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 Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b03052 • Publication Date (Web): 12 Sep 2017 Downloaded from http://pubs.acs.org on September 16, 2017

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Directed Two-Photon Absorption in CdSe Nanoplatelets Revealed byk -Space Spectroscopy Jan Heckmann,

†, §

Riccardo Scott,

Nina Owschimikow,

†

Anatol V. Prudnikau,

Mikhail Artemyev,

Nicolai Grosse,

†Institute

†, §

†

‡

‡

Artsiom Antanovich,

Juan I. Climente,

and Alexander W. Achtstein

¶

Ulrike Woggon,

‡

†

∗,†

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 Química Física i Analítica, Universitat Jaume I, E-12080, Castelló de la Plana, Spain

§R.

Scott and J. Heckmann contributed equally to this work

E-mail: [email protected]

Fax: +49(0)30 31421079

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 1

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to one-photon absorption. The observed directional TPA is shown to be related to fundamental band anisotropies of zincblende CdSe and the ultra-strong anisotropic connement. We recover the internal transition dipole distribution and nd that this directionality arises from the intrinsic directionality of the underlying Bloch and envelope functions of the states involved. We note that the photo-emission 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, nd employment as ecient logic AND elements in integrated photonic devices, or directional photon converters.

Keywords 2D k -Space Spectroscopy, 2D Semiconductors, CdSe Nanoplatelets, Angle-Dependent Twophoton Absorption, Transition Dipole Distribution, Bright plane Two-photon absorber nano-crystals are rapidly gaining interest for their diverse applicability to nonlinear gain media, 1 power limiting, photo-dynamic two-photon (TP) cancer therapy, 2 and bio-labeling. 3 In addition, confocal two-photon imaging, which combines high spatial resolution and deep tissue penetration, greatly benets in-vivo cell and animal imaging. 4,5 Further applications have been demonstrated in micro-fabrication, lithography, polymerization, data storage, and spectroscopy. 68 Thus ecient 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 nano-particles like CdSe or CdS dots and rods, 1012 or typical organic dyes with TPA cross sections up to ∼ 103 − 105 GM (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 low-dimensional systems of those materials. 1518 2


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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 for both (degenerate) twophoton 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 investigation 19 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 connement lifts the degeneracy of the heavy hole ( hh) and light hole (lh) in zincblende systems, subsequently leading to the loss of isotropy. Dierences 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-o (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 connement-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 Zincblende (ZB) CdSe nanoplatelets, covered with oleic acid ligands,

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a)

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b)

c) Excitation

z

Emission

ky

Air

θout

Nanoplatelet with Ligands Substrate

Matching Fluid

High-NA Objective

Backaperture

1

-1

kx

wave-vector

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 nano-platelets. 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 reection occurs for wave-vectors larger than the wave-vector in the surrounding air, (k/k0 > 1). Variation of the excitation wave-vector in the back aperture enables 2D k -space spectroscopic measurements. c) Typical emission pattern for oriented nano-platelets. The kx - and ky -cuts can be referred to as p- and s-polarized emission, respectively. were deposited on 170 µm thick fused silica substrates by a Langmuir technique. This produces a monolayer of at-lying nanoplatelets which are oriented on the substrate (xy-plane), 19,20 see inset in Fig. 1 a. The sample can thus be modeled as a three layer system consisting of the glass substrate, an eective medium of a monolayer of nanoplatelets and their ligands, and air (Fig. 1 b). 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 Fig. 1 a), yielding an aspect p ratio AR = Lz / Lx · Ly = 0.10. For TP excitation we use a Ti:Sapphire laser (FWHM 150 fs, 75.4 MHz, 0.8 kW/cm 2 CW-equivalent 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

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objective shown in Fig. 1 b). 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 information about the samples and the experimental conguration 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, Fig 1 c. To determine the distribution of transition dipoles involved in the emission processes, we introduce a polarizer into the detection path. The cuts in Fig. 1 c) 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 eld and the wavevector coincide in the plane of incidence) in order to excite IP as well as OP dipoles as a function of the incoming wavevector kxin . 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 coecient A (proportional to the TDM |µ2 |) and the normalized density of (photon) states ρ˜. For modeling the radiation of IP and OP transition dipoles in our platelet monolayer on SiO 2 we use a normalized eective optical density of states

ρ˜IP,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 dened as ρ0 (ω) = ω 2 (εµ)3/2 /π . The radiative rate is then given by 2

|µ(ω)| with ε = εr 0 . 24 The total radiative decay rate and the Γr (ω) = A(ω)˜ ρ(ω) = ρ(ω) πω 3¯ h ε

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TDM are both decomposed into IP and OP components using the related Einstein coecients

AX (ω) = ρ0 (ω)

πω Nr,X |µX (ω)|2 3¯ hε

(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 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 ρ˜(ω) relates the IP and OP transition dipoles to the radiative rates. For p-polarization

ρ˜(ω) can also be decomposed into IP and OP components. Due to Lorentz reciprocity 25 we can treat the absorption and emission process equivalently in this formalism and relate them to ρ˜(ω) in the absorbers/emitters environment. We can model the (p-polarized) k -vector dependent one-photon absorption/emission characteristics as the superposition of IP and OP absorption/emission rates:

h i p p p S1P (ω, k ) = C ρ ˜ (ω, k )A (ω) + ρ ˜ (ω, k )A (ω) x x IP x OP (1) A IP OP

(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 (2)

(2)

TPA process with eective Einstein coecients AIP and AOP . To account for the nonlinearity 26 in modeling the TPA process, the calculated intensities from eq. 2 have to be squared:

STp P A (ω, kx )

h i2 (2) (2) p p = C(2) ρ˜IP (ω, kx )AIP (ω) + ρ˜OP (ω, kx )AOP (ω) .

(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 dened by the experimental setup or literature values (listed in the SI) except for the ratio 6


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(a)

(b) Two-photon

One-photon

excitation

excitation

1

1 Norm. absorption (a.u.)

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0

0 0.0

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1.0

1.5

0.0

0.5

1.0

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in

in

kx / k0

kx / k0

100% IP dipoles

Isotropic (67% IP 33% OP)

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) the same for one-photon excitation at 415 nm, using eq. 2. The curves are normalized to their area. of IP and OP oriented transition dipoles. Fig. 2 a 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 kxin /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. Fig. 2 b 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 dier only slightly from the curves in Fig. 2 b due to diering material dispersion eects at dierent energies. The maxima in the pure IP dipole absorption patterns (for TPA and 1PA) correspond to modes beyond the angle of total internal reection (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 eld component in the z-direction and is not expected to interact with the light eld around the TIR angle. On the other hand pure OP dipoles can only interact with the light eld in

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1.5

Two-photon excitation

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1

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1.0

Model for 95% IP After two-photon excitation

0.5

After one-photon excitation 95% IP

0.0 0.0

0.5

1.0

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out

Emission kx

/k0

Figure 3: Comparison of k -resolved two- and one-photon excitation of oriented nano-platelets 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(kxout ) (horizontal) for every excitation wave vector ( kxin , vertical). The emission at kxout /k0 = 1.25 (vertical cross section, dashed line) and t 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 (heavy-hole) emission at 512 nm I(kxout ) (taken from horizontal cross-sections in (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). 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 dierence between the maximum and the local maximum (at

|kx |/k0 = 0, see Figure 2). It also increases with increasing OP contributions. The dierences between the calculated two-photon and one-photon k -space excitation curves are related to the quadratic nature of eq.3. Furthermore, the glass substrate and eective medium of the nanoplatelet monolayer and ligands have dierent dispersion relations at 830 nm 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

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related emission patterns are shown in Fig. 3. Experimental 2D k -space spectra are shown as false color plots in Figure 3 (a) and (b) (left). They are obtained by plotting the cut in x direction of the back aperture image, see Figure 1 (c), for every excitation angle kxin . Vertical cross sections in these 2D k -space maps are shown on the right in gure 3 (a) and (b). A proxy for the excitation wavevector ( kxin ) dependent absorption, they can be modeled with eq. 3 for two-photon absorption, panel (a) and eq. 2 for one-photon absorption, panel (b). The contributions of IP and OP transition dipoles involved in TPA and 1PA processes are obtained by tting the experimental results with a least squares routine. IP contributions are given in Figure 3 (a) 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 reects 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 uncertainity 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 gure 3 (c). We observe a strong decrease of emitted intensity at kxout /k0 = 1. As discussed in gure 2 only OP dipoles can radiate here to the far-eld leading to the characteristic minimum. In fact the t 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 nding results in strongly directed emission in z-direction, perpendicular to the nano-platelet 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. The 95% IP emission transition dipole fraction also conrms the nearly perfect parallel

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orientation of the platelet lm, seen in Figure S1 of the SI. A considerable amount of platelets tilted with respect to the SiO 2 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 right hand false color plots (model) in Figure 3 (a) 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 zincblende 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, dier. 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 dierent dipole distributions, we need to focus on the optical selection rules involved in one- and two-photon absorption processes. For convenience of the analysis, we use Bloch's theorem to factorize the wave function of 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 dened within each unit cell. 28 f can be calculated using eective 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 = fνjj ,nj uj , where nj denotes the n-th state with the envelope function point symmetry νj . A rectangular nanoplatelet belongs

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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 Fig. 4(a) we schematically plot the two lowest envelope states. The (nodeless) ground state has

νj = Ag (totally symmetric), while the rst excited state (one node on the x − z plane) has νj = B2u .

(a)

(b)

(c) Td*

E1/2

D2h

=fAg ucb

Energy cb

s

k hh lh

G

(px,py)

z y

x

E5/2

(px,py,pz)

so

=fAg uhh

(px,py,pz)

fAg

fB2u

ucb

=fB2u ucb

uhh

=fB2u uhh

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 Td∗ ) 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. As for the Bloch functions uj , nanoplatelets have 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 11


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harmonics of atomic orbitals {s} ({px , py , pz }) form bases of such irreducible representations, it is customary to use them in order to represent conduction (valence) band |uj i states. Once spin-orbit interaction relevant in CdSe is considered, the corresponding group is Td∗ . The conduction band evolves into E1/2 and |ucb i is found to preserve s-orbital symmetry. In turn, the valence band splits into hh and lh subbands (G symmetry) plus so subband (E5/2 symmetry), see Fig. 4(b). It follows that |uhh i has mixed px and py symmetry, while |ulh i and |uso i have mixed px , py and pz character. 28 Here x and y are in-plane directions, while z is the [001] direction (platelet thickness direction). To illustrate this point we schematically plot the ucb and uhh functions in Fig. 4(b). We recall that the complete wave functions Ψj are obtained as the product of envelope and Bloch parts. Fig. 4(c) 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:

ha|~p|bi = hfa |hua |~p|ub i|fb i = hfa |fb i hua |~p|ub i + hfa |~p|fb i hua |ub i.

(4)

The rst summand is nite for interband transitions ( a 6= b) and the second one for intraband ones (a = b). We rst 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 = |hfνcbcb ,ncb |fνjj ,nj i hucb |µ|uj i|2 , with

j = hh, lh, so. The envelope integral provides selection rules via the symmetry of the envelope function ν , since only transitions fullling 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 hucb |µ|uhh i is non-zero only for (x, y)-polarized light and emission takes place predominantly orthogonal to the NPL surface. Therefore, a hh exciton Ψ = Ψcb Ψhh forms an in-plane electronic dipole. Identied as the bright plane, it is observed as a 95% IP transition

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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 i) 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 |ii to a nal state |f i takes place via a virtual non-resonant state. The eciency of this process is proportional to the sum of perturbational terms involving all possible intermediate states |mi. 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 the contributing linear combinations. dipole contribution for nanoplatelet emission in Figure 3 (c), after both o-resonant linear or two-photon excitation. By contrast, cb-lh and cb-so band transitions have nite dipole projection along z , so that hucb |µ|ulh i (hucb |µ|uso i) is nite 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 5 (a). 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 3 (b). With the one-photon absorption understood, we now discuss the two-photon absorption

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in Figure 5 (b). The rate of a TPA process is given by second order Fermi's golden rule:

WT P A

2 2π X X hf |~e · p~|mi hm|~e · p~|ii = δ (Ef − Ei − 2hν) h ¯ i,f m Em − Ei − hν

(5)

where |ji is the wave function and Ej the energy in the initial ( j = i), the intermediate (j = m), and the nal (j = f ) state. Here, ~e is the light polarization vector. In our degenerate two-photon excitation it is identical for both photons. p~ 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 nal states ( f ) are in the conduction band, represented by the outer summation over i and f in eq. 5. Intermediate states for two-photon transitions are virtual, non-resonant, states, which are short-lived and hence energetically not well dened. The transition probability of such processes is proportional to the sum of perturbation terms involving all possible intermediate states |mi (summation over m in eq. 5). It can thus be decomposed into a linear combination of transitions between real states. Fig. 5 (b) 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 between hh, lh, so and the cb take place. There are also pathways with one intraband and one interband transition connecting the initial and nal state (Figure 5). Hence the related transition dipole moments in Eq. 5 and their matrix elements may involve not only the rst summand of Eq. 4, but also the second one. The applicable polarization selection rules for the transition come either from the Bloch functions (rst 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.

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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 dierent 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 dierences near the laser energy (1.5 eV at 830 nm), where the intermediate state |mi is near the band edge. Because of the strong connement of NPLs along the [001] direction, holes near the top of the valence band have almost exclusively hh character. 2931 Therefore, the highest probability transitions involve hh-cb excitations, as shown in Fig. 5 (b). The corresponding transition dipoles are optically in-plane (IP) oriented leading to the observed strong anisotropy in the two-photon absorption. Hence the intrinsic orientation of the transition dipoles can be largely attributed to the Bloch functions of the bands involved in the inter-band 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 signicant. In this case the interband transition selection rule allows OP dipoles. However, the second step of the TPA path is a cb-cb intra-band transition, for which OP oriented dipoles are forbidden by envelope point symmetry (see SI), as illustrated in Fig. 5 (b). 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 Fig. 5 (b). The observed TP absorption anisotropy at 830 nm reects the high, yet not exclusive, probability of transitions with optically in-plane orientation that are involved in TPA excitation of oriented nano-platelets. Based on the considerations above, we can reason that

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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 (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. depending on the laser energy this anisotropy changes. For the minimal energy for a twophoton 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 uorescence signal IT P L can be related to the TDM distribution and the local eld factors (the dielectric contrast to the surrounding). By expressing the electric eld amplitude of the exciting photon eld in polar coordinates we can obtain IT P L as a function of the incident angle θin . Since IT P −P L ∝ σ (2) I 2 and I 2 constant in angle, IT P −P L and σ (2) have identical angular distribution functions. This yields an analytical expression of the angular dependence of the TPA cross section as a function of IP and OP contributions and local eld factors. 16


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The TPA cross sections of CdSe nanoplatelets of dierent sizes have been reported in ref. 9. There the nanoplatelets were measured in a randomly orientated ensemble in solution, i.e. with dielectric contrast to the surrounding (oleic acid). From the volume scaling given in D E (2) ref. 9 we can approximate the angularly averaged TPA cross section σOA of the NPL size Ω

used here to be ∼ 3 · 10 GM. This corresponds to the integral of the angular distribution 5

(2)

σOA (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, i.e. the electronic contribution to the TP absorption pattern. To the best of our knowledge this is the rst time the TPA cross section is quantied as a function of incidence and the excitation solid angle. For the sake of clarity gure 5 shows σ(θ, φ) for xed 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 (2)

(2)

in-plane oriented ensemble of our platelets in an isotropic medium. σOA and σCdSe denote, respectively, the distributions of the TPA cross section of platelets with and without dielectric contrast to CdSe, i.e. 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 eciency 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 k -vectors. The k -resolved two-photon excitation spectra calculated in Fig. 2 (a) and measured in Fig. 3 (a) represent the two-photon 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

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induced PL is then determined by the overlap integral of R(k) and Ir2 (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 reection |k/k0 | < 1, i.e. for small angles, see also Fig. 2 (a). 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 ecient than the isotropic case. Even at NA=1.49 this directionality yields 25% more ecient TPA. A detailed discussion is found in the SI section 7. The phasematching condition between the fundamental and frequency doubled beam in autocorrelators based on second harmonic generation (SHG) limits their spectral bandwidth. A typically used beta barium borat (BBO) crystal, for instance, has acceptance angles of about 0.2◦ for an ecient conversion of 800 nm light. 32 Autocorrelators based on TPA and two-photon induced photoluminescence are not subject to phasematching conditions. 33 The conversion eciency per interaction length of two-photon induced PL using a monolayer of oriented nanoplatelets and that of SHG 34 is in the same order of magnitude. The employment of oriented nanoplatelets in TPA based autocorrelators would benet 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 6 (b), they have a much greater acceptance angle, for example σ (2) /Ω decreases to half its value at an incident angle of θin ≈ 30◦ degrees. An enhanced conversion eciency 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, e.g. in integrated optics as a photon redirector.

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Conclusion We have shown, that in contrast to linear absorption, continuum two-photon absorption in oriented CdSe nano-platelets 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 ultra-strong anisotropic connement 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 rst time to reconstruct the angular 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 ecient 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 e.g. for two-photon imaging applications, for high eciency logic elements in integrated photonics such as logic AND devices, or directional photon converters.

Supporting Information Available Details to the synthesis, k -space resolved setup and analysis, power-dependent data as well as theoretical considerations.

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Acknowledgement R.S., U.W. and A.W.A acknowledge DFG grants WO477-1/32 and AC290-1/1 and 2/1, J.H. CRC 787. J.I.C. acknowledges support from MINECO project CTQ2014-60178-P and UJI project P1-1B2014-24, M.A. from the CHEMREAGENTS program, A.A. from BRFFI grant No. X16M-020. The authors declare no competing nancial interest.

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Directed Two-Photon Absorption in CdSe Nanoplatelets Revealed by

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