Supplementary Information Charge transport and ... · Scattered X-rays were collected by a Pilatus 200K detector at a distance of 902 mm with an exposure time of 3 s. Supplementary
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Supplementary Information
Charge Transport and Localization in Atomically Coherent Quantum Dot Solids
Kevin Whitham1, Jun Yang2, Benjamin H. Savitzky3, Lena F. Kourkoutis2,4, Frank Wise2, Tobias Hanrath5
1Materials Science and Engineering, Cornell University, Ithaca, NY 14853 2School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853 3Department of Physics, Cornell University, Ithaca, NY 14853 4Kavli Institute for Nanoscale Science, Cornell University, Ithaca, NY 14853 5Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853
Calculation of Nanocrystal Diameter and Distribution
Following purification in a nitrogen glovebox, a solution was prepared for absorbance measurement. The
purified nanocrystal solution was diluted by vacuum drying 100 µL from hexane and redissolving in 3 mL
tetrachloroethylene. The solution was sealed in a quartz cuvette before removal from the glovebox. Background
contributions from the instrument, air, and solvent were accounted for by measuring a baseline spectrum with neat
tetrachloroethylene.
The mean diameter and distribution were calculated following the method of Moreels et. al.1
Supplementary Information Figure S1 shows the lowest energy exciton absorption feature. We determined the peak
location and peak width by fitting a Gaussian function after subtracting a linear background from the high energy
tail to the low energy tail of the peak. The peak location of 1821 nm or 681 meV gives a mean NC diameter of
6.5 nm using the empirical relation by Moreels et. al. 𝐸𝐸! = 0.278 + (0.016𝑑𝑑! + 0.209𝑑𝑑 + 0.45)!! where 𝐸𝐸! is the
mean energy of the lowest energy exciton in eV and 𝑑𝑑 is the NC diameter in nm. The standard deviation of the NC
diameter was calculated by converting the standard deviation of the excitonic peak using the same equation. The
standard deviation of the Gaussian peak is 76.5 nm giving energies of 710.8 meV and 653.5 meV. These energies
give diameters of 6.08 nm and 6.92 nm respectively and therefore the standard deviation of the NC diameter is
0.42 nm.
Charge transport and localization in atomicallycoherent quantum dot solids
The nanocrystal diameter distribution was also determined by fitting the form factor measured by small-
angle X-ray scattering (SAXS) from the as-synthesized NCs in solution. The SAXS data was acquired using an
environmentally controlled sample chamber at the D1 station of the Cornell High Energy Synchrotron Source
(CHESS). The sample chamber consisted of an aluminum enclosure with two kapton windows mounted on a 4-axis
goniometer. A teflon block with a 1 cm x 1 cm x 5 mm rectangular well was filled with anhydrous ethylene glycol.
A separate solvent reservoir inside the enclosure was filled with 1 mL of hexane before sealing the chamber. The
sample chamber was positioned such that the X-ray beam grazed the apex of the curved ethylene glycol liquid
surface. The enclosure was purged with helium to decrease oxygen and then left static to allow hexane in the
reservoir to reach equilibrium vapor pressure at the ambient temperature of 23 C. A 25 μL droplet of a 5 μM
concentration of PbSe NCs in hexane was then deposited on the ethylene glycol liquid surface using a 100 μL air-
tight syringe though a septum on top of the enclosure.
The scattered X-rays were recorded by a Pilatus 200k detector positioned 902 mm from the sample and
calibrated using a silver behenate standard and the attenuated direct beam. Incident radiation had a flux of ����
photons/s mm2 and a wavelength of 0.1157 nm. The image was corrected for dark current. We used the software
package Fit2D to azimuthually integrate the image shown in Supplementary Figure S2(a) over the azimuthal range
indicated. The integrated intensity was fit using a spherical form factor with a Gaussian distribution of the sphere
radii2. The form factor of a sphere with radius � is given by ������� ��
���
� �������������������
�����. The distribution
of the NC radii with an average radius �� and standard deviation � was modeled as ���� � �
������
��������
���.
Figure S1 | Absorbance spectrum of PbSe nanocrystal colloidal solution. The mean nanocrystal diameter andstandard deviation were found by fitting a Gaussian function to the first excitonic peak after subtracting a linear background.
Determination of Superlattice Disorder from X-ray Scattering
To quantify disorder of the superlattice we measured grazing incidence small-angle X-ray scattering
(GISAXS) from the nanocrystal film on the surface of the field-effect transistor used for the transport measurements
shown elsewhere in this work. The sample was prepared for GISAXS after transport measurements by dissolving
Figure S2 | Nanocrystal size and distribution from X-ray scattering. a, Small angle X-ray scattering of a colloidal solution of PbSe nanocrystals. Color values are arbitrary units and scaled logarithmically. b, Azimuthally integratedintensity from the outlined area of the scattering image. Lines show form factor functions for a sphere, a cube, and atruncated cube, see text for details.
Determination of Superlattice Disorder from Microscopy
To quantify superlattice disorder we analyzed radial distribution functions g(r) of nanocrystal positions in
TEM images. We considered two models to fit g(r): with and without correlation. Uncorrelated disorder is modeled
by equation (1), where the coherence between particles is independent of the interparticle distance.
� � � ���
���
�
�������
� ��� ������
������ � ���
� � � ���
���
�
��� � � � �������
� ��� ������
���� � � � ���� � ���
Paracrystalline disorder is described by equation (2), in which coherence decreases with interparticle
distance6. In both equation (1) and equation (2) L is the superlattice constant, r is the radial distance, σ is the
standard deviation of the nearest neighbor distance, and m,n are the indices for the two-dimensional lattice vectors.
Because oriented attachment requires the correlated motion of NCs, it is not surprising that the measured g(r) is
better described by equation (2). In Figure S4 we show fits using equation (1) and equation (2) for comparison. The
fitting was calculated by setting the superlattice constant L to 6.6 nm, the average nearest neighbor distance between
NCs in the image. The reflection of the superlattice {10} planes in the GISAXS image at 0.98 nm-1 gives
L = 6.4 nm, a difference of 2 Å compared to the TEM image, less than a single Pb-Se bond length. We find the
standard deviation of the nearest neighbor distance σ = 0.22 nm, or 3.4% relative to the superlattice constant.
Figure S3 | Paracrystalline superlattice structure analysis by GISAXS. a, Scattering from a NC superlattice on a field effect transistor. The color scale is logarithmic with arbitrary units. The Yoneda band is marked by red dashed lines.b, Scattered intensity along the Yoneda band. Solid lines show calculated intensity from a square lattice using aparacystal disorder model or a Debye-Waller disorder model.
Determination of epitaxially connected nanocrystal diameter distribution
Measuring NC diameters from TEM images is commonly performed using image analysis software
packages such as ImageJ. However, these packages are not reliable if the boundary of the NC is not distinct such as
when NCs are joined by an epitaxial bridge. The diameters could be determined by measuring each NC individually
by hand, but this is not feasible when analyzing dozens of images with thousands of NCs per image. Additionally,
hand measurement introduces user bias. Therefore, we wrote an automated image analysis algorithm to measure the
diameter of the NC cores from a TEM image of an epitaxial superlattice.
The main operations performed by the algorithm are threshold and watershed. Briefly, the algorithm first
performs a threshold operation to convert the greyscale image to a binary image based on the distribution of pixel
values. The threshold operation attempts to separate NC sample pixels from empty background pixels. The binary
image is then converted to a linear scale image where the value of each pixel represents the Euclidean distance from
that pixel to the nearest background pixel hereafter called a distance image. The sample pixels are then separated
into groups, each group representing a NC. The groups are determined by a watershed operation on the distance
image. The location of each NC is then found by calculating the center of mass of the group of pixels belonging to
each NC pixel group. The diameter of each NC is calculated as the average of the major and minor axes of an ellipse
fit to the group of pixels belonging to each NC. The threshold operation is repeated, this time after dividing the
original greyscale image into sections. Each section is a rectangle with dimensions some multiple of the average NC
Figure S4 | Analysis of the superlattice radial distribution function. a, Data measured from TEM imageanalysis overlaid with the calculated radial distribution function of a paracrystalline square lattice with latticeconstant of 6.6 nm and standard deviation of the nearest neighbor distance of 0.22 nm. b, The same data overlaid with the best fit of a square lattice radial distribution function with static disorder. The lattice constant is 6.6 nm and the standard deviation of the nearest neighbor distance is 0.38 nm.
diameter found after the first threshold operation. Running the threshold operation on each section rather than the
entire image improves the quality of the binary image by compensating for non-uniform illumination in a bright-
field image or non-uniform scattering due to sample thickness variation in a dark-field image. The watershed
operation is repeated as before on a distance image generated from the binary image sections to refine the NC
locations and diameters. The output of the algorithm on a dark-field scanning TEM image is shown in Figure S5.
We find the average diameter of the 1,693 NCs in the image shown in Supplementary Figure S5 to be
5.75 nm with a standard deviation of 0.19 nm. This is comparable to the average NC diameter and standard
deviation (6.1 nm ± 0.3 nm) used to calculate the form factor fit to GISAXS data from an epitaxially connected
superlattice (see Supplementary Figure S3).
Determination of Epitaxial Connection Width and Connectivity
We measured the distribution of epitaxial connection widths using microscopy images. We used both
atomically resolved dark field STEM and bright field TEM images of monolayer samples. We used the software
package ImageJ to measure the width of 124 atomically resolved epitaxial connections by hand, drawing a line
across each connection at the narrowest part. In Supplementary Figure S6 we show each connection and a histogram
of the measurements. We find the mean connection width to be 2.9 nm with a standard deviation of 0.68 nm. We
also analyzed lower magnification bright field TEM images for better statistics. Supplementary Figure S7 shows
such an image with 10,122 epitaxial connections.
Figure S5 | Measurement of NC diameter from a transmission electron micrograph. a, An annular dark-fieldscanning transmission electron micrograph of a NC superlattice. b, The location and diameter of 1,693 nanocrystals inthe image was calculated using in-house code. The NC locations and diameters are represented by red circles overlaidon the image. c, A histogram of the diameters found in the image, with an average of 5.75 nm and a standard deviation
To measure the large number of connections in the bright field image, we wrote a software routine. First we
generate a Voronoi diagram from the NC locations. The line segments that make up the cells of the Voronoi diagram
intersect the epitaxial connections midway between each nearest neighbor pair of NCs. To measure the width, we fit
an approximately square function to the pixel values along the Voronoi diagram lines, as shown in Supplementary
Figure S8. We use a sum of Gaussian functions instead of a true square function because the function must be
differentiable for the least-squares minimization fitting routine. The fit function is given by
���� � ����������������
�
��������������� where �� is the midpoint of the connection and � is the connection width. The
parameters ���� are found by minimizing the sum of the residuals ���� � ���� where ���� is the intensity at pixel
�.
Figure S6 | Distribution of epitaxial connections from an atomically resolved image. a, Annular dark field scanningtransmission electron microscope image of a monolayer superlattice overlaid with lines marking the widths of 124 epitaxial connections. b, Histogram of the epitaxial connection widths shown in the image. The mean width is 2.9 nm witha standard deviation of 0.68 nm.
Figure S7 | Distribution of epitaxial connections from TEM. A bright field transmission electron micrograph of a monolayer superlattice. The location and width of 10,122 epitaxial connections are indicated by solid lines. Upper insetshows a magnified view of the region outlined with a black rectangle. The inset histogram shows an average width of3.26 nm with a standard deviation of 0.48 nm.
Figure S8 | Automated analysis of epitaxial connection width. a, Line profile of pixel intensities across an epitaxialconnection. Blue markers are pixel intensity values, red line is a fit to the pixel values. b, The epitaxial connection underanalysis. The image is a bright field TEM image that has been inverted such that the background is black and the NCs are white. The blue markers indicate the pixels analyzed. c, The red line indicates the location and width of the epitaxialconnection from the fit.
Transmission electron microscopy is useful to determine the lateral structure, but does not give quantitative
information normal to the plane of the sample. The samples studied are thinner than the extinction length of the
electrons, therefore quantitative measure by the Kossel-Möllenstadt technique is not possible. Relative thickness
(monolayer, bilayer, etc.) can be determined by scattered electron intensity, but absolute thickness determination is
not straightforward. Furthermore, gathering sufficient statistics on the thickness over a large area by microscopy is
challenging. Therefore we compared GISAXS data to calculated scattering intensity by monolayers, bilayers, etc.
Figure S9 | Connectivity of a square superlattice. Solid green lines indicate epitaxially connected neighbors, reddashed lines indicate unconnected neighbors. Of the 13,937 nearest neighbors analyzed, 3,815 are not connected,yielding a connectivity of 73%.
Figure S10 | Estimation of superlattice thickness by GISAXS. a-c,e-g, Calculated scattered intensity fromsuperlattice models one layer (1L) to six layers (6L) thick. Color scale is logarithmic. d, Intensity along the {10} Bragg rod for 1-6 layers and 10 layers (2D image not shown) compared to experimental data (Exp.). h, Measured scatteredintensity. The black region is from a beam-stop. Color scale is logarithmic.
Calculation of Field Effect Mobilities and Hysteresis
Field effect mobilities were calculated from source-drain current vs. gate voltage in the linear regime using
the square law relationship that assumes the charge density can be approximated by a 2D sheet at the dielectric-
semiconductor interface: � � ��� ������ where �� � ��� ��� is the transconductance, ��� are the channel
length (100 μm) and width (0.3 cm), ��� is the specific capacitance of the SiO2 gate dielectric, and �� is the source-
drain voltage. The specific oxide capacitance was calculated to be 1.7 x 10-8 (F/cm2) using the parallel plate
approximation ��� � ���� � where �� is the vacuum permittivity, and �� is the relative dielectric constant of SiO2
(3.9), and � is the oxide thickness (200 nm). The hole and electron mobilities at 245 K were calcualted using the data
in Supplementary Figure S11 to be 0.54 cm2/Vs and 0.2 cm2/Vs respectively.
Figure S11 | Transistor transport characteristics. a, Source-drain current versus source-drain voltage of a nanocrystal superlattice field-effect transistor at 245 K. b, Same data as panel (a) on a logarithmic scale. c, Source-drain currentversus gate voltage. Data marked by open circles was acquired while sweeping the gate voltage from -40 V to 40 V. Datamarked by crosses was acquired while sweeping the gate voltage from 40 V to -40 V. The sweep rate was 1 V/ms. d, The absolute difference between the current measured during forward and reverse direction gate scans.
Figure S12 | Encapsulation of FET device and transport characteristics. a, Cross section schematic illustration ofthe FET device. b, Transfer curves from one device before and after encapsulation measured at 295 K showingambipolar transport and reduced hysteresis after encapsulation. The channel dimensions were 3 mm x 0.1 mm. Thesource-drain voltage was 1 V. The SiO2 dielectric layer thickness was 200 nm. c, Molecular structure of the encapsulation layer monomer, pentaerythritol-tetrakis(3-mercaptopropionate) (tetrathiol), and the cross-linking agent, 1,3,5-triallyl-1,3,5-triazine-2,4,6(1H,3H,5H)-trione (TATATO). d, Transfer curves of a NC FET device measured at 295 K after coating with TATATO or tetrathiol independently without UV exposure. Channel dimensions were 3 mm x 0.1 mm. The source-drainvoltage was 1 V. The SiO2 dielectric layer thickness was 200 nm.
For a wavefunction to be delocalized, the fractal dimension must be at least 1. There is a one to one correspondence
between fractal dimension and wavefunction coherence length. Localization can be expressed as a length in
dimensionless units of a0. A completely localized system with fractal dimension of zero corresponds to a
localization length of one.
The fitted lowest conduction band (CB) and highest valence band (VB) for a square superlattice calculated
with the parameters summarized in Table 1 is plotted in Supplementary Figure S13. The coupling energy
corresponds to an epitaxial connection width of 2.45 nm or eight Pb-Se bonds. The truncation factor is 0.45, i.e., the
epitaxial connection width is 45% of the NC diameter.
Compared with the results of Kalesaki et al.17 the effective Hamiltonian captures the essential features of
the band-structure. From the fitting parameters, it is also clear that coupling between the same valley is an order of
magnitude stronger than in different valleys.
Figure S13 | Lowest conduction band and highest valence band of a square superlattice. a,c,Reproduced from literature, calculated by an atomistic method. b,d, Calculated using the effective four-bandHamiltonian. The zero energy reference is the valence band maximum of bulk PbSe.
In the atomistic calculation17, it was stressed that the sign of the coupling strength ��� depends on whether
the number of biplanes in the NCs of the superlattices is even or odd, (also see Table 1) and that affects whether the
conduction band minimum and valence band maximum are at the � or the � point. However, the sign of the basis
function of the tight-binding model is not fixed. In a square or cubic superlattice, one can choose a particular basis
set so that the wavefunction of every other NC switches sign, and thus the sign of the coupling strength ��� and ���
reversed. This is equivalent to moving the Brillouin zone center from � to � in the extended zone scheme.
There is a one to one correspondence between fractal dimension and coherence length. Supplementary
Figure S15 shows the coherence length vs. fractal dimension �� for 400 different combinations of disorder and
energy calculated in a 2D square lattice using a single-band model. The fractal dimension is calculated in the same
way; the coherence length is calculated using a previous method20. It is clear that all these points lie on a single
universal curve, which shows that coherence length and fractal dimension describe the same thing (degree of
delocalization) and they have a one to one correspondence. For a fractal dimension of ���, the coherence length (the
characteristic decay length of the wavefunction) is only 2; if the fractal dimension is 0.3, coherence length increases
to �; if the fractal dimension is 0.7, the coherence length increases to 10. The non-linear relationship of coherence
length on fractal dimension highlights the sensitivity of wavefunction localization to the amount of disorder.
Table 1 | Parameters used in the fitting of square lattice for 4.89 nm, 4.28 nm, and 6.53 nm diameter PbSe NCs.All units are eV. Notes: (a) interpolated11 (b) extrapolated17.
Figure S14 | Density of states calculated from the tight-binding model. Connectivity is the percentage of nearestneighbor pairs in the sample that are coupled by an epitaxialconnection. Error bars are the standard deviation of five MonteCarlo calculations.
Figure S15 | Localization length vs. fractal dimension. All points lie on a universal curve, shown by the solid line.
Figure S16 | Scaling of localization length with disorder parameters. a-c, Calculated fractal dimension (d2) and localization length for states in the 1Se or 1Sh bands. a, Effect of NC size disorder with three values for the standarddeviation of the distribution of NC 1Se or 1Sh energies. The epitaxial connection disorder was 0.92 nm. The connectivity was 90%. b, Effect of epitaxial connection width disorder, with three values for the standard deviation of the epitaxialconnection width. The NC size disorder was 11 meV. The connectivity was 90%. c, Effect of connectivity disorder, withthree values for the percentage of nearest neighbors that are energetically coupled by an epitaxial connection. The NC size disorder was 11 meV. The epitaxial connection width disorder was 0.92 nm.
1. Moreels, I. et al. Composition and size-dependent extinction coefficient of colloidal PbSe quantum dots. Chemistry of Materials 19, 6101-6106 (2007).
2. Lazzari, R. IsGISAXS: a program for grazing-incidence small-angle X-ray scattering analysis of supported islands. Journal of Applied Crystallography 35, 406-421 (2002).
3. Evers, W. H. et al. Low-dimensional semiconductor superlattices formed by geometric control over nanocrystal attachment. Nano Lett 13, 2317-2323 (2013).
4. Hendricks, R. W., Schelten, J. & Schmatz, W. Studies of voids in neutron-irradiated aluminium single crystals: II. Small-angle neutron scattering. Philosophical Magazine 30, 819-837 (1974).
5. Smilgies, D.-M., Heitsch, A. T. & Korgel, B. A. Stacking of hexagonal nanocrystal layers during Langmuir Blodgett deposition. The Journal of Physical Chemistry B 116, 6017-6026 (2012).
6. Vogel, W. & Hosemann, R. Evaluation of paracrystalline distortions from line broadening. Acta Crystallogr A Cryst Phys Diffr Theor Gen Crystallogr 26, 272-277 (1970).
7. Smilgies, D.-M. Scherrer grain-size analysis adapted to grazing-incidence scattering with area detectors. Journal of applied crystallography 42, 1030-1034 (2009).
8. Shklovski, B. I. & Efros, A. L. Percolation theory and conductivity of strongly inhomogeneous media. Soviet Physics Uspekhi 18, 845 (1975).
9. Mott, N. F. Conduction in glasses containing transition metal ions. Journal of Non-Crystalline Solids 1, 1-17 (1968).
10. Kang, I. & Wise, F. W. Electronic structure and optical properties of PbS and PbSe quantum dots. JOSA B 14, 1632-1646 (1997).
11. Allan, G. & Delerue, C. Confinement effects in PbSe quantum wells and nanocrystals. Physical Review B 70, 245321 (2004).
12. Skal, A. S. & Shklovsky, B. I. Mott’s Formula for Low-Temperature Jump Conductivity. Fizika Tverdogo Tela 16, 1820-1822 (1974).
13. Luther, J. M. et al. Schottky solar cells based on colloidal nanocrystal films. Nano Lett 8, 3488-3492 (2008). 14. Landolt-Börnstein - Group III Condensed Matter (Springer-Verlag GmbH, Heidelberg, 1998). 15. Romero, H. E. & Drndic, M. Coulomb blockade and hopping conduction in PbSe quantum dots. Physical
Review Letters 95, 156801 (2005). 16. Grove, A. S., Deal, B. E., Snow, E. H. & Sah, C. T. Investigation of thermally oxidised silicon surfaces using
Figure S17 | Transfer curves measured before and after 500electrical measurements during temperature scans from 85 Kto 350 K over 106 hours. The electron mobility changed from0.5 cm2/Vs to 0.3 cm2/Vs, the hole mobility changed from 0.4 cm2/Vs to 0.6 cm2/Vs.
metal-oxide-semiconductor structures. Solid-State Electronics 8, 145-163 (1965). 17. Kalesaki, E., Evers, W. H., Allan, G., Vanmaekelbergh, D. & Delerue, C. Electronic structure of atomically
coherent square semiconductor superlattices with dimensionality below two. Phys. Rev. B 88, (2013). 18. Marko, P. Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia. arXiv preprint
cond-mat/0609580 (2006). 19. Mirlin, A. D. Statistics of energy levels and eigenfunctions in disordered systems. Physics Reports (2000). 20. MacKinnon, A. & Kramer, B. The scaling theory of electrons in disordered solids: additional numerical
results. Zeitschrift für Physik B Condensed Matter 53, 1-13 (1983). ����