# Utilizing multimodal microscopy to reconstruct Si/SiGe interfacial atomic
disorder and infer its impacts on qubit variability

Luis Fabián Peña∗ Sandia National Laboratories, Albuquerque NM, 87185 USA
Current affiliation: Centre of Excellence for Quantum Computation and
Communication Technology, School of Physics, University of New South Wales,
Sydney, NSW 2052, Australia  Justine C. Koepke  Sandia National Laboratories,
Albuquerque NM, 87185 USA  J. Houston Dycus  Advanced Microscopy, Eurofins EAG
Materials Science, Raleigh, NC USA  Andrew Mounce  Center for Integrated
Nanotechnologies, Sandia National Laboratories, Albuquerque NM, 87185 USA
Andrew D. Baczewski  Center for Computing Research, Sandia National
Laboratories, Albuquerque NM, 87185 USA  N. Tobias Jacobson  Center for
Computing Research, Sandia National Laboratories, Albuquerque NM, 87185 USA
<EMAIL_ADDRESS>Ezra Bussmann  Center for Integrated Nanotechnologies,
Sandia National Laboratories, Albuquerque NM, 87185 USA<EMAIL_ADDRESS>
###### Abstract

SiGe heteroepitaxial growth yields pristine host material for quantum dot
qubits, but residual interface disorder can lead to qubit-to-qubit variability
that might pose an obstacle to reliable SiGe-based quantum computing. We
demonstrate a technique to reconstruct 3D interfacial atomic structure
spanning multiqubit areas by combining data from two verifiably atomic-
resolution microscopy techniques. Utilizing scanning tunneling microscopy
(STM) to track molecular beam epitaxy (MBE) growth, we image surface atomic
structure following deposition of each heterostructure layer revealing
nanosized SiGe undulations, disordered strained-Si atomic steps, and
nonconformal uncorrelated roughness between interfaces. Since phenomena such
as atomic intermixing during subsequent overgrowth inevitably modify
interfaces, we measure post-growth structure via cross-sectional high-angle
annular dark field scanning transmission electron microscopy (HAADF-STEM).
Features such as nanosized roughness remain intact, but atomic step structure
is indiscernible in $1.0\pm 0.4$ nm-wide intermixing at interfaces. Convolving
STM and HAADF-STEM data yields 3D structures capturing interface roughness and
intermixing. We utilize the structures in an atomistic multivalley effective
mass theory to quantify qubit spectral variability. The results indicate (1)
appreciable valley splitting (VS) variability of roughly $\pm$ $50\%$ owing to
alloy disorder, and (2) roughness-induced double-dot detuning bias energy
variability of order $1-10$ meV depending on well thickness. For measured
intermixing, atomic steps have negligible influence on VS, and uncorrelated
roughness causes spatially fluctuating energy biases in double-dot detunings
potentially incorrectly attributed to charge disorder. Our approach yields
atomic structure spanning orders-of-magnitude larger areas than post-growth
microscopy or tomography alone enabling more holistic predictions of disorder-
induced qubit variability.

Quantum dot, qubit, SiGe, valley splitting, STM, MBE

##  I Introduction

Nanoelectronic devices using Si/SiGe heterostructures to host quantum dot
qubits offer robust coherence, one-/two-qubit gate fidelity, and compact
device footprints compatible with Si foundry processing.[1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12] With the ultimate goal of monolithic Si integration, recent
qubit research primarily utilizes epitaxial single quantum well
heterostructures depicted schematically in Fig. 1 (a).[13, 14, 15, 16, 17, 18,
19, 20] Briefly, typical qubit heterostructure material comprises a strained-
Si (s-Si) well layer pseudomorphically lattice-matched in-plane with
surrounding relaxed Si1-xGex, $x\sim 0.3$.[18, 19] Leading qubit varieties
consist of two or three coupled electrostatic dots, depicted as harmonic wells
in Fig. 1 (a), confining one or a few electrons vertically in the s-Si well by
type-II band offsets, Fig. 1 (b), and laterally by voltages on nanoscale metal
gates [top Fig. 1 (a)].[21, 20] Heterostructure growth by chemical vapor
deposition (CVD) and molecular beam epitaxy (MBE) yields suitable qubit
environments in the s-Si well with figures-of-merit including low metal-
insulator percolation e- densities (<$10^{11}$ cm-2), and minimal nuclear spin
background via 28Si (spin-free) isotopic enrichment (>$99.9\%$).[17, 18, 19,
20] Consequently, this material has enabled leading Si-based qubit technology
demonstrations, e.g., coupling multiple high-fidelity qubits and rudimentary
quantum error correction.[7, 8, 9, 10, 11, 12] Investigation and understanding
of salient future scale-up challenges including expected variability over
qubit ensembles is timely.[22, 23]

Residual Si/SiGe interfacial atomic structure disorder is one cause for qubit
variability.[24, 25, 26, 27] In contrast to the ideal of flat interfaces and
abrupt potentials, realistic structure includes disorder, inset right side
Fig. 1 (a), and resulting variability in qubit confinement potentials, bottom
Fig. 1 (b).[2] Disorder-induced qubit variability might result from intimate
contact between dot electron wave functions and disordered interfaces
exhibiting (1) random intermixing between miscible Si and Ge and (2) growth
roughness of interfaces between the Si and SiGe layers. Factor (1) results in
broader interface barrier potential, bottom Fig. 1 (b), and consequent dot-to-
dot variability in the valley character of low-lying orbitals, impacting both
valley splitting (VS) and inter-dot tunnel or exchange couplings used to drive
logic gating operations.[28, 24, 29] Factor (2) modifies the quantum well
width, $w$, in Fig. 1 (b), resulting in appreciable variation of inter-dot
energy biases that might otherwise be attributed to charge disorder. For qubit
operation protocols that may depend on tight control of quantum dot energy
offsets, such as “conveyor mode” shuttling, this might serve as a nuisance
source of variation that is similar in effect to but distinct in origin from
charge disorder.[30, 31]

Valley splitting is a focal point for experiments, theory, and simulations
because experiments indicate VS variability in the range $20-300$ $\mu$eV.[15,
32, 33, 34, 35, 36, 37, 27, 10] This variability is significant because it is
comparable to the energies separating typical qubit basis states. So, valley
states are a degree-of-freedom that can act as a potential leakage channel or
be harnessed into new forms of qubits.[15, 32, 33, 34, 35, 36, 38, 37, 27] In
either case, it is useful to understand and quantify VS variability.

To understand and discover control strategies for qubit-to-qubit VS
variability, empirical pseudopotential, tight-binding, and effective mass
calculations have been useful tools.[2, 31] Early work assessed VS variability
starting from principled assumptions that as yet unresolved embedded interface
structure consists of features, e.g., discrete mono-/bilayer- atomic-step
roughness, seen on s-Si and SiGe growth surfaces with local miscuts, Fig. 1
(c).[39, 40, 41, 42, 43, 2] More recently, near-atomic-resolution studies
using atom-probe tomography (APT) supplemented with high-angle annular dark
field scanning transmission electron microscopy (HAADF-STEM) imaging over the
10-nm-scale, show gradual interface transitions, Fig. 1 (b-c), and diffuse
alloy disorder essentially ruling out abrupt stepped interfaces. Reported
interface widths span 0.7-1.0$\pm$0.3 nm, i.e. several atomic layers, with
alloy number fluctuations adding variability.[44, 45, 27] Hence, recent
theory, simulation, and experiment focus primarily on VS variability owing to
random alloy disorder and alloy fluctuations.[24, 23, 27, 46]

In contrast to VS variability, which is a consequence of the particular
atomic-scale alloy disorder realized in the vicinity of any given quantum dot,
orbital level variability in response to disorder is comparatively
straightforward to understand as a consequence of varying well width, $w$,
Fig. 1 (b).[2] Well width is expected to vary owing to undulations, e.g.,
local growth roughness, at each interface. Prior work with hard X-ray nanospot
diffraction shows roughly periodic (few-hundred-nm wavelength) lateral
undulations of well width at a few atomic layer amplitude.[47] Such long-
period undulation is unlikely to be connected with (Å-scale) alloy disorder
and was attributed to epitaxial growth roughness, Fig. 1 (b). Overall, the
available data hints at a qualitative structural description of the s-Si well
and interfaces including longer-period undulations (roughness) convolved with
diffuse interface broadening (intermixing), as depicted in the perspective
view in Fig. 1 (c).

Anticipating appreciable qubit-to-qubit variability owing to contributions of
roughness and alloy disorder, a strategy embraced in recent works is to
engineer ensemble distributions, e.g., VS distributions, over many qubits by
targeted manipulation of ensemble disorder.[24, 23, 27, 46] For example,
precision placement of inherently-disordered layers of Ge in or near the well
is found to amplify VS.[23, 27] To advance this strategy, some recent theory
implementations (atomistic tight binding, empirical pseudopotential, effective
mass theory) have been developed to predict structure-VS relationships from
atomistic materials descriptions capturing specific disorder realizations.[23,
27, 48] The calculations demand accurate ensemble descriptions of buried
interfaces over volumes encountered by numerous qubits sampling multiple forms
of disorder. Comprehensive 3D multiscale ensemble descriptions capturing both
longer-ranged undulations convolved with interfacial alloy intermixing, Fig. 1
(c), are intractable for individual local probing methods, e.g., HAADF-STEM
and APT, Fig. 1 (d), owing to limited sample volumes and image convolution
effects, indicating that a combination of techniques sampling across atomic-
to-micrometer interface disorder realizations will yield more complete
structural descriptions.

In this manuscript, we describe a multimodal microscopy approach to
reconstruct 3D atomic structure at Si/SiGe heterointerfaces. Then we use the
structures to model dot-to-dot (qubit) variability of VS and orbital levels
(detunings). The results characterize interface undulation, alloy disorder,
and resulting variability of dot spectral properties over areas (>$1\mu$m2)
characteristic of multi-qubit devices. Utilizing scanning tunneling microscopy
(STM) to track molecular beam epitaxy (MBE) growth, we image surface atomic
structure over micron-square areas following deposition of each
heterostructure layer. STM indicates Å-to-nanometer roughness of s-Si and
relaxed Si0.7Ge0.3 growth surfaces that subsequently become buried interfaces.
The finished heterostructure interfaces are imaged using cross-sectional
HAADF-STEM. For s-Si, STM shows atomically flat growth surfaces with Poisson-
random terrace sizes cascading along the (local) miscut.[49] By contrast, 2D
adatom island nucleation, stacking, and resulting nanometer-sized undulations
dominate SiGe surfaces. Si-Ge miscibility and established intermixing pathways
render uncertain what surface structures survive burial during overgrowth
($T=550^{\circ}$C). Post-growth HAADF-STEM imaging near regions probed by STM
reveals that nanometer-sized SiGe roughness survives largely intact, while
Å-sized s-Si step-terrace structure is lost in diffuse $1.0\pm 0.4$ nm-wide
interfaces. Interfaces in STM and HAADF-STEM are reconciled assuming
established intermixing mechanisms, supporting an overall structure
description intractable to prior reported approaches using individual local-
probe techniques, e.g., HAADF-STEM or APT alone. Notably, our STM and HAADF-
STEM data reveal interface roughness autocorrelation lengths ($40-107$ nm)
that are longer than the entire span ($\sim 30\times 30$) of post-growth
microscopy or tomography structural data used in related prior works.[44, 27]
We use our structure data to calculate dot-to-dot variability including well
thickness variation-induced inter-dot energy biases and valley splitting
statistics. We predict a spread of electronic conduction band valley
splittings of $0-200\mu$eV, which is consistent with available experimental VS
measurements on qubits.[15, 32, 33, 34, 35, 36, 37, 27, 10] Also, we use our
data to estimate inter-dot bias spatial variability owing to interface
roughness modulating quantum well confinement along the growth axis by up to
an appreciable tens of meV for typical well thicknesses ($5-10$ nm). We are
not aware of such a broad-range ensemble structure-properties description at
the atomic resolution limit in both measurement and modeling having been
reported previously. Both model findings related to VS and orbital variability
have significance for understanding performance limits in quantum computing
applications.

Figure 1: (a) A schematic of the heterostructure in this work consisting of a
s-Si well layer embedded within SiGe layers. The inset (within the dashed
rectangle) depicts intermixing and roughness disorder near interfaces. (b) An
energy $E$ diagram indicating that a quantum well forms in the s-Si layer
owing to conduction band edge offset from adjacent SiGe layers, and the
confinement potential width, $w$, and transition length ($\sim 4\tau$) are
shaped by growth roughness and intermixing. Generally, interface roughness is
not strictly correlated between interfaces and intermixing may be superposed
to yield (c) complex 3D interface structures that are intractable for (d) the
highest spatial resolution post-growth measurement techniques, such as
transmission electron microscopy (TEM) and APT owing to, e.g., limited
practical measurement volumes smaller than established roughness
autocorrelation lengths, along with probe convolution effects, e.g., averaging
of structure information along the TEM beam path through a cross-sectional
slice.

##  II Experiment, SiGe growth & atomic structure measurements

Our STM growth study follows a layer sequence in Fig. 2 (a). Supplementary
Information (SI) S1 describes material preparations, the growth process, and
STM data acquisition and analysis (see Figs. S1.1-1.4). Si/SiGe
heterostructure MBE on relaxed, epi-ready, Si0.7Ge0.3 virtual substrates
started by depositing a $70$ nm-thick SiGe regrowth layer surface shown in STM
images in Fig. 2 (b-c). The regrowth surface undulates at the nanometer scale
(RMS roughness $\sim 0.5$ nm) and sparse metastable artifacts resembling pits
appear occasionally, Fig. 2 (c). Similar pits are associated with threading
dislocations terminating at the surface, although we see no indication of
dislocations reaching the heterostructure layers in post-growth HAADF-STEM
images.[50, 51, 52]

Next, a $\sim$15 nm thick s-Si well was deposited on the SiGe regrowth layer.
Comparing STM data, Fig. 2 (b-c), the well’s dominant qualitative surface
features are totally different from the regrowth surface. Moreover, it is
evident that the layout of atomic steps at the s-Si well surface differs from
the uniform staircase-like distribution commonly utilized in prior models.[2].
The well surface is dominated by atomic steps with extreme fluctuations, which
is an indicator of competition between underlying strain fields, step
energies, and surface stress that manifests step oscillations, and even
bunching in equilibrium, which enhances roughness.[53, 54, 51, 55, 56, 57, 55]
As the average terrace width decreases, the likelihood of step bunching
increases.[58]

A smoothing effect was observed after Si epitaxy, consistent with its inherent
step-flow growth mechanism at $T\geq 550^{\circ}$C that reduces surface
roughness (similarly observed by Baribeau and Kuan et al.).[59, 60] Contrary
to historical model assumptions, for low miscut samples (miscut$\sim
0.5^{\circ}$), we observed no correlation between atomic-scale surface
structure from the underlying substrate and the top of the quantum well, Fig.
2 (c).[2] This indicates that initial surface topography does not translate to
subsequent layers, and new material does not grow conformally but instead is
governed by each layer’s differing growth dynamics. For example, pit-like
artifacts observed on the regrowth layer did not translate to the s-Si well,
indicating that they are metastable in SiGe layers. These observations are
captured quantitatively in a structure analysis via $z$ (height)
autocorrelation functions (see SI1 Fig. S1.5 and S1.6). Autocorrelation
functions indicate significantly different surface character for the SiGe
regrowth versus the s-Si well surfaces, both in terms of root mean square
(RMS) roughness (0.54 versus 0.18 nm, respectively) and correlation lengths
(45 nm versus 107 nm, respectively), indicating no evident relationship of
surface morphology. The apparent lack of correlation from interface to
interface is most qualitatively consistent with Evans et al.’s X-ray data.[47]
Notably, uncorrelated interface behavior indicates that well thickness is
likely to vary spatially, leading to variability of qubit (dot) orbital
levels.[47]

Following well growth, a $45$ nm-thick SiGe buffer layer was deposited. Buffer
surface structure, Fig. 2 (c), resembles the SiGe regrowth layer, e.g., there
are surface undulations and some sparse (metastable) pit-like artifacts.[61]
An increase in surface roughness compared to the well is evident (and
summarized in SI1 Fig. S1.6).[62, 63] Finally, a 3-nm-thick Si cap was
deposited as a seed layer for subsequent atomic layer deposition of a
dielectric for metal gate isolation. It is important to point out that this
final surface is not evidently correlated with the surface structure of the
s-Si well, i.e., the random atomic step order differs, Fig. 2 (b). Thus, post-
growth surface measurements, e.g. atomic force microscopy, of the final
surface will not correlate very closely with surface structure at the buried
s-Si well or other layers.

Figure 2: The heterostructure layer sequence. Each column of this figure is to
be read from bottom to top and shows: (a) a schematic indicating
heterostructure layer sequence, (b) STM images showing surface structure upon
layer completion just prior to overgrowth and heterointerface formation, (c)
STM images in smaller areas showing some key atomic structure features. Note
that the area of the images in column (c) is orders of magnitude greater than
the interface area typically sampled by techniques such as APT and HAADF-STEM.
STM images were acquired at 3V/0.5 nA tunnel current. The data has been
plotted in a coordinate system defined by principal crystal directions, as
indicated, to enable accurate, oriented, geometric comparisons to HAADF-STEM
data.

So far, we have described growth surfaces observed via STM after each new
heterostructure layer is added, and just prior to overgrowth and
heterointerface formation. Since surface dynamics including step mobility and
intermixing are integral to growth, it is nearly certain that surfaces that we
observed with STM change through interface formation.[62, 64, 65, 66]

To compare STM surface structure with buried interface structure, we utilize
post-growth cross-sectional HAADF-STEM scanning transmission electron
microscopy along [110] directions. Details of HAADF-STEM lamella preparation,
measurement, and analysis are described in Supplementary Information (SI) S2.
The cross-sectional lamella imaged here is cut from an interface region that
is near (within same square millimeter) but not identical to STM data in Fig.
2 (b). The lamella is wedge-shaped tapering from roughly $20-120$ nm-thick as
assessed by scanning transmission electron microscopy electron energy loss
spectrometry (STEM-EELS)(see SI2 Fig. S2.1). Fig. 3 (a) shows a HAADF-STEM
image of the s-Si well interface with adjacent SiGe layers. HAADF-STEM
intensity, $I$, is a probe of nuclear charge, $Z$, with $I\sim Z^{1.8}$, so
the image contrast is an indicator for Ge content in each atomic column along
the electron beam path.[44] To emphasize nanosized interface features, the
HAADF-STEM data is plotted in Fig. 3 (b) with a $\sim$1:10 vertical stretch
(see SI2 Fig. S2.2 and related discussion). Note that exaggerated vertical
scaling is typical for all STM data to emphasize atomic steps and roughness,
$cf$ x versus z-scales in Fig. 2 (b). The HAADF-STEM images reveal: (1) that
nanosized SiGe regrowth surface undulations survive overgrowth/burial, and (2)
as anticipated, the s-Si well surface is relatively flat with some small
undulations ($\sim$1 nm) reminiscent of waviness due to step-density
fluctuations and bunches observed in STM data, Fig. 2 (b-c).

As a reference for comparing STM and HAADF-STEM image features, we have
plotted several STM line traces from both the $[110]$ and $[\bar{1}10]$
directions, Fig. 3 (c), encompassing roughly the same interface area ($\sim
900$ nm length $\times 47$ nm thickness, into the plane of the page) as the
HAADF-STEM, and it is clear that nanoscale roughness at buried interfaces is
similar to the STM growth surface roughness. Consistent with STM observations,
the most significant observation from Fig. 3 (a-b) is that the well thickness
varies appreciably owing to roughness of the SiGe regrowth relative to the
nearly atomically flat Si.

In addition, we have calculated and compared autocorrelation functions
describing the STM (growth surface) and HAADF-STEM (post-growth interface)
structure (SI2 Fig. S2.3). Consistent with the visually apparent similarity in
the surface and interface data, the surface/interface autocorrelation traits
(RMS roughness and correlation length) are similar, except for a significantly
smaller post-growth correlation length (41 nm versus 107 nm) and an increased
roughness (0.25 nm versus 0.18 nm) observed for the s-Si well top interface.
We attribute this decrease in correlation length and increase in roughness to
disordering effects of intermixing described in detail next.

In contrast to our STM data, atomic-resolution HAADF-STEM does not show any
atomically abrupt interfaces or atomic step-terrace structure, but rather
$\sim$1 nm-wide gradual diffuse interfaces in several atomic-resolution images
of various lamella thicknesses. Fig. 3 (d) shows one example of an atomic-
resolution HAADF-STEM image, and Fig. 3 (e) shows atomic detail of the gradual
Si-SiGe transitions typical of our HAADF-STEM data, as well as other recent
atomic-resolution HAADF-STEM and APT.[44, 27] From our STM spatial
distribution of steps (See SI1 Fig. S1.7-1.9), we would expect to observe
atomically-abrupt terraces (0.134 nm wide interfaces) over terrace regions
with a finite probability of capturing at least a few steps in an area the
size in Fig. 3 (e). Instead, as indicated in Fig. 3 (f), we observe smooth
interface transitions spanning several atomic layers with smooth transitions
reasonably estimated by sigmoid curves (see SI2 Figs. S2.4-2.8 for analysis
details). The metric that we use to measure the interface width is $4\tau$,
where $\tau$ is the sigmoid width parameter. The parameter $4\tau$ measures
the distance for $\sim 0.12-0.88$ of the full transition, and is a commonly
utilized and easy-to-automate tool to estimate HAADF-STEM and APT interface
widths.[44, 27]

Figure 3: (a) Post-growth cross-sectional HAADF-STEM image of the
heterostructure. Note: STEM-EELS indicates cross-section thickness (normal to
page) ranges from 20-50 nm thick from left-to-right. (b) HAADF data plotted
with $\sim 10:1$ vertical:horizontal scaling highlights nanosize structure
similar to (c) STM data plotted on the same scale, in the same crystal-
oriented coordinate system, along $x_{i=1,2}=$ [110]-equivalent directions.
Comparing HAADF-STEM images with STM data shows nanosized growth surface and
buried interface features are very similar. (d) Atomic-resolution HAADF-STEM
of the s-Si well, (e) cropped to show heterointerfacial atomic details
indicating that the interfaces are gradual and intermixed at the atomic scale.
Here, STEM-EELS (SI2) indicates 22 nm lamella thickness. (f) Plots of column
intensity across the interface span approximately $7\pm 3$ atomic layers on
average across the image, after correcting for HAADF-STEM beam spread within
the solid. The method of interface width measurement is indicated in SI2.

Si and Ge intermixing between layers during overgrowth is a plausible cause
for nanometer-wide interfaces in HAADF-STEM. This interpretation is consistent
with established Si-Ge bulk miscibility accessed via thermally-activated
surface and near-surface intermixing pathways likely to be operative at $T\leq
550^{\circ}$C.[62, 64, 65, 66] However, preexisting tilts and nanosized-
roughness present on growth surfaces in STM, Fig. 2 (b-c), and subsequently in
HAADF-STEM, Fig. 3 (a-b), both contribute to the apparent cross-sectional
interface width since the HAADF-STEM beam interacts with and convolves
embedded structure along its path through the lamella.

In order to discern alloy intermixing-induced broadening requires deconvolving
it from other factors that potentially contribute to apparent interface width,
e.g. preexisting crystal tilts and roughness seen in STM, in Fig. 2 (b-c). We
take an approach outlined in Fig. 4 (a-b). In principle, preexisting roughness
contributions can be deconvolved by performing HAADF-STEM with sufficiently
thin lamella thicknesses that only the local alloy distribution contributes to
the interface width. To estimate the scale for which roughness and tilt
contributions to the width become negligible, we take an approach of measuring
the HAADF-STEM cross-section interface width as a function of a few lamella
thicknesses (22, 45, and 120 nm, see SI2 Fig. S2.4) and comparing the
interface width values to contributions from preexisting tilts and roughness
calculated from STM data. The approximate HAADF-STEM sample volumes are
depicted in Fig. 4 (a). To get an ensemble measurement of the interface width
from each HAADF-STEM image area, we apply an image segmentation and sigmoid
fitting routine, shown schematically in Fig. 4 (a), to extract an interface
width, $4\tau$, from every atomic plane in all atomic-resolution HAADF-STEM
images [see SI2 Fig. S2.8]. For conciseness, we use a mean $4\tau\pm\sigma$,
where sigma denotes the standard deviation from the mean for each lamella
thickness.

Next, we estimate interface width contributions due to preexisting surface
roughness and tilts in STM data. We use a metric $\Delta
z=\mathrm{max}(z)-\mathrm{min}(z)$, that we refer to as the range function,
calculated over intervals, $\Delta x_{i=1,2}$ along $(110)$-equivalent
directions in the STM data, i.e. along the same directions probed by HAADF-
STEM. The concept of the STM tilt and roughness characterization for
comparison to HAADF-STEM is shown in Fig. 4 (b). We take $\Delta x_{i=1,2}$ to
span the HAADF-STEM lamella thickness range. By rastering the $\Delta
x_{i=1,2}$ window over micron-sized STM data, Fig. 4 (c), a micron-scale
characteristic ensemble description for $\Delta z$ is calculated along $\Delta
x_{i=1,2}$ with results shown in Fig. 4 (d-e).

Finally, we place bounds on the interface intermixing length, $L$, based on
our data in Fig. 4. In Fig. 4 (d) and (e), we see that only two HAADF-STEM
interface widths from the thinnest ($22$ nm and $45$ nm-thick) lamella for the
s-Si well surface are clearly distinguishable from the cloud of $\Delta z$
values. By contrast, all HAADF-STEM interface widths for the SiGe regrowth
surface fall within the range of nanosized roughness and tilt observed in STM
data. As we noted earlier, nanosized roughness (undulation) of SiGe surfaces
is evident in post-growth HAADF-STEM data, cf. Fig. 3 (a), and it is probable
that we are observing these preexisting features in the HAADF-STEM for all
lamella thickness. From the thinnest samples (22 and 45 nm thick) of the s-Si
well interface we measure $4\tau=1.2\pm 0.5$ nm and $1.0\pm 0.3$ nm
respectively, while growth roughness and tilt contributions are on the order
of $0.04-0.08$ nm, and $0.1-0.25$ nm, respectively depending upon direction.
Since the STM surfaces are atomically flat, and $\Delta z$ in STM is largely
dominated by the tilt [i.e. overall slope Fig. 4 (c)] $\Delta z$ increases
linearly with $\Delta x_{i=1,2}$ and contributes directly to interface
thickness. However, for approximate intermixing length bounds, we take the
difference (in quadrature) $L=[4\tau^{2}-\Delta z^{2}]^{1/2}$, where the
$\Delta z$ values $(0.04-0.25~{}\mathrm{nm})$ contribute negligibly, and we
conclude that $0.9$ nm $<L<1.2$ nm ($\pm 0.4$ nm) or $7-9(\pm 3)$ layers.

Figure 4: Analysis of interface width in cross-sectional HAADF-STEM images.
(a) Schematic view of our technique to measure interface width showing the
approximate volumes imaged with the electron beam directed along $\Delta x$
through lamella thicknesses of 22, 45, and 120 nm. Each image is discretized
by atomic columns and a measure taken of Z (atomic number) contrast, all
atomic columns are fit using an automated sigmoid-fitting process, then the
value $4\tau$ (see SI2) is taken as a measure of interface width. (b)
Schematic showing our approach for estimating an interface-width contribution,
$\Delta z$, due to roughness in STM data. (c) Two orthogonal views of the STM
data used to calculate ensemble roughness, plotted in the crystal-oriented
coordinates with (001) as the z-axis and (110)-directions aligned to $\Delta
x_{i=1,2}$. A single line is plotted in black to indicate directional
relationships in the two orthogonal views. (d) s-Si well-to-buffer interface
width vs. lamella thickness ($\Delta x$) characterized by HAADF-STEM $4\tau$
and surface width contributions, $\Delta$z, from STM data along $\Delta
x_{i=1,2}$ orientations. (e) SiGe regrowth-to-Si well interface width vs.
lamella thickness ($\Delta x$) characterized by HAADF-STEM 4$\tau$ with
$\Delta z$ from STM data.

Si and Ge are fully bulk miscible, and there are a few surface/near-surface
mechanisms that are likely to be active allowing Ge to intermix with the Si
over multiple atomic layers, as well as allowing Si transport upward from the
original interface.[67, 64, 62, 68, 69, 65, 70, 71, 66] First, there is a
near-surface enhanced interstitial mechanism of Uberuaga et al. that allows
transport of appreciable Ge up to 4 atomic layers below the original Si growth
surface for $T\leq 500^{\circ}$C.[64] Second, intuitively consistent with bulk
Si-Ge miscibility, SiGe alloys form a 2D surface wetting layer on Si surfaces
by purely surface atomic exchange diffusion processes, leading to an upward
exchange of atoms from surface lattice sites to the supersaturated gas of
adatoms involved in growth, and ultimately into subsequent atomic layers as
they nucleate. Surface exchange diffusion is rapid at $T>90^{\circ}$C for Si
and Ge and anticipated to contribute significantly to intermixing at
increasing temperatures.[62, 68, 66] If atomic exchange-diffusion promotes Si
upward with a probability $p\sim 0.5$, then we anticipate additional Si to be
distributed upward to the nth layer above the original growth surface with a
probability $~{}p^{n}$, such that layers above the original Si surface become
Si-rich in a diminishing, roughly geometric, progression and plausibly
contribute a few additional intermixed layers to the observed 4$\tau$.[62, 68,
66] Hence, we conclude that the intermixing length $7-9$ atomic layers is at
least possible as an outcome of prior established surface intermixing
processes during growth. Finally, note that broadening effects are not limited
to the upper interface, rather they are definitively resolvable and
quantifiable there for thinner lamella (22, 45 nm thick).

##  III Interface disorder model & resulting qubit spectral variability

Given these STM and HAADF-STEM observations, we propose a model for atomic
structure for the well where the mean position $\bar{z}(x_{1},x_{2})$ of each
interface at a given location $(x_{1},x_{2})$ is set by the STM data and the
elemental identity at each lattice site along atomic columns across the
interface is determined by drawing from the sigmoid distribution with a width
set by HAADF-STEM data. In our atomistic multi-valley effective mass theory
simulations, for any given alloy realization we construct a bulk silicon
(diamond) lattice encompassing the simulation domain and then update the Si or
Ge identity of each lattice site according to the above distribution. These
structures are available from the authors on request.

Interface topographic data from both HAADF-STEM (Fig. 3 (a-b)) and STM (Fig. 3
(c)) indicate that the well thickness varies significantly across the sample,
on the scale of a few nm. For example, the well width in the HAADF-STEM data
(Fig. 3 (a-b)) fluctuates by a root mean square deviation of 1.7 nm. The well
thickness sets the energy scale of confinement along the growth axis of an
electron in a quantum dot. For a given quantum dot this amounts to an overall
offset that, for a double- or multi-quantum dot system, will manifest as an
inter-dot energy offset (detuning) bias. To model the consequences of well
thickness variation, we solve the one-dimensional Schrödinger equation for
various well thicknesses, including the potential induced by the conduction
band offsets between the s-Si well and SiGe layers as well as interface
thickness $4\tau$, assumed here to be 1 nm (Fig. 5 (a)). The confinement
energy as a function of well thickness is shown in Fig. 5 (b). We find that
the scale of variation of vertical confinement depends significantly on the
mean well thickness, with thinner wells manifesting much larger fluctuations
in vertical confinement. To estimate the effect of well thickness variation on
the detuning bias offset between nearby double quantum dots, we use the
measured variation of well thickness shown in Fig. 5 (c) and assumption of a 5
nm average well thickness to find the detuning bias variation of Fig. 5 (d)
for quantum dots having a nominal 80 nm center-to-center separation. We
account for the finite size of the dots by performing a Gaussian convolution
over the x-y variation of well thickness with a standard deviation of 15 nm, a
typical quantum dot size. For relatively thin wells ($\sim 5$ nm), this
simulated level of detuning bias variation would correspond to significant
offsets in, for example, the voltage bias on applied gate electrodes required
to induce an inter-dot transition of an electron. Such variation may otherwise
be attributed to charge disorder, but we point out here that well thickness
variation may be another source of bias variation to consider. An open
question to be addressed in future work is how thick a well must be for the
well thickness variation to reach the scale of what we have observed in our 15
nm thick sample. If growth of the s-Si well is closer to conformal for thinner
interfaces, we would expect that variation of well thickness may be
correspondingly reduced.

Figure 5: Influence of interface topography variation and confinement along
the growth axis. (a) Example one-dimensional (valley-free) Schrödinger solve
for the ground state, illustrating quantum confinement along the growth axis
(z-axis) of the well. Here, the quantum well is 5 nm thick, with an
intermixing length of 4$\tau=$1 nm. (b) Confinement energy as a function of
well thickness for 4$\tau=$ 1 nm (inset) Distribution of growth-axis
confinement energies for the measured interface topography, assuming three
different average well thicknesses. (c) Well thickness as a function of x,y
position in the measured sample, assuming an average thickness of 5 nm and
topography measured for the 15 nm sample. A pair of quantum dots 30 nm in
diameter and 80 nm apart is denoted in gray to give a sense of scale. (d)
Calculated effective detuning bias between dots 80 nm apart in a 5 nm well due
to spatial variation of the growth-axis confinement energy.

To probe disorder impacts on valley splitting, we perform multi-valley
effective mass theory simulations of single-electron quantum dots in the
presence of atomistic disorder corresponding to specific alloy realizations.
Our simulation method incorporates detailed Bloch functions derived from
density functional theory (DFT)[72] and treats each Ge atom in the simulation
domain explicitly as a repulsive localized defect potential (see Methods
section for more details). In Fig. 6 (a), we show the distribution of valley
splitting as a function of interface width $4\tau$ for a 5 nm thick well, for
three different cases of step structure. In these calculations, we assume
harmonic confinement in the $x$-$y$ plane corresponding to a 1.5 meV orbital
splitting. We consider a step oriented along the $[110]$ axis that is $m$
atomic monolayers thick and passing through the center of the quantum dot. For
the case of zero intermixing (perfectly abrupt interface), we find that the
presence of the step modulates the valley splitting significantly, though the
valley splitting remains relatively high ($\sim 1$ meV) on average. However,
as the interface width grows the influence of the step rapidly vanishes, with
even a relatively abrupt interface of $4\tau=0.5$ nm exhibiting negligible
step-induced modulation of valley splitting.

Next, we explore how the valley splitting depends on well thickness. In Fig. 6
(b), we show how the valley splitting statistics depend on well thickness in
the presence of an m-monolayer atomic step through the center of the dot, in
the case of an interface width of $4\tau=1$ nm. Well thickness clearly has a
significant influence over valley splitting, with thinner wells clearly
preferable to thicker wells and the presence of the few-monolayer step having
minimal influence over the valley splitting distribution. In Fig. 6 (c), we
show the electronic wave function for a 5 nm well, along with a simulated
HAADF-STEM image of Ge alloying analogous to Fig. 3 (d). These simulations
emphasize the critical importance of small well thickness and relatively
abrupt interfaces in ensuring large valley splitting, while for realistic
intermixing lengths the influence of few-monolayer steps is modest.

Figure 6: Simulations of quantum dot valley splitting in the presence of alloy
disorder and interfacial steps, assuming in-plane harmonic confinement
corresponding to an orbital splitting of 1.5 meV. (a) For a 5 nm thick well,
distributions of valley splitting as a function of interface width
(intermixing length) 4$\tau$ in the presence of either no step (m=0), a single
monoatomic layer (m=1), or two monoatomic layer (m=2) steps through the middle
of the dot oriented along the [110] crystallographic axis. The solid curves
are best-fit Rice distributions.[27] (b) Valley splitting statistics as a
function of well thickness for each of these step configurations, (c) Example
simulation of quantum dot probability density $|\Psi|^{2}$ for a 5nm well,
superimposed on a simulated HAADF-STEM image of Ge concentrations for a
lamella 20 nm thick parallel to the [110] plane.

##  IV Summary

We have presented an analysis of interfacial atomic structure disorder for
s-Si quantum wells bounded by Si0.7Ge0.3 layers. The technique combines two
forms of atomic-resolution data: in-operando/in-situ STM interleaved with MBE
growth, and post-growth HAADF-STEM. STM is applied to image surfaces of each
heterostructure layer immediately prior to subsequent overgrowth (interface
formation). The characteristics of s-Si surfaces include atomic steps
cascading essentially monotonically along local miscuts indicative of a
predominantly step-flow (Frank-van der Merwe) growth mode with little evidence
for nucleation. The resulting atomic “staircases” with Poisson-distributed
terrace widths reflect tendencies for step-bunching and atomic-step meandering
attributable to elastic effects and growth step-attachment dynamics.[49] By
contrast, SiGe has drastically different features dominated by a 2D
nucleation, island-stacking, growth mode (qualitatively Stranski-Krastanov-
like) leading to nanosized roughness and undulations.[73] The s-Si versus SiGe
surfaces reflect different growth dynamics and correlation between adjacent
interfaces is negligible, explaining prior observations of thickness
undulations in buried s-Si wells.[47] Post-growth cross-sectional HAADF-STEM
measurements indicate that nanoscale traits of flat s-Si versus undulating
SiGe survives growth intact, but that interfaces appear broadened by
$\sim$1.0$\pm 0.4$ nm ($\sim$4-10 atomic layers). We rule out growth roughness
as a cause for the observed breadth. Instead, interface breadth is due to
intermixing reflecting Si-Ge miscibility that is accessible by surface/near-
surface atomic kinetic paths.[64, 62, 65, 66] Consistent with STM and HAADF-
STEM observations, we propose a simple overall atomic structure for the well
where the mean position $\bar{z}(x_{1},x_{2})$ of each interface at a given
location $(x_{1},x_{2})$ is set by the STM data and the elemental identity at
each lattice site along atomic columns across the interface is set by drawing
from a sigmoidal distribution. Our model extends beyond prior interface
structure descriptions by more than two orders-of-magnitude in area
($<$100$\times$100 nm2 to $>$1$\times 1$ $\mu$m2). Notably, we find surface
(STM) and interface (HAADF-STEM) roughness autocorrelation lengths ($45$-$107$
nm) that are up to 3 times larger than the dimensions of data (APT and HAADF-
STEM) used in other recent studies, underlining that our approach reveals
broader information.[44, 27]

Finally, we utilize our structures to estimate growth-axis confinement energy
and valley splitting variability for quantum dots in the s-Si layer.
Confinement energy variability is calculated using a straightforward effective
mass theory solution for the ground state confinement energies which vary in
response to spatial undulations of well width resulting from uncorrelated
roughness at Si/SiGe surfaces. We find that this interfacial roughness leads
to appreciable confinement energy variability in our simulations of tens of
meV for a 5-nm-thick well. This implies that each dot must be uniquely tuned
adding complexity to e.g. shuttling, requiring coordinated manipulation of the
electrostatic potential landscape. Valley splitting varies appreciable, e.g.
in the range of 0-200 $\mu$eV for a 5-nm-thick well with $4\tau=1$ nm
interface width. This significant VS variability presents similar measurement
challenges. Mitigation strategies for VS variability have been proposed, e.g.
positioning Ge layers strategically in the well or near interfaces to
regularize valley-state phases and break degeneracies.[2, 23, 27] We
anticipate our structure data and model to be useful in understanding outcomes
of such strategies.

##  V Acknowledgement

We thank R. E. Butera and C. J. K. Richardson of University of Maryland, C.
Barry Carter, Emeritus Professor U. of Connecticut, and Tzu-Ming Lu at Sandia
National Laboratories for thought-provoking discussions. This work was
performed at the Center for Integrated Nanotechnologies, an Office of Science
User Facility operated for the U.S. Department of Energy (DOE) Office of
Science. Research supported as part of $\mu$-ATOMS, an Energy Frontier
Research Center funded by the U.S. Department of Energy (DOE), Office of
Science, Basic Energy Sciences (BES), under award DE-SC0023412 (data analysis
and manuscript preparation). Sandia National Laboratories is a multi-mission
laboratory managed and operated by National Technology and Engineering
Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell
International, Inc., for the U.S. DOE’s National Nuclear Security
Administration under contract DE-NA-0003525. This paper describes objective
technical results and analysis. Any subjective views or opinions that might be
expressed in the paper do not necessarily represent the views of the U.S.
Department of Energy or the United States Government.

##  VI Author Contributions

L. F. Peña: investigation, formal analysis, writing - review and editing; J.
C. Koepke: investigation, writing - review and editing; A Mounce: software; A.
D. Baczewski: software, writing - review and editing; N.T. Jacobson:
theoretical analysis, software, writing - original draft, funding acquisition,
project administration; E. Bussmann: conceptualization, writing - original
draft, funding acquisition, project administration.

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