear optical properties and their anisotropy in semiconductors and their nanostructures. The extracted ratio of in-plane and out-of-plane transition dipoles was combined with the measured absolute TPA cross sections. This allows for the first time to reconstruct the angular 6327

DOI: 10.1021/acs.nanolett.7b03052 Nano Lett. 2017, 17, 6321−6329

Letter

Nano Letters

(7) Wu, E.-S.; Strickler, J. H.; Harrell, W. R.; Webb, W. W. Twophoton lithography for microelectronic application. SPIE Proceedings, Vol.1674; Optical/Laser Microlithography V; 1992. (8) Cumpston, B. H.; Ananthavel, S. P.; Barlow, S.; Dyer, D. L.; Ehrlich, J. E.; Erskine, L. L.; Heikal, A. A.; Kuebler, S. M.; Lee, I.-Y. S.; McCord-Maughon, D.; et al. Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication. Nature 1999, 398, 51−54. (9) Scott, R.; Achtstein, A. W.; Prudnikau, A.; Antanovich, A.; Christodoulou, S.; Moreels, I.; Artemyev, M.; Woggon, U. Two Photon Absorption in II−VI Semiconductors: The Influence of Dimensionality and Size. Nano Lett. 2015, 15, 4985−4992. (10) Li, X.; van Embden, J.; Chon, J. W. M.; Gu, M. Enhanced twophoton absorption of CdS nanocrystal rods. Appl. Phys. Lett. 2009, 94, 103117. (11) Achtstein, A. W.; Hennig, J.; Prudnikau, A.; Artemyev, M. V.; Woggon, U. Linear and Two-Photon Absorption in Zero- and OneDimensional CdS Nanocrystals: Influence of Size and Shape. J. Phys. Chem. C 2013, 117, 25756−25760. (12) Achtstein, A. W.; Antanovich, A.; Prudnikau, A.; Scott, R.; Woggon, U.; Artemyev, M. Linear Absorption in CdSe Nanoplates: Thickness and Lateral Size Dependency of the Intrinsic Absorption. J. Phys. Chem. C 2015, 119, 20156−20161. (13) Makarov, N. S.; Campo, J.; Hales, J. M.; Perry, J. W. Rapid, broadband two-photon-excited fluorescence spectroscopy and its application to red-emitting secondary reference compounds. Opt. Mater. Express 2011, 1, 551−563. (14) Pawlicki, M.; Collins, H.; Denning, R.; Anderson, H. TwoPhoton Absorption and the Design of Two-Photon Dyes. Angew. Chem., Int. Ed. 2009, 48, 3244−3266. (15) Hutchings, D. C.; Van Stryland, E. W. Nondegenerate twophoton absorption in zinc blende semiconductors. J. Opt. Soc. Am. B 1992, 9, 2065−2074. (16) Mahan, G. D. Theory of Two-Photon Spectroscopy in Solids. Phys. Rev. 1968, 170, 825−838. (17) Rothenberg, E.; Ebenstein, Y.; Kazes, M.; Banin, U. TwoPhoton Fluorescence Microscopy of Single Semiconductor Quantum Rods: Direct Observation of Highly Polarized Nonlinear Absorption Dipole. J. Phys. Chem. B 2004, 108, 2797−2800. (18) Shimizu, A.; Ogawa, T.; Sakaki, H. Two-photon absorption spectra of quasi-low-dimensional exciton systems. Phys. Rev. B: Condens. Matter Mater. Phys. 1992, 45, 11338−11341. (19) Scott, R.; Heckmann, J.; Prudnikau, A. V.; Antanovich, A.; Mikhailov, A.; Owschimikow, N.; Artemyev, M.; Climente, J. I.; Woggon, U.; Grosse, N. B.; et al. Directed emission of CdSe nanoplatelets originating from strongly anisotropic 2D electronic structure. Nat. Nanotechnol. 2017, DOI: 10.1038/nnano.2017.177. (20) Gao, Y.; Weidman, M. C.; Tisdale, W. A. Weidman CdSe Nanoplatelet Films with Controlled Orientation of their Transition Dipole Moment. Nano Lett. 2017, 17, 3837−3843. (21) Lieb, M. A.; Zavislan, J. M.; Novotny, L. Single-molecule orientations determined by direct emission pattern imaging. J. Opt. Soc. Am. B 2004, 21, 1210−1215. (22) Schuller, J. A.; Karaveli, S.; Schiros, T.; He, K.; Yang, S.; Kymissis, I.; Shan, J.; Zia, R. Orientation of luminescent excitons in layered nanomaterials. Nat. Nanotechnol. 2013, 8, 271−276. (23) Taminiau, T. H.; Karaveli, S.; van Hulst, N. F.; Zia, R. Quantifying the magnetic nature of light emission. Nat. Commun. 2012, 3, 979. (24) Loudon, R. The Quantum Theory of Light, 3rd ed.; Oxford Univ. Press: Oxford, 2000. (25) Brown, S. J.; Schlitz, R. A.; Chabinyc, M. L.; Schuller, J. A. Morphology-dependent optical anisotropies in then-type polymer P(NDI2OD-T2). Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 94, 165105. (26) Gryczynski, I.; Gryczynski, Z.; Lakowicz, J. R. Two-Photon Excitation by the Evanescent Wave from Total Internal Reflectio. Anal. Biochem. 1997, 247, 69−76.

distribution of the TPA cross section from a semiconductor nanocrystal. It has further been shown that an oriented ensemble of directional TP absorbers is fundamentally more efficient than a random oriented ensemble of the same absorbers or that of quantum dots which additionally have much lower cross sections. A new class of directional two-photon absorbers with high cross sections and in addition directed emission, CdSe nanoplatelets may be used, for example, for two-photon imaging applications, for high efficiency logic elements in integrated photonics such as logic AND devices, or directional photon converters.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.7b03052. Details of the synthesis, k-space resolved setup and analysis, power-dependent data as well as theoretical considerations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: +49(0)30 31421079. ORCID

Mikhail Artemyev: 0000-0002-6608-0002 Alexander W. Achtstein: 0000-0001-8343-408X Author Contributions

R.S. and J.H. contributed equally to this work. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.S., U.W., and A.W.A. acknowledge DFG grants WO477-1/32 and AC290-1/1 and 2/1 and J.H. from CRC 787. J.I.C. acknowledges support from MINECO project CTQ201460178-P and UJI project P1-1B2014-24, M.A. from the CHEMREAGENTS program, and A.A. from BRFFI grant no. X16M-020.

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REFERENCES

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DOI: 10.1021/acs.nanolett.7b03052 Nano Lett. 2017, 17, 6321−6329