supplementary information charge transport and ... · scattered x-rays were collected by a pilatus...

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

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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.


3

The calculated intensity given by �� was fit to the measured intensity by

including a scaling factor � to account for the photon flux and a constant background factor �� to account for diffuse

scattering. The best fit is shown in Supplementary Figure S2(b) with values �� . This gives

an average NC diameter of 6.2 nm with a standard deviation of 0.44 nm, in good agreement with values calculated

from the absorbance spectrum (6.5 ± 0.42 nm).

Although nanoparticles are often fit using spherical form factors, it is known that 6.2 nm PbSe NCs are

faceted3. However, our experimental GISAXS configuration was optimized for structure analysis and limited to

2.5 nm-1. Therefore we do not have enough data to discriminate between spherical or other form factors. The

spherical form factor is a good representation of the form factor for the scattering angles relevant for superlattice

structure analysis. Regardless, we present the results comparing cubic and truncated cube form factors below for

completeness.

The form factor of a cubic scatter is given by ��

�� where � is the length of one side of the cube and �� are the components of the scattering vector

along the three axes of the cube2. We calculated scattering by cubes in solution by integrating the form factor over

all orientations according to ��

��

��

�

��

�

where � and � are the angles between the scattering vector and the x and z axes of the cube respectively and �� is

the distribution of NC sizes as for spheres.

In Supplementary Figure S2 we show the calculated scattering intensity by a cube with length 4.7 nm. A

slightly better fit was found with a truncated cube, where the eight corners of a cube are removed. This approximates

a NC with eight {100} and eight {111} facets. The form factor for a truncated cube is given by

��

�� where �� is the form factor of a cube and �� is the

form factor of a tetrahedron with sides of length �� 4. The eight tetrahedral form factor functions represent the eight

truncated corners of the cube. These are given by

��

��

��

��

��

��

��

�

��


4

��

� ��

��

��

��

��

��

� ��

��

� ��

��

��

��

��

The parameters used to calculate the form factor in Supplementary Figure S2 for the truncated cube were:

length � � � nm, � � �� nm and truncation parameter � � ��.

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.


5

away the polymer encapsulation layer in chloroform. The sample was mounted on an open stage (in air) attached to

a four-axis goniometer at the D1 station of the Cornell High Energy Synchrotron Source. The sample was positioned

for grazing incidence with an incident angle of 0.25 degrees. The incident radiation flux was of the order

1012 photons/(s mm2) with a wavelength of 0.1157 nm. Scattered X-rays were collected by a Pilatus 200K detector at

a distance of 902 mm with an exposure time of 3 s. Supplementary Figure S3 shows the recorded image. We

analyzed a line-profile of the intensity along the Yoneda band. To slightly improve the signal to noise, we averaged

the intensity in the vertical direction over 8 pixels or 0.08 nm-1 about the Vineyard peak.

Because the incident angle was sufficiently above the critical angle (the reflected intensity was about 2% of

the incident intensity) and because the NC layer is thin, we can use the quasi-kinematic approximation instead of the

distorted wave Born approximation. We calculated scattered intensity using both the local monodisperse

approximation and the decoupling approximation. We find the decoupling approximation gives a better fit to

experimental data. This is consistent with our assumption that there is a random spatial distribution of different size

and shape NCs. We calculated the scattered intensity from a 2-D powder of a simple square superlattice, with a

uniform distribution of in-plane orientations. The calculated intensity is given by:

��

� �� where � is the scattering vector, �� is the

background intensity, �� is the Vineyard factor at the Yoneda band where �� , � is a scaling factor

to account for the incident photon flux, �� is the form factor, and �� is the structure factor.

The spherical form factor in three dimensions is given by2

��

��

��

��

where � is the sphere radius, �� are the components

of the scattering vector in the reference frame of the sample and ��

�. The form factor was calculated to

account for a distribution of NC sizes, such that ��

��

��

�

��

�

and ��

��

��

��

�

�

where � is the angle between the in-plane

component of the scattering vector �� and the x-axis of the sample reference frame. The NC radius distribution is

given by ��

��

��

��.


6

Two models for the structure factor were compared. Disorder of the Debye-Waller kind, also called static

disorder, describes structures where long range order is preserved, but the scattering objects are randomly displaced

from the lattice points according to some probability distribution. The complete structure factor is given by

�� . The Bragg peaks are given by ��

��

��

where �� is the multiplicity of the �� reflection located at ��, and the half width at half maximum ��

where � is the average superlattice grain size, and the Scherrer constant � � ��5. The Debye-Waller factor is given

by �� where � is the standard deviation of a normal distribution of NC displacement2.

Paracrystalline disorder describes structures where long range order is compromised by correlation between

the displacement of adjacent scatters6. The 2D paracrystal structure factor for a square lattice is given by2

�� where � � ��

�� and ��

� � ��.

Disorder is introduced by the term �, which is the standard deviation of the Gaussian distribution describing the

nearest neighbor distance. The effect of the finite size of superlattice grains is included by the term � � where � is

the average nearest neighbor distance and � is the average grain size. To account for the 2D powder-like scattering,

we average over all grain orientations in the sample plane. The full structure factor is therefore given by ��

�

��

��

� where � is the angle between �� and the x-axis of the superlattice.

We find better agreement with the paracrystal model than with the Debye-Waller model, as shown in

Supplementary Figure S3b. The Debye-Waller model is unable to account for the increasingly broad higher order

peaks nor the diffuse scattering at small ��. We find satisfactory agreement with equal scattering contributions from

all grain orientations, as expected for random in-plane grain orientations where the average grain size is small (μm2)

compared to the size of the sampled area (mm2). Parameters that give the best fit curves shown in Supplementary

Figure S3 are, for the Debye-Waller model: � � �� nm, � � �� nm and � � �� nm, and for the paracrystal

model: � � �� nm, � � �� nm and � � �� nm. For both models the best fit was found with mean NC radius

�� nm and standard deviation�� . The paracrystalline disorder parameter � � �� nm is comparable

to the value �� nm found by fitting the radial distribution function from TEM images (see Supplementary

Figure S4).


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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.


8

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.


9

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

f 0 19


10

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.


11

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.


12

Estimation of Superlattice Thickness by GISAXS

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%.


13

Using parameters determined from analysis of the in-plane structure and form factors (see Supplementary

Figure S3), we calculated the full two-dimensional scattering images using the software package BornAgain

(version 1.4.0). The model consisted of spherical particles with diameter 6.1 ± 0.3 nm arranged in a square 2D

paracrystalline superlattice with superlattice constant 6.4 ± 0.3 nm and average grain size of 100 nm. The

superlattice layers were positioned on a substrate of 200 nm SiO2 on Si (infinite thickness). The incident angle was

0.25 degrees, as in the experimental setup. Angular divergence and energy bandwidth of the beam were neglected as

these are small in our experimental setup7. Scattering was calculated using the decoupling approximation and the

distorted-wave Born approximation (DWBA). Supplementary Figure S10 shows calculated scattering intensity from

monolayer to six-layer superlattice models. The superlattice layers were positioned 6.4 ± 1.0 nm apart vertically.

We found this model unable to exactly replicate the measured scattering data, however we can estimate the

thickness based on features in the scattered intensity normal to the sample. In Supplementary Figure S10 we show

line profiles of calculated and measured intensity along the {10} Bragg rod at �� 0.98 nm-1. In the experimental

data we observe a sharp feature near the Vineyard peak and several more peaks of increasing width and rapidly

decreasing intensity. Though the calculated intensity does not reproduce the rapidly decreasing intensity, we note

that a single sharp peak is only observed for superlattices more than 3 and less than 10 layers thick. As the number

of layers increases, interference causes the peak to become sharper. However, as thickness continues to increase, the

number of peaks at small ��increases, leading to a disappearance of the single peak. We therefore estimate the

average thickness to be more than 2 but less than 10 NC layers.


14

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.


15

Percolation Theory Interpretation of Charge Transport

Our analysis of conductance vs. temperature follows that of Efros and Shklovskii8. It is a generalization

using percolation theory of the variable range hopping equation derived by Mott9. Here we replace Mott’s

assumption of a constant density of states with one appropriate for the energetic and spatial distribution of states

gleaned from analysis of NC superlattices. With a realistic density of states, the analysis proceeds exactly as by

Efros and Shklovskii8, although the solution becomes non-trivial, therefore we used numerical solvers.

Briefly, the analysis starts as with any percolation problem, by defining the bonding criterion. We use the

bonding criterion and the density of states to express the percolation network density. This density only includes

sites that are likely to participate in electron transport at a certain temperature and chemical potential. It is obvious

that the minimum density required for percolation depends on the localization length of the electron wavefunction.

Below we will show how the localization length is calculated from the network density at a specific temperature and

chemical potential.

To solve for the localization length we need an explicit value for the network density at some temperature

and chemical potential. This value comes from the transition between nearest neighbor hopping (NNH) and variable

range hopping (VRH). As temperature decreases, the network density decreases because the probability for hopping

between sites separated by a large energy difference relative to kT, decreases exponentially. The transition from

nearest neighbor hopping (NNH) to variable range hopping (VRH) occurs at a temperature where the network

density reaches a critical value. The critical value is available in percolation theory literature for various lattice and

continuum networks8. In the following we describe in detail the steps outlined above.

The probability for charge transfer between two NCs (i and j) scales exponentially with distance ��, and

energy ��,, according to ��

��

��

�� where � is called the bonding criterion9. We assume the wavefunction

of an electron (hole) in the quantized 1Se (1Sh) state is approximately spherical10, with a decay length of �. The

important conclusion of percolation theory is that there is an abrupt change in the probability of transitions between

NC sites (i,j) at �� .


16

The density of the network of states that satisfy the inequality �� is given by a dimensionless number

��, which can be calculated by integrating the density of states function over a characteristic energy range and

normalizing by the square of a characteristic length, equation (3). We use a model density of states based on our

analysis of the superlattice structure. The density of states is given by equation (4) where N2D is the 2D density of

NC sites in the square superlattice and � is the standard deviation of energy levels caused by disorder of the NC

diameters and the epitaxial connections.

��

��

(3)

� � ��

��

��

�� (4)

From TEM image analysis and X-ray scattering we measure the NC diameter to be 6.1 ± 0.3 nm. We used

the empirical relation by Moreels et. al.1: �� where �� is the first exciton

energy and � is the NC diameter in nm, to determine the distribution of energy levels. This gives energies of

732 meV and 687 meV for NCs ±1 standard deviation (5.8 nm and 6.4 nm respectively). The difference of 45 meV

is the difference between a 1Se level one standard deviation above the mean 1Se level and a 1Sh level one standard

deviation below the mean 1Sh level. Therefore the standard deviation of a 1Se or a 1Sh energy level is

45/4 = 11 meV. The empirical relation given by Allan and Delerue11 gives a similar value of 13 meV.

The distribution of isolated NC energy levels should be convolved with the distribution of coupling

energies due to different epitaxial connection widths. For 6.1 nm diameter NCs joined by an epitaxial connection

nine Pb-Se bonds wide (2.8 nm), the coupling energy is 12 meV. The distribution of epitaxial connection widths is

normal with a standard deviation of three Pb-Se bonds (0.92nm). If we linearly extrapolate the coupling energy over

this range, the standard deviation is 4 meV. Convolving this distribution with the isolated NC energy level

distribution gives a total standard deviation of �� meV.

We are interested in �� at temperature Tc, therefore we integrate over an energy range defined by the largest

possible energetic transition at temperature Tc, i.e., �� . The characteristic length scale is the largest

possible hopping distance, i.e., ��

�. Integrating the density of states about the chemical potential �, we find a


17

relationship between ��, �, ��, and�� given in equation (5). For a random network, the critical value of the network

density was calculated to be 6.9 by Skal et. al.12.

��

�

��

��

��

� ��

��

� � (5)

Next we find � corresponding to each measured gate voltage. We define zero chemical potential as the

average 1Se (1Sh) site energy. The expectation value of the nearest neighbor hopping energy �� depends on the

position of the chemical potential. We numerically evaluated the relationship between nearest neighbor hopping

energy and chemical potential using equations (6-8). Equation (7) defines the probability distribution of the energies

�� and ��, with the same standard deviation as in equation (4). Equation (8) defines the transition energy �� between

sites i,j relative to the chemical potential �. Equation (6) was integrated numerically over the range � � ��. The

numerical result was approximated as a power law, given by equation (9) and shown in Figure 4. Using measured

nearest neighbor hopping energies ��, we thus calculated the position of the chemical potential for electrons and

holes at all measured gate voltages.

�� (6)

� � ��

� ��

��

�� (7)

��

�� (8)

��

�

�� (9)

��

� ��

��

(10)

We introduce equation (10) by realizing that at Tc, the hopping distance is approximately the nearest

neighbor distance, and the hopping energy is approximately the nearest neighbor hopping energy �� . Rearranging

equation (9), we substitute for � in equation (5) and solve equations (5) and (10) numerically to find the localization

length � for each measured gate voltage.

We now estimate the uncertainty in the analysis of temperature dependent conductance by the percolation

interpretation outlined above. Experimental sources of uncertainty include the transition temperature ��, the

distribution of NC energy levels �, the nearest neighbor hopping energy �� , and the nearest neighbor distance


18

��. The uncertainty in �� is the largest source of error on the calculated localization length. This error, on the order

of 10 K, leads to an uncertainty in the localization length �, of up to 1 nm. Other parameters have a smaller

significance.

The localization lengths estimated from conductance measurements by the above method are qualitatively

consistent with an interpretation based only on the change in conductance with temperature in the VRH regime. In

the case of Efros-Shklovskii variable range hopping, the range of values for the static dielectric constant that give

localization lengths consistent with our calculated range of 5 nm to 10 nm is 30 to 19. This is reasonable given that

the static dielectric constant was calculated to be 12 for PbSe NCs in a matrix of alkane ligands whereas the value

for bulk PbSe is near 20013,14. Therefore a value moderately higher than 12 is reasonable for an all inorganic PbSe

NC solid.

Consideration of Coulomb Blockade

Coulomb blockade could play an important role if the charging energy is greater than the thermal energy.

We can estimate the charging energy as �� 15. The self-capacitance is given by �� ,

where � is the dielectric constant of the PbSe superlattice and � is the NC radius. The coupling capacitance between

NCs is given by � � �� where � is the length of the tunnel barrier. We estimate � = 0.3 nm by

the difference between the superlattice constant 6.4 nm and the average NC diameter 6.1 nm. The factor of 4

accounts for coupling between the 4 nearest neighbors in a 2D square superlattice. We can use the value of 12 found

by Luther et. al.13 for ligand passivated NCs as a lower bound for the effective dielectric constant in a PbSe NC

solid. This gives us �� = 5.5 meV, which is smaller than the thermal energy at 85 K, 7.3 meV. If we use a more

realistic estimate of 20 for the dielectric constant of an all inorganic PbSe NC array, the charging energy decreases

to 3.4 meV. Therefore we do not consider Coulomb blockade to be an important factor in this temperature range.

Validity of Analyzing Transport using a Monolayer Model

In this work we compare our experimental results to 2D models for charge transport and electronic

structure. Although we estimate the average thickness of the superlattice to be less than 10 NC layers (64 nm), we

assume that charge transport is confined to the first NC layer. This is due to the electric field created by the gate

voltage. Because the data show ambipolar transport, we assume the Fermi level is near mid-gap and any doping


19

caused by mid-gap states is small. The transistor therefore operates in accumulation mode. We estimate the depth of

the charge accumulation layer away from the oxide interface by16: � � ��

�� . The Debye screening length is given by ��

�

� where � is the vacuum

permittivity, �� is the relative dielectric constant of the semiconductor, � is the Boltzmann constant, � is the

temperature, � is the charge of the electron, and �� is the intrinsic doping concentration. The potential at the oxide-

semiconductor interface, � is calculated from the gate voltage by: ��

�� where �� is the oxide thickness (200 nm) and �� is the relative dielectric constant of the oxide (3.9

for SiO2). Using conservative values for gate voltage (22 V), intrinsic carrier concentration (3.8x1012 cm-3, or one

electron per 106 NCs), and the dielectric constant (20), the effective charge layer is 2.5 nm. Only if the relative

dielctric constant of the superlattice exceeded 50 would the charge layer exceed the size of one NC layer.

�


20

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.


21

We make a simple estimation of the carrier density (electrons per NC) from the FET capacitance. Again

using the 2D charge sheet approximation, and assuming there are no fixed charges in the dielectric layer or at the

dielectric-semiconductor interface, the number of electrons per NC is given by � � �� where

�� = 1.7 x 10-8 (F/cm2), the cross-sectional area of a NC is �� , �� is the gate voltage, and � is

the electron charge in Coulombs. We estimate � is 0.7 to 1.2 at gate voltages of 22 V to 40 V, respectively.


22

FET Device Architecture and Encapsulation Effects

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.


23

Tight-Binding Calculation

The mini band structure of a perfect superlattice of PbSe NCs has been calculated before using atomistic

tight-binding calculations17. However this method is computationally infeasible to study a large superlattice with

disorder. Here, we fit the calculated band structure using an effective multi-band tight-binding Hamiltonian. We

treat each NC as an atom with 4 states, which correspond to the electron states originating from the four L-valleys.

� �

��

��

� � � �

��

��

��

��

� ��

��

��

� ��

��

� ��

��

��

��

��

��

� � ��

� � ��

Here �� depends on the relative direction of vector connecting the valleys and the direction of the bond,

and is �� if they are parallel, and �� if they are perpendicular. We then use this effective Hamiltonian to study

the wavefunction localization with disorder. Two major sources of disorder are present in this structure: the

fluctuation of the diameter of each NC and the fluctuation of the width of the epitaxial connection between NCs.

The diameter fluctuation is modeled as a Gaussian distribution of the energy levels. For a first order approximation,

all four states in the NC are shifted together, thus accounting for the size dependence of the quantum confinement

effect but not the valley coupling effect. The disorder of the epitaxial connection width causes fluctuation of the

coupling strengths between NCs, and is proportional to the width of the epitaxial connection17. We limit the

percentage of coupled nearest neighbors to 80%, 90%, or 95% to reflect the percentage of missing epitaxial

connections in our samples. The remaining percentage of nearest neighbor coupling energies are set to zero.

Supplementary Figure S14 shows densities of sates calculated for 80%, 90% and 95% connectivity.

We use the inverse participation ratio �� and its scaling with system size to analyze the

degree of wavefunction delocalization18,19. If the wavefunction is completely delocalized, �� where � is the

system size, � is the system dimension and equals 2 in our case; if the wavefunction is completely localized, �� .

Thus, by fitting �� , we obtain a fractal dimension �� that directly represents the degree of localization.


24

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.


25

� ��

��

��

��

��

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.


26

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.


27

Relative Influence of Disorder Parameters on Localization Length

To investigate the relative influence of each disorder parameter on the localization length, we varied the

standard deviation of the NC diameter, the standard deviation of the epitaxial connection width, and the percentage

of nearest neighbors coupled by epitaxial connections (the connectivity). The results are shown in Supplementary

Figure S16. We express the standard deviation of the NC diameter in terms of the corresponding standard deviation

of the 1Se or 1Sh energy level. We show that decreasing NC size disorder from the experimentally realizable value

of 11 meV by 50% or 90% to 5 meV or 1 meV increases the localization length by about 4 or 10 times respectively.

Similarly, decreasing the standard deviation of the epitaxial connection width by 50% or 99.97% would increase the

localization length by about 4 or 25 times respectively. Improving the connectivity would have less of an effect on

the localization length, whereby increasing the connectivity from 90% to 95% or 99.9% would increase the

localization length by about 1.2 or 3 times respectively.


28

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.


29

Electronic Stability and Aging

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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.


30

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supplementary information charge transport and ... · scattered x-rays were collected by a pilatus...

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