# Solving Complex Nanostructures With Ptychographic Atomic Electron Tomography

Philipp M. Pelz<EMAIL_ADDRESS>Department of Materials Science and
Engineering, University of California Berkeley, Berkeley, CA 94720  The
Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California
94720, United States  Sinéad Griffin  The Molecular Foundry, Lawrence Berkeley
National Laboratory, Berkeley, California 94720, United States  Materials
Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California
94720, United States  Scott Stonemeyer  Materials Sciences Division, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, United States  Kavli
Energy NanoSciences Institute at the University of California at Berkeley,
Berkeley, California 94720, United States  Department of Chemistry, University
of California at Berkeley, Berkeley, California 94720, United States
Department of Physics, University of California at Berkeley, Berkeley,
California 94720, United States  Derek Popple  Materials Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, United
States  Kavli Energy NanoSciences Institute at the University of California at
Berkeley, Berkeley, California 94720, United States  Department of Chemistry,
University of California at Berkeley, Berkeley, California 94720, United
States  Department of Physics, University of California at Berkeley, Berkeley,
California 94720, United States  Hannah Devyldere  Department of Materials
Science and Engineering, University of California Berkeley, Berkeley, CA 94720
Peter Ercius  The Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, United States  Alex Zettl  Department of Materials
Science and Engineering, University of California Berkeley, Berkeley, CA 94720
Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, United States  Kavli Energy NanoSciences Institute at the
University of California at Berkeley, Berkeley, California 94720, United
States  Department of Physics, University of California at Berkeley, Berkeley,
California 94720, United States  Mary C. Scott  Department of Materials
Science and Engineering, University of California Berkeley, Berkeley, CA 94720
The Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, United States  Colin Ophus<EMAIL_ADDRESS>The Molecular
Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720,
United States

###### Abstract

Transmission electron microscopy (TEM) is a potent technique for the
determination of three-dimensional atomic scale structure of samples in
structural biology and materials science. In structural biology, three-
dimensional structures of proteins are routinely determined using phase-
contrast single-particle cryo-electron microscopy from thousands of identical
proteins, and reconstructions have reached atomic resolution for specific
proteins. In materials science, three-dimensional atomic structures of complex
nanomaterials have been determined using a combination of annular dark field
(ADF) scanning transmission electron microscopic (STEM) tomography and
subpixel localization of atomic peaks, in a method termed atomic electron
tomography (AET). However, neither of these methods can determine the three-
dimensional atomic structure of heterogeneous nanomaterials containing light
elements. Here, we perform mixed-state electron ptychography from 34.5 million
diffraction patterns to reconstruct a high-resolution tilt series of a double
wall-carbon nanotube (DW-CNT), encapsulating a complex $\mathrm{ZrTe}$
sandwich structure. Class averaging of the resulting reconstructions and
subpixel localization of the atomic peaks in the reconstructed volume reveals
the complex three-dimensional atomic structure of the core-shell
heterostructure with $17\text{\,}\mathrm{pm}$ precision. From these
measurements, we solve the full $\mathrm{Zr_{11}Te_{50}}$ structure, which
contains a previously unobserved $\mathrm{ZrTe_{2}}$ phase in the core. The
experimental realization of ptychographic atomic electron tomography (PAET)
will allow for structural determination of a wide range of nanomaterials which
are beam-sensitive or contain light elements.

Knowledge of the three-dimensional atomic structure of natural and
manufactured materials allows us to calculate their physical properties and
deduce their function from first principles. Because of this, methods for
atomic structure determination have been a key area of research in the
biological and physical sciences, including x-ray crystallography [1] and
nuclear magnetic resonance spectroscopy [2]. For crystalline samples, micro-
crystal electron diffraction provides data of similar quality to X-ray
diffraction for solving structures [3]. More recently, cryo-electron
microscopy has become the dominant method for atomic structure determination
of molecules, either from ensembles fulfilling the single-particle assumption
[4], or by cryo-electron tomography and subsequent subtomogram averaging [5].
However, these methods require averaging of many near-identical structures.
One method which is capable of solving structurally and chemically
heterogeneous nanostructures is atomic electron tomography (AET) using
scanning transmission electron microscopy [6, 7, 8, 9, 10, 11]. However, the
dark field imaging method used in these AET studies produces very little
contrast for light elements, and requires too much electron dose to be used
for beam-sensitive samples. Phase contrast imaging such as the method used in
cryo-electron microscopy overcomes these obstacles, and has recently been
extended to include multiple scattering [12] and demonstrated experimentally
at sub-nm resolution [13]. A STEM-based phase contrast method which is capable
of removing residual aberrations of the probe and enhancing the resolution is
ptychography [14, 15, 16], recently demonstrated with deep sub-Ångstrom
resolution [17, 18]. Here, we use ptychographic atomic electron tomography
(PAET) to solve the 3D atomic structure of a zig-zag double wall-carbon
nanotube (DW-CNT) encasing a structurally complex Zr-Te shell and a previously
unseen Zr-Te core structure.

Figure 1:  Ptychographic atomic electron tomography (PAET) of a complex
nanostructure. (a) A converged electron beam is scanned over a nanoscale
sample which is tilted around the $\mathrm{\alpha}$ axis of a high-precision
tomography holder. Four-dimensional scanning diffraction datasets are recorded
at every tilt angle using a direct electron detector. (b) Reconstructed
complex wavefunction of an example STEM probe. (c) Examples of phase-contrast
projection images, reconstructed using mixed-state electron ptychography. (d)
High-SNR class averages calculated from the images in (c). (e) Average
diffraction pattern of all 36 tilt angles with some scattering angles labeled.
(f) Two orientations of the 3D reconstructed volume from the 2D unit cell
averages in (d). All real space scales bars are
$10\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, scale bar in (e) is
$0.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{-1}$.

Low-dimensional van der Waals (vdW) materials such as the transition metal di-
and tri-chalcogenide families or CNTs exhibit a range of desirable properties
that often emerge as the material thickness reaches the one- or two-
dimensional thickness limit [19, 20, 21]. In order to synthesize otherwise
unstable vdW materials, encapsulation inside CNTs has been developed as a
strategy to stabilize quasi one-dimensional structures [22, 23, 24] of
transition metals. The encapsulation approach protects the interior structure
from oxidation, and can result in chiral structures [21, 23] or fillings with
high aspect ratio [25] that exhibit properties differing drastically from
their bulk counterparts. While electron microscopy has played an important
role in the identification and characterization of the encapsulated phases,
the detailed 3D structure of the interior nanowire is not always clear. For
example, the encased material’s structure often exhibits a dependence on the
diameter of the nanotube [22], and can form complex three-dimensional
structures inside the CNT, such as core-shell structures [26] and multilayer
moiré structures [27]. In the latter cases, the atomic structure cannot be
uniquely determined from a single projection, and three-dimensional imaging is
paramount for structure-function determination.

The experimental setup for PAET in the STEM is shown in Fig. 1a. A converged
electron probe is raster-scanned over the sample, with one diffraction pattern
recorded at every probe position using a fast-framing direct-electron detector
operating at $87\text{\,}\mathrm{kHz}$. The nanotube is tilted around its axis
using a specialized dual-axis tomography holder, and at each tilt angle the
scanning diffraction measurements are repeated. We reconstruct the partially
coherent illumination wave function from each four-dimensional dataset, where
the dominant mode of an example probe’s complex wavefunction is shown in Fig.
1b. The probe wavefunctions and their positions are reconstructed jointly with
the sample object wave. Fig. 1c shows the phase of the object wave from three
of the experimental projections. The strong contrast of the carbon atoms in
the 2D projections allows us to determine a zig-zag nanotube configuration and
the dimensions of the semi-minor and semi-major axes of the elliptical DW-CNT
(see Methods Section .7), information unobtainable from the ADF signal at the
same electron dose.

Figure 2:  DW-CNT-encapsulated $\mathrm{Zr_{11}Te_{50}}$ sandwich structure.
(a) atomic model overlaid over vertical slice through the reconstructed
volume. (b) Three-dimensional model of the $\mathrm{Zr_{11}Te_{50}}$ structure
with folded-out one-dimensional $\mathrm{ZrTe_{5}}$ chains. (c) Left panel:
side view of four $\mathrm{ZrTe_{5}}$ chains building one side of the sandwich
structure. Right panel: Front view of four $\mathrm{ZrTe_{5}}$ chains of one
side of the sandwich structure. (d) Front view of the full atomic model
showing the elliptical CNTs and the spatial extents of their semi-minor and
semi-major axes. (e) Side view of the DW-CNT with zig-zag configuration. (f)
Side view of the $\mathrm{ZrTe_{2}}$ core structure and the $\mathrm{Te}$
chains.

To increase the signal-to-noise ratio of the phase-contrast projections, and
enable determination of the 3D structure of the center of the tube even with a
limited experimental tilt range, we computed class averages along the
nanotube, with 3 classes shown in Fig. 1d which correspond to the images in
Fig. 1c. The size of these class images were chosen to include 11 repeats of
the core structure, and approximately 23 repeats of the DW-CNT structure. This
corresponds to a rational approximate of the irrational ratio of the repeat
length for these two incommensurate structures. Fig. 1e shows the average
diffraction pattern of the 38 phase-contrast projections, which displays a
high degree of periodicity along the tube. The
$1\text{/}2.13\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{-1}$ reflection of the
graphene lattice is clearly visible, as well as the 1st and 2nd diffraction
order of the Zr-Zr and Te-Te lattice spacing of
$1\text{/}3.95\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{-1}$ and
$1\text{/}1.975\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{-1}$. The unit-cell
averaging procedure yielded a high-SNR image of the core structure, while the
incommensurate $2.13\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ spacing of the
graphene lattice in the carbon nanotube is highly suppressed.

We have reconstructed the 3D volume shown in Fig. 1f, from the 2D class images
in shown in Fig. 1d, using the methods described in Methods Sec. .5. From this
reconstruction, we can clearly identify the outer 2 shells which correspond to
the DW-CNT, which have been colored in blue. We can also see the complex Zr-Te
interior structure, colored in green and gold. This structure is highly
periodic along the tube axis, but does not possess a periodic crystal
structure in the plane perpendicular to the nanotube axis. Instead, the Zr-Te
structure possesses a distinct core and shell structure, with a high degree of
variation in the structural sub-units making up each. Most atoms in the Zr-Te
core and shell can be directly resolved, while some atoms close to the edges
along the missing wedge direction appear as elongated columns. From the
reconstructed volume, we have measured the 3D positions of 578 Zr and Te atoms
to subpixel precision using the procedures developed in AET routines [6, 12].
The chemical species of the Te and Zr atoms were determined by analyzing the
number of nearest-neighbors and the local coordination (Methods section .7).

Figure 3:  First principles calculations of Zr11Te50. (a) DFT optimized
Zr11Te50 structure. (b) Crystal structure of $Cmcm$ ZrTe5 (c) Fully optimized
inner ZrTe2 structural unit d) Calculated electronic band structure of $Cmcm$
ZrTe5. e) Calculated electronic band structure of inner ZrTe2 structural unit.
For both band structures, the Fermi level is set to 0 eV and is marked by the
green dashed line.

The configuration and elliptical dimensions of the DW-CNT were determined by
first excluding chiral CNTs based on the Fourier spectrum of the projections
shown in Fig. 1e. The armchair configuration was excluded based on the
observation of a strong reflection at
$1\text{/}2.13\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{-1}$ in the direction
of the nanotube, which is present only in zig-zag nanotubes where the
$2.13\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ spacing is oriented across the
nanotubes. The dimensions of the CNTs were determined by maximizing the
agreement of the atomic coordinates with the reconstructed projected intensity
of the CNTs from the volume in Fig. 1f. We determined best-fit chiral indices
of $\mathrm{(50,0)}$ and $\mathrm{(39,0)}$ for the outer and inner CNT.

Fig. 2a shows the full atomic model overlaid over a 2D slice through the
reconstructed phase-contrast volume. Fig. 2b shows a three-dimensional
rendering of the structure, with the outermost Zr-Te shells peeled back to
display the local coordination. Inside the DW-CNT, eight one-dimensional
$\mathrm{ZrTe_{5}}$ clusters encapsulate a three unit-cell wide
$\mathrm{ZrTe_{2}}$ structure. The outer $\mathrm{ZrTe_{5}}$ clusters are
split into two groups of 4 $\mathrm{ZrTe_{5}}$ sub-units, shown in Fig. 2c.
Each of these clusters has a 1D chain structure along the nanotube axis, with
the same orientations [28]. The 4 groups are circumscribed inside the inner
CNT wall, with an angle of approximately $26^{\circ}$ between each cluster and
a radius of $14\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ to the central Zr
atom. The overall geometric arrangement of the full structure is shown in Fig.
2d and e, with the DW-CNT major and minor axes lengths labeled.

On the top and bottom of the encapsulated Zr-Te structure, there are two
single-atom Te chains. One-dimensional Te-Te chains have been observed before
as stable structures encapsulated in CNTs [29]. The relative position of the
Te-Te chains and the $\mathrm{ZrTe_{2}}$ core structure are shown in Fig. 2f.
The $\mathrm{ZrTe_{2}}$ is a previously unobserved phase of Zr-Te, which we
examine in more detail in the following section. The overall structure has a
stoichiometry of $\mathrm{Zr_{11}Te_{50}}$, and a striking elliptical
structure which may be caused by the intercalation of the ZrTe structures into
the DW-CNT.

To investigate the stability and electronic structure of our proposed
$\mathrm{Zr_{11}Te_{50}}$ structure, we have performed first principles
calculations, as described in Methods Sec. .9. The resulting optimized atomic
structure is depicted in Fig. 3a. As previously discussed, the outer tube is
well described as one-dimensional chains of ZrTe5 which consist of face-
sharing Zr-Te polyhedra where each Zr is 8-coordinated with Te. These
8-coordinated Zr-Te chains are distorted versions of those that make up bulk
$Cmcm$ ZrTe5 crystal where each Zr has three pairs of identical Zr-Te bond
lengths, with the two remaining Zr-Te bonds being different. The coordination
of the outer ring of Zr in our Zr11Te50 is structurally the same with slight
deviations from the “paired” Zr-Te bond lengths (e.g. 2.955 Å -vs- 2.960 Å).
We present the calculated electronic band structure of the bulk ZrTe5 from
which our outer structure is derived in Fig. 3d – we confirm the previously-
reported topological insulating phase with the band inversion near to $\Gamma$
[30].

The inner core ZrTe2 structure presents an intriguing case for structural
analysics. We find two different coordinations for Zr in the innermost layer.
The central Zr is 8-fold coordinated with Te, with the other two Zr sites
being 6-fold coordinated with Te. We next isolated this ZrTe2 structural unit
and performed a full structural optimization to remove the influence of any
confinement/pressure from the sandwich structure, with the result depicted in
Fig. 3c. This new ZrTe2 structure adopts the space group Pmmm, and forms a
thin two-dimensional slab. In fact, the ground state polymorph, $1T$-ZrTe2
(space group $P\overline{3}m1$), has only recently been grown by molecular
beam epitaxy [31]. This $1T$ phase is 6-fold coordinated and forms a van-der-
Waals two-dimensional structure with measurements and theory suggesting it is
a Dirac semimetal [31]. The innermost Zr coordination is more unusual. While
this Zr site is 8-fold coordinated as in the ZrTe5 chains, it now forms a
regular octahedron where each Zr-Te bond length is the same. The calculated
electronic band structure of this new structural phase is shown in Fig. 3e.
Our $Pmmm$ ZrTe2 phase is a metal with primarily hole pockets at the Fermi
level – the low-dimensional nature of the structure is also confirmed with the
flat bands in the X to S directions (short axis). Finally, using symmetry
indicators we predict this new phase to be a Dirac semimetal, akin to its
stable $P\overline{3}m1$ polymorph [31].

In summary, we have experimentally demonstrated atomic structure determination
of a complex nanomaterial by phase-contrast ptychographic atomic electron
tomography. In addition to showing contrast improvement compared to AET for
the weakly-scattering carbon atoms, PAET simultaneously recovers partial
coherence effects, probe positions and probe aberrations present in the
experiment. This in turn allows focusing at lower resolution and an overall
lower dose due to less pre-exposure irradiation of the sample. Further
improvements of the experimental protocol will allow skipping the unit-cell
averaging step in the reconstruction and enable imaging of single light atoms
at atomic resolution. We have used PAET to solve the structure of a double-
walled carbon nanotube encapsulating a complex Zr11Te50 nanowire structure,
which contains both previously observed linear chain structures of ZrTe5 and
Te-Te, and a previously unobserved ZrTe2 structure. Density functional
calculations both confirm the stability of our Zr-Te model structure, and
elucidate its electronic properties. We expect that PAET will find widespread
application for solving the structures of complex materials which contain
light elements, weakly-scattering structures, and beam-sensitive materials.

## Methods

###  .1 Sample preparation

Encapsulated $\mathrm{Zr_{11}Te_{50}}$ species are synthesized within CNTs
using protocols similar to those for the growth of confined
$\mathrm{TaTe_{2}}$, $\mathrm{NbSe_{3}}$, and $\mathrm{HfTe_{3}}$ structures
[27, 21, 22]. The $\mathrm{Zr_{11}Te_{50}}$ species can be synthesized
following the synthesis of $\mathrm{ZrTe_{3}}$, where stoichiometric
quantities of Zr powder and Te shot ($~{}450\text{\,}\mathrm{mg}$ total),
$1\text{\,}\mathrm{mg}2\text{\,}\mathrm{mg}$ of end-opened multi-walled CNTs
with an inner diameter ranging from $1.0\text{\,}\mathrm{nm}$ to
$10.0\text{\,}\mathrm{nm}$ (CheapTubes, 90% SW-DW CNT), and
$5\text{\,}\mathrm{m}\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$(ampoule
volume) of $\mathrm{I_{2}}$ are sealed under vacuum
($1.33\text{\times}{10}^{-6}\text{\,}\mathrm{mbar}$) in a quartz ampoule. The
ampoule is heated in a single-zone furnace at
$550\text{\,}\mathrm{\SIUnitSymbolDegree}$ for 3-5 days, cooled to
$350\text{\,}\mathrm{\SIUnitSymbolDegree}$ over 3 days, then cooled to room
temperature over 1-2 days. The Zr-Te filled CNTs are dispersed in isopropyl
alcohol by bath sonication for 1 hour and drop-cast onto lacey carbon TEM
grids for subsequent electron microscopy analysis.

###  .2 Data Acquisition

A tomographic tilt series was acquired from a $\mathrm{Zr_{11}Te_{50}}$ DW-CNT
using the TEAM 0.5 microscope and TEAM stage [32] at the National Center for
Electron Microscopy in the Molecular Foundry. Before the tilt series, the
sample was beam showered for 30 minutes. We recorded four dimensional-scanning
transmission electron microscopy (4D-STEM) datasets [33] with full diffraction
patterns over$1600\text{\times}600$ probe positions at each tilt angle. The
diffraction pattern images were acquired with the 4D Camera prototype, in-
house developed in collaboration with Gatan Inc., a direct electron detector
with $576\text{\times}576$ pixels and a frame rate of
$87\text{\,}\mathrm{kHz}$ [34], at 80 kV in STEM mode with a
$25\text{\,}\mathrm{mrad}$ convergence semi-angle, a beam current of
$40\text{\,}\mathrm{pA}$, estimated from 4D camera counts, a real-space pixel
size of $0.4\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, and camera reciprocal
space sampling of $173.6\text{\,}\mathrm{\SIUnitSymbolMicro rad}$ per pixel.
These settings amounted to an accumulated fluence of
$1.79\text{\times}{10}^{4}\text{\,}\mathrm{\text{ $e$
}}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}^{2}$ per projection and
$6.28\text{\times}{10}^{5}\text{\,}\mathrm{\text{ $e$
}}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}^{2}$ for the whole tilt series. The
tilt series was collected at 38 angles with a tilt range of +$63$ to $-58$
degrees. To minimize the total electron exposure, focusing was performed at a
resolution of 80 kX before switching to high magnification for data
collection.

###  .3 Ptychographic reconstruction

The raw 4D-STEM datasets of size $650\text{\,}\mathrm{GB}$ per tilt angle and
$22.75\text{\,}\mathrm{TB}$ in total were electron-counted using the open-
source stempy software [35] on the Cori supercomputer at NERSC and saved in a
sparse linear-index encoded electron event representation (EER), using
$~{}6.5\text{\,}\mathrm{GB}$ storage per tilt. Crop-outs containing only the
scan positions covering the $\mathrm{Zr_{11}Te_{50}}$ DW-CNT for all tilts
were determined from the manually aligned annular bright-field images of the
tilt series for ptychographic reconstruction. Initial guesses for the defocus
aberration for each tilt were manually obtained with an interactive real-time
implementation [36] of the single-sideband ptychography (SSB) method [37]. The
data in EER format was further preprocessed by centering the bright-field
disk, symmetric cropping to a maximum detector angle of 2$\alpha$, with
$\alpha$ the semi-convergence angle, and binning to a detector size of
$88\text{\times}88$ pixels. The final cropped and preprocessed data of the
whole tilt series, compressed with the gzip compression algorithm, had a total
size of $12.3\text{\,}\mathrm{GB}$. Phase-contrast images, probe positions and
a low-rank approximation of the partially coherent illumination were jointly
reconstructed using 115 iterations of the Least-Squares Maximum Likelihood
(LSQML) method with gradient-based scan-position correction [38, 39], with the
parameters in Suppl. Table II. The maximum thickness for a sample to fulfill
the projection approximation of ptychography for a numerical aperture of
$25\text{\,}\mathrm{mrad}$ is $9\text{\,}\mathrm{nm}$. The largest dimension
of the DW-CNT along the beam direction is $3.9\text{\,}\mathrm{nm}$, it is
therefore within the limit of the projection approximation for single-slice
ptychography. Due to the slight tilt of the nanotube, the projected thickness
reaches $6\text{\,}\mathrm{nm}$ for some projections. For the best
reconstruction quality of projections where the nanotube is tilted along the
beam direction, we reconstruct slices along the nanotube separately with
different probes.

###  .4 Preprocessing for phase-contrast tomography

Since low spatial frequencies are weakly transferred in noisy 4D-STEM datasets
collected close to the focus condition [40], the ptychographic tilt-series
reconstructions display characteristic halos around the nanotube. With the
prior knowledge that the nanotubes are suspended over vacuum, we manually
create vacuum masks around the nanowire and set the vacuum phase equal to the
lowest phase value of the halo close to the nanotube. We then performed a
first manual alignment using the Tomviz tomography software [41]. Subsequently
we scaled all projections with a scalar value to have the same integrated
phase shift along the direction perpendicular to the tilt axis. Tomviz was
also used for visualization of the reconstructed volume.

###  .5 Tomographic reconstruction

From the initially-aligned tilt series a 3D reconstruction was performed using
the fast adaptive shrinkage-thresholding algorithm [42], an accelerated
gradient algorithm with adaptive stepsize for faster convergence. To compute
the forward- and backward projections, we used the generalized ray transform
interface of the Operator Discretization Library [43] in a 3D parallel-beam
Euler geometry with an GPU-accelerated backend of the ASTRA tomography toolbox
[44]. To increase the accuracy of the projections, we used the trilinear
interpolation feature of the ASTRA library to compute the forward and inverse
ray-transforms. To minimize the translational and angular misalignments, we
adopt the following procedure. We perform a grid search for a range of global
tilt axis rotation errors in the range of -2 to 6 degrees. For each of these
global rotation axis guesses, we perform a local refinement by an iterative
projection matching approach [45] with simulated annealing. In this approach,
all three Euler angles are varied by a randomly picked value in the range of
$-1$ to $1$ degrees and the calculated projection error compared with the
current projection error after a full reconstruction. The lowest-error angles
are then used as new initial angles for the next tomographic reconstruction.
Initially the reconstruction was performed on the full 2D projections. After a
reconstruction, the auto-aligned projections were checked for unresolved unit-
cell hops. These unit-cell hops in the tilt series were fixed manually in
tomviz [41], and then another round of algorithmic auto-alignment was
performed. This process of manual alignment and auto-alignment was repeated
for 16 rounds until no further improvements could be made. The 2D class
averaged projections were then manually aligned to the final aligned full
projections to avoid unit-cell hops. Using this approach, the reconstruction
with the 2D class averages converged to a minimum R-factor of
$5.4\text{\,}\mathrm{\char 37\relax}$ for a global tilt axis rotation of 6.3
degrees. The reconstructed volume from the full 2D projections is
$27.7\text{\times}11.6\text{\times}11.6\text{\,}\mathrm{nm}$ in size, while
the volume reconstructed from 2D class averages is
$5.88\text{\times}5.3\text{\times}5.3\text{\,}\mathrm{nm}$ in size.

###  .6 Atom tracing

The 3D atomic positions of the Zr and Te atoms were determined using the
following procedure developed in [12]. (I) all local intensity maxima were
identified from the 3D reconstruction and added to a candidate list. From the
initial candidate list, peaks which were within a minimum distance of
$2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ of a higher-intensity peak were
deleted. (II) The initial list of peak positions was refined by fitting a 3D
Gaussian function to each peak after subtracting neighboring peaks within a
maximum radius of $4.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. Using this
initial atom candidate list, we added, refined, and merged new unidentified
peaks for 4 iterations in the following order: (III) Subtract the fitted
Gaussians of all current peak candidates from the reconstruction volume. (IV)
Add new candidate peaks over an intensity threshold of $50$ to the candidate
list. (V) Refine the positions of all atom candidates as in (II) for 4
iterations. (VI) Merge peaks that are closer than a minimum distance of
$2.2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ after position refinement. (VII)
Refine the positions of all atom candidates as in (II) for 4 iterations.
(VIII) go to (III) if iterations not done. (IX) A final set of 4 positions
refinement iterations as in step (II) yielded the set of coordinates of 714
candidate atoms. The rotation of the nanotube with respect to the beam
propagation normal plane was determined as
$5.2\text{\,}\mathrm{\SIUnitSymbolDegree}$ and the candidate atoms were then
rotated onto this normal plane. Atomic columns along the tube were identified
by projecting the volume along the tube and 2D peak finding. Atom candidates
were that were farther than $2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ from
the next column were snapped onto the column in the projection along the tube,
and duplicate atoms were removed. This procedure resulted in 578 candidate
atoms.

###  .7 Atomic model construction

From the candidate atoms, we have constructed the Zr-Te model structure. We
note that for most sites, it was not possible to directly determine the atomic
species from the reconstructed 3D phase-contrast signal at the experimental
signal-to-noise ratio. The prototype 4D camera detector is mounted off-center
relative to the ADF detectors, such that the simultaneous ADF signal could not
be used for tomographic reconstruction. We expect better chemical contrast in
future experiments with both ADF and 4D-STEM signals [46, 16]. Chemical
species of all but 2 atomic columns in the nanotube could be determined from
the coordination environment with the following procedure. For each atom
candidate, we determined the number of nearest neighbors (NNs) in a shell of
$3.1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ radius. As candidate stable
structures we considered all known stable Zr and Te containing structures,
including ZrTe2, ZrTe3 and ZrTe5. In those structures, the Zr atoms are bonded
to six or eight Te atoms, while the Te atoms can be bonded in different
geometries to 2-4 Zr atoms. We therefore created two classes for atoms with
less than four neighbors and those with more than four neighbors. The result
of this classification is shown in supplementary Fig 7. We then assigned the
columns with majority of atoms with more than 4 NNs as Zr atoms and the
columns with majority of atoms in the class with less than 4 NNs as Te atoms.

This procedure left two atomic columns in the center of the nanotube that
could not be assigned based on NNs. These are marked with circles in
supplementary Fig 7. Those two ambiguous site identities were tested using DFT
calculations, where in all cases we used the lower energy stoichiometries.
From the chemical classification based on NNs and the candidate atomic
positions, the 8 outermost ZrTe5 units and their positions / angles were
identified. Next, we identified two isolated chains of atoms following a
zigzag pattern at the top and bottom of the tube as positioned in Fig. 2.
These sites were assumed to be composed of Te atoms, due to previous
observations of Te chains encapsulated in carbon nanotubes [29], and the
presence of weakly-bound Te chains in the ZrTe5 structural units. Finally, the
core was determined to be composed of a central Zr atom with 8 Te neighbors,
and two partially-coordinated Zr atoms, forming a small 2D ribbon of ZrTe2
with a previously unreported structure.

The DW-CNT configuration was determined in the following way. CNTs are
characterized by their chiral indices $\mathrm{(n,m)}$. Chiral nanotubes can
be excluded for our sample because the diffraction pattern of chiral nanotubes
follows a Bessel function radial symmetry [47] with oscillating layer lines.
In the diffraction patterns of the experimental projections in supplementary
Fig. 1 (insets) it can be seen that the
$1\text{/}2.13\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{-1}$ graphene
reflection is not an oscillating layer line. This leaves armchair
configuration $\mathrm{(k,k)}$ or zig-zag configuration $\mathrm{(k,0)}$
nanotubes as possible options. In armchair configuration nanotubes, the
$2.13\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ spacing lies across the nanotube
and can be suppressed by material in the tube, while in zig-zag configuration
the $2.13\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ spacing lies along the tube
and is strongly visible at the edges of the tube where it is overlapped by no
other material. This is the case in our sample, and we therefore exclude the
armchair configuration, which leaves us to determine the chiral indices
$\mathrm{k_{1}}$ and $\mathrm{k_{2}}$ of the outer and inner nanotube, and the
determination of the semi-minor and semi-major axis dimensions of the
nanotube. We determined the chiral indices by maximizing the the intensity of
the pixels sampled by CNT atomic coordinates overlaid over the 2D projection
of the reconstructed volume along the tube, as shown in supplementary Fig 7.
The sampled intensity was maximized for chiral numbers $\mathrm{k_{1}=50}$ and
$\mathrm{k_{2}=39}$, with semi-minor axes of
$16.7\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and
$13.0\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and semi-major axes of
$19.6\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and
$15.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. The only parameter that could
not be determined uniquely was the relative rotation of the CNTs to each
other, as the 2D unit-cell averaging procedure reduced the atomic contrast for
the CNT. As such, the relative rotation of the CNTs shown in Fig. 2 is only
one possible configuration.

###  .8 Estimation of Precision

To calculate the precision of our position measurements, we first investigated
if the nanotube can be modeled with a linear model or if a full multislice
treatment is necessary. To this end, we simulated one projection both with a
linear forward model as in [10] and with a multislice simulation incorporating
all partial spatial and temporal coherence effects, followed by mixed-state
ptychographic reconstruction and unit cell averaging. First, we created 12
different atomic models where the Zr and Te atoms and atom positions are
identical, but the carbon nanotube is displaced by
$2.13\text{/}12\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ every time, such that
the averaging removes atomic contrast of the nanotube. We then fitted a linear
model to this average structure by determining the H- and B-factors that
minimize the R-factor between model and experimental reconstruction. This
linear model is shown in the middle panel of supplementary Fig. 4 and achieves
and R-Factor of $12\text{\,}\mathrm{\char 37\relax}$. Then, we performed
multislice 4D-STEM simulations using the PRISM algorithm [48] in the abTEM
simulation code [49], incorporating partial spatial coherence with a FWHM
source size of $80\text{\,}\mathrm{pm}$ and partial temporal coherence with a
chromatic defocus spread of $6\text{\,}\mathrm{nm}$, as calculated from the
chromatic aberration coefficient and energy spread of the TEAM 0.5 microscope
at $80\text{\,}\mathrm{keV}$, and with the experimental dose of
$1.72\text{\times}{10}^{4}\text{\,}\mathrm{\text{ $e$
}}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}^{2}$ per projection.

We then performed mixed-state ptychographic reconstruction using the
parameters in supplementary Table II, and averaged the resulting phase-
contrast images. The resulting 2D class average is shown in the right panel of
supplementary Fig. 4. We see that the ptychographic reconstruction from full-
multislice simulated data is overall sharper than the experimental
reconstruction, and the overall R-factor between experiment and mixed-state
ptychographic reconstruction from multi-slice simulation is
$32\text{\,}\mathrm{\char 37\relax}$. Possible reasons for the better quality
of the reconstruction from simulated data, even including all partial
coherence and dose effects, are imperfect detector quantum efficiency and
modulation transfer function of the camera, which was not modeled in the
simulations, and slight differences in the structure along the tube.
Especially amorphous carbon wrapped around the tube, which was very helpful
for alignment of the projections, causes resolution loss compared to the
simulated structure.

Because of the better agreement of the linear model with the experiment, and
the much lower computational complexity required for simulation, we performed
subsequent tomographic reconstructions from model-generated projections with
the linear model. We generate 28 projections with the linear forward model as
above and reconstruct the volume with the same algorithm and alignment
strategy as the experimental dataset, described in Section .5. We then trace
the atomic coordinates in the reconstructed volume with the same parameters as
the experimental projections as described in Section .6, resulting in 575
traced atomic coordinates. The traced coordinates are then aligned to the
experimental coordinates with the iterative closest point algorithm, and atoms
obtained from simulation paired with the nearest traced atom from experiment.
We then determine the displacement between experimental and simulated,
retraced atoms. supplementary Fig. 6a shows a histogram of the RMS
displacement of all 575 paired atoms, with a mean position error of
$17\text{\,}\mathrm{pm}$ and median position error of
$10\text{\,}\mathrm{pm}$. supplementary Fig. 6b shows the spatial distribution
of the displacement errors. It can be seen that Zr & Te atoms close to the top
and bottom of the nanotube, where the missing wedge artefacts are strongest,
have the highest position error.

###  .9 Density functional theory calculations

Previous first principles calculations on ZrTe5 have emphasized the importance
of accurately treating dispersive interactions to reproduce experimental
structural parameters, with a comparison given in the SI, and the resulting
electronic and topological properties [50]. Because of this, we first
performed structural optimizations on bulk ZrTe5 to benchmark our choice of
exchange-correlation functional with dispersive corrections. We find that the
optB86b-vdW functional gives lattice parameters that are less than 1%
different than those measured by powder diffraction, a significant improvement
over PBE which gives an error of over 9% for the $b$ lattice parameter. (See
further details in the SI).

Owing to the prohibitively large unit cell size of the full Zr-Te/CNT
structure depicted in Fig. 2b which comprises over 5000 atoms for full ab
initio treatment, we instead perform calculations of the inner Zr-Te sandwich
structure. To simulate the confining influence of the DW-CNT around the Zr-Te
sandwich, we fix the outermost Te atoms during the structural optimization,
while allowing the remaining Zr and Te positions to relax. We perform fixed-
volume calculations for several values of the short a lattice parameter,
finding a minimum at 3.97 Å, very close to the value of 3.95 Å suggested from
our diffraction analysis. As a test of the stability of the Zr11Te50 structure
without the confining influence of the CNTs, we also performed a fixed-volume
structural optimization allowing all atomic positions to relax without
including van-der-Waals corrections (i.e. using the PBE exchange correlation
functional). With PBE only, we find the Zr11Te50 structure expands, which is
primarily due to the separation between the structural units increasing and
not changes in bond lengths or configurations, as we would expect without the
dispersive corrections.

Data and Code Availability: The data and code used to produce the plots within
this work will be released on the repository Zenodo upon publication of this
preprint.

Acknowledgments: We would like to thank Gatan, Inc. as well as P. Denes, A.
Minor, J. Ciston, J. Joseph, and I. Johnson who contributed to the development
of the 4D Camera.

Competing Interests: The authors declare no competing interests.

Funding Information: PP and MCS are supported by the Strobe STC research
center, Grant No. DMR 1548924. CO acknowledges support from the Department of
Energy Early Career Research Award program. Work at the Molecular Foundry was
supported by the Office of Science, Office of Basic Energy Sciences, of the
U.S. Department of Energy under Contract No. DE-AC02-05CH11231. SS, DP and AZ
acknowledge support from the Director, Office of Science, Office of Basic
Energy Sciences, Materials Sciences and Engineering Division, of the US
Department of Energy under Contract No. DEAC02-05-CH11231: within the
sp2-Bonded Materials Program (KC-2207) which provided for high resolution TEM
imaging; and within the van der Waals Heterostructures program (KCWF16) which
provided for synthesis of the encapsulated material, and the National Science
Foundation under Grant No. DMR-1807233 which provided for preliminary
conventional TEM imaging. This research used resources of the National Energy
Research Scientific Computing Center (NERSC), a U.S. Department of Energy
Office of Science User Facility located at Lawrence Berkeley National
Laboratory, operated under Contract No. DE-AC02-05CH11231 using NERSC award
ERCAP0020898 and ERCAP0020897.

Author contributions: S.S. and D.P. prepared and screened CNT-encapsulated Zr-
Te samples under supervision of A.Z. P.E. developed parts of the 4DCamera
control and data management software. P.M.P. carried out the 4D-STEM
experiments, assisted by P.E. M.C.S. performed tilt axis alignment
measurements. P.M.P. implemented the data preprocessing codes and adapted the
mixed-state ptychography reconstruction for 4DCamera data. P.M.P. and H.D.
performed mixed-state ptychography reconstructions. P.M.P. developed the
tomographic reconstruction codes and performed tomographic reconstructions.
C.O. developed elliptical tube fitting codes, template matching, and unit cell
extraction codes. P.M.P and C.O performed unit cell averaging and atom
tracing. P.M.P. performed multi-slice simulations and precision estimation.
C.O, P.M.P., D.P. discussed and solved the atomic structure. S.G. performed
DFT calculations, using input models created by C.O. The study was conceived
by P.M.P. under discussion with M.C.S, C.O. and S.S. P.M.P., C.O. M.C.S and
S.G. wrote the manuscript with input from all authors.

## Supplementary Material

Microscope Voltage | $80\text{\,}\mathrm{keV}$
---|---
Electron gun | S-FEG
Source size (FWHM) | $0.8\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
Cc | $0.6\text{\,}\mathrm{mm}$
Defocus spread (FWHM) | $6\text{\,}\mathrm{nm}$
Convergence semi-angle | $25\text{\,}\mathrm{mrad}$
Depth of focus | $9\text{\,}\mathrm{nm}$
Detector | 4D Camera @ $87\text{\,}\mathrm{kHz}$
Detector pixel size |  $4.14\text{\times}{10}^{-3}\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{-1}$ / $173.6\text{\,}\mathrm{\SIUnitSymbolMicro rad}$
Detector pixel size (binned) |  $0.0272\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{-1}$ / $1.136\text{\,}\mathrm{mrad}$
Detector outer angle | $50\text{\,}\mathrm{mrad}$
Reconstruction pixel size | 0.397 Å
Energy filter | No
Number of projections | 36
Tilt range | $-53.9\text{\,}\deg$
| $65.2\text{\,}\deg$
Total recorded patterns | $34\,560\,000$
Manually selected patterns | $8\,423\,259$
STEM step size | $0.397\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
STEM dwell time | $11.49\text{\,}\mathrm{\SIUnitSymbolMicro s}$
Probe current | $40\text{\,}\mathrm{pA}$
Electron fluence accumulated | $6.28\text{\times}{10}^{5}\text{\,}\mathrm{\text{ $e$ }}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}^{2}$
Electron fluence per projection | $1.72\text{\times}{10}^{4}\text{\,}\mathrm{\text{ $e$ }}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}^{2}$
Avg. electrons per pattern | $2726$
Table 1:  Experimental parameters for PAET. Figure 1:  Ptychographic electron tomography tilt series of the the multi-walled carbon nanotube filled with $\mathrm{ZrTe_{x}}$. The 36 phase-contrast projection images with a tilt range from $-53.9$ to $65.2$ degrees (shown at bottom right of each panel) were measured with the 4D Camera and reconstructed using the LSQML algorithm. The total electron dose of the experiment is $6.28\text{\times}{10}^{5}\text{\,}\mathrm{\text{ $e$ }}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}^{2}$. The inset shows the power spectrum of each projection. Figure 2:  Tilt series of unit-cell averaged ptychographic phase-contrast images from $-53.9\text{\,}\mathrm{\SIUnitSymbolDegree}$ to $37.2\text{\,}\mathrm{\SIUnitSymbolDegree}$. Nprobes | 4
---|---
Nmodes | 4
method | MLs
Niter | 115
probe_pos_search | 10
beta_object | 1
beta_probe | 1
likelihood | amplitude
grouping | 200
apply_subpix_shift | true
variable_probe | true
variable_probe_modes | 1
variable_intensity | false
beta_LSQ | true
apply_multimodal_update | false
delta_p | 0.1
probe_inertia | 0.1
W | 0
object initialization | constant phase
probe initialization | defocus from interactive
| single-sideband reconstruction [36]
Table 2:  Reconstruction parameters for LSQML reconstruction. Figure 3:
Reconstructed probe wave function for each tilt. Amplitude is shown using
color saturation, while phase of each pixel is given by the hue. Figure 4:
Left panel: 2D class average obtained from experimental mixed-state
ptychography reconstruction. Middle: 2D unit-cell average obtained from 12
atomic models where the carbon nanotube was shifted by multiples of
$17.75\text{\,}\mathrm{pm}$ along the tube direction. Right Panel: 2D unit-
cell average obtained from 12 ptychographic reconstructions of 4D-STEM
datasets simulated with the PRISM algorithm and partial spatial and temporal
coherence effects accommodated. The carbon nanotube was shifted by multiples
of 17.75 pm along the tube direction. Scale bar: $2\text{\,}\mathrm{nm}$
Figure 5:  Reconstructed projections a) before, b) during and c) after the
tilt series, close to $0\text{\,}\mathrm{\SIUnitSymbolDegree}$ tilt. c) has a
slightly different viewing angle due to tilt hysteresis. Only minor damage to
the DW-CNT is visible after the tilt series is taken, indicated by the red
arrow in c). Figure 6:  a) Histogram of the distance between experimentally
determined coordinates and coordinates obtained from traced coordinates from a
simulated tilt series with the determined atomic model. Mean position error is
$17\text{\,}\mathrm{pm}$ and median position error is
$10\text{\,}\mathrm{pm}$. b) Spatial distribution of the position errors of
the Zr & Te atoms, viewed along the nanotube. Figure 7:  Projection of the
core of the nanotube along the nanotube axis. Overlayed are the color-coded
traced atomic positions depending on the number of the nearest neighbors
(NNs). Atomic columns with predominantly less or equal 4 NNs were assigned as
Te, while columns with predominantly greather than 4 NNs were asssigned Zr.
The chemical species of the circled atoms was determined by choosing the
lowest-energy configuration as determined by density functional theory.

### Density Functional Theory calculation details

All DFT calculations were carried out with the Vienna Ab initio Simulation
Package (vasp) [51, 52, 53] using projector augmented wave (PAW)
pseudopotentials [54, 55]. Zr(4s, 4p, 5s, 4d) and Te(5s, 5p); electrons were
treated as valence , and the wave functions of the system were expanded in
plane waves to an energy cutoff of 600 eV. Gamma-centered k-point grids of
$14\times 14\times 4$, $12\times 12\times 1$, and $12\times 12\times 1$ were
used for Brillouin Zone sampling of the bulk ZrTe5, innercore Zr-Te tube, and
central Zr-Te structure respectively. Dispersive corrections were accounted
for using the optB86b-vdW of Klimeš et al. [56] which gave calculated lattice
parameters for bulk ZrTe5 in the $Cmcm$ space group that are very close to
those measured at 10 K [57]. Spin-orbit coupling was not included in the
structural optimizations as it was found to have very little effect on the
structural parameters for bulk ZrTe5, however it was included self-
consistently in the electronic structure calculations. The electronic
convergence criteria set to $10^{-7}$ eV and the force convergence criteria
set to 0.002 eV / Å.

Topological characterization using symmetry indicators and DFT calculated band
structures was carried out using symtopo [58]. Previous work on ZrTe5
discussed how the choice of exchange-correlational functional is essential for
the accurate description of its structure, and hence electronic and
topological properties [57, 59, 50]. We find that the optB86b-vdW of Klimeš et
al. [56] gives lattice parameters extremely close to those measured in
experiment [57].

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