LETTER pubs.acs.org/NanoLett
Far-Infrared Absorption of PbSe Nanorods Byung-Ryool Hyun,*,† A. C. Bartnik,† Weon-kyu Koh,§ N. I. Agladze,‡ J. P. Wrubel,‡ A. J. Sievers,‡ Christopher B. Murray,§,|| and Frank W. Wise† School of Applied and Engineering Physics and ‡Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14850, United States § Department of Chemistry and Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States )
†
bS Supporting Information ABSTRACT: Measurements of the far-infrared absorption spectra of PbSe nanocrystals and nanorods are presented. As the aspect ratio of the nanorods increases, the Fr€ohlich sphere resonance splits into two peaks. We analyze this splitting with a classical electrostatic model, which is based on the dielectric function of bulk PbSe but without any free-carrier contribution. Good agreement between the measured and calculated spectra indicates that resonances in the local field factors underlie the measured spectra. KEYWORDS: PbSe, nanorod, far-infrared absorption, Fr€ohlich mode, local field factor
I
nterest in semiconductor nanostructures has been stimulated by their novel optical and electronic properties associated with quantum confinement effects, and their potential technological importance for applications (e.g., in lasers and solar cells1,2). The size-dependent properties of spherical nanocrystals (NCs) have been studied extensively, while more recently, scientific and technological interest in shape-dependent optical properties of nanostructures is emerging with the advent of colloidal CdS,3 CdSe,4 PbS,5 and PbSe6 nanorods (NRs). Compared to NCs, NRs exhibit larger absorption cross sections, strongly suppressed Auger processes,7,8 faster radiative decay rates,9 larger Stokes shifts,4,10 linearly polarized photoluminescence,10,11 and faster carrier relaxation.12 Compared to the major efforts devoted to the electronic and optical properties, the vibrational modes of nanostructures have received relatively little attention. However, the vibrational modes also reflect the size and shape of a nanostructure. In addition to their intrinsic importance, the vibrations play a major role in electron dynamics. Successful development of devices will require a more complete understanding of carrier relaxation mechanisms, which in turn requires accurate characterization of the phonon modes. As the shape of a nanostructure changes from 0D to 1D, the vibrational modes change. The modes of NRs1315 and nanowires (NWs)16 have been studied with Raman scattering measurements. Observed Raman peaks or features have been assigned to longitudinal optical modes, surface optical modes, and radial breathing modes.14,17 Shape-dependent phonon modes are expected to manifest themselves dramatically in far-infrared (FIR) absorption.18,19 Because of its large polarizability, the primary vibrational mode seen in the FIR absorption of small r 2011 American Chemical Society
ionic crystals is the electrostatic mode, or so-called Fr€ohlich mode, which corresponds to a uniform polarization of the crystal.20,21 Because this mode satisfies electrostatic boundary conditions, it is strongly shape-dependent18,19 and thus is expected to depend on the aspect ratio of small anisotropic particles. The FIR absorption of anisotropic small particles has been calculated18,22 but, as yet, there has been no systematic experimental result that demonstrates the expected shape dependence of the Fr€ohlich mode. This deficiency arises primarily from the lack of high-quality anisotropic particles with different aspect ratios. With recent advances in the synthesis of colloidal NRs with precisely controlled sizes, FIR measurements can now test the validity of theoretical models. Here we present measurements of the FIR absorption spectra of colloidal semiconductor NRs. The spectra of PbSe NRs exhibit clear splitting of the Fr€ohlich mode as the aspect ratio increases. A simple electrostatic model accounts for the experimental spectra. Implications of the FIR spectra will be discussed. Lead-salt semiconductors have unusual dielectric properties (e.g., ε¥ = 23.423 and ε0 = 25024 for PbSe), along with dispersive optical phonons. These properties make them ideal for investigation of the vibrational modes of nanostructures.25,26 PbSe NCs and NRs were synthesized as reported in ref 27 and ref 6. Figure 1 shows the transmission electron microscope (TEM) images of the PbSe NCs and NR sample NR-4, and are representative of all samples. The diameters and lengths of NCs and NRs used in this work determined from the TEM images are listed in Table 1. The Received: April 3, 2011 Revised: May 17, 2011 Published: May 31, 2011 2786
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Figure 1. Typical TEM images of PbSe (a) NC and (b) NR-4.
materials
diameter (D) [nm]
length (L) [nm]
aspect ratio L/D
PbSe NC PbSe NR 1
5.0 4.4
N/A 19
1 4.3
PbSe NR 2
3.9
17
4.4
PbSe NR 3
4.0
20
5.0
PbSe NR 4
4.0
37
9.0
samples cover the range of aspect ratios that have been synthesized to date. Figure 2a shows typical visible/near-infrared absorption spectra of the PbSe NCs and NRs dispersed in tetrachloroethylene (TCE). The Fourier transform FIR spectra of PbSe NCs and NRs dispersed in hexane were measured at room temperature. Hexane is used because it is transparent between 15 and 350 cm1. NC and NR solutions are loaded into a homemade 2.54 mm thick cell with 6 μm thick Mylar film windows. The measured FIR spectra of the same samples are displayed in Figure 2b. The FIR spectrum of the PbSe NCs has a single peak centered at 134 cm1. As the aspect ratio increases, the peak splits into two slightly asymmetric peaks. As the aspect ratio varies from 4.3 to 9, the high-frequency peak shifts from 140 to 144 cm1, while the low-frequency peak moves from 105 to 81 cm1. The measured peak positions are summarized in Figure 4 below (symbols). We analyze the FIR spectra using two approaches within the electrostatic approximation, in which the wavelength is much larger than the particle size. In the first approach, the vibrational modes are calculated, and the absorption cross section is determined from the modes. An alternative approach is to consider Rayleigh scattering from the composite nanostructure-host medium with a known dielectric function. The two methods produce absorption peaks at identical frequencies, but the latter directly produces the line shape. Because it is more revealing to compare spectra to our measured absorption data, we will focus on the latter method. An established way to model electrostatics in an anisotropic (rod-shaped) structure is to approximate it by a prolate spheroid, where the aspect ratio corresponds to the ratio of the ellipsoidal semiaxes.28 By solving a simple set of equations, the optical phonon mode frequencies can be obtained analytically (see the Supporting Information). The absorption spectrum
can be obtained by calculating the FIR properties of a composite medium consisting of isolated and randomly oriented rods. For this calculation, three quantities are needed: the host dielectric constant, the frequency dependence of the dielectric function, and the local field factor of the NCs. As will be explained below, resonances in the FIR spectra are determined primarily by the local field factors at the Fr€ohlich frequency.22,29,30 In the Rayleigh scattering limit, the local field factor directly determines the FIR absorption spectrum of NCs or NRs dispersed in dielectric hosts. The absorption coefficient of the NC-host composite is given by31,32 Rabs ¼ f
n 4πk nH λ
∑i jfLF, i j2
ð1Þ
(see the Supporting Information) where f is the volume filling fraction, λ is the wavelength of the incident light, nH is the refractive index of the host medium (1.37 for hexane), and n þ ik is the complex refractive index of the NR. fLF,i is the local field factor, which relates the applied electric field to the internal field within the high-dielectric-constant NCs when immersed in a lowdielectic-constant medium (εH = n2H) and i corresponds to x, y, and z Cartesian coordinates. fLF is a function of the aspect ratio (L/D) and the dielectric contrast εNR/εH, where εNR (= (n þ ik)2) is the dielectric function of the nanoparticles. In a rodshaped structure, the local field factor depends on direction, parallel or perpendicular, to the rod axis. The local field factors for a prolate spheroid can be expressed28,33 as fLF, ¼ )
Table 1. Diameters, Lengths, and Aspect Ratio of PbSe NCs and NRs
1 εNR ðωÞ εNR ðωÞ 1 η0 ½ð1 η20 Þcoth1 ðη0 Þ þ η0 εH εH
ð2Þ 1 εNR ðωÞ fLF, ^ ¼ 2 2 þ 1 η0 ½ð1 η20 Þcoth1 ðη0 Þ þ η0 εH
ð3Þ where η0 = A/(A2 1)1/2 and A = a/c is the ratio of the major to minor axis of the spheroid. To calculate fLF, we need to know the dielectric function of a PbSe NR, which we assume is that of bulk PbSe. This function has been measured experimentally and fit to 2787
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Figure 2. (a) Optical absorption spectra of PbSe NCs (solid red line) and PbSe NRs (solid blue and green lines) in TCE. The diameters and lengths are determined from their TEM images. (b) FIR spectra of PbSe NC, NR-3, and NR-4 samples in hexane.
Figure 3. (a) n (real part of refractive index, squares) and (b) k (imaginary part of refractive index, circles) versus wavelength for bulk PbSe. The blue lines are fits of eq 4 with ε¥ = 23.4,23 ε0 = 250,24 ωTo = 45 cm1,23 ωp = 880 cm1,23 Γ = 20 cm1, and γ = 40 cm1. The green lines are calculated without the free-carrier contribution.
Figure 4. Fr€ohlich phonon frequencies of PbSe NRs versus aspect ratio. The blue and green lines are the frequencies obtained from the dielectric function of eq 4 with and without free carrier absorption, respectively. The open circles are from the measured FIR spectra of PbSe NCs and NRs.
a classical oscillator model: ωp ðε0 ε¥ Þω2To þ 2 2 2 ωTo ω iΓω ω iγω 2
εpbSe ðωÞ ¼ ε¥ þ
ð4Þ
where ε¥ and ε0 are the high-frequency and static dielectric constants. Of course, the optically active phonons underlie the dielectric response. The second term of eq 4 represents the contribution of the optical phonons and the third term is due to
free carriers, which cannot be neglected in the bulk lead salts. ωTo and ωp are the transverse optical (TO) phonon and plasma frequencies of bulk PbSe, respectively, and Γ and γ are the corresponding damping rates. The fits (solid and dotted blue lines) agree very well with the experimental real and imaginary refractive indices of bulk PbSe,23,34 as seen in Figure 3. From the complex dielectric function of PbSe, the local field factors of PbSe NRs are calculated for different aspect ratios. (See Figure S-1 in the Supporting Information.) As the aspect ratio increases from 1, the vibrational resonance splits, into one resonance corresponding to polarization parallel to the NR axis that moves to lower frequency, while a resonance for the perpendicular polarization moves to higher frequency. Incorporating this into eq 1, we expect absorption spectra with two peaks at the frequencies shown as blue lines in Figure 4. The frequencies exhibit the qualitative trends of the experimental results but are quantitatively inaccurate; the calculated frequencies are systematically about 30% higher than the measured values. In general, there are negligible free carriers in nanostructures in equilibrium. The energy gap of the NRs is ∼1 eV, compared to 0.3 eV in the bulk material, and the nanostructures have few impurities that could act as dopants. Considering this, we eliminate the plasma term from the dielectric response. The resulting resonant frequencies (green lines in Figure 4) agree very well with the measured values. Figure 5a,b shows the frequency dependence of the local field factors parallel and perpendicular to the long axis of the rod for this revised model. 2788
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Figure 5. Local field factors of PbSe NC and NR for the indicated aspect ratios, calculated neglecting the free-carrier contribution. (a,b) The local field factors parallel and perpendicular to the long axis, respectively. The values are nearly constant for frequencies above 300 cm1 .
Figure 6. The calculated (red line) and measured (blue line) FIR absorption spectrum of PbSe NRs with aspect ratio of 4.3. For the calculation, eq 1 is used with the dielectric function that neglects free carriers.
Using the revised model, we calculate the FIR absorption spectra of PbSe NRs from eq 1. The calculated spectra are broadened by assuming that the length of the NRs has a Gaussian size distribution of ΔL/L = 20%, which is typical for our synthesis, while the diameters have a much smaller variation around 5%. Calculated and measured spectra for NRs with aspect ratio of 4.3 are compared in Figure 6. (See the Supporting Information for the spectra of other sizes.) While the peak locations and widths are reproduced, the model consistently underestimates the relative height of the high-frequency peak. It is difficult to explain this within the framework of our model, which has no free parameters. It seems to imply that the predicted ratio of local field factors is overestimated by a factor of ∼1.5, but if true then the peak locations would also shift. So, it is likely that some other effect is causing this difference. We conjecture that the assumption of a true cylinder shape, rather than a spheroid, might produce better agreement for the relative peak heights. The good agreement between the measured FIR spectra of PbSe NRs and a calculation based on the dielectric function of the bulk phonons is convincing evidence of the splitting of the Fr€ohlich mode as the nanostructure goes from spherical to rodshaped. There are several implications of these results. Calculations of the modes of nanostructures that account for the detailed electromagnetic and mechanical boundary conditions depend on the length and diameter individually.35,36 We find that the simple classical model, which only depends on the aspect ratio of the structure, accounts for the measured spectra. A second point is
that the relationship (if any) of the dielectric “constant” of a nanostructure to that of the parent bulk material is an open question.3739 These results imply that the dielectric function of the nanostructure is well-approximated by that of the bulk material apart from any contribution of free carriers; resonance structure in the FIR spectrum arises from resonances in the local field factor. In principle, if the nanostructures are monodisperse with known sizes, this may be a way to extract the dielectric function of the nanostructure. The inferred function could then be compared to first-principles calculations of the dielectric function. In conclusion, the FIR absorption spectrum of PbSe nanorods exhibits splitting with increasing aspect ratio of the nanorod. This is the first demonstration of the expected splitting of the Fr€ohlich mode in anisotropic nanostructures. A classical electrostatic model that depends only on the aspect ratio of the structure accounts for the experimental results.
’ ASSOCIATED CONTENT
bS
Supporting Information. Additional information. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT This work was supported by the Cornell Center for Materials Research (CCMR) with funding from the Materials Research Science and Engineering Center program of the National Science Foundation (cooperative agreement DMR 0520404) and in part by Award No. KUS-C1-018-02 made by King Abdullah University of Science and Technology (KAUST). C.B.M. and W.-k.K. acknowledge financial support from the NSF through DMS-0935165. This research was partially supported by the Nano/Bio Interface Center through the National Science Foundation (NSEC DMR08-32802). Authors N.I.A., J.P.W., and A.J.S. were supported by DMR-0906491 and DOE No. DE-FG02-04ER46154. ’ REFERENCES (1) Klimov, V. I.; Mikhailovsky, A. A.; Xu, S.; Malko, A.; Hollingsworth, J. A.; Leatherdale, C. A.; Eisler, H.-J.; Bawendi, M. G. Science 2000, 290, 314. 2789
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