Including surface ligand effects in continuum elastic models of nanocrystal vibrations

The photophysical properties of semiconductor nanocrystals, also known as quantum dots (QDs), are affected by the underlying vibrations of the atomic lattice.^1–3 Unlike their bulk counterparts, semiconductor nanocrystals include low frequency, confined vibrational modes associated with particle distortions, such as the symmetric or ellipsoidal stretching. The characteristics of these collective vibrational modes are known to depend on QD size, shape, composition, and heterostructure. These dependences can be understood in terms of elastic continuum models.^4–14 Recently, however, it has been observed that these low frequency vibrational modes also depend on the identity of the ligands that passivate the surface of the QDs,¹⁵ presenting a challenge to existing theories. In this article, we show that the effect of ligand identity on the low frequency vibrations of QDs can be understood in the context of traditional continuum elastic theories by designing the boundary conditions to mimic viscoelastic properties of the ligand shell. This theory, as we show, is able to predict the result of a broad range of experiments, where existing approaches fail.

The prevailing methods for computing the vibrational properties of QDs are based on a theoretical model in which the nanocrystal is described as a linear elastic field whose properties are assumed to be identical to that of a bulk crystal. The size and shape of this field determine the spectrum of quantized low frequency collective vibrations or acoustic phonon modes. Lamb¹⁶ first formalized this model by applying the Cauchy-Navier equation,¹⁷ 

to a perfectly spherical elastic particle, where u is the displacement field, and $vl$ and $vt$ are longitudinal and transversal sound velocities, respectively. The solution to this equation, expressed in terms of eigenmodes, is $u(r,t)=∑iui(r)exp(−iωit)$⁠, where $ωi$ describes the corresponding frequency of the i-th eigenmode. Lamb’s model predicts that frequencies of the lowest energy vibrational modes, e.g. the radial breathing mode and ellipsoidal mode, scale inversely with QD radius, R.¹⁶ This scaling relationship has enabled the detailed assignment of Raman spectra for a wide range of different nanoparticles.^{4,6–8,12,13,18}

The proportionality constant associated with this 1/R scaling relationship depends on the choice of boundary conditions. Limiting cases include free boundary conditions (as in Lamb’s original model), where the stress, $σ$⁠, is equal to zero at the surface, and rigid boundary conditions, where the displacement, u, is equal to zero at the surface. In some cases, such as modeling QDs embedded in a glass matrix, the boundary conditions are treated as a dissipative bath.⁸ In the absence of a well controlled series of experiments, however, the appropriate choice of the boundary condition has remained ambiguous.

Experiments have shown that in some cases, the scaling of low frequency vibrational modes deviates from the 1/R prediction of elastic continuum models.^11,15,19,20 These deviations were found to be more pronounced for smaller quantum dots (⁠$R≲2$ nm), which has led to speculations about the origin of these deviations. Possible interpretations include ligands affecting the boundary condition of the QD surface¹¹ and structural relaxation occurring at the surface that may lead to size-dependent elastic moduli.¹⁹ Recent ligand substitution experiments on both spherical QDs and 2D nanoplatelets suggest, however, that mass loading of the nanocrystal surface is the dominant cause of these observed deviations.^15,21

Here, we show that traditional elastic continuum models can be modified to describe these reported attenuation of the acoustic vibrational energies through ligand substitutions. By doing so, we successfully describe small-R deviations in the scaling of low frequency vibrational modes, along with their dependence on changes in ligand identity. Furthermore, we show that this model is capable of accurately predicting the vibrational resonances of more complex spherical heterostructures, including core/shell QDs.

In Sec. II, we describe a general framework for including the boundary effect of ligands in elastic continuum models. We then compare the results of this model to a series of low-frequency Raman scattering experiments carried out on QDs with varying core size and ligand identity.

The Raman active modes of a homogenous elastic sphere are types of pure compressional modes, as given by

where

and $Ylm(𝜃,ϕ)$ is the spherical harmonic function, A and B are constants, and j_l and y_l are spherical Bessel functions of the first and the second kinds, respectively. Among these compressional modes, only the radial breathing (l = 0) and ellipsoidal (l = 2) modes are Raman active²² and are illustrated in Fig. 1(a). In particular, the lowest frequency radial breathing mode (l = 0 and m = 0) has been the focus of many experimental investigations,^9,11,19,23 including our recent study of CdSe QDs.^15,20 In this manuscript, we focus specifically on modeling this radial breathing mode. Thus, for notational simplicity, we will denote the mode and its frequency without subscripts as u(r, t) and $ω$⁠, respectively.

For the eigenmode corresponding to the lowest energy breathing mode, only the radial component of the displacement field is non-zero, which we denote as

where $q≡ω∕vl$⁠. The strain tensor, $𝜀=12[∇u+(∇u)T]$⁠, associated with Eq. (4) has only three non-zero components, $𝜀rr=∂ur∕∂r$ and $𝜀𝜃𝜃=𝜀ϕϕ=ur∕r$⁠. Based on the isotropic linear elastic stress-strain relation, the normal stress on the spherical surface is given by

where $σ$ is the stress tensor and $Q≡ω∕vt$⁠. Here constants A and B are determined in order to satisfy the boundary conditions.

We model the low frequency acoustic properties of an isolated inorganic core with a core-only model. In this model, the stress and displacement fields of a semiconductor nanocrystal are characterized simply by its radius, R, sound velocities, $vl$ and $vt$⁠, and density, $ρ$⁠. These properties, along with a description of the boundary conditions, combine to determine properties of the radial breathing mode of the nanocrystal. Two limiting cases are the free boundary, defined by the vanishing normal stress at the surface, $σrr(R,t)=0$⁠,¹⁶ and the rigid boundary, defined by the vanishing displacement at the surface, u_r(R, t) = 0.⁴ In either case, this model predicts a Raman frequency shift that is given by $ν=ω2πc=αvl2πc1R$⁠, where c is the speed of sound and $α≡qR$ is a dimensionless frequency that depends on the details of the boundary condition. Specifically, the value of $α$ is given by the characteristic root of the secular equation $f(α)=0$⁠, where

for the case of the rigid boundary condition and

for the case of the free boundary condition.

The core-ligand model extends the previous model to include the effects of surface ligands on the vibrations of the inorganic core. Particularly, this model describes the vibrations of the QD core against a compressible ligand layer of finite thickness with a fixed outer boundary. We model the core (C) as a sphere of radius R_C that is nested within the center of spherical ligand shell (L) with radius R_L such that the thickness of the ligand shell is given by d_L = R_L − R_C. As illustrated in Fig. 1(c), the acoustic properties of the core (as specified by $ρ(C)$⁠, $vl(C)$⁠, and $vt(C)$⁠) are defined separately from that of the ligand layer (as specified by $ρ(L)$⁠, $vl(L)$⁠, and $vt(L)$⁠). We set a matching boundary condition at the core-ligand interface (i.e., at r = R_C) and a rigid boundary condition at the outer ligand boundary (i.e., at r = R_L). Formally, these boundary conditions are represented as

Additionally, we impose a condition B = 0 on $ur(C)$ and $σrr(C)$ to prevent their divergence at the center of the core. By modeling the ligand layer as a solid elastic material with prescribed equilibrium shell thickness, d_L, and density, $ρ(L)$⁠, we describe the ligand layer acting as a time varying mass load that compresses or stretches the QD core over time. The elastic ligand layer attenuates the resonant vibrational frequency of the QD core in a manner that obeys the above boundary conditions.

We further extend the previous model to describe QDs with core-shell structure and surface ligands. As illustrated in Fig. 1(d), this core-shell-ligand model includes a spherical inorganic core (C) of radius R_C, embedded within a spherical inorganic shell of radius R_S. The inorganic core-shell structure is then embedded within a spherical ligand shell (L) of radius R_L. The acoustic properties of each layer are defined separately as $ρ(j)$⁠, $vl(j)$⁠, and $vt(j)$⁠, where the superscript j = C, S, or L, for the core, shell, or ligand materials, respectively. We assume all the layers in the onion-like structure vibrate as a single composite structure. The boundary conditions for this system are

As in the core-ligand model, we constrain B = 0 for $ur(C)$ and $σrr(C)$⁠.

For this class of models (i.e., the core-ligand and the core-shell-ligand models), the set of boundary conditions gives rise to a system of linear equations that can be solved to compute the frequency $ω$⁠. In  Appendices A and  B, we describe the details of these systems of equations for the core-ligand and the core-shell-ligand models, respectively.

We validate our model by comparing model predictions to Raman spectra taken on QD thin films. Specifically, we have synthesized thin films of CdSe QDs capped with octadecylphosphonic acid (ODPA) native ligands through a standard hot-injection method.²⁴ We have collected the Stokes and anti-Stokes Raman spectra between 10 and 100 cm⁻¹ (about 0.3–3 THz) using a 785 nm narrow linewidth laser as the nonresonant Raman pump. By varying the size of the QDs in each film, we are able to measure how acoustic vibrations depend on QD size. The Raman spectra reveal a monotonic redshift in both the l = 0 and l = 2 phonon features with increasing nanoparticle size (see Fig. S9a of the supplementary material), as has been observed previously.^5,9,15,20 As Fig. 2 illustrates, the scaling of vibrational energies with QD size is consistent with a core-only model for relatively large cores (⁠$R≳2$ nm).

The core-only model predicts a linear relationship between $ν$ and 1/R, with a slope that depends on the details of the boundary conditions. For comparison to experiment, we treat the CdSe core as a uniform material with the elastic parameters of bulk wurtzite CdSe: $ρ(C)=5.81$ g/cm³, $vl(C)=3559$ m/s, and $vt(C)=1607$ m/s.^25–27 As Fig. 2 illustrates, we observe that for either choice of boundary conditions the core-only model tends to overpredict the value of the phonon energy for a given QD size. Notably, the model with the free boundary condition is in much better agreement than that of the rigid boundary condition, suggesting that shape distortions are fundamental to the low frequency vibrations of QDs.

To model the effects of ligands on the vibrational energies of QDs, we apply the core-ligand model to the same system, retaining the same bulk CdSe wurtzite parameters to describe the QD core. We assume that the ligands comprise a shell of constant thickness, d_L = R_L − R_C, and uniform density, $ρ(L)$⁠, regardless of QD size. We derive the ligand layer thickness to be d_L = 1.3 nm based on X-ray scattering measurements of the interparticle spacing of similar sized QDs capped with oleic acid (OA),²⁸ which has similar hydrocarbon length to ODPA. Specifically, we find that the size-dependent l = 0 phonon energies of ODPA capped QDs synthesized by the standard hot-injection method and those measured for OA capped QDs synthesized by the seeded growth method have comparable energies (see Fig. S7 of the supplementary material). Likewise, we assume that the density of the ligand shell is comparable to that of hydrocarbon-based organic polymers (e.g., polyethylene), which is about $ρ(L)=0.9$ g/cm³.²⁹ 

After fixing $ρ(L)$ and d_L, we parameterize the longitudinal sound velocity of the ligand layer, $vl(L)$⁠, by fitting the core-ligand model to experimental data. We find that the fitted value of $vl(L)=2424±70$ m/s for the ODPA sound velocity is in remarkably good agreement with the experimentally measured value of 2430 m/s for high-density polyethylene,²⁹ a similarly saturated hydrocarbon material.

In addition to the longitudinal sound velocity, it could be possible to determine the transversal sound velocity of the ligand shell by fitting the core-ligand model to the size-dependent l = 2 energy data (see Fig. S9 in the supplementary material). For most of our QDs samples, however, the l = 2 Raman features tend to appear in regions with energies below $∼10$ cm⁻¹ where the Raman scattered light is blocked by a volume holographic grating notch filter that has been used to suppress the Rayleigh peak.

Additional details on the data fitting procedure and ligand parameter estimations used in the core-ligand model are described in Sec. S6 of the supplementary material.

The low frequency vibrations of QDs depend on the identity of ligands that passivate their surfaces. We analyze this dependence by applying our model to a series of measurements carried out on films comprised of CdSe cores with fixed size but with different ligands. Here we consider the case where ligand exchange has been carried out to generate CdSe QDs with straight-chain alkanethiols of variable length, ranging from hexanethiol (C6) to octadecanethiol (C18).¹⁵ We assume that the ligand density is constant, $ρ(L)=0.9g/cm3$⁠, for all lengths of alkanethiol ligands. We assign ligand layer thickness, d_L based on X-ray scattering studies of similar sized PbS nanocrystals,²⁸ whose values are tabulated in Table I. The only fitting parameter is, therefore, the longitudinal sound velocity of the ligand layer, $vl(L)$⁠.

Figure 3(a) shows that the core-ligand model captures most of the observed variation in acoustic mode energy with changing ligand length. The grey lines in this figure delineate the core-ligand model fits to the hexanethiol [C6 in Fig. 3(a)] and octadecanthiol [C18 in Fig. 3(a)] data series. For these data, the fitted ligand sound velocities compare favorably to earlier mechanical measurements by DelRio et al.³⁰ for linear alkanethiols self-assembled on gold surfaces [Fig. 3(b); see Sec. S6 of the supplementary material]. The close agreement between the sound velocities extracted from our model and previously measured elastic properties of surface-bound molecules in self-assembled monolayers (SAMs) implies a similarity in the elastic behavior of organic molecules bound in either flat (SAM) or curved (QD) geometries.

Furthermore, we relate $vl(L)$ to the elastic stiffness tensor, C, of the ligand shell, which is an analog of the spring constant in Hooke’s law for multidimensional systems. In our model, the elastic stiffness moduli C₁₁ and C₁₂ relate the normal stress, $σrr$⁠, to the stress components by $σrr=C11𝜀rr+C12(𝜀𝜃𝜃+𝜀ϕϕ)$⁠. In particular, for an isotropic material, C₁₁ is the elastic stiffness modulus that describes the compression along the longitudinal axis and is related to the sound velocity by $C11=vl2ρ$⁠. Thus, variations in $vl(L)$ with respect to the ligand length imply that the elastic stiffness modulus also depends on the ligand length. Our computed values of $C11(L)$ are listed in Table I. Calculated stiffness moduli of various ligands (⁠$C11(L)=0.8$–5 GPa) are comparable to values that have been estimated for other colloidal nanocrystal assemblies using atomic force microscopy (AFM)³¹ as well as those of self-assembled monolayers on gold substrates³⁰ and organic polymers.²⁹ 

The highly luminescent QD materials used in light-emitting applications nearly always include an epitaxially grown shell of a higher bandgap semiconductor around the central core.³² To investigate how this inorganic shell impacts the acoustic phonon spectrum of core-shell heteronanocrystals, we have synthesized a set of nanocrystals with a constant size CdSe core and variable thickness CdS shell (see Sec. S3 of the supplementary material). The low-frequency Raman spectra of these particles reveal a monotonic redshift in acoustic phonon frequencies with increasing shell thickness [Fig. 4(a)]. We also observe an additional peak at about twice the frequency of the l = 0 peak, which has been identified through our previous temperature dependent measurements as a higher spherical harmonic—the n = 2, l = 0 eigenmode—rather than a two-phonon scattering event.²⁰ 

We model these experiments with the core-shell-ligand model, using parameters for bulk CdS (wurtzite CdS bulk parameters: $ρ(S)$ = 4.82 g/cm³, $vl(S)=4289$ m/s, and $vt(S)=1870$ m/s)^25–27 to describe the properties of the inorganic shell. We find that without any further fitting of the ligand parameters, the model agrees well with the experiment. The ability to transfer the ligand parameters across different models suggests that the nature of the chemical linkage at the core/ligand interface, i.e., ligand-CdSe versus ligand-CdS, does not significantly affect the contribution of the ligands to low frequency vibrational modes.

The observation that the low frequency vibrational modes of small QDs deviate from the 1/R scaling of Lamb’s model has been interpreted to indicate a breakdown of the assumptions that underlie traditional elastic continuum theories. It has thus been suspected that small QDs (i.e., with radius $R∼2$ nm) are not well approximated as uniform elastic spheres, due to, for instance, microscopic effects such as surface-specific lattice restructuring and collective ligand dynamics. Our model highlights that these types of effects are unnecessary to explain the small QD scaling deviations. In fact, these deviations are entirely consistent with the framework of elastic continuum theory. As we have demonstrated, the failure of Lamb’s model is to omit the elastic effects of the ligands that passivate the surface of the semiconductor nanocrystal. These effects are negligible when the mass of the inorganic core is much larger than that of the ligand shell, but they become increasingly significant as the size of the nanoscrystal is reduced and becomes comparable to that of the ligand shell thickness.

Our model implies a physical description of QD films in which an array deformable inorganic nanocrystals is embedded within a uniform ligand matrix. This description emerges primarily from the requirement of rigid boundary conditions at the exterior of the ligand layer. As illustrated in supplementary material, the use of free ligand boundary conditions in our model yields breathing mode frequencies that are significantly lower than those from experimental measurements (see Fig. S8). This suggests that in QD films the ligand shell volume does not undergo significant shape distortions but, rather, is constrained by the surrounding environment. This is more consistent with a model in which adjacent ligand shells solublize each other to form a partially inter-digitated background ligand environment.^33,34

Further details about the nature of the background ligand environment can be derived by analyzing the line shape of the lowest frequency Raman mode. There are many possible contributions to this line shape, including QD size polydispersity, multiphonon effects, and anharmonic coupling at the QD surface.^35,36 Raman spectra of core-only dots whose phonon energies have been plotted in Fig. 2 (see Fig. S9a of the supplementary material for the corresponding Raman spectra) reveal that the linewidth includes contributions that vary with CdSe core size. In particular, Fig. 5 reveals that the lowest frequency peak narrows with increasing core size in a manner that scales inversely with the radius of the core. This scaling suggests that the spectral narrowing may be due to ligands exerting an acoustic impedance effect. Such an effect has been observed previously for nanoparticles embedded in a glass matrix.^5,8 Unfortunately, our purely elastic model lacks any information about spectral line shapes. To address this effect, we thus modify our model to describe the ligand layer as a dissipative medium rather than an elastic solid layer. As described in more detail in  Appendix C, this dissipative ligand model damps the acoustic vibration of the elastic core. This model yields a finite (homogeneous) linewidth that can be compared directly to the experimental results plotted in Fig. 5.

As illustrated in Fig. 5, the dissipative ligand model predicts approximately the correct scaling for linewidth but underpredicts its value by about a factor of two. We attribute this apparent discrepancy to the presence of inhomogeneous broadening (due to variations in QD size/shape), which is not included in our model. We note that previous measurements in our lab have shown that for a given QD thin film, the linewidth is approximately invariant to temperature over the range 77-300 K.²⁰ These experiments allow us to estimate the contribution to the l = 0 linewidth from inhomogeneous broadening as a function of QD size. Based on this, we assume a Gaussian inhomogeneous broadening arising from 10% polydispersity, which yields predicted total linewidths that are consistent with experimental observations.

While we have invoked a dissipative ligand model and QD polydispersity to explain the size-dependent Raman linewidth, this observation can also be due to changes in the environment that surrounds the outer surface of the elastic ligand layer.^37,38 For instance, the influence of temperature and local dielectric environment on surrounding ligands have been known to cause fluctuations in the positions of the ligands,^39,40 which may in turn give rise to changes in the density of the ligand shell surrounding each QD. Changes in the density of the ligand shell will likely cause variations in phonon energies even for monodisperse QDs.

In this manuscript, we have demonstrated that the anomalous scaling of low-frequency vibrational energies that have been observed for small QDs is consistent with a simple elastic continuum description of the nanocrystal. Here we have extended a traditional elastic continuum model (Lamb’s model) to include the elastic response of the ligand layer that passivates the nanocrystal surface. We have highlighted that the effects of this ligand layer are negligible for larger QDs, but plays an important role in tuning the acoustic properties of QDs that are smaller than about 2 nm in radius. We have demonstrated that our extended elastic continuum model is in quantitative agreement with Raman measurements carried out on a variety of QDs with different sizes, surface ligands, and compositions.

By comparing to experiments, we have shown that our elastic continuum model can be used to estimate the acoustic properties of the ligand layer. In fact, we have found that the sound velocity, $vl(L)$⁠, of the ligand layer that surrounds QDs is nearly identical to that of analogous bulk organic polymer systems and self-assembled monolayers. This finding implies that the elastic properties of a given type of ligands bonded to nanostructures could be universal, enabling to transfer ligand material parameters across various materials in both bulk-phase and nanocomposites.

See supplementary material (link) for details of nanocrystal synthesis and characterization, fitting procedures, and modeling size-dependent phonon frequencies.

This work was supported by the U.S. National Science Foundation Division of Chemistry and Division of Materials Research under Award No. 1452857. E.M.Y.L. and A.J.M. have been partially supported by the U.S. National Science Foundation Graduate Research Fellowship Program under Grant No. 1122374.

The set of boundary conditions in Eq. (8) yields a system of linear equations for the constants A_C, A_L, and B_L in the form of Mx = 0, where

and M is a 3 × 3 matrix whose non-zero elements M_ij’s are

where $αij=qiRj$ and $qi=ω/vl(i)$ for i, j denoting the type of material, e.g., (C) = core and (L) = ligand.

Having an additional inorganic shell layer compared to the core-ligand model leads to a total of five unknown constants, which are again determined by solving the equation Mx = 0, where

and M is a 5 × 5 matrix whose non-zero elements M_ij’s are

where (S) = shell.

Treatment of organic ligands as a dissipative medium that damps vibrational modes originating from an inorganic core (e.g., CdSe) with radius R allows sound waves to be transmitted to the surrounding medium (⁠$r>R$⁠) but not reflect back toward the core. Thus for $r>R$⁠, we replace spherical Bessel functions with the spherical Hankel function of the first kind, $hl(1)$ in Eqs. (4) and (5) such that

With continuous stress and displacement boundary conditions at r = R, we solve for the non-linear system of equations, f(x) = 0, to calculate the complex-valued frequency, $ω=ω1+iω2$⁠,

where

M is a 2 × 2 matrix with elements,

The real part, $ω1$⁠, is the phonon energy, and the imaginary part, $ω2$⁠, is the phonon half-width at half-maximum of a Lorentzian homogenous broadening.⁵ Using the octadecylphosphonic acid (ODPA) ligand parameters (⁠$vl(L)=2424$ m/s and $ρ(L)=0.9$ cm³/g) and assuming $vt(L)/vt(C)≈0$⁠, we find that the phonon energy is the same as that of the free-boundary core-only model (Fig. 2) while the homogenous linewidth is proportional to 1/R as shown in Fig. 5.