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March 21, 2014

C 2014 American Chemical Society

The “Surface Optical” Phonon in CdSeNanocrystalsChen Lin, David F. Kelley, Mikaela Rico, and Anne Myers Kelley*

Chemistry and Chemical Biology, University of California, Merced, 5200 North Lake Road, Merced, California 95343, United States

The resonance Raman spectra of bulkCdSe and many other semiconduc-tors are dominated by the longest-

wavelength longitudinal optical (LO) pho-nons. While the details of the spectradepend on resonance condition, samplepreparation, and polarization,1 in CdSe, theLO phonon mode generally appears as anarrow feature at 212�213 cm�1.1�3 TheRaman spectra of CdSe nanocrystals (NCs)show a similar feature, shifted several cm�1

to lower frequency from the bulk value4�11

and attributed to the corresponding long-wavelength optical phonon of the nano-crystal. The LO phonon also appears inother spectroscopies that involve exciton�phonon coupling including single-particleand line-narrowed emission and excitationspectra12�17 and time-domain pump�probe and photon echo experiments.18�23

Closer examination of Raman spectra ob-tained with a high signal-to-noise ratio re-veals an additional, low-frequency shoulderon the LO phonon. This shoulder is seenquite consistently in nanocrystals with dif-ferent sizes and shapes and with differentsurface chemistries6,24�36 and also in manysemiconductor materials other than CdSe.

It was assigned as a “surface optical” (SO)phonon by comparison with predictionsbased on dielectric continuum theoryfor the optical phonon modes of ioniccrystals.37,38 Although alternative explana-tions involving only the dispersion of the LOphonons have been presented,39 the SOassignment seems to have been widelyaccepted and there have been a numberof efforts to correlate the frequency and/orintensity of this mode with surface proper-ties of core�shell nanocrystals such as shellthickness and degree of alloying at thesurface.6,26,27,33,40 However, the phononmode assignments seem tenuous giventhat surface modification often affectsthe LO phonon about as much as the“SO”.6,26�28,35 The frequencies of the surfacemodes of a small spherical nanocrystal aregiven by dielectric continuum theory as37

ΩSO ¼ ΩTOε0l þ εM(l þ 1)ε¥l þ εM(l þ 1)

� �1=2

l ¼ 1, 2, 3::: (1)

where ΩTO is the frequency of the bulktransverse optical (TO) phonon, ε0 and ε¥are the static and high-frequency dielectric

* Address correspondence toamkelley@ucmerced.edu.

Received for review February 11, 2014and accepted March 21, 2014.

Published online10.1021/nn5008513

ABSTRACT The origin of the ubiquitous low-frequency shoulder on the longitudinal optical (LO) phonon

fundamental in the Raman spectra of CdSe quantum dots is examined. This feature is usually assigned as a

“surface optical” (SO) phonon, but it is only slightly affected by modifying the surface through exchanging

ligands or adding a semiconductor shell. Here we present excitation profile data showing that the low-

frequency shoulder loses intensity as the excitation is tuned to longer wavelengths, closer to resonance with

the lowest-energy 1Se�1S3/2 excitonic transition. Calculations of the resonance Raman spectra are carried out

using a fully atomistic model with an empirical force field to calculate the phonon modes and the standard

effective mass approximation envelope function model to calculate the electron and hole wave functions.

When a force field of the Tersoff type is used, the calculated spectra closely resemble the experimental ones in showing mainly the higher-frequency LO

phonon with 1Se�1S3/2 resonance but showing intensity in lower-frequency features with 1Pe�1P3/2 resonance. These calculations indicate that the main

LO phonon peak involves largely motion of the interior atoms, while the low-frequency shoulder is more equally distributed throughout the crystal but not

surface-localized. Interestingly, very different results are obtained with the widely used Coulomb plus Lennard-Jones force field developed by Rabani, which

predicts far more disordered structures and more localized phonon modes for the nanocrystals compared with the Tersoff-type potential.

KEYWORDS: surface optical phonon . electron�phonon coupling . empirical force field . cadmium selenide . nanocrystal .quantum dot

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constants of bulk CdSe, εM is the static dielectric con-stant of the surrounding medium, and l is the angularmomentumquantumnumber. With a few exceptions,33

the frequency of the SOmode is found to shift by far lessthan predicted upon replacing the organic ligands witha semiconductor shell,6,28,29 changing the mediumrefractive index from about 2 to about 9. This indicatesthat thepurely ionic, dielectric continuummodel is not agood approximation for CdSe NCs.Most Raman studies of the SO mode have been

performed using a single excitation wavelength that iswell to the blue of the lowest excitonic absorptionpeak. High-quality CdSe nanocrystals are highly fluor-escent, and obtaining a Raman spectrum becomesvery difficult as the excitation is tuned to longerwavelengths. (This is not a limitation for phononspectra obtained by Fourier transformation ofpump�probe time-domain data, and the band-edgeexcited spectra obtained in ref 18 show little or nocontribution from the low-frequency shoulder.) How-ever, many different transitions contribute to the ab-sorbance of CdSe nanocrystals at shorter excitationwavelengths,41 and the modes contributing to differ-ent parts of the optical phonon band may havedifferent degrees of resonance enhancement fromdifferent excitonic transitions. In our previous excita-tion profile study on CdSe nanocrystals,11 we focusedon the integrated intensity of the sharp peak plus itsshoulder, but we did note that the “LO phonon” bandbecame narrower at longer excitation wavelengthsapproaching the lowest-energy excitonic transition.In this work, we carefully examine the excitationwavelength dependence of the relative intensities ofthe sharp, higher-frequency peak and the broader,lower-frequency shoulder for spherical CdSe nano-crystals with different surface treatments. We thencompare these to Raman spectra calculated on reso-nance with different electron�hole states using pho-non modes generated from two different empiricalpotentials for CdSe: the widely used Coulomb plusLennard-Jones force field of Rabani,42 and a modifica-tion of the covalent Tersoff-type potential developedby Benkabou et al.43

EXPERIMENTAL RESULTS

Resonance Raman excitation profile measurementswere carried out on four different samples: CdSesynthesized through the standard method (expectedSe-rich surface) without fluorescence quenching, asimilar sample with fluorescence quenching by ligandexchange with hexadecanethiol, CdSe with a Cd-richsurface and hexadecanethiol quenching, and CdSe/ZnSe core�shell with hexadecanethiol quenching. Thenormalized absorption spectra of the four samples areshown in Figure 1.Although extensive curve fitting was carried out on

all data, the results aremost clearly conveyed by simply

examining plots of the normalized optical phononregion of each sample as a function of excitationwavelength (Figure 2). In all four samples examined,the optical phonon spectra show a moderately stronglow-frequency shoulder (SO band) when excited atshort wavelengths, then undergo a fairly sudden nar-rowing (reduction in the intensity of the shoulder) atexcitation wavelengths longer than a certain value. Forthe unquenched Se-rich sample (absorptionmaximum546 nm), the narrowing occurs between 476.5 and488.0 nm (2671 to 2177 cm�1 above the λmax). For thequenched sample (absorption max 561 nm), it occursbetween 496.5 and 514.5 nm (2316 to 1611 cm�1

above the λmax). For the Cd-rich sample (absorptionmax 568 nm), the narrowing occurs between 514.5 and532.0 nm (1830 to 1191 cm�1 above the λmax). For thecore�shell nanocrystals (absorption max 570 nm), itoccurs between 514.5 and 532 nm (1892 to 1253 cm�1

above the λmax). Although the instrumental resolutionnarrows slightly from 458 to 532 nm excitation, thisdoes not account for the sudden narrowing of theRaman spectra of different samples at different excita-tion wavelengths.These observations strongly suggest that the pho-

non mode(s) making up the low-frequency shoulderare relatively insensitive to the nature of the NC surfaceand obtain relatively greater resonance enhancementfromhigher-lying excitonic states (e.g., 1Pe�1P3/2) thando the mode(s) making up the higher-frequency LOphonon. To understand the origins of these effects, weturn to analysis of the calculated phonon modes andresonance Raman spectra.

COMPUTATIONAL RESULTS

Comparison of Force Fields. The Rabani force fieldcontains solely two-body terms and includes long-range Coulombic interactions, while the Tersoff forcefield includes three-body terms but has no long-rangeinteractions. Despite these large differences, bothforce fields do a reasonably good job of reproducingmany of the physical properties of bulk CdSe. The

Figure 1. Optical absorption spectra of the four CdSenanocrystal samples.

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Rabani potential gives lattice constants of a = 4.38 Åand c = 6.96 Å, while the Tersoff potential gives a =4.35 Å and c = 7.10 Å (experimental values, a = 4.34 Åand c= 7.02 Å).42 The bulkmodulus calculated from theRabani potential is 45 GPa, while the Tersoff potentialgives 48GPa (experimental value 53GPa).42 The Rabanipotential gives frequencies of 219 and 205 cm�1 forthe highest-frequency LO phonon at the Γ point and atk = 0.5 along the A direction, respectively, while theTersoff potential gives 213 and 211 cm�1; the experi-mental values taken from Figure 1 of ref 44 appearto be about 207 and 199 cm�1. All in all, crystallo-graphic and LO frequency results for the bulk materialat 1 atm give little reason to prefer one potential overthe other.

The two potentials do, however, make very differ-ent predictions for the TO phonon frequencies. Thehighest frequency TO phonon is calculated at153 cm�1 with the Rabani potential, giving an LO/TOfrequency ratio of 1.43. In contrast, the Tersoffpotentialgives degenerate LO and TO modes, so LO/TO = 1.00.The experimental result from inelastic neutron scat-tering44 is LO/TO = 1.19. This value is consistent withthe Lyddane�Sachs�Teller (LST) relation,45 ε0/ε¥ =ωLO2 /ωTO

2 , where ε0 (9.3) and ε¥ (6.25) are the staticand high-frequency dielectric constants, respective-ly.46 The LST relation gives LO/TO = 1.22. This analysisindicates that the Rabani potential overestimates theCoulombic contribution to the potential. The Tersoff

potential is the other extreme;it has no Coulombiccontribution to the potential.

Because Coulombic interactions are much longer-ranged, the Rabani potential gives a much largerchange in geometry between the bulk crystal andthe nanocrystal than does the Tersoff potential. Thisis manifested most obviously in the energy minimiza-tions, where the change in energy between bulk crystalatomic positions and equilibrium positions for a 3.2 nmdiameter spherical crystal is more than 100 eV with theRabani potential and only about 2 eV with the Tersoffpotential. The final, optimized structures deviate muchmore from the bulk crystal geometry with the Rabanipotential and are also considerably more disordered.This can be seen most clearly by simply looking at thepositions of the atoms along the z axis (the c directionof the crystal), shown in Figure 3 for one representativestructure. The high degree of disorder in the Rabanistructure is hard to reconcile with the observation ofclean atomic planes in high-resolution TEM images ofhigh-quality CdSe nanocrystals.47 (Note that these arezero-temperature calculations and greater disorderwould be observed at room temperature.) On the basisof these considerations, we conclude that the Tersoffpotential better describes the bonding in CdSe quan-tum dots. We note that the different amounts ofdisorder also manifest themselves in the form of thecalculated phonon modes. In a perfect bulk crystal at 0K, all of the phonon modes are completely delocalized

Figure 2. Normalized resonance Raman spectra of the four CdSe samples in the optical phonon region. The rising intensity atthe high-frequency side of the plots is from the tail of the chloroform solvent peak at 261 cm�1.

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throughout the crystal. The Tersoff-optimized nano-crystals similarly have most of their calculated phononmodes delocalized throughout the crystal; however, inthe Rabani-optimized NCs, many of the phonons aremostly localized to just a small group of a dozen or soatoms because of the disorder-induced symmetrybreaking.

Calculated Raman Spectra. Figure 4 shows Raman spec-tra calculated on resonance with the 1Se�1S3/2 andwith the 1Pe�1P3/2 (z-component) transitions for boththe Tersoff and Rabani potentials. These spectra areeach averages of 12 slightly different structures. TheTersoff potential gives a sharp LO phonon peak withtwo very weak low-frequency shoulders for 1S-reso-nant excitation, while 1P-resonant excitation showsthe same sharp LO peak plus a considerably strongerlow-frequency shoulder. The frequency separation be-tween peak and shoulder (about 11 cm�1) and therelative intensities of the two features in the 1P-reso-nant spectrum are in reasonably good agreement withthe experimental data of Figure 2. The ensemble-averaged spectra calculated from the Rabani potential,

in contrast, are much broader and show both a veryextended low-frequency shoulder and a couple ofweak high-frequency shoulders. Neither the 1S-reso-nant nor the 1P-resonant spectra are very similar tothe experimental data. For the remainder of the ana-lysis, we focus on spectra calculated with the Tersoffpotential.

Figure 5 compares the spectra calculated on reso-nance with four different excitonic states: 1Se�1S3/2,both z- and x-components of the 1Pe�1P3/2, and1Se�2S3/2. The two P-type excitations give very similarspectra that closely resemble the experimental ones,while the 1Se�2S3/2 and 1Se�1S3/2 resonant spectraare nearly identical. The P�P resonant spectra appearto consist of a sharp, strong high-frequency line and abroader, weaker low-frequency shoulder, but in fact,both features have contributions from a number ofdifferent phonon modes. Figure 6 shows calculatedspectra for both 1Se�1S3/2 and 1Pe�1P3/2 resonancefor two different individual structures and then thedecomposition of those spectra into contributionsfrom individual normal modes. When each frequency

Figure 3. Histogramof atomic positions along the z axis for one representative structure following energyminimizationwitheither the Rabani (left) or Tersoff (right) potential.

Figure 4. Resonance Raman spectra calculated for an ensemble of ∼3.2 nm diameter CdSe NCs using the Tersoff potential(left) and the Rabani potential (right), on resonance with either the 1Se�1S3/2 transition (black) or the z-component of the1Pe�1P3/2 transition (red).

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is assigned a 4 cm�1 Lorentzian line width, the under-lying modes become unresolved. Each of the 12

structures shows both the sharper peak and the low-frequency shoulder, but different structures do giveslightly different frequencies and relative intensities forthe two components, and ensemble averaging pro-duces the somewhat less resolved spectra shown inFigure 5.

In order to address the characterization of the low-frequency shoulder as a “surface”mode, the followingquantity was calculated for each phononmode of eachof the 12 structures:

Ri ¼ 1ravg

∑j

Di(j)r(j)

∑j

Di(j)(2)

where r(j) is the radial distance of atom j from theorigin, Di(j) is the absolute displacement of atom j innormal mode i, and ravg = (Σjr(j))/Natom is the averageradial distance from the origin. With this definition,Ri = 1 if the displacements are, on average, distributed

Figure 5. Resonance Raman spectra calculated for an en-semble of ∼3.2 nm diameter CdSe NCs using the Tersoffpotential and assuming resonance with the four differentexcitonic transitions indicated.

Figure 6. Top: calculated resonance Raman spectra (Tersoff potential) for two separate nanocrystal structures with both1Se�1S3/2 and1Pe�1P3/2 resonance. Bottom: individual phononmodes thatmakeup each spectrum. The height of eachbar isproportional to its calculated Raman intensity, and all modes having at least 10% the intensity of the strongest line areplotted.

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equally among atoms independent of their distancefrom the origin. A “surface mode” should have Ri > 1,while a mode localized to the interior of the NC hasRi < 1. In the continuum limit, the summation in eq 2can be replaced by an integral and a mode localized atthe surface gives Ri = 4/3. Figure 7 shows a scatter plotof the correlation between this quantity and thefrequency of the mode for 3.2 nm NCs. The phononmodesmaking up the LO phonon, near 205 cm�1, havea large density of states and are mostly localized to theinterior of the NC. The modes assigned to the SOphonon at 185�200 cm�1 have a relative radial valuethat is close to 1.0 and are therefore neither surface-nor interior-localized. Interestingly, the modes in the140�160 cm�1 region and above 210 cm�1 do involvepreferential motion of the surface atoms, but thesemodes are not calculated to have significant Franck�Condon activity nor are they observed in the experi-mental resonance Raman spectra.

Although the lower-frequency modes have a smalldensity of states, there is considerable density of statesfor the modes above 210 cm�1. Although calculatedto have very small electron�phonon couplings andtherefore not observed in the Raman spectrum, thesemodes have Ri values of about 1.2, approaching thesurface limiting value of 4/3. These are calculatedmodes that one could reasonably describe as beingsurface phonons. To further describe the atomic mo-tions associatedwith all of the phononmodes, we havecalculated the extent to which thesemotions are radialor tangential to the particle surface. Specifically, foreach mode we calculate

Ci ¼ ∑j

jv̂j, i 3 r̂jjjvBj, ij∑j

jvBj, ij

0BB@

1CCA (3)

where vBj,i is the displacement of atom j in mode i, v̂j,i isthe unit vector along the direction of the displacementof atom j in mode i, and r̂j is the unit vector along the

radius of atom j. The quantity in parentheses weightseach dot product by the relative contribution of thatatom's motion to the normal mode. If all of the dotproducts are zero (pure tangential motion), Ci = 0. If allof the dot products are 1 or �1 (pure radial motion),then Ci = 1. The scatter plot of these radial componentsis also shown in Figure 7. The LO phonons result inatomic motions that are, on average, isotropic;theradial character is very close to 0.50. In contrast,the high-frequency surface modes have a small radialcomponent, as low as 0.20. This indicates that in thesemodes the atoms move primarily parallel to thesurface.

DISCUSSION

Although there have been a number of large-scaleatomistic simulations of phonon modes in other typesof semiconductor nanocrystals,48�53 nearly all discus-sions of phonons in CdSe nanocrystals have referred todielectric continuum models such as described in refs37 and 38. Fonoberov and Balandin38 show plots of thephonon potentials of representative modes for sphe-rical wurtzite NCs that are helpful in visualizing boththe “confined” and the “interface” (surface) modesobtained from this treatment. Both types of modeshave well-defined angular momenta (l and m) asexpected for a spherically symmetric structure. Realnanocrystals, however, are composed of a finite num-ber of discrete atoms and cannot be perfectly sphe-rical, and while we have tried to construct our modelNCs to be as nearly spherical as possible, the resultingphonon modes do not have well-defined angularmomenta. We have calculated the projection of theangular dependence of the atomic displacements ontoeach spherical harmonic up to l = 4, and we find thatthe largest projection has an absolute magnitude of nomore than 0.5, and much smaller for most modes. Thatis, most of the phonon modes have an angular depen-dence that is a superposition ofmany different Yml , withno single one being dominant. Thus, there does not

Figure 7. Quantity Ri (eq 2) and density of states versusmode frequency for all of the phonon modes in the 100�230 cm�1

range for all 12 structures (left panel) and the radial character of each of the modes (eq 3, right panel).

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appear to be any clear mapping of the phonon modespredicted by dielectric continuum theory onto thoseobtained fromour atomistic simulations for NCs having∼600 atoms.Neither the experimental results (ours and those of

other workers) nor our calculations give much supportfor characterizing the Raman shoulder at 185�200 cm�1 as a surface mode. Rather, this shoulderappears to originate from a collection of optical pho-nons having nodal patterns that are preferentiallyenhanced by resonance with the 1Pe�1P3/2 and/orother transitions that occur in the wavelength regionwhere most Raman spectra are excited, considerablyabove the first 1Se�1S3/2 excitonic absorption peak.The fact that all of our samples show a strong reductionof intensity in the low-frequency shoulder as theexcitation wavelength is tuned to longer wavelengths,independent of surface chemistry or the presence of asemiconductor shell, strongly suggests that theshoulder is not specifically associated with the sur-face but rather reflects the spatial distribution of theelectron�hole wave functions with which the Ramanexcitation is resonant.In bulk CdSe, the highest-frequency phonon is the

LO phonon at the Γ point. The LO phonons with largerwavevectors, that is, more nodes in the phonon wavefunction, have slightly lower frequencies. The 1S elec-tron and hole wave functions also have no nodes andare largely confined to the interior of the nanocrystal,and we calculate that 1Se�1S3/2 resonant excitationenhances primarily the NC phonons whose frequen-cies are similar to that of the bulk k = 0 mode and, asindicated in Figure 7, involve mainly motions of theinterior atoms. The 1P electron and hole functions havean angular node and have more charge densitynear the surface of the NC, and excitation on resonancewith the 1Pe�1P3/2 exciton enhances both the high-frequency, bulk-like phonons (the “LO” phonon) and acollection of slightly lower-frequency modes that areless localized to the NC interior and presumably havemore nodes in their wave functions (the SO phonon).When valence-band mixing is included, both the 1S3/2and the 2S3/2 hole functions have no nodes (whensummed over all four M components), and althoughthey have rather different radial distributions, bothgenerate nearly the same resonance Raman spectrum(Figure 5). In contrast, the nS hole functions calculatedwithout S�Dmixing have (n� 1) radial nodes, and thecalculated 1Se�2S3/2 and 1Se�3S3/2 resonant Ramanspectra using these pure nS functions are dominatedby the low-frequency SO band. These observations canbe generalized to conclude that the more nodes theelectron and/or hole wave functions contain, the morelower-frequency optical phonons will be resonantlyenhanced. We note that valence-band mixing hasa large effect on the 2S hole function but only asmall effect on the 1S; the 1Se�1S3/2 resonant Raman

spectrum calculated using pure 1S envelope func-tions differs only slightly from that calculated fromthe S�Dmixed functions. Similarly, we find that addingsmall amounts of 1F radial function to the 1P holefunction changes the wave function only slightly andslightly increases the intensity in the low-frequencyshoulder.A full understanding of why the modes in the

185�200 cm�1 range gain intensity with shorter-wavelength excitation is hampered by the complexityof the absorption spectrum in this region. Norris andBawendi41 identified at least five transitions within thefirst∼4000 cm�1 above the 1Se�1S3/2 absorption, ten-tatively assigned as 1Se�2S3/2, 1Se�1S1/2, 1Pe�1P3/2,1Se�2S1/2, and 1Pe�1P5/2/1Pe�1P1/2, along with anunassigned underlying continuum absorption. Ourprevious excitation profile study11 required inclusionof four transitions above the 1Se�1S3/2, not countingthe fine-structure splitting. While the transitions thatinvolve no change in principal quantum number(1S�1S and 1P�1P) are expected to carry most ofthe absorption oscillator strength, the contribution ofeach state to the resonance Raman spectrum alsodepends on other factors, chiefly the Huang�Rhysfactors and the electronic dephasing rates. However,independent of the exact composition of the absorp-tion spectrum, higher-energy excitations will generallyinvolve electron and/or hole wave functions that havemore nodes, and thesemore structured excitonic wavefunctions couplemore strongly to the lower-frequencycomponents of the optical phonon band.Figure 7 indicates that the modes in the 185�

200 cm�1 range are not particularly surface-localized.This figure, however, could be misleading in that itshows all of the calculatedmodes, many of which havelittle or no Raman intensity. Another way to gauge thecontribution of motions of surface atoms to the Ramanspectrum is shown in Figure 8. Here we compare the1Pz�1Pz resonant spectrum calculated in the usualway (equivalent to the red curves in Figure 4, left,and Figure 5) with the same spectrum calculated underthe artificial assumption that the electron and holewave functions are confined to a radius that is 75% ofthe true radius. In this way, motions of the surfaceatoms contribute negligibly to the Raman intensities.Removing the surface atom contributions reduces theintensity of the low-frequency shoulder with P-reso-nant excitation but does not eliminate it. In contrast,the higher-frequency, sharper feature is hardly af-fected by removing surface atom contributions.The calculated resonance Raman spectra are not

expected to agree quantitatively with experimentbecause of a variety of approximations that have beenemployed. As mentioned above, there is an inconsis-tency in considering the Raman enhancement to arisepurely from the changes in the interatomic Coulombicforces upon electron�hole pair creation when the

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ground-state force field does not include explicit Cou-lombic interactions. The assumption that the reso-nance Raman intensities are proportional to Δ2ω2 isvalid only for certain ranges of resonance conditionsand electronic dephasing times. In real systems, reso-nance with a single excitonic transition as implied inFigure 5 can rarely be achieved. Multiple excitonictransitions will usually contribute to the enhancement,withweights that depend on a variety of photophysicalparameters and experimental conditions, and the con-tributions from the different resonant states must beadded at the amplitude level before squaring to getthe observed Raman intensity, resulting in interferenceeffects among excitonic transitions.11 Along similarlines, the contributions from each of the angularmomentum fine-structure components of each exci-ton should also be included separately in calculatingthe resonance Raman amplitude. We have chosen tokeep this analysis as simple as possible in order toavoid having to choose values for a large number ofpoorly determined additional parameters. We believethat the qualitative conclusions reached through thissimple analysis are valid, but quantitative agreementwith experiment would require much more detailedcalculations on specific systems similar to those pre-sented in ref 11.The above data and analyses give a self-consistent

picture to explain the optical phonon Raman lineshapes in CdSe NCs based on an effective mass envel-ope function model for the electron and hole wavefunctions and an empirical interatomic potential toobtain the structures and phonon modes. An impor-tant caveat is that the success of this approach de-pends critically on the type of interatomic potentialemployed. The widely used Coulomb plus Lennard-Jones potential of Rabani does not predict Raman lineshapes that closely resemble the experimental ones forany choice of excitonic resonance, as shown in Figure 4.

The large amount of disorder present in NC structuresminimized with GULP and the Rabani potential (e.g.,Figure 3) appears inconsistent with the observation ofwell-defined atomic planes in HRTEM images as well assharp X-ray diffraction peaks from high-quality nano-crystals. The Tersoff potential contains no explicitcharges on the atoms, and while the parameters ofthat potential can mimic the Coulombic contributionto the short-range interactions, there should also belong-range Coulombic forces that are not captured bythis potential. It should also be noted that our calcula-tions are carried out on bare NCs without ligands orsolvent. Inclusion of a dielectric medium surroundingthe NC will reduce the long-range Coulombic interac-tions and might produce better structures with theRabani potential.

CONCLUSIONS

The primary conclusion of this study is that the sharp“LO phonon” at 205�208 cm�1 and the shoulder at185�200 cm�1 both represent a superposition ofcontributions from a number of different phononmodes of the CdSe nanocrystal. The modes contribut-ing to the sharp peak mostly involve motions of theatoms in the interior of the NC, while the modescontributing to the low-frequency shoulder involveatomic motions more equally distributed throughoutthe NC but are not well-described as surface phonons.Resonance with the lowest-energy 1Se�1S3/2 exci-tonic transition preferentially enhances the higher-frequency interior-localizedmodes,which have the great-est spatial overlap with the nodeless, interior-localized1S electron and hole wave functions. Resonancewith higher-energy excitonic transitions such as the1Pe�1P3/2 provides more enhancement of the lower-frequency components. Apparent correlations be-tween alloying at the core�shell interface and thefrequency or intensity of the low-frequency shoulder

Figure 8. Raman line shapes (Tersoff potential) for 1S�1S and 1Pz�1Pz resonant excitation calculated in the usual way (blackcurves) and calculated assuming that the electron and hole wave functions are confined to 75% of the full NC radius (redcurves).

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should be interpreted with care because they mayreflect mostly changes in the extent of resonance with

different excitonic transitions as the absorption spectrashift relative to a fixed Raman excitation wavelength.

EXPERIMENTAL METHODSSynthesis of Semiconductor Nanocrystals. The procedures for

sample preparation were similar to those described in refs 54and 55. In a typical CdSe (Se-rich) synthesis, cadmium oxide(CdO, 26 mg), oleic acid (OA, 0.4 mL), and octadecene (ODE,3 mL) were added into a 25 mL three-neck flask, purged withnitrogen, and heated to ∼250 �C until the solution was clear.This Cd precursor solution was then cooled to room tempera-ture, and 1 g of octadecylamine (ODA) and 0.4 g trioctylpho-sphine oxide (TOPO) were added. It was again purged withnitrogen and reheated to 280 �C for the injection of the Seprecursor (78 mg of Se, 1 mL of ODE, and 0.6 mL of trioctyl-phosphine (TOP)). The temperature after injection dropped toabout 252 �C, and the reaction continued for about 1min beforebeing stopped. The resulting CdSe nanocrystal solution wasthen extracted by hexane/methanol (1:1 v/v) two times, and theorganic layer was separated and heated to 75 �C to remove theresidual hexane and methanol for Raman measurement orfurther treatment. Growth at an initial nucleation temperatureof 280 �C is known to yield the wurtzite crystal structureexclusively; the zinc blende structure requires nucleation at amuch lower temperature.56�59

For the growth of the ZnSe shell, 1 mL of Zn solution (0.2 M,with 162.74 mg of ZnO, 2.52 mL of OA, 7.5 mL of ODE) and0.5 mL of Se precursor solution (0.2 M) were added dropwiseinto the purified CdSe core solution (∼10�7 mol CdSe in∼3 mLof ODE) at 235 �C, and the total shell growth time was about7min. For the synthesis of Cd-rich CdSe, instead of using Zn andSe precursors as for ZnSe shell growth, ∼0.95 mL of Cdprecursor (0.1 M) was added dropwise to the CdSe solution at185 �C. The total reaction time was ∼9 min. Fluorescence ofsamples was quenched by mixing 0.2 mL hexadecanethiol and2 mL of toluene with 1 mL of purified QD solution. The mixturewas stirred at 82�100 �C for ∼2 h under nitrogen until theligand exchange was complete.

Raman Spectroscopy. The Raman measurements, includingdepolarization ratios, were carried out as described in ref 11.All samples were dissolved in chloroform at optical densitiesranging from∼0.38 to ∼0.78/mm. These are higher concentra-tions than used in ref 11 because we wished to minimizeinterference from solvent peaks and were not concerned withobtaining accurate absolute scattering cross sections. The laserpower at the sample was around 0.5 mW. All measurementswere performed on samples that were continuously translatedto prevent local heating or accumulation of photoproducts. Signalwas integratedon thedetector for 60�120 sbeforebeing readout,and 5�30 such integrations (10�30 min total) were summed toobtain the spectrumof each sample. At the fixedphysical slit widthemployed, the spectral resolution was about 8�11 cm�1 over therange of excitation wavelengths from 532 to 458 nm.

Computational Methods. Two different empirical interatomicpotentials were employed. Nearly all prior molecular dynamicscalculations on bulk or nanosized CdSe have employed theRabani two-body potential,42 which consists of a Lennard-Jonesterm plus a Coulomb term:

VRabani ¼ ∑i∑j>i

qiqjrij

þ 4εijσij

rij

!12

� σij

rij

!6( )(4)

The sums in eq 4 run over all pairs of atoms. The parameters ofthe potential were taken directly from ref 42 and are given inTable 1. This potential assumes that the bonding is mostly ionic;the Coulombic interactions are dominant, and there is nodependence on bond angles.

Although the Rabani potential is widely used, our initialresults indicated an apparently unphysical amount of structuralreorganization in small nanocrystals compared with the bulk.Therefore, we also tried a potential of the Tersoff form, which is

generally used for crystals in which the bonding is largelycovalent. This potential contains no explicit Coulombic interac-tions but has both two-body and three-body terms:

VTersoff ¼ ∑i∑j>i

fc(rij)[Aexp( �λ1rij) � Bbijexp( �λ2rij)] (5)

where

fc(rij) ¼1, r < R � D

12� 12sin

π

2(r � R)

D

� �, R � D < r < RþD

0, r > RþD

8>>><>>>:

(6)

and

bij ¼ (1þ βnξnij )�1=(2n) (7)

ξij ¼ ∑k 6¼i, j

fc(rij)g(θijk ) (8)

g(θ) ¼ 1þ c2

d2� c2

d2 þ (h � cosθ)2(9)

The parameters of this potential were slightly adjusted fromthose originally given by Benkabou43 to better reproduce thelattice constants, phonon frequencies, and bulkmodulus of bulkCdSe. The final parameters employed are given in Table 2.

Model nanocrystals were constructed as follows. The atomiccoordinates of a bulk lattice were first generated using thelattice constants that result from a bulk calculation using thedesired potential (Rabani or Tersoff). An origin for the NC wasthen selected, either at the center of a unit cell or not. A radiuswas chosen, and all atoms outside that radius were eliminated.Any atoms having fewer than two nearest neighbors (defined asatoms of the opposite type within a distance of 2.9 Å) weresequentially eliminated from the structure. This resulted innearly spherical crystals that were in some cases stoichiometric

TABLE 1. Parameters of the Rabani Potential (eq 4)a

parameter q σ/Å ε/meV

Cd 1.18 1.98 1.4477Se �1.18 5.24 1.2840

a Combining rules: εij = (εiεj)1/2 and σij = (σi þ σj)/2.

TABLE 2. Parameters of the Tersoff Potential (eqs 5�9)

parameter value

A/eV 5214B/eV 239.5λ1/Å

�1 3.1299λ2/Å

�1 1.7322β 1.5724 � 10�6

n 0.78734c 100390d 16.217h �0.57058R/Å 3.175D/Å 0.15

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and in other cases had a small excess of Cd or Se (nomore than 6“extra” atoms out of ∼600 total) depending on the choice oforigin and radius. These structures were then used as input for acalculation of the phonon eigenmodes and frequencies viathe general utility lattice program (GULP)60 following energyminimization. In principle, this should produce all real phononfrequencies, but in practice, there were often a few small(few cm�1) imaginary frequencies because of imperfect energyminimization. For each type of calculation, 12 slightly differentstructures were generated by varying the origin and radius bysmall amounts (e10% and less than a bond length, respectively),and the spectra of the 12 were averaged to give the reportedspectra. A single potential function was assumed for all atomsof a given type, whether surface or interior. All calculations werecarried out at zero temperature.

In order to calculate the Raman spectrum for each structure,electron and hole wave functions were calculated using theparticle-in-a-sphere envelope function approach as describedpreviously.61 The potential for both electrons and holes wasassumed to step up by 4 eV at the edge of the NC, defined as thedistance from the origin of the most distant atom. The electronand hole effective masses were taken to be 0.11 and 0.42 me,respectively. Valence-bandmixing, whichmixes a small amountof the spatial wave function with angular momentum l þ 2 intothe envelope function with nominal angular momentum l , wastaken into account for the 1S3/2 and 2S3/2 hole states asdescribed in ref 61. The 1P states were treated as pure l = 1envelope functions because of the greater complexity involvedin including the valence-band mixing62 (see Discussion). Oncethe wave functions were calculated, the changes in chargedistribution produced by electron�hole pair formation werecalculated by assigning each Cd (Se) atom a charge equal to thesquare of the electron (hole) wave function at that point inspace, normalizing the total charge to unity. The Huang�Rhysfactor for each phonon mode i is given by Si = Δi

2/2, where Δi,the displacement between ground and excited electronicstate minima along dimensionless coordinate i in the linearelectron�phonon coupling limit, is given by dEeg/dqi = pωiΔi,where Eeg is the energy difference between excited and groundstates. Eeg was calculated as the difference between the Cou-lombic energy when the atoms have their ground-state charges(taken as the Rabani charges of (1.18) and that when theatomic charges have been modified by electron�hole pairformation.61 This purely Coulombic mechanism of electron�phonon coupling is perhaps inconsistent with the purely cova-lent Tersoff potential (the band gap transition is nominally a Se4P�Cd 5S transition). However, for the ground-state structureand phonon frequencies, the relative coupling strengths ofdifferent phononmodes do not depend on the values assumedfor the ground-state charges.

The resonance Raman spectra were calculated by assumingthat the Raman intensity in mode i is proportional to Δi

2ωi2, the

slope of the excited-state potential energy surface along theground-state dimensionless phonon coordinate, and assigninga Lorentzian line width of 4 cm�1 (fwhm) to each phonontransition. I ∼ Δ2ω2 is usually a good approximation for low-frequency vibrations and unstructured electronic spectra. Onlythe Raman fundamentals were calculated.

In order to determine the influence of surface atoms on thephonon modes at different frequencies, the Raman intensitieswere calculated using two different assumptions about theextent of the electron and hole wave functions: either theireffective radius is equal to the actual radius of the nanostruc-ture, or their effective radius is 75% of this value. In the lattercase, an artificial assumption that is not intended to representphysical reality, the wave functions do not extend to the surfaceof the nanocrystal and motions of surface atoms do not con-tribute to the Huang�Rhys factors, so any surface-localizedmodes will make a very small contribution to the calculatedRaman spectrum.

Conflict of Interest: The authors declare no competingfinancial interest.

Acknowledgment. This work was supported by NSF GrantNo. CHE-1112192.

Supporting Information Available: Resonance Raman spectraextended into the optical phonon overtone region for thesamples shown in Figure 2. This material is available free ofcharge via the Internet at http://pubs.acs.org.

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