Prashant K. Jain et al- Near-Field Manipulation of Spectroscopic Selection Rules on the Nanoscale

8/3/2019 Prashant K. Jain et al- Near-Field Manipulation of Spectroscopic Selection Rules on the Nanoscale

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Supplementary Online Material for

Near-Field Manipulation of Spectroscopic Selection Rules on

the Nanoscale

Prashant K. Jain, 1,2 ,3 Debraj Ghosh,1, 2 Roi Baer, 4 Eran Rabani,3, 5 and A. Paul

Alivisatos 1, 2

1. Department of Chemistry, University of California, Berkeley, California 94720, USA2. Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California

94720, USA

3. Miller Institute for Basic Research in Science, University of California, Berkeley,California 94720, USA

4. Fritz Haber Center for Molecular Dynamics, The Chaim Weizmann Institute of Chemistry,The Hebrew University of Jerusalem, Jerusalem 91904 Israel

5. School of Chemistry, The Sackler Faculty of Sciences, Tel Aviv University, Tel Aviv 69978,Israel

Absorption Spectrum

The calculation of the absorption spectrum is based on linear response. To lowest order in

the amplitude of the incoming field,, one can show that the absorption rate is given by:

where and are the electron and hole eigenvalues, respectively, and and are the corresponding single particle wave functions, which are the solutions of the real-space pseudopotential model (see below). The above expression reduces to the well-known

dipolar absorption spectrum when :

Electronic Structure Model

The electronic structure of the nanorods was described within the real-space screened

pseudopotential method1. The local screened pseudopotentials used in the results shown

here were fitted to reproduce the experimental bulk band-gap and effective masses

neglecting spin orbit coupling. The form of the pseudopotential is given in Ref. 1 along

with the parameterization for CdSe. For CdS, we have preformed a new parameterization

similar to the one describe in Ref 1. The parameters will be given in a forthcoming

publication2. Ligand potentials were used to represent the passivation layer.


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We studied a series of nanorods with varying aspect ratios ().These were:Cd256Se(S)255 , Cd548Se(S)547 , Cd840Se(S)839 , Cd1132Se(S)1131 ,Cd1424Se(S)1423 , Cd1716Se(S)1715 , Cd2008Se(S)2007 , Cd2154Se(S)2153 , and Cd2300Se(S)2299 . Grid size ranged from for the smallestnanorod to

for the largest nanorod, depending on its aspect ratio. Thediameter of the nanorods was smaller than the experimental value to reduce thecomputational cost. For CdSe and CdS the diameter was and ,respectively. In Fig. S1 we show the structure of CdSe and CdS nanorods with an aspect

ratio .

Figure S1. Images of CdS (upper panel) and CdSe (lower panel) nanorods used in the atomistic calculations with an aspect

ratio of .To obtain the lowest transitions and calculate the absorption cross section, we have used the

filter diagonalization approach3

to filter states near the conduction and valance band edge.

Filtering about states was sufficient to obtain converged results for the first peak in theabsorption spectrum (see below). The quasi-particle gaps calculated for the above nanorods

are summarized in Table S1. Within the accuracy of the calculation, we find that above the quasi-particle gap is independent of the length of the nanorod . CdSe (eV) CdS (eV)1 2.44 3.27

2 2.30 3.13

3 2.28 3.11

4 2.27 3.10

5 2.26 3.10

6 2.26 3.10

7

2.26 3.098 2.26 3.09

CdS

CdSe


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Near-field calculations

In order to obtain the near field generated by the metal tip, we follow standard procedures

based on solving the relevant Maxwell equations:

where is the spatial electric density at frequency and is the electric displacement, related linearly to the to the electric field through thedielectric medium with constant . In typical cases the transverse component of theelectric field is negligible and so where is the electricpotential and thus:

( )

Figure S2. Real and imaginary parts of the dielectric constant

and

for

silver (black curves) and gold (red curves).

When the external field is homogeneous, one has and so the potential is , where as : ( )


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For a sphere of radius and dielectric constant embedded in an infinite medium ofdielectric constant and an external field in thez direction, this equation has asimple solution, outside the sphere:

And inside the sphere:

( )with

In Fig. S2 we plot

and

for gold and silver, assuming the surrounding medium

has a refractive index of 1.45.

Single band effective mass

While the single band effective mass model is a crude approximation to the electronic

structure of the nanorod, it provides means to analyze the results based on the atomistic

calculations in simple terms. Within the single band effective mass model, one assumes that

the electron (hole) wave functions are given by a product of and envelop function and a

Bloch function:

where is the envelop function and is the Bloch function for the valance () andconduction () bands. The matrix element determining the selection rules is given by:

Where is the electron charge, ,

the sum is over all unit cells, and the remaining integral is over a unit cell. Using the conventional

assumptions of the effective mass model and defining ||,we find:


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||

||

where we used the approximation:

| | | |

| |The integration over the unit cell can be simplified to yield:

|| ||( ) ( )||

Inserting this into the expression for we find:

|| ( )||

|| ( )||

and

||

( )||

|| ( )||

where4

The dipolar term () leads to strict selection rules where the quantum numbers of the electronand hole are preserved upon excitation resulting from the above overlap integral. This is no longer

the case in the presence of a near field, where the leading term

|| does not vanish even when the electron and holequantum numbers in the longitudinal direction differ.


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Convergence tests

Figure S3: Absorption cross section of a CdSe nanorod with an aspect ratio of for different number of filteredstates near the band edge.A typical spectrum of a CdSe nanorod with aspect ratio of is shown in Fig. S3 for anear field generated from a gold nanoparticle of diameter . The calculation showthat the first peak in the absorption spectrum is converged with approximately 300 filtered

states.

Exciton-Plasmon Couplings

It is possible that the blue-shift in the excitonic peak of the semiconductor nanorod results

in whole or part from a coupling between the transition dipole moments of the exciton of

the semiconductor nanorod and the plasmon oscillation5. We performed electrodynamic

simulations of the extinction spectra of a semiconductor nanorod with and without an Au

tip to check for this possibility. The discrete dipole approximation6

method was employed

with 1nm inter-dipole spacing and all other parameters as described above. The

semiconductor nanorod was represented by a 4nm diameter cylinder with hemispherical

end caps resulting in a total length of 60nm.

The dielectric function for the semiconductor was generated from the real-space

pseudopotential model, naturally taking into account quantum confinement effects. The

complex dielectric function is given by:


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where is the volume of the nanoparticle, is Plancks constant divided by , and arethe electrons mass and charge, respectively, and is the vacuum permittivity. In the resultsshown below, eV is the broadening parameter, is the energy differencebetween an electron in state (conduction electron) and an electron in state (valance electron),and

is the oscillator strength given by:

|| ||,where, as before, is the dipole operator, | and are the conduction and valancesingle particle wave functions, respectively, which are the solution of real-space pseudopotential

model.

For convergence purposes, we subtract the zero frequency dielectric const function from the above

equation and add the experimental value :

The results for CdSe nanocrystals of varying diameters are shown in Fig. S4. These are in good

agreement with other calculations for CdSe nanocrystals7, and seem to converge to the results

known for bulk CdSe8.

Figure S4. Real and imaginary part of the dielectric function calculated from the real-space pseudopotential model for

CdSe nanocrystals of varying diameters. The value of is subtracted from the real part.The dielectric function for a semiconductor nanocrystal with a lowest exciton transition

around 2.7 eV, close to the experimental situation, was employed in our discrete dipole

approximation simulations. In the case of the Au tipped nanorod, a 10nm sphere in direct


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contact with one of the hemispherical end-caps was included in the target structure. The

bulk dielectric function without any size correction was used for the 10nm Au nanoparticle.

A value of 1.45 was used for the refractive index of the medium. Two different polarization

directions were employed, one along the long axis of the nanorod and the other along the

short axis.

In Fig. S5 we plot the simulated extinction spectra. It is clear that the coupling between thetransition dipole moments of the exciton and the plasmon oscillation leads to a negligible

shift in both polarizations. Thus, the observed experimental shifts in the optical spectra oftipped CdS nanorods cannot result from exciton-plasmon couplings, within the model

described above.

Figure S5. Simulated extinction spectrum of a semiconductor nanorod (red) with and (black) without an Au nanoparticle

tip. The excitation polarization direction is (top) along the nanorod long-axis and (bottom) perpendicular to the nanorodlong-axis. The dotted green line is a guide to the eye showing the peak position of the first exciton transition.

Band BendingAn alternative source that may lead to the observed spectral shifts is due to band bending. In order

to equilibrate the chemical potentials of the metal and semiconductors, charges will move from the

semiconductor to the metal if its Fermi energy is above that of the metal and vice versa. The

migration of charges leads to a field across the interface that causes the bands of the semiconductor

to bend9. This bending may also lead to spectral shifts.

In Fig. S6 we plots the absorption spectrum of a CdSe nanorod for two aspect ratio and . Two limiting cases of bending are shown, positive and negative. Bending is modelled by addinga constant energy () to the ligand potentials on one size of the nanorod. This leads to anaddition field at the nanorods edge of (), used to represent the field generated at the metal-semiconductors junction. As clearly can be seen, band bending leads to a red shift in the optical

spectrum in contrast to the blue shift observed experimentally. Moreover, the magnitude of the shift

is quite small, approximately 1nm for the fields described above.

If bending was a source of spectral shifts and if it would lead to blue shifts (which it does not), then

one would expect the magnitude of the shift to saturate when the Fermi energy of the metal

nanoparticle approaches the bulk value. This typically occurs at nanoparticle size below 200 atoms.


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However, the fact that the spectral shift continue to grow with the plasmon intensity is another

indication that band bending is not a significant source of spectral shifts.

Figure S6. Absorption cross section of CdSe nanorods with an aspect ratio of

(left panel) and

(right panel).

References

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Rabani, E., J. Comp. Phys. 2002, 180, 256.2Rabani, E., unpublished.

3Wall, M. R.; Neuhauser, D., J. Chem. Phys. 1995, 102, 8011.

4Haug, H.; Koch, S.W., Quantum theory of the optical and electronic properties of semiconductors,

World Scientific, 2004.5 Shaviv, E.; Schubert, O.; Alves-Santos, M.; Goldoni, G.; Di Felice, R.; Vallee, F.; Del Fatti, N.;

Banin, U.; Sonnichsen, C., ACS Nano, 2011, 5 4712.6

Kelly, K. L.; Coronado, E.; Zhao, L. L.; Schatz, G. C., J. Phys. Chem. B, 2003, 107, 668; Yang,W. H.; Schatz, G. C.; Van Duyne, R. P. J. Chem. Phys. 1995, 103, 869; Purcell, E. M.;Pennypacker, C. R. Astrophys. J. 1973, 186, 705;7Alves-Santos, M.; Di Felice, R.; Goldoni, G., J. Phys. Chem. C, 2010, 114, 3776.

8Kim, Y. D.; Klein, M. V.; Ren, S. F.; Chang, Y. C.; Luo, H.; Samarth, N.; Furdyna, J. K.; Phys.

Rev. B 1994, 49, 7262.9Nozik, A.J.; Memming, R. J. Phys. Chem. 1996, 100, 13061.

Prashant K. Jain et al- Near-Field Manipulation of Spectroscopic Selection Rules on the Nanoscale

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