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Electrons, phonons and their interaction in semiconductor nanocrystal quantum dots:

Electrons, phonons and their interaction in semiconductor nanocrystal quantum dots Mikhail I. Vasilevskiy Centro de Física Universidade do Minho, Braga, Portugal Laboratoire de Physique des Milieux Denses, Institut de Chimie, Physique et Matériaux, Paul Verlaine Université, Metz, 26.10.2010

Universidade do Minho:

Universidade do Minho Braga Guimarães UM

Centre of Physics of the University of Minho (CFUM):

Centre of Physics of the University of Minho (CFUM)

Staff and facilities (January 2010):

Staff and facilities (January 2010) 59 permanent members including 47 staff members with PhD ≈ 32 full time equivalent 9 full time experienced full time researchers contracted in 2008/09 (5 year contracts in the frame of the Ciência 2007-08 National Program) 13 Post-Docs (some in cooperation with other R&D centres) ≈ 10 full time equivalent ≈ 35 PhD, MSc and other research students 33 research laboratories (24 in Braga, 9 in Guimarães) Scientific equipment worth more than 4 M€ ( various thin film growth facilities, transmission, Raman, FTIR spectroscopies, time-resolved techniques with femtosecond laser, etc .)

Major Research Fields:

Major Research Fields Condensed Matter Physics Thin Solid Films: Physics and Applications Molecular and Atomic Physics Optics and Optometry

Research Groups:

Research Groups Atomic, Molecular and Optical Physics (FAMO) Cooperative Phenomena in Dielectrics (FCD) Computational and Theoretical Physics (GFCT) Physics of the Nano-Crystalline Materials (FMNC) Functional Coatings (GRF) Complexity and Electronic Properties (GCEP ) Optics and Vision Science (OCV) MV

The story of Nanocrystal QD’s:

The story of Nanocrystal QD’s Evident Technologies:


Outline Introduction Nanocrystal quantum dots Electron, hole and exciton states Optical phonons confined in a spherical QD Confined optical phonon modes FIR absorption in QD’s Exciton-phonon (ex-ph) interaction and Raman scattering Ex-ph interaction mechanisms Resonant Raman scattering (RRS): calculated and experimental results, discussion of different QD materials Polaron effect Conclusions on phonons and ex-ph interaction Nanocrystalline Si inclusion in a-Si:H matrix: is it a QD? Introduction/Motivation Theory: localisation by random barrier Experimental testing

Introduction: quantum dots (QDs) :

Introduction: quantum dots (QDs) Semiconductor quantum dots (QDs) are systems of dimensions as small as 10 – 1000 nm that contain a small controllable number (1-100) of electrons. [ L. Jasak, P. Hawrylyak, A. Wojs, “Quantum Dots”, Springer, 1997 ] In fact, nanocrystal QDs are as small as 3-5 nm in size.

Nanocrystal QDs: How it began…:

Nanocrystal QDs: How it began… In the 80-th of the last century it was understood that the variety of colours of silica glasses doped with CdS x Se 1-x (commercially produced by Schott Glass and Corning and used as filters and sunglasses ) is determined not only by the chemical composition but also by the size of the inclusions  CdS x Se 1-x particles embedded in silica are QDs! [Al. L. Efros & A. L. Efros, Sov. Phys. Semicond. 16 , 772 (1982)] [L. E. Brus, J. Chem. Phys. 79 , 5566 (1983)] In 1932 H. P. Rocksby discovered that the red or yellow colour of some SiO 2 based glasses is related to micro-scopic inclusions of semicondutors CdSe e CdS . [J. Soc. Glass Tech. 16 , 171 (1932)] It was demonstrated the possibility to grow nanocrystals of a range of II-VI and I-VII semiconductors ( CdS, CdSe, CuBr, CuCl ) inside a SiO 2 glass matrix. [A. I. Ekimov et al, Solid State Communications 56 , 921 (1985)]

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Fusion with silica (bulk); Growth in porous matrices like zeolites; Ion implantation of the semiconductor material components into silica and subsequent annealing in order to achieve crystallisation; Magnetron sputtering and laser ablation of targets composed of semiconductor material and silica or another dielectric; Chemical synthesis in colloidal solutions using metal-organic precursors (“wet chemistry methods ”). Growth techniques: InP NC QDs revealed by high resolution TEM [A. Rogach] Advantages: approximately spherical shape; controllable size(+/- 5-10%); high crystalline quality and efficient light emission. Desadvantage: Necessary to passivate the NC surface. Nanocrystal QDs: Fabrication

Nanocrystal QDs: 3D quantum confinement:

Nanocrystal QDs: 3D quantum confinement For a NC of a typical semiconductor (like CdS, CdSe, InP, CuCl) embedded in a typical dielectric (like SiO 2 ), there are large (several eV) barriers for both electrons and holes at the NC/matrix interface. Dielectric Semi-conductor EMA theory with infinite barriers -> good description of NC QDs

Spherical QD: Electron and hole states :

Spherical QD: Electron and hole states Particle in a spherical box: R Conduction band electron Valence band hole = a spin J =3/2 particle [Al. L. Efros et al , PRB 46 , 7448 ( 1992 ) ; PRB 54 , 4843 ( 1996 )] .

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Exciton (e-h bound state) can be created by a photon 1p e 1P 3/2 2 2S 3/2 3 0 E Electron states Hole states 1s e 1S 3/2 1 h ν Exciton Strong confinement: QD Spherical QD: Exciton states Exciton ground state In the limit of large spin-orbit splitting (e.g. CdSe), the exciton ground state is an octet denoted

Optical phonons confined in a spherical QD:

Optical phonons confined in a spherical QD

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Interesting physics. Possibilities to check the applicability of bulk crystal lattice dynamics to nanocrystals containing just hundreds of atoms, to study effects of spatial quantisation on phonons. Spectroscopy of Raman scattering on phonons confined in QDs is useful for characterisation (QD size, crystalline quality). Electron-phonon interaction is extremely important for various optical processes in QDs (relaxation of electrons excited to upper allowed levels, fine structure of QD photoluminescence spectra, etc ). Are its mechanisms the same as in the bulk materials? Motivation: Why study phonons in QDs?

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Basic equations: Poisson equation for the electrostatic potential (  ) and a phenomenological equation for the relative displacement [E. Roca, C. Trallero-Giner and M. Cardona, PRB 49 , 13704 (1994)] : Confined optical phonon modes

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Basic equations: Poisson equation for the electrostatic potential (  ) and a phenomenological equation for the relative displacement [E. Roca, C. Trallero-Giner and M. Cardona, PRB 49 , 13704 (1994)] : Confined optical phonon modes Bulk

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Boundary conditions: All components of the displacement vanish, while  and the normal component of the electric displacement are continuous at the surface. Confined optical phonon modes Solution for optical phonon modes in a nanosphere: Four types of solution of the continuum model equations, determined by 3 “potentials”, χ 1 , χ 2 and ψ : The functions χ 1 and ψ obey a Helmholtz equation, while χ 2 obeys the Laplace equation.

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Confined optical phonon modes LO-TO-SO phonon modes in a nanosphere ( ν = { l p , m p , n p }) l p =0 l p =2 CdSe [A. G. Rolo and M. I. Vasilevskiy, J. Raman Spectroscopy 38 , 618 (2007)] Their frequencies are determined by a characteristic equation, The successive roots of this equation, for a given l p , are labelled by the radial quantum number, n p . The eigenmodes are degenerate with respect to m p and denoted by .

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FIR response of a single spherical QD l p = 1, n p = 1, 2,… CdSe NC R =125nm Only when the QD radius exceeds 1.5-2 nm, the QD response becomes dominated by modes with the frequencies close to  F . They constitute only a small portion of all l p = 1 phonon states. [M.I. Vasilevskiy, PRB 66 , 1 95326 ( 2002 )] Density of l p =1 states

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Effective dielectric function ( ε *) of a composite containing QD’s N – QD concentration, f=N V QD - volume fraction of QD’s FIR response of a QD ensemble Low concentration limit : modified Maxwell-Garnett model High concentration limit : Bruggemann model ε s – QD dielectric function.

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FIR absorption of a QD ensemble: experimental results (2nm) The theoretically predicted fine structure is observed for highly uniform CdSe QDs of just 1 nm in radius, while only the single (Fr öhlich-type) mode is seen for larger CdTe QDs. [M.I. Vasilevskiy etal, pss (b) 224 , 599 (2001); A.G. Rolo etal, pss (b) 229 , 433 (2002)]

Exciton-phonon interaction and Raman scattering:

Exciton-phonon interaction and Raman scattering

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Exciton-optical-phonon interaction B. Optical deformation potential (ODP) mechanism A. Fröhlich mechanism [A. Dargys, Semicond. Sci. Technol. 20 , 733 (2005)]

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One-phonon Raman scattering w ν  I  S Scattering matrix element for a phonon mode ν :

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One-phonon Raman scattering w ν  I  S Scattering matrix element for a phonon mode ν : Scattering cross-section :

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Raman selection rules: symmetry analysis Fr öhlich ODP

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Calculated RRS spectra Excitation in resonance with 1s e 1S 3/2 state Polarised scattering ( e I || e S ) [A. G. Rolo, M. I. Vasilevskiy, M. Hamma, C. Trallero-Giner, PRB 78 , 081304 (2008)]

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Calculated RRS spectra Calculated RRS spectra of CdSe QDs: ( a ) with < R >=2.2 nm (dispersion +/- 10%), e I || e S , excitation as indicated; ( b ) < R >=1.2 nm (+/- 10%), two polarisations, λ exc = 325 nm. The inset shows the relative contribution of different phonon modes. The ODP mechanism contribution is negligible. [A. G. Rolo and M. I. Vasilevskiy, J. Raman Spectroscopy 38 , 618 (2007)] CdSe

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InP Calculated and experimental RRS spectra of InP QDs of two different radii as indicated on the plots (size dispersion +/- 5%), for two polarisations; λ exc = 488 nm. The inset shows the relative contribution of different phonon modes. The ODP interaction mechanism is almost as important as the Fr öhlich one . The “anomalous” TO-type mode is naturally explained by the ODP interaction mostly with l p =3 modes with large transverse displacement component. [A. G. Rolo, M. I. Vasilevskiy, M. Hamma, C. Trallero-Giner, PRB 78 , 081304 (2008)]

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Discussion of RRS 1. The relative importance of the two ex-ph interaction mechanisms: The ODP contribution increases for smaller QD radii. Indeed, the “anomalous” TO-type Raman peak is more intense for smaller InP NCs. It is observed for InP NCs because of the higher ODP interaction constant (35 eV for InP, against 9 eV for CdSe) and lower ionicity of InP compared to the II-VI-s. Among the II-VI-s, the (usually overlooked) ODP mechanism can be relevant for small CdTe QDs (high ODP interaction constant). 2. If the ODP interaction mechanism is relevant, then what? The ODP mechanism involves phonon modes distributed over the whole reststrahlen band of the underlying bulk material. Then: (i) This opens a possibility to probe bulk TO phonon dispersion curves through the lineshape of the TO-type Raman band of small QDs. (ii) This should result in a denser exciton-polaron spectrum, which should facilitate its relaxation via interaction with acoustic phonons  faster optical response of the QD . Relative importance of the ODP/ Fröhlich interactions versus QD radius for several materials. Note that the curves for InAs and InP nearly coincide. [M. I. Vasilevskiy and C. Trallero-Giner, PSS (B) 247 , 1488 (2010)]

Polaron effect on the Raman scattering in QD’s:

Polaron effect on the Raman scattering in QD’s

What is polaron?:

What is polaron? An electron (or a hole) interacting with the gas of phonons forms a new stationary state of a mixed nature. This mixed state can be described by a quasiparticle called polaron . In bulk semiconductors, the polaron energy spectrum differs from the bare electron one only by a small downward shift and a slightly heavier mass. However, one should expect more dramatic changes in quantum dots , because of the enhanced coupling between confined electrons and phonons, the importance of the reversibility of phonon-mediated electronic transitions.

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Polaron effect in a two level system With just two localised levels, new polaron effects arise due phonon-mediated inter-level coupling (non-adiabaticity). They give rise to Rabi-type splitting. Two-level system: [ M. I. Vasilevskiy et. al ., P RB 70, 035318 (2004)] Bare states

Polaron effect on resonant Raman scattering:

Polaron effect on resonant Raman scattering W I W S Σ Initial state : a photon ( Ω I ) and a QD with a phonon state { m ν } Intermidiate state : the QD in a non-stationary polaron state Σ Final state : a photon ( Ω S ) and the QD with a phonon state { m ν ´ } Perturbation theory Polaronic picture w n 1 W I w n k W S w n 2 Σ

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Multi-phonon Raman spectra of CdSe QDs Perturbation theory Polaron approach Experimental data h Ω I = 2.572 eV <R> = 1.9 nm T = 77 K m lh / m hh = 0.17 [provided by A.I. Ekimov ] The high intensity of the second-order peak cannot be explained unless the polaron effect is taken into account. [ R. P. Miranda, M. I. Vasilevskiy, and C. Trallero-Giner, P RB 74, 115317 (2006)]

Summary on phonons and ex-ph interaction - I:

Summary on phonons and ex-ph interaction - I For nearly spherical nanocrystal QDs made of the most typical II-VI materials (CdSe, CdS, CdTe), quantised FIR-active phonon modes with angular momentum l p = 1 have been predicted and observed experimentally. As the QD size grows, the multimode structure in the FIR absorption disappears and the spectra show a single peak at the Fröhlich frequency. Parallel-polarised resonance Raman scattering spectra are due to l p =0 and l p =2 phonon modes, while only the latter contribute to the cross polarisation scattering (depolarisation ratio ~ 0.3). The dominating scattering mechanism is the Fr őhlich-type exciton-phonon interaction. The most intense modes have frequencies just slightly (few cm -1 ) below ω LO . Higher order confined phonon modes, although cannot be resolved in the experimental spectra because of the NC size distribution, are responsible for the asymmetrically broadened lineshape of the Raman band.

Summary on phonons and ex-ph interaction - II:

Summary on phonons and ex-ph interaction - II The strong exciton-phonon interaction via ODP is responsible for two intense modes observed in the Raman spectra of InP QDs, contrary to just one (close to ω LO ) as typical for II-VI NCs. The “anomalous” Raman peak appearing in the vicinity of ω TO is mostly due to l p =3 phonons with large transverse displacement component. Such a TO-type scattering is also relevant for InAs and CdTe dots. Enhanced exciton-phonon interaction leads to the formation of polaron in typical QD’s. Both intra- and inter-level couplings are important, leading to a rather complex polaron energy spectra. One of the consequences of the polaron effect is the enhancement of the higher-order Raman scattering.

NC-Si inclusion in a-Si:H matrix: Is it a quantum dot? :

NC-Si inclusion in a-Si:H matrix: Is it a quantum dot?

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Cross-sectional TEM image of an amorphous silicon film containing NC-Si inclusions, deposited on a glass substrate by magnetron sputtering. The NC size ranges from 1 to 3 nm. [M. Losurdo et al , Appl. Phys. Lett. 82 , 2993 (2003)] Introduction/Motivation Nanocrystalline (NC) Si inclusions in a-Si:H matrix [B. Rezek et al , Nanotechnology 20 , 045302 (2009)] Regular NC arrays produced by solid-state crystallization of a-Si:H Since apparently no considerable regular potential barrier exist at the NC/matrix interface, does such a NC have the properties of a quantum dot ? Do such NCs produce any (NC size dependent) discrete electron and/or hole states? Can these states be distinguished from those due to the amorphous matrix?

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CB edge: remains very close to the bulk c-Si one [G. Allan et al , Phys. Rev. B 57 , 6933 (1997)] VB edge: varies from -0.19 eV to 0.36 eV, with respect to the bulk c-Si one, depending on the H concentration (a negative value means VB edge higher in a-Si) [G. Allan et al , Phys. Rev. B 57 , 6933 (1997); M. Peressi et al , Phys. Rev. B 64 , 193303 (2001)] Experimentally measured line-ups between a-Si:H and c-Si: published values range from 0 to 0.7 eV depending on the hydrogen concentration (10-15%) [M. Tossolini et al , Phys. Rev. B 69 , 075301 (2004)] Band offsets at c-Si /a-Si interface

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Theoretical approach Envelope Function Approximation Schr ődinger-type equation for the envelope function, F , can be derived using the k ·p method: The differential operator H ij is represented by: 2 × 2 matrix in the X-point of the conduction band, 3 × 3 matrix in the Γ -point of the valence band (neglecting the spin). V ij are the matrix elements of the difference between the electron potential energy in the amorphous region and in the perfect crystal (inside the NC). They vanish inside the NC and are some random numbers outside the NC. Isotropic approximation for the H ij and V ij operators. Both operators are diagonal with respect to the sub-band indices. Then the equations for each sub-band are decoupled. No regular barriers between the NC and the matrix, i.e. < V >=0. Simplifying assumptions:

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V is a random potential produced by disorder: Consider a spherically-symmetric system of radius ( a+L ) in d- dimensional space ( d =1,2,3), described by the Schrödinger-type equation for the envelope WF: Model system and basic equations Standard ansatz for the envelope wave-function ( zero angular momentum ): NC Matrix 2 a L

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Equation defining the resonance states: Electron localization in the NC as a consequence of its reflection by the random barrier Formal solution of the Schr ö dinger equation (0< x < a ) : Limiting case : a totally reflecting random barrier:

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How much does the reflection coefficient change if the barrier becomes thicker? Parameterization : Scaling equations L L+dL

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Weak localization limit ( L<<l 0 ): Fokker-Planck approximation Localization length: Strong localization limit ( L>>l 0 ):

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Discretization of the scaling equations: L=N · Δ x, Δ x – spatial step Numerical study

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Numerical results The total and local (NC) densities of states: Dimensionless energy: The density of the sates (left) and the reflectivity R =| r | 2 , (right) for a typical realisation of a 1D random barrier (NC slab) . Strong localization limit for E < 1 . d= 1 --- g(E) --- g a (E)

Si-NC/a-Si:H system:

Si-NC/a-Si:H system Radial distribution function (RDF): a) crystalline Si; b) amorphous Si The amplitude of the dispersion D 0 , of the random potential acting on the electrons can be estimated from the relative broadening, δ R 2 = Δ R 2 / R 2 , of the radial distribution function for a-Si:H (2-nd coordination): D 0 ≈ ( a c · δ R 2 ) 2 , where a c is the CB bulk deformation potential.

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Zero-angular-momentum electron states in spherical NCs embedded in a-Si matrix Total (black) and local NC ( red ) DS calculated for conduction band electrons. The total size of the system is 40 nm. The results were averaged over ≈ 10 3 Monte-Carlo generated realisations of disorder. Resonant states which are quasi-localized in Si NC – it is a QD!

Experimental testing: NC-Si/a-Si:Er system:

Experimental testing: NC-Si/a-Si:Er system Er 3+ 1.54 μ m emission was excited resonantly , through Si NCs (632 nm, close to the NC PL peak ), and also non-resonantly, through the a-Si matrix (514.5 nm). It can be seen that for the resonance excitation, the PL intensity is some 100 times higher, which can be explained by the existence of long-lived states in the Si NCs and an efficient energy transfer from these states to the erbium ions. Using Er 3+ photoluminescence (PL) as a marker to probe the nature of the NC states:

Summary on Si NCs in a-Si:H matrix:

Summary on Si NCs in a-Si:H matrix Electron confinement by random barriers: Anderson localization of electronic states due to the strong reflection of the de Broglie wave from random barrier with a spatially-dependent correlation function has been studied. One can expect that there are long-lived states, quasi-localized in Si NCs embedded in an amorphous silicon matrix. (ii) We observed a resonant enhancement of the Er 3+ emission in the nc-Si/a-Si:Er system. This PL emission can be mediated by a Förster-type energy transfer from the Si NCs to the Er 3+ ions. This confirms the existence of the long-lived states in the NCs. It can make Si NCs in a-Si matrix behave as QDs! Message: Can it be shown by means of direct, realistic (e.g. DFT) calculations?


Collaboration Rafael P. Miranda, Anabela G. Rolo, M. Fátima Cerqueira Centro de Física, Universidade do Minho, Braga, Portugal Andrey L. Rogach Department of Physics and CeNS, Ludwig-Maximilians-Universität, Munich, Germany Sergio S. Makler Instituto de Física, Universidade Federal Fluminense, Niterói - RJ, Brazil Enrique V. Anda Departamento de Física, Pontifícia Universidade Católica, Rio de Janeiro - RJ, Brazil Carlos Trallero-Giner Department of Physics, University of Havana, Cuba Vladimir A. Burdov and Arkady M. Satanin Faculty of Physics, N. I. Lobachevskii State University, Nizhnii Novgorod, Russia Support Fundação para a Ciência e a Tecnologia (FCT), Portugal

Thank you!:

Thank you!

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Exciton-optical-phonon interaction Matrix elements (Fröhlich interaction)

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Hole-optical-phonon interaction Matrix elements Dependence of the ex-ph coupling parameter (Fröhlich interaction) for modes with n p =1, 2 on the QD radius for a CdSe dot. [ R. P. Miranda, M. I. Vasilevskiy, and C. Trallero-Giner, P RB 74, 115317 (2006)] Fröhlich ODP

Pontos quânticos: NCs produzidos por via química:

Pontos quânticos: NCs produzidos por via química Vários materiais semicondutores: II-VI, III-V, IV-VI Tamanho dos NCs 2 – 10 nm Possibilidade de fabricar estruturas do tipo “ core-shell ”: PQs com: CdSe core ZnS shell Evident Technologies: Espectros de luminescência de NCs de alguns semicondutores com tamanhos diferentes ( T =300 K). [R. J. Warburton, Contemp. Phys. 43 , 351 (2002)]

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