Electronic Structure and Spectroscopy of Cadmium Telluride Quantum

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NANO LETTERS

Electronic Structure and Spectroscopy of Cadmium Telluride Quantum Wires

2008 Vol. 8, No. 9 2913-2919

Jianwei Sun,† William E. Buhro,† Lin-Wang Wang,‡ and Joshua Schrier*,‡ Department of Chemistry and Center for Materials InnoVation, Washington UniVersity, St. Louis, Missouri 63130-4899, and Computational Research DiVision, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720 Received June 17, 2008; Revised Manuscript Received July 28, 2008

ABSTRACT The size-dependent electronic structure of CdTe quantum wires is determined by density functional theory using the local density approximation with band-corrected pseudopotential method. The results of the calculations are then used to assign the size-dependent absorption spectrum of colloidal CdTe quantum wires synthesized by the solution-liquid-solid mechanism. Quantitative agreement between experiment and theory is achieved. The absorption features comprise transitions involving the highest 25-30 valence-band states and lowest 15 conduction-band states. Individual transitions are not resolved; rather, the absorption features consist of clusters of transitions that are determined by the conduction-band energy-level spacings. The sequence, character, and spacing of the conduction-band states are strikingly consistent with the predictions of the simple effective-mass-approximation, particle-in-a-cylinder model. The model is used to calculate the size dependence of the electron effective mass in CdTe quantum wires.

Introduction. We report that the diameter-dependent absorption features from colloidal CdTe quantum wires (QWs) correlate quantitatively with the optical transitions predicted by high-level electronic-structure calculations. The general picture of QW electronic structure that emerges from the excellent agreement of experiment and theory is conceptually consistent with the simplest model for two-dimensional (2D) quantum confinement. In recent years, several theoretical studies of the electronic and optical properties of II-VI1-3 and III-V4-12 semiconductor QWs have appeared. Some of these have reported calculated optical spectra; however, to our knowledge, only one direct comparison of calculated and experimental spectra has been previously reported.1 In that case, the theoretical spectra of CdSe QWs were compared to the experimental spectra obtained from long CdSe quantum rods (QRs). In contrast, studies of the electronic structure of semiconductor quantum dots (QDs) have benefited greatly from comparisons of calculated energy-level diagrams and calculated optical spectra with experimental spectroscopic data, generally from photoluminescence excitation (PLE) spectroscopy.13-18 Furthermore, related comparative studies for semiconductor QRs have revealed, among other insights, the geometric requirements for the crossover from a dark to bright exciton ground state.19-21 Consequently, efforts to correlate experimental and theoretical studies of QWs will be useful for characterizing * Corresponding author. E-mail: [email protected]. † Washington University. ‡ Lawrence Berkeley National Laboratory. 10.1021/nl801737m CCC: $40.75 Published on Web 08/13/2008

 2008 American Chemical Society

their electronic structures and the size dependences of their optical properties. We report herein that the experimental absorption spectra of CdTe QWs are dominated by the first 15 conduction-band (CB) energy levels determined by local density approximation with band-corrected pseudopotential (LDA + C) densityfunctional-theory calculations.22 The results establish that the absorption features observed experimentally are not discrete transitions but rather clusters of closely spaced transitions that are grouped by the energy spacing of the CB levels. Furthermore, we show that these CB energy levels follow the sequence predicted by the simple effective-mass-approximation, particle-in-a-cylinder (PIC) model, by assigning symmetries to the LDA + C electron-density distributions. The PIC model is used to determine the size dependence of the electron effective mass (m*e ) and to confirm that the low resolution of the experimental spectra is due to the QW size distribution and spectroscopic method employed rather than to closely spaced QW energy levels. The close correspondence of the LDA + C results and the simple PIC model provides a conceptually attractive framework for analyzing the electronic structure of semiconductor QWs. Materials and Methods. Absorption Spectroscopy. Colloidal CdTe QWs were prepared by a previously reported bismuth-catalyzed solution-liquid-solid (SLS) method.23 The absorption spectra of the QWs were collected at room temperature from toluene dilutions of the as-prepared samples. To obtain the positions of absorption features, spectra were fit using a modification of a method reported

previously.24,25 Briefly, each spectrum was converted from a wavelength to an energy scale, and then fit with one exponential (for the background) and multiple Gaussian functions (for the features) using Origin 7.5 software (www.OriginLab.com). The number of Gaussian functions used was determined by the number of absorption features observed in the spectra. The fitting procedure yielded the center energy of each absorption feature and the error in the center energy. Computational Methods. The absorption spectrum was calculated using the LDA + C method as described previously for CdTe QDs and zinc-blende QWs,22 performed with a 35 Ryd planewave energy cutoff. To sample the Brillouin zone of the QW, we explicitly calculated 5 k points along the c axis of the QW and then linearly interpolated the energies and transition dipole matrix elements for 27 intermediate points. The imaginary part of the dielectric function was calculated with a constant, 20 meV Gaussian broadening and is proportional to the optical absorption spectrum as discussed in ref 26. The local density approximation (LDA) is well-known to underestimate semiconductor band gaps, by a material-dependent constant.27 The LDA + C approach attempts to obtain the correct effective masses, and as a result, the calculated band gap is not exact. Although the resulting bulk band gap is larger than the original LDA band gap, it is still smaller than experiment.22 Our LDA + C bulk band gap for zinc-blende (ZB) CdTe is 1.12 while the experimental value is 1.56. Therefore, a constant shift of 0.44 eV was added to the calculated QW single-particle band gap to correct for this bulk difference. The calculated bulk wurtzite band gap is 0.05 eV larger than that calculated for ZB CdTe, in agreement with previous allelectron, relativistic, linearized augmented plane wave (LAPW) calculations.28 In addition to this constant shift of 0.44 eV necessary to correct the LDA + C single-particle band gap, the excitonic binding energy, which is enhanced by the dielectric mismatch between the QW and its surrounding environment,29 must also be incorporated. We consider two approaches for estimating this contribution: (1) For a given QW diameter, the excitonic energy contribution will be approximately constant for all of the excited states in that QW. Thus, by plotting excited-state energies relative to the energy of the first excited state, as we have done in Figures 2, 3, and 5, the exciton-binding energy is effectively canceled. (2) The final optical band gap is modified by two contributions with opposite signs (in addition to the (LDA + C) + 0.44 eV single-particle band gap). While the exciton-binding energy due to the Coulomb interaction leads to a reduction in the band gap, there is also a polarization term that increases the band gap (a quasi-particle band gap). Shabaev and Efros calculated the size dependence of these two terms for CdSe QRs using an effective-mass approach,29 (corresponding to the difference between the dotted line and the lower solid red line in Figure 2 of ref 29). We extrapolated their curves to accommodate the 10.2 nm diameter QW. Assuming the CdSe and CdTe exciton-binding energies to be similar, the total effect is to lower the single-particle (LDA + C) + 0.44 2914

Figure 1. Representative TEM images (a-c) and UV-vis absorption spectra (d) of CdTe QWs of various diameters (dwire). (a) dwire ) 5.3 ( 1.1 nm ((20.8%), (b) dwire ) 7.3 ( 1.0 nm ((13.7%), and (c) dwire ) 10.2 ( 1.7 nm ((16.7%).

eV band gap by 0.08 eV, 0.07 eV, and 0.06 eV, respectively, for the 5.3 nm, 7.3 nm, and 10.2 nm diameter QWs considered in the present work. Therefore, we have added 0.36 eV, 0.37 eV, and 0.38 eV to the LDA + C band gaps for the 5.3 nm, 7.3 nm, and 10.2 nm diameter QW results (respectively), shown in Figures 4, 6, 7. The precise values of these small excitonic energy corrections shift the x axis values of the calculated data points in Figure 4, the energy spacing between the conduction and valence bands (but not the relative spacings within the conduction or valence band) in Figure 6, and the y axis values of the data points in Figure 7 but otherwise have no substantial quantitative effect on our conclusions. Results and Discussion. Experimental Absorption Data. We previously reported the synthesis of colloidal CdTe QWs having lengths of several micrometers and well-controlled diameters of 5.3-10.2 nm, within the strong confinement regime.23 The diameter distributions, characterized as (1 standard deviation and expressed as a percentage of the mean diameter, were in the range of 12-20%, which were narrow for semiconductor QWs grown by the SLS mechanism.30 The TEM images in Figure 1a-c are representative of the general quality of these CdTe wires. Our prior study examined the absorption spectra of the CdTe wires, but analyzed only the diameter dependence of the lowest-energy feature, and therefore the effective band gap, in the wires. However, Figure 1d shows that the absorption spectra are richly structured, and contain several higher-energy absorption features. Characterizing those higher-energy features is a major goal of this work. The spectra were decomposed by nonlinear least-squares fitting, which allowed extraction of at least four absorption Nano Lett., Vol. 8, No. 9, 2008

Figure 2. Experimental absorption spectra (upper panels) and calculated absorption spectra with average polarization (lower panels) of CdTe QWs with diameters of (a) 5.3 nm, (b) 7.3 nm, and (c) 10.2 nm. Fitted spectra (barely visible orange curves) are shown behind the corresponding experimental spectra. Individual fitting functions (one exponential function for the background and multiple Gaussian functions for the absorption features) of each fitted spectrum are also plotted (the absorption features are colorcoded).

Figure 3. Calculated imaginary dielectric constant (optical absorption) as a function of energy, for (a) 5.3 nm, (b) 7.3 nm, and (c) 10.2 nm diameter wurtzite CdTe QWs. The solid line shows the average (disordered) polarization; the dotted and dashed lines show the contribution from the momentum matrix elements perpendicular and parallel to the wurtzite c axis (i.e., the growth axis). The numbered arrows indicate the proposed correspondence to the experimental absorption envelopes depicted in Figure 2 and plotted in Figure 4.

envelopes from each spectrum. Three examples of the fitted spectra and extracted features, individually modeled as Gaussian peaks, are shown in Figure 2. The remainder is included in Supporting Information as Figure S1. In all cases, the lowest-energy feature was fit as a reasonably narrow Gaussian, whereas the higher-energy features were generally much broader (as a result of fitting uncertainties and the numbers of transitions contained within the features; see below). We next sought to compare the experimental data with calculated absorption spectra. Calculated Absorption Data. Figure 3 shows the calculated optical absorption (proportional to the imaginary part of the

dielectric constant, ε2) for 5.3 nm, 7.3 nm, and 10.2 nm diameter wurtzite CdTe QWs. The value of the imaginary part of the dielectric constant in the calculated energy range was comparable to the bulk measurements of Cardona.31 Excellent agreement between the experimental spectra and the corresponding calculated spectra with average polarization is evident in Figure 2, for the first four fitted absorption envelopes at each diameter. The higher resolution of the calculated spectra resulted from two factors: (1) Only a finite number of k points along the wire axis were sampled, thus artificially limiting the broadening of the peak energies; and (2) the experimental spectra were further broadened by the

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Figure 4. Comparison of the experimentally observed absorption features (open symbols) and the calculated absorption features (solid symbols). The same color scheme is used as in Figure 3.

Figure 5. Calculated optical absorption for transitions at the Γ point (solid curve) for the 5.3 nm diameter wurtzite CdTe QW. The vertical lines indicate the individual transitions, which are summed and broadened to produce the total curve. We have color-coded the transitions making the largest contributions, matching the colorcoding used in Figures 2-4, and labeled the states involved in these transitions. The dotted curve shows the sum of the contributions from all 31 k points, for comparison.

diameter distributions of the samples. However, the clusters of features evident in the calculated spectra fell nicely within the broad Gaussian functions used to fit the experimental spectra. As the QW diameter increased, the resolution of the calculated peaks decreased, leading to broadened absorption features. To quantitatively compare the calculated and experimental results, the relative energies of the excited-state envelopes from experiment (open symbols) and theory (filled symbols) were plotted versus the first excited-state energy (Figure 4). The resolved peaks in a relevant cluster of peaks in the calculated spectra were averaged for comparison to the experimental Gaussian-fitted envelopes. Peaks subjected to averaging are indicated by the designations in Figure 3 (e.g., 4a, 4b, 4c, etc.). The near overlap of the calculated and experimental curves in Figure 4 confirmed the excellent agreement. The theoretical and experimental curves were separated by only 35-90 meV, with the greatest separation for the highest-energy absorption envelope (the blue envelope labeled 4 in Figure 3). An assignment of the QW transitions 2916

Figure 6. (a) Energy-level diagram for the 5.3 nm diameter wurtzite CdTe QW, with the transitions contributing to the broad absorption features identified by color-coded arrows matching Figures 2-5. (b) Plots of the electron- and hole-density distributions for the conduction band (CB) and valence band (VB) states. Boxes indicate pairs of energetically degenerate levels. States that do not make major contributions to the optical absorption in Figure 5 are omitted.

contributing to these absorption envelopes was next undertaken. Assignment of Absorption Features. The calculated absorption spectra shown in Figures 2-4 consist of an integral of the k points along the nanowire axis. To facilitate the identification of the strongest contributions to each of the absorption peaks, we used the Γ-point wave function to determine the specific transitions, for each of the three diameters investigated theoretically. The explicit case of the 5.3 nm diameter QW is presented in more detail in Figure 5 and Figure 6. For all three sizes of the QWs, we found that the first excitation into the CB1 state more substantially involved the VB2 state than the VB1 state. The second excited peak was found to be a contribution predominantly from the VB5 and VB6 states into the degenerate CB2 and CB3 states (as well as a few other smaller contributions that varied for the three sizes examined here). These first two sets of peaks are consistent with the results of previous empirical pseudopotential calculations of transitions in CdSe Nano Lett., Vol. 8, No. 9, 2008

Figure 7. Plots of the CB-state energy (En,l) of CdTe QWs of 2 . The two 1 Φ states for all various diameters at the Γ point vs φn,l e three diameters and the two 2Πe and two 1Γe states for the 10.2 nm diameter QW are not strictly degenerate (in the LDA + C results). Consequently, two data points are plotted for each of these pairs, one open point and one closed point, which overlap significantly. The solid and dashed lines are the quadratic and linear fits to the energy levels, respectively. Inset: Plot of electron effective mass vs quantum-confinement energy. The linear fit yields a y intercept of (0.098 ( 0.005)m0, which is close to the bulk value of 0.095m0.

QWs.1 The third excited peak always contained a contribution into the degenerate CB4 and CB5 states, and to a lesser extent into the CB6 state. Finally, the fourth excited peak was the sum of many small peaks, without any particularly dominant contribution for all sizes of the QWs. While we considered a large number of states in our calculations, we noted that in all of the QWs the largest values of the momentum matrix elements (and hence the strongest contributions to the absorption spectrum) resulted from transitions between the highest 25-30 VB and lowest 15 CB states for the energy range considered here. We also note that the peak position obtained using only the Γ-point imaginary dielectric function (ignoring the other k points completely) was within 60 meV of the peak position obtained by including all of the 31 k points as discussed above; as is revealed by comparing the solid (Γ point only) and dashed (all k points) absorption spectra in Figure 5. Both of these factors suggest that future semiquantitative assignments could be made on the basis of substantially fewer calculations than in the present work. QW Electronic Structure. The LDA + C electron and hole density distributions for the CB and VB states participating in the transitions of the 5.3 nm QW are shown in Figure 6b. The sequence of the (electron) CB energy levels is strikingly consistent with that predicted by the simple PIC model (see Supporting Information). According to this model, the CBstate energies En,l are expressed by eq 1, where d is the QW diameter, m*e is the electron effective mass, Eg,bulk is the bulk CdTe band gap (1.606 eV at 2 K),32 and φn,l is the nth root (zero) of the lth-order cylindrical Bessel function.29,33 En,l )

h2φ2n,l 2π2m/e d2

+ Eg,bulk

(1)

The predicted PIC sequence is thus 1Σe < 1Πe < 1∆e < 2Σe < 1Φe < 2Πe < 1Γe, which corresponds to CB1 < Nano Lett., Vol. 8, No. 9, 2008

(CB2,CB3) < (CB4,CB5) < CB6 < (CB8,CB9) < (CB12, CB13) < (CB14,CB15). (For the spherical symmetry of QDs, these states would be labeled 1Se < 1Pe < 1De < 2Se < 1Fe < 2Pe < 1Ge, by analogy to atomic term symbols. However, the corresponding Greek characters are used by convention in cases of cylindrical symmetry.29) In the PIC model, Σ states are singly degenerate, and non-Σ states are doubly degenerate,34 which faithfully reproduces the LDA + C calculations, except for CB8 and CB9 (1Φe), which are energetically close (9 meV) but nondegenerate. Note that the expected radial nodes are present in the 2Σe (CB6) and 2Πe (CB12, CB13) density distributions. In general, the electron-density distributions are nearly idealized representations of the expected PIC CB states. Interruptions in the predicted PIC sequence are evident in the insertions of the CB7 and CB10, CB11 states, which do not belong. Examination of the electron-density distributions for CB7 and the doubly degenerate CB10, CB11 states establishes their 1Σ and 1Π symmetry, respectively (Figure S2). Their energetic placement requires a different electron effective mass than that for the continuous series described above and indicates that CB7, CB10, and CB11 are intruder states that have folded into the Γ-point sequence from a separate valley of the band structure. In the 7.3 nm diameter QW, these intruder 1Σe and 1Πe states are intercalated into the Γ-point sequence at different positions, at CB11 and CB14, CB15 respectively, demonstrating their different energy-diameter scaling and confirming the different electron effective mass at their corresponding k point. Projections of the Bloch function for these states show that they originate along the Γ-Μ line in the bulk wurtzite Brillouin zone, analogous to previously calculated results for zinc-blende quantum wires.35 Consequently, having a non-Γ k value, they are optically inactive (note the absence of transition arrows involving CB7 and CB10, CB11 in Figure 6a), and thus the intruder states do not contribute to the calculated absorption spectra. In comparison to the CB-state electron-density distributions, the VB-state hole-density distributions (Figure 6b) are more complex, as a result of VB mixing of the heavy-hole and light-hole states (derived from the bulk A and B VBs).13,21 Consequently, the VB (hole) states generally have a hybridized character, rather than being pure Σ, Π, ∆, and so forth. Significantly, the VB states are much more closely spaced than are the CB states, because of the greater hole effective masses. Therefore, the comparatively large spacing of the CB levels dominates the absorption spectra, such that the first absorption envelope constitutes a cluster of transitions terminating in CB1, the second in (CB2, CB3), the third in (CB4, CB5) and CB6, and the fourth in (CB8, CB9), (CB12, CB13), and (CB14, CB15). The absorption envelopes are thus defined by the spacing of the CB states. Energy Scaling of CB States. The PIC model asserts that the CB energy levels should scale linearly with φ2n,l (eq 1). Recognition of the nonparabolic CB dispersion adds a quadratic term (φ4n,l) to the energy dependence as shown in eq 2, where R is the diameter-dependent isotropic nonparabolicity constant (see Supporting Information for the deriva2917

tion of eq 2).36,37 We note that eq 1 in ref 29 provides an alternative means to account for conduction-band nonparabolicity. The CB-state energies (with respect to the top of the bulk VB) for the 5.3, 7.3, and 10.2 nm diameter QWs are plotted versus φ2n,l in Figure 7, along with both linear (by eq 1) and quadratic (by eq 2) fits to the data (eqs 3-5). Again, a close correspondence is evident between the results of the LDA + C calculations and the simple PIC model. Gentle curvature is present in all three sets of theoretical data (Figure 7), which is well fit by the eq 2 quadratic term (eqs 3-5). The curvature (and thus the quadratic term) decreases with increasing QW diameter, indicating that eq 1 becomes a better approximation at larger diameters. The curves should extrapolate to the bulk band gap of 1.606 eV (2 K), and all come close to this value: 1.58, 1.55, and 1.58 eV for the 5.3, 7.3, and 10.2 nm diameter QWs, respectively. En,l(d) )

h2φ2n,l

+ 2π2m/e d2

16Rφ4n,l d4

+ Eg,bulk

(2)

5.3 nm QW: En,l ) (-1.63 × 10-4 eV)φ4n,l + (2.73 × 10-2 eV)φ2n,l + 1.58 eV (3) 7.3 nm QW: En,l ) (-9.28 × 10-5 eV)φ4n,l + (1.82 × 10-2 eV)φ2n,l + 1.55 eV (4) 10.2 nm QW: En,l ) (-4.30 × 10-5 eV)φ4n,l + (1.08 × 10-2 eV)φ2n,l + 1.58 eV (5)

Size-Dependent Electron EffectiVe Mass. Quantum confinement increases the effective masses of carriers,36 and prior studies have confirmed that m*e increases as QW diameters decrease.5,6,9,37,38 The size dependence of m*e in CdTe QWs can be extracted from the slopes of the En,l versus φ2n,l plots in Figure 7 (see eq 1). As noted above, these plots exhibit slight curvature, and so we have calculated m*e values using the coefficients of the φ2n,l terms in eqs 3-5 in place of the slopes from the linear fits. These values range from 0.20m0 for the 5.3 nm diameter QW to 0.14m0 for the 10.2 nm diameter QW. Interestingly, the Figure 7 inset shows that the m*e values scale linearly with the QW confinement energy (∆Eg) and extrapolate to m*e ) (0.098 ( 0.005)m0 at ∆Eg ) 0, which is close to the bulk value of 0.095m0 (at ∼2 K).39 We note that the good fits of eqs 2-5 to the LDA + C results, evident in Figure 7, do not indicate that simple effective mass theory can be used to predict quantitatively the CB-state energies for QWs. Calculation of CB-state energies by eq 2 requires knowledge of the diameterdependent electron effective mass and isotropic nonparabolicity constant, which must be obtained by retroactive analysis of data from atomistic calculations, as in the present work. 1Πe-1Σe Spacing. As established above, the absorption features we observed experimentally resulted from clusters of closely spaced transitions rather than from individual transitions. In contrast, prior studies of the optical spectra of CdSe,13 CdTe,14 InAs,15 and InP16 QDs successfully resolved the individual transitions. We suggested above that the CdTe QW absorption features were broadened by the size distribu2918

tions, which are wider than those achieved for QDs. Additionally, the prior QD studies used PLE spectroscopy, which provides inherently higher spectral resolution than does the standard absorption spectroscopy used here. However, we now consider if the energy-level spacings in CdTe QWs are considerably smaller than those in the corresponding QDs, which would also result in lower spectral resolution. To investigate the comparative level spacings in CdTe QWs and QDs, we use the difference in the 1Σe (1Se) and 1Πe (1Pe) energies as representative. The simple PIC model above predicts a ratio (1Πe - 1Σe)QW/(1Pe - 1Se)QD of 0.86, independent of diameter (assuming equal QW and QD diameters and m*e values; see Supporting Information). Correspondingly, the LDA + C calculations for CdTe QWs reported here and for CdTe QDs reported previously22 produce a ratio (1Πe - 1Σe)QW/(1Pe - 1Se)QD of 0.87 at QD and QW diameters of 5.3 nm (Figure S3). Therefore, the CB levels in CdTe QWs and QDs are similarly, although not equally, spaced. The result suggests that the lower resolution of the current experimental results is due to the broader diameter distributions and the spectroscopic method employed. Conclusion. The close agreement between the calculated and the experimental absorption spectra for CdTe QWs corroborates the electronic structure depicted in Figure 6. The absorption features observed are largely determined by the energy spacing of the CB states. The sequence, character, and spacing of the CB states are remarkably consistent with the PIC model, which is the simplest model for 2D confinement in QWs. The VB states are complicated by VB mixing, and thus do not submit to this simple analysis. Prior theoretical studies have shown that the CB structure in QDs40-42 and QRs29 is analogously simple and well-described by particle-in-a-sphere or PIC models, respectively. Here, we demonstrate such a correlation explicitly for QWs and present it in greater detail than the previous studies. The close correspondence of the LDA + C results and the PIC model confirms the latter as a useful first-order picture for the CB structure in semiconductor QWs. Acknowledgment. J. Sun and W.E.B. thank Professor Richard A. Loomis for helpful discussions, and the National Science Foundation (Grant CHE-0518427) for support. L.-W.W. and J. Schrier are supported by DMSE/BES/SC of the U.S. Department of Energy under Contract Nos. DE-AC02-05CH11231 and used the resources of the National Energy Research Scientific Computing Center. Supporting Information Available: Additional fitted absorption spectra, derivation of the PIC CB sequence, electron-density distributions for CB7 and (CB10, CB11) states, and the PIC and LDA + C 1Πe-1Σe spacing calculations. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Li, J.; Wang, L.-W. Nano Lett. 2003, 3, 101–105. (2) Xia, J. B.; Zhang, X. W. Eur. Phys. J. B 2006, 49, 415–420. (3) Huang, S.-P.; Cheng, W.-D.; Wu, D.-S.; Hu, J.-M.; Shen, J.; Xie, Z.; Zhang, H.; Gong, Y.-J. Appl. Phys. Lett. 2007, 90, 031904. Nano Lett., Vol. 8, No. 9, 2008

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