Quantum-Dot Size and Thin-Film Dielectric Constant: Precision

Dec 22, 2014 - D.D.W. and P.R.B. gratefully acknowledge support from the John and Fannie Hertz Foundation, and P.R.B. acknowledges support from the ...
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Letter pubs.acs.org/NanoLett

Quantum-Dot Size and Thin-Film Dielectric Constant: Precision Measurement and Disparity with Simple Models Darcy D. W. Grinolds,† Patrick R. Brown,‡ Daniel K. Harris,§ Vladimir Bulovic,∥ and Moungi G. Bawendi*,† †

Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States § Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States ∥ Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States ‡

S Supporting Information *

ABSTRACT: We study the dielectric constant of lead sulfide quantum dot (QD) films as a function of the volume fraction of QDs by varying the QD size and keeping the ligand constant. We create a reliable QD sizing curve using small-angle X-ray scattering (SAXS), thin-film SAXS to extract a pairdistribution function for QD spacing, and a stacked-capacitor geometry to measure the capacitance of the thin film. Our data support a reduced dielectric constant in nanoparticles. KEYWORDS: Dielectric constant, quantum dot, PbS, SAXS, capacitance

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imaginary components of the absorption,7,8 and few experimental measurements of the dielectric constant of a single PbS thin film have been reported9,10 using CELIV or Schottky devices in reverse bias. In this work, we use a stacked-capacitor geometry to measure the dielectric constant of thin films of a size series of QDs using tetrabutylammonium iodide, a ligand that is commonly used in QD photovoltaic devices. The advantage of this capacitor technique is that it is not sensitive to electronic interface effects such as Fermi-level pinning and depletion regions. In our method, the QD film is deposited onto an HfO2 insulating oxide and covered by a film of parylene, with an ITO or Al electrode on either side of the insulating materials. We measure the capacitance of the stack and model it as three capacitors in series. We isolate the capacitance of the QD film by subtracting the effects of the capacitance of the HfO2/parylene control device. Our experiments yield values of dielectric constants as a function of the volume fraction of QDs in the thin films. We find that our data do not fit within any simple model that applies the bulk dielectric constant of PbS. We suggest that surface and other size effects play a role in altering the dielectric constant of the individual QDs, as has been observed in the nanoscale Si/SiO2 system11−13 and supported by previous theoretical and computational studies.14−20 QD Sizing Curve. The first step in calculating the volume fraction of QDs in a thin film is to precisely determine the size of the QDs themselves. To do this, we use a small-angle X-ray

n quantum dot (QD) photovoltaics and light-emitting diodes, the dielectric constant is an important parameter in device design but it is not well measured. In this work, we perform a thorough study of the dielectric constant of lead sulfide (PbS) QD films by varying the QD size and keeping the ligand constant. We measure the volume fraction of these films by measuring the QD size and spacing in the film. We use a capillary small-angle X-ray scattering (SAXS) technique to create a reliable QD sizing curve, SAXS measurements of thin films to extract a pair-distribution function for QD spacing, and a stacked-capacitor geometry to measure the capacitance and determine the film dielectric constant. Generally, the dielectric constant of a composite film is estimated by a volume-fractionweighted average of the component materials or another effective-medium theory (Maxwell-Garnett, Bruggeman).1 Because bulk PbS has a high static dielectric constant of 169,2 there is a large contrast between component dielectric constants in PbS QD thin films. This high bulk dielectric constant also suggests that increasing the volume fraction of QDs, for example, by shortening the spacing between QDs by attaching a shorter ligand or using larger QDs in the films, could result in a large increase in film dielectric constant. This change in QD density would then permit longer depletion regions and enable increased thin-film absorption and photovoltaic efficiency. Device design relies on an accurate dielectric constant calculation; errors in this value can result in errors in device architecture, for example, to determine pillar spacing in a lateral heterojunction architecture.3 Most literature reports of QD thin-film dielectric constants are estimated using an effective-medium theory.4−6 Two reports of PbS-QD film dielectric constants have been derived using the real and © XXXX American Chemical Society

Received: June 28, 2014 Revised: December 4, 2014

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experimental data are shown to the right of the curve, in addition to the average diameter of the corresponding simulation curve. We find these average diameters to range from 2.97 to 5.69 nm (see Supporting Information). We use these data to establish a sizing curve with a second-order polynomial fit, such that the (average) size of subsequent samples can be determined using only the location of the first exciton peak. Figure 2 shows this sizing curve alongside other

scattering (SAXS) method where a concentrated QD solution in octane is placed air-free into a capillary tube. We match the experimental low-angle signals with computed scattering of a simulated PbS QD that takes into account the lattice and individual atomic scattering factors.21 Because this measurement takes place in solution such that the interparticle spacing is both variable and outside of the region probed, this sizing method only measures X-ray scattering of individual QDs rather than interparticle interactions. The analysis method is the same as that used for QD self-scattering of X-rays in Murray et al.21 We note that a high degree of monodispersity in the QD samples is necessary for this measurement to ensure a high quality of fit between the experimental and simulated scattering and that the simulation assumes a spherical QD shape (which may vary as a function of synthesis method). We chose this sizing method over the more common transmission-electronmicroscope (TEM) sizing method because it requires no user input or thresholding, is not sensitive to systematic TEM magnification or tilt calibrations, and demonstrates a higher precision. We used this sizing technique on QDs with a range of first-exciton peak wavelengths from 844 to 1425 nm (Figure 1a), where the samples remained air-free from synthesis to absorption measurement. Figure 1b shows the X-ray intensity as a function of angle (2θ) and the matching simulation curve for each of ten differently sized samples. The location of the first exciton peak (λabs) and the emission peak (λPL) from the

Figure 2. Sizing curve plotted with literature sizing values. This solution-SAXS sizing method maps a given diameter to a larger bandgap than other methods22−27 or a given peak location to a larger diameter. This result suggests that other methods may underestimate the size of the QD particles, which is consistent with previous observations of a poorly resolved surface layer in TEM images of CdSe QDs that is resolved in X-ray analysis.28

literature values for sizing. This solution-SAXS sizing method maps a given diameter to a larger bandgap than other methods22−27 or a given peak location to a larger diameter. This result suggests that other methods may underestimate the size of the QDs, which is consistent with previous observations of a poorly resolved surface layer in TEM images of CdSe QDs that is resolved in X-ray analysis.28 QD Spacing in a Film. The second step to measuring the volume fraction of QDs in a close-packed film is to measure the center-to-center distance between particles. We performed SAXS measurements on spin-cast thin films of QDs of a range of sizes, which had all undergone a solid-state ligand exchange with tetrabutylammonium iodide (TBAI), a process commonly employed for photovoltaic QD applications.29 The SAXS data for the TBAI-treated films are shown in Figure 3a. The most prominent peak of the SAXS data shifts to smaller 2θ angles with larger QD size. The smaller peak at higher 2θ angles in each of the four curves matches the location of the solutionSAXS peaks for the corresponding QD size. Though it is tempting to apply a standard Bragg conversion of 2θ to distance and identify the peak d-position as the nearest neighbor distance, correct determination of the nearest-neighbor distance requires extracting the pair-distribution function21 because the particles are not arranged in a crystalline lattice. The calculation of the pair-distribution function takes into account the particle geometry and form factor. The pair-distribution function was calculated from these data using the program GNOM.30 Figure 3a shows the SAXS data (open circles) and their GNOM fits (solid lines) of four samples ranging in size (2.92, 3.58, 4.49, and 5.78 nm, with first exciton peaks at 841, 966, 1155, and 1407 nm in solution), as determined using the solution-SAXS sizing curve from the previous section. The minor peaks at larger 2θ values are self-scatter peaks like those shown in Figure 1. Figure 3b shows the pair-distribution functions, P(r), as

Figure 1. Development of the PbS QD sizing curve. (a) Absorption and photoluminescence spectra for all sizing samples. (b) Solution SAXS data and simulated fits for all sizing samples. Absorption and photoluminescence peak positions are noted in addition to the matching simulated diameter. B

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profilometry. By incorporating the known QD film thickness and device area, and subtracting the contributions of the HfO2 and parylene layers, the capacitance of a sample yields the dielectric constant of its QD film. Figure 4a shows the dielectric

Figure 3. (a) Raw film SAXS data (open data points) and GNOM fit (solid black line). The minor peaks at larger 2θ values are self-scatter peaks like those shown in Figure 1. (b) Extracted pair-distribution function for fits in (a) using GNOM. Error bars are calculated by GNOM. The peaks of the P(r) curves are the center-to-center spacings for nearest neighbors and other features of the curves are a result of self-scattering and limited measurement range.

Figure 4. Measurement and evaluation of the film dielectric constant. (a) Dielectric measurements as converted from capacitance as a function of frequency, after control subtraction. The bands of each color denote measurements of different chips with the same size of QDs used in fabrication. The width of the color band indicates the deviation of calculated dielectric constants based on the thickness variation of the QD film for each chip (the thickness variation is significantly larger than the measurement error in thickness). (b) Impedance phase angle as a function of frequency to show strong capacitive behavior over a wide frequency range. The inset shows the layered structure of the sample device (left) and control device (right).

determined by GNOM, as labeled by their λabs. The P(r) values are normalized and offset on the y axis for clarity. The peaks of the P(r) curves in Figure 3b are the center-to-center spacings for nearest neighbors and other features of the curves are a result of self-scattering and artifacts of the limited measurement range. The center-to-center spacings extracted from the pairdistribution function (4.5, 4.6, 5.3, and 6.4 nm) are consistently larger than the size of the particles determined from the sizing curve. Dielectric Measurement Using Stacked Architecture. The capacitance of the QD film was measured by AC impedance spectroscopy using a stacked-capacitor geometry so that 1 1 1 1 = + + C tot C HfO2 CQDfilm Cparylene =

dHfO2 A ϵHfO2

+

dQDfilm A ϵQDfilm

+

constants of the films as a function of frequency for the size series of QDs. The bands of each color denote measurements of different chips with the same size of QDs used in fabrication. The width of the color band indicates the deviation of calculated dielectric constants based on the thickness variation of the QD film for each chip (the thickness variation is significantly larger than the measurement error in thickness). The capacitance (and therefore also the extracted dielectric constant for a given film thickness) is constant across several orders of magnitude of frequency. Figure 4b shows the impedance phase angle of the stack containing the QD layer as a function of frequency, which is relatively constant at frequencies lower than 104 Hz and very near to the pure capacitive phase angle of 90 deg. Discussion. To convert the QD size and spacing data into volume fraction, we assume that the QD-ligand units are randomly close packed. Mathematically

d parylene A ϵparylene

(1)

where C is capacitance, d is film thickness, A is device area, ϵ is the dielectric constant, and the subscripts denote the material to which these values belong. This architecture was engineered to have a high-dielectric oxide and thin insulating layers (∼25−50 nm) to maximize the relative contribution of the QD film layer to the total capacitance. A control device with only HfO2 and parylene was also measured and the inverse of its capacitance was subtracted from the inverse of the total capacitance to yield the inverse of the QD film capacitance. The thickness (in the range of 100−200 nm) of the QD film is measured using a profilometer and the device area (4.86 mm2) is known from the mask geometry and confirmed via optical microscopy and

η = packing fraction ×

4 3 πr 3 QD QD − QD spacing 3 4 π 3 2

(

)

(2)

where η is the QD volume fraction and rQD is the QD radius determined from the sizing curve in Figure 1. C

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The reported value for the dielectric constant of the QD film is found by averaging the values measured between 1000 Hz and 10 000 Hz because the phase angle is consistently flat and less noisy in this frequency range. We note that the experimental uncertainty is dominated by chip-to-chip variations rather than frequency or thickness variations within a single chip. Figure 5 compares the dielectric constant data

ϵ=

β 2 + 8ϵ1ϵ2 )

β = (3η1 − 1)ϵ1 + (3η2 − 1)ϵ2

(6)

where ϵ is the dielectric constant of the composite, ϵ1 is the dielectric constant of the inclusion material, ϵ2 is the dielectric constant of the matrix, η1 is the volume fraction of the inclusion material, and η2 is the volume fraction of the matrix (η2 = 1 − η1 for the two-component case considered here). Each of these simple effective-medium models is predicated on the local field being the same throughout the particle.1 This may be an incorrect assumption for close-packed QD films on the scale of the relevant polarizations because the high surfaceto-volume ratio results in a much higher relative number of polarized QD-ligand bonds at the surface than in conventional theories. The linearity of the data suggests that some volumeweighted theory may be appropriate, but the shallower slope implies that the effective QD dielectric constant is smaller than that of the bulk. The deviation from simple models is an indicator of an effectively lower dielectric constant of PbS when it is in nanoscale structures instead of in the bulk. This sizeregime-dependent dielectric constant of the particles in the film is expected based on experimental measurements11−13 and theoretical and computational studies14−20 primarily in the nanoscale Si/SiO2 system, for which both band gap expansion and surface reconstruction have been discussed as explanations for the dielectric constant reduction. Conclusion. We have provided a reliable measure of QD size using solution SAXS, of QD−QD spacing using the pairdistribution function from thin-film SAXS data, and of QD-film dielectric constant using a stacked capacitor geometry and have found that the dielectric constant is not accurately predicted by simple effective-medium theories. We propose that the dielectric constant of a QD film is strongly affected by the nanoscale nature of the material due to surface and other size effects. We conclude that no simple effective-medium theory model can be employed for these nanoscale films using the bulk PbS dielectric constant and that a size-regime-dependent dielectric constant like those observed and discussed in Si/ SiO2 work is supported by this experimental study. This conclusion has implications in device architecture design; higher film dielectric constants allow for increased depletionregion depth. Because the intrinsic dielectric constant of QDs is limited by the nanoscale nature of the material, higher-dielectric ligand matrices may enable higher film dielectric constants and consequently deeper depletion regions. By identifying the inadequacy of common effective-medium models for the dielectric constant and providing a method to empirically determine a film dielectric constant, we hope to aid both theorists and experimentalists in understanding the nanoscale physics and engineering and improving QD optoelectronic devices. Experimental Methods. PbS QD Synthesis and Purification. PbS QDs were synthesized according to a previously published procedure.33 The relative amount of oleic acid and octadecene in the initial four-necked flask was varied to create different QD sizes. For the smallest particle sizes, the injection temperature was lowered from 150 to 120 °C or 90 °C. Air-free cannula-transfer techniques were used to transport the crude

Figure 5. Dielectric measurements compared with several effectivemedium models. The data do not fit to these simple effective-medium models with a PbS dielectric constant of 169, as in the bulk. The linearity of the data suggests that some volume-weighted theory may be appropriate, but the shallower slope implies that the effective QD dielectric constant is smaller than that of the bulk.

from our stacked capacitor method against three simple models using the bulk dielectric constant of PbS, ϵ = 169 and a matrix dielectric constant of 2, which is a reasonable approximation for a combination of air and any residual organics. Volumeweighted effective-medium theory predicts a linear relationship between the volume fraction occupied by QDs, a slope that reflects the value of the PbS static dielectric constant (169 at 300 K),2 and a small y intercept set by the static dielectric constant of the matrix surrounding the QDs ϵtotal = (ϵQD − ϵmatrix )*ηQ + ϵmatrix

1 (β + 4

(5)

(3)

Volume-weighted effective medium theory significantly overestimates the dielectric constant. Maxwell−Garnett effective-medium theory is frequently referenced for nanoparticle film dielectric calculations.4,5,13,31 Its functional form weights the dielectric constant of the surrounding medium more heavily than the dielectric constant of the inclusion material ϵ −1 ϵ−1 = η1 1 ϵ+2 ϵ1 + 2 (4) where ϵ is the dielectric constant of the composite, ϵ1 is the dielectric constant of the inclusion material, and η1 is the volume fraction of the inclusion material. Maxwell−Garnett effective-medium theory significantly underpredicts the dielectric constant, which is not surprising because it is optimized for very dilute solutions of particles or inclusions.1 Bruggeman effective-medium theory is better suited for highvolume fractions because it incorporates a threshold volume fraction and material parity and reaches the bulk value at a volume fraction of unity,1 but this theory also clearly does not match the shape or values of the measured dielectric constants. The characteristic equations of Bruggeman effective-medium theory are D

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precise distance value was calibrated using a silver-behenate standard in relevant geometry (a powder on a glass surface for film samples and powder coating a glass capillary for solution samples) and matching its circularly integrated peak positions to standard values. After loading each QD sample, the position was adjusted to optimize the amount of signal and data was then collected for 30 min. The GADDS program was used to perform chi integration (circular for solution samples and semicircular for film samples). Capacitor Measurement. AC impedance spectroscopy was performed using a Solartron 1260 impedance analyzer. No DC voltage was applied, the AC voltage was 10 mV, and the frequency was varied. The system was nulled using an evaporated gold film for the short-circuit current and without a chip in the setup for the open-circuit voltage. The impedance testing setup is housed in a dark box with low-noise cables in a glovebox.

solution into a N2 glovebox without exposure to air. Within 1 h after synthesis, the crude solution was purified by adding either a butanol/methanol mixture or acetone and was centrifuged at 3.9 krpm for 3 min. The supernatant was discarded and the precipitate was dissolved in a minimal amount of hexane for storage (in the glovebox and with anhydrous solvents). Further purification was performed more immediately prior to the various measurements. For the solution SAXS measurements, a very careful size-selective precipitation was performed in a glovebox using butanol or, in some cases, a butanol/methanol mixture. The supernatant of the first size-selective crash out of storage was precipitated separately, such that there are two solutions of similar size obtained from each synthetic batch. Two total precipitations were performed on each initially stored solution and the final precipitate was dissolved in a minimal amount of octane for solution measurements. For the dielectric devices and film SAXS measurements, the solutions were precipitated from the storage solution in a volumetric ratio of 4:2:1 QD solution:butanol:methanol. Two precipitations were performed on each initially stored solution, and the final precipitate was dissolved in octane at a concentration of ∼50 mg/mL for deposition. SAXS Sample Preparation. All sample preparation was performed in a nitrogen glovebox. For each solution SAXS sample, a capillary (0.5 mm, glass number 50 from Hampton Research) was snapped toward one end in the thin section, and the thin end of the capillary was dipped in a concentrated QD solution (in octane). The capillary was removed from the solution before the meniscus reached the widened portion of the capillary and was tipped horizontally to leave a gas bubble at either end of the capillary. The thin end of the capillary, while still being held horizontally, was pressed into a slab of Critoseal and twisted to seal the thin end of the capillary. A small fragment of Critoseal was packed into the wide, top end of the capillary using tweezers; the press-and-twist method was deemed ineffective because it induced a pressure in the capillary that dislodged the sealant of the thin end. The sealed capillary tubes were removed from the glovebox and the SAXS measurement was performed within several hours. For each film SAXS sample, six layers of QDs were deposited using the following procedure for each layer: deposit 10 μL of QD solution on silicon wafer fragment (blow-dried with nitrogen prior to use), spin at 1000 rpm for 10 s, deposit ∼100 μL of solution of 10 mg/mL TBAI in methanol, wait 30 s, spin at 1300 rpm for 5 s to remove the treatment solution, deposit ∼100 μL of methanol, and spin at 1300 rpm for 5 s to remove the methanol rinse solution. Film samples were removed from the glovebox and measured within several hours. Stacked Capacitor Device Fabrication. Patterned ITO was purchased from Thin Film Devices in Anaheim, CA. The ITO was cleaned by sequential sonication in water/detergent, DI water, acetone, and isopropanol and blow dried with nitrogen. HfO2 (25 nm) was deposited via sputtering. Six layers of QDs were deposited as described for SAXS film sample preparation in a glovebox using TBAI ligand treatment. The samples were transferred without air exposure to the parylene deposition system, where 25 nm of parylene was deposited via chemical vapor deposition. An aluminum electrode was deposited by thermal evaporation. SAXS Measurement. Small-angle X-ray scattering measurements were made on a Bruker D8 with GADDS and 2D detector SAXS attachment and 0.05 mm monocap, with copper Kα radiation. The detector distance was set at ≥40 cm and the

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ASSOCIATED CONTENT

S Supporting Information *

Solution SAXS simulations and data with size-matching information, film SAXS data, and values for samples used in stacked-capacitor measurements. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was primarily supported by Samsung SAIT. D.D.W. and P.R.B. gratefully acknowledge support from the John and Fannie Hertz Foundation, and P.R.B. acknowledges support from the National Science Foundation. Pearl Donohoo-Vallett assisted with the iterative algorithm and wrote a portion of the MATLAB code for the solution sizing background and scaling to match simulated and experimental curves. Dave Strasfeld was a part of the idea conception for the measurement of QD film dielectric constants. Ankit Rohatgi, the author of WebPlotDigitizer32 greatly eased the collection of literature data for sizing curve comparison. Scott Speakman provided valuable discussion and experimental assistance for both the solution and film SAXS measurements.

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REFERENCES

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