# Consensus Driven Learning

Kyle L. Crandall
Center for High Assurance Computer Systems
U.S. Naval Research Laboratory
Washington, DC 20375
<EMAIL_ADDRESS>
&Dustin Webb
AI Influx, LLC
Salt Lake City, Utah 84109
<EMAIL_ADDRESS>

###### Abstract

As the complexity of our neural network models grow, so too do the data and
computation requirements for successful training. One proposed solution to
this problem is training on a distributed network of computational devices,
thus distributing the computational and data storage loads. This strategy has
already seen some adoption by the likes of Google and other companies. In this
paper we propose a new method of distributed, decentralized learning that
allows a network of computation nodes to coordinate their training using
asynchronous updates over an unreliable network while only having access to a
local dataset. This is achieved by taking inspiration from Distributed
Averaging Consensus algorithms to coordinate the various nodes. Sharing the
internal model instead of the training data allows the original raw data to
remain with the computation node. The asynchronous nature and lack of
centralized coordination allows this paradigm to function with limited
communication requirements. We demonstrate our method on the MNIST, Fashion
MNIST, and CIFAR10 datasets. We show that our coordination method allows
models to be learned on highly biased datasets, and in the presence of
intermittent communication failure.

##  1 Introduction

With the increase in popularity of Machine Learning (ML), and as it is applied
to more and more applications, the models being trained to solve these
problems grow more and more complex and require more and more data to train
on, and more and more computational power to process this data through the
complex models. This growth, coupled with growing concerns about data security
have made distributed learning methods more appealing as they allow for an
array of compute nodes to work on the same problem simultaneously, reinforcing
the learning done by other nodes. This creates an environment with
significantly more scaleability compared to fully centralized methods. An
increase in the number of compute nodes allows for an increase in the amount
of data being processed at once, as the each node would maintain its own local
dataset. Some early work in distributed learning introduced methods such as
HOGWILD! [1] and DOGWILD! [2]. These methods involve many different compute
nodes training a model, occasionally overwriting the weights in a central,
master model, and occasionally overwriting their own weights with the ones on
the central model in the case of HOGWILD!, or overwriting and copying each
other as is the case in DOGWILD!. As Google began to explore this problem,
they developed a series of asyncronous learning techniques, including their
Asynchronous Advantage Actor Critic (A3C) method [3]. In these methods,
compute nodes accumulate updates, then periodically push them to a global
model, then copy that model and begin accumulating more updates. In 2019, they
proposed an overarching framework for this sort of approach they refer to as
Federated Learning (FL). In this framework, models are sent to remote compute
nodes, they are trained on local data on those nodes, then the updates are
sent back to the master node, which aggregates the data from the various
compute nodes [4]. These methods provide advantages in scalability, and only
rely on local data by sharing the current iteration of the model in training
rather than the raw data. One of the major attractions of such methods is that
they adress contemporary concerns about data security [4].

While these methods are great for the large scale ML applications that are
common these days, we are particularly interested in their use with mobile
teams of robots. With the exception of DOGWILD!, all the above methods rely on
a central parameter server that tracks the current “truth” of the model.
Maintaining a central model allows for good testing and verification of the
model during training. This is undesirable in the case of a mobile robotic
team, as it makes the individual robots dependent on a communication channel
to the central server to get the latest version of the complete model, which
puts severe restrictions on where the robots can move, especially if
communication infrastructure does not exists, say in a disaster zone after an
earthquake or hurricane. In addition, these methods, especially the only
decentralized model, DOGWIlD!, also require robust, high bandwidth
communication, which, is not always in field robotics.

In this paper, we propose an algorithm we call CL Consensus Learning (CL)
algorithm. While the previously discussed algorithms, perform well in
industrial and laboratory conditions where high fidelity communication between
compute nodes is a reasonable assumption, CL is designed to work with minimal
communication requirements. This algorithm is inspired by Distributed
Averaging Consensus (DAC) algorithms, a class of algorithms run over a network
of nodes, each with a value. The average of these values is calculated in a
distributed manner by sharing individual estimates of the average repeatedly
until they converge to a consensus. Much of the earlier work in DAC algorithms
focused on averaging constant values. Of particular interest to us were
methods using asynchronous updates [5]. More recent work has focused on time
varying signals in continuous time [6][7], but much of this work requires
synchronized, continuous updates. Using similar methods, our algorithm reaches
a consensus on the model being trained without any centralized coordination.
We implement our algorithm without the extreme bandwidth requirements of
methods like DOGWILD!, and even show that it is robust to communications
failures.

In this paper, we give an overview of our proposed method, and outline its
theoretical advantages and limitations. We then present an example case using
this method to train a model on the MNIST dataset. We perform several
variations to the training setup to demonstrate the strengths and limitations
of our method. We show that allowing the nodes to communicate allows them to
achieve better performance overall than when they do not, and that this method
is robust to various communication failures. We also show that this method
allows the model to learn well, even when local datasets are extremely biased.
We then repeat all of these experiments on the Fashion MNIST and CIFAR10
datasets, and compare the results from all three sets of experiments.

##  2 Consensus Learning Algorithms

CL algorithms run on a network of compute nodes with some ability to
communicate with each other. Each compute node uses a standard training method
with local data to train, and a consensus algorithm to coordinate the training
across the full network.

Our proposed algorithm is inspired by the DAC algorithm presented in [5]. This
algorithm has a node send a copy of its estimate to another node, the
receiving node takes a weighted average of its estimate and the received
estimate, then sends the deltas back to the originating node. The originating
node then subtracts the deltas from its estimate completing the update. The
results in the two estimates moving towards each other, and the mean between
the two values in preserved. If a node is currently in the middle of an
update, that node rejects any additional update requests. Our algorithm is
based on this method, however rather than ignoring incoming updates while
busy, they are stored in a buffer, and local updates to the values are
performed using traditional training methodology.

Our algorithm assumes ML ML problem set up as follows:

  * •

a model $y=\phi\left(x|p\right)$ where $y$ are the labels, $x$ are the inputs,
and $p$ are the parameters.

  * •

a loss function $\lambda=L\left(x,y\right)$

  * •

a training function $p_{k+1}=\texttt{trainOnBatch}\left(p_{k},x,y\right)$

  * •

a training dataset with a function that returns a batch of data to train on
$x,y=\texttt{getBatch}\left(\right)$

  * •

the parameters $p$ are initialized to some value $p_{i0}$. This value can be
different for each node.

We also require a communications structure to facilitate the sending of
messages between nodes. We assume this structure has the following
capabilities for a given node $i$:

  * •

$C_{it}$, a set of all nodes to whom a communication channel exists at time
$t$

  * •

$U_{i}$, an input First In, First Out Buffer (FIFO Buffer)

  * •

$\texttt{sendTo}\left(m,j\right)$, a function that will place message $m$ into
$U_{j}$ if $j\in C_{it}$.

  * •

$m,j=\texttt{recvNext}\left(U_{i}\right)$ a function to retrieve the next
message in $U_{i}$ and its source node, or returns $\emptyset$ if
$U_{i}=\emptyset$.

<<<<<<< HEAD Because our algorithm uses a similar strategy as the DAC
algorithm in [5] to update the weights across pairs of nodes, there are two
types of messages being sent between nodes. To signal the type of data a
message contains, it is prepended with either a WEIGHTS or DELTAS flag.
======= Because the consensus algorithm uses two different types of messages,
each message has a flag at the beginning to specify the type. Specifically a
STATE flag to signal the message contains the current estimate of the
averages, and an UPDATE flag that indicates the message contains an update to
be applied to an internal state. >>>>>>> dustin

Our CL algorithm is broken into two distinct phases, the local learning phase
where the nodes internal model is updated based on local training data, the
asynchronous update phase where the nodes update each other to reach towards a
consensus on the model. The local learning phase will perform localized
training on $N_{i}$ batches of data, then move on the the asynchronous update
phase. This phase will start by sending the node’s current weights to $M_{i}$
random nodes in $C_{it}$. Then the node will process all the messages in its
communication buffer $U_{i}$, then it will return to the local learning phase.
Each node will alternate through these different phases until the learning
task is deemed complete. Algorithm 1 outlines this process in pseudo code.

1 $p_{i}\leftarrow p_{i0}$;

2 while  _running_ do

// Local Learning phase

3 for  _$j\leftarrow 0$ to $N_{i}$_ do

4 $x\leftarrow$ getBatch(__) ;

5 $p_{i}\leftarrow$ trainOnBatch(_$x,p_{i}$_);

6

7 end for

// Asynchronous Update phase

8 for  _$j\leftarrow 0$ to $M_{i}$_ do

9 packet $\leftarrow\left(\texttt{WEIGHTS},p_{i}\right)$;

10 $c\leftarrow$ randSelect(_$C_{it}$_);

11 sendTo(_packet ,$c$_);

12

13 end for

14 while  _$U_{i}\neq\emptyset$_ do

15 packet,$c\leftarrow$ recvNext(_$U$_);

16 if  _packet $\left[0\right]==\texttt{WEIGHTS}$_ then

17 $p_{j}\leftarrow$ packet $\left[1\right]$;

18 $\Delta p\leftarrow\gamma_{i}\left(p_{j}-p_{i}\right)$;

19 packet $\leftarrow\left(\texttt{DELTAS},\Delta p\right)$;

20 sendTo(_packet ,$c$_);

21 $p_{i}\leftarrow p_{i}+\Delta p$;

22

23 else if  _packet $\left[0\right]==\texttt{DELTAS}$_ then

24 $\Delta p\leftarrow$ packet $\left[1\right]$;

25 $p_{i}\leftarrow p_{i}-\Delta p$;

26

27 end if

28

29 end while

30

31 end while

Algorithm 1 CL Algorithm for Node $i$

##  3 Experimental Results

We performed several experiments to demonstrate the utility of this algorithm,
and to explore its strengths and limitations, by training DNN Deep Neural
Network (DNN) on the MNIST dataset as it is a common toy case that
demonstrates how our method works quite nicely. In the first set of
experiments, the number of updates that are sent out each iteration are
varied. In the second set, we trained the DNN on locally biased datasets, and
varied the level of bias. In the last set of experiments, we explored a case
where communication is unreliable between nodes by dropping packets at a
varying rate. In all sets of experiments, we compare the results to a control
where the DNN is trained on a the complete dataset using traditional, non-
decentralized methods, we refer to this case as the monolithic model. Because
MNIST is a widely considered to be a toy problem, we also performed these same
tests on the Fashion MNIST dataset [8] and the CIFAR10 dataset [9] to show
that it works when confronted with more complex problems as well.

For the MNIST and Fashion MNIST datasets, we trained a DNN that is MLP Multi-
Layered Perceptron (MLP). This DNN had a single hidden layer with 72 neurons
using Rectified Linear Unit (ReLU) activation, and an output layer with 10
neurons using a SoftMax activation. For the CIFAR10 dataset, we trained CNN
Convolutional Neural Network (CNN) with two convolutional layers identifying
16 and 32 features with 3x3 pixel windows, ReLU activation, and max pooling
after each layer (This network architecture was derrived from the tensorflow
[10]). The network was then flatened, and a 128 neuron dense layer was applied
with ReLU activation and dropout. Finaly a 10 neuron dense layer was used as
the output layer with SoftMax activation. In all the experiments, the DNN was
trained using a Sparse Categorical Cross-Entropy cost function along with an
Adam Optimizer [11]. Each compute node also has a copy of the same validation
and test sets so the results are comparable.

###  3.1 Data Sharing

In this experiment, the DNN described above was trained on the MNIST dataset
distributed over 8 nodes, with each node having a local dataset that is 1/8th
of the full dataset. We vary the number of times a copy of the weights are
sent out each cycle ($M_{i}$). We do a control test where $M_{i}=0$, and
compare it to cases where $M_{i}=1,2,$ and $5$. We also compare these results
to the monolithic model results.

|
---|---
(a) | (b)
Figure 1: Learning performance during first set of experiments on the MNIST
Dataset where $M_{i}$ was varied. Metrics were evaluated on a validation set.
Subfigure (a) shows the loss, and (b) shows the accuracy. The metrics were
averaged over all the nodes.

Figure 1 shows the metrics taken during these tests. In the case where
$M_{i}=0$, the nodes seemed to learn on their local datasets for a short time,
then quickly begin to memorize this dataset (as indicated by the upward trend
in the validation loss signal that begins around epoch 20). When we examine
the actual weights, we find that each node learns a distinct model (which is
expected given randomized weight initialization). In this case, these models
were not able to achieve the level of performance as the monolithic model.

When we allow the nodes to update each other, the models are able to achieve
similar performance to the monolithic model. However, it took significantly
longer to to achieve this performance. In addition, when examining the weights
of the final model, we found that the nodes all learned more or less the same
model, as we expect from employing the DAC algorithm. This indicates that the
sharing of the model in our proposed algorithm allows relevant information in
the local dataset to make its way to the other nodes resulting in a model that
performs more like it was trained on the global dataset.

The last key observation of this experiment is that when $M_{i}\neq 0$, the
value of $M_{i}$ did not seem to dramatically effect the performance of the
algorithm. This means that we can limit the number of updates to reduce the
amount of network traffic necessary to run this algorithm. Though on
unreliable networks, multiple updates might by necessary to increase the
probability of an update making it to its destination, as discussed later.

###  3.2 Training on Locally Biased Data

To explore the ability of this method to effectively share information between
nodes, we considered a case where nodes are given very biased local data while
the global dataset remains unbiased. To create these biased data sets, we
first sorted the MNIST Dataset by label. Given a mix rate $0\leq r_{m}\leq
100$, we created 10 datasets where for dataset $i=0...9$, we took $r_{m}$% of
the data points labeled $i$ and distributed them among the other 9 datasets,
leaving $\left(100-r_{m}\right)\%$ of the data with label $i$ to be added to
dataset $i$. This resulted in 10 datasets with a variable level of bias
regulated by $r_{m}$. We setup 10 nodes to train on these datasets using the
proposed CL algorithm, and ran this test for mix rates at intervals of 10%
from 10% to 90%.

|
---|---
(a) | (b)
Figure 2: Learning performance during second set of experiments on the MNIST
Dataset where the DNN is trained on locally biased datasets. Metrics were
evaluated on a validation set. Subfigure (a) shows the loss, and (b) shows the
accuracy. The metrics were averaged over all the nodes.

Figure 2 shows the results of these tests. Not shown in this figure are the
results when using a mix rate of 0%, in this case, the learning failed, giving
an accuracy of around 10%, which is what we would expect of a model trained on
a dataset consisting of only one label. However, with a very low mixing rate
of even 10%, the nodes were able to learn to recognize inputs with labels not
prominent in their own local set. This does come at a price however, the
learning on highly biased datasets takes longer to converge than better mixed
datasets.

###  3.3 Unreliable Communication

In this last experiment, we considered cases where communication is not
reliable, and packets are dropped. As was shown in the first set of
experiments we presented, increasing $M_{i}$ does not seem to improve the
performance of the algorithm under ideal conditions. However, when we consider
the possibility of the dropped packets, increasing $M_{i}$ could be used to
increase the probability that a packet will make it through. What is more
concerning to us is if this algorithm is still valid if the first packet makes
it to its destination, but the reply packet gets dropped. We ran the same
scenario that we ran as part of the first set of experiments, but instead of
$M_{i}$, we vary the probability a reply will get dropped. We use dropout
rates of 0%, 25%, 50%, 75%, and 100%.

|
---|---
(a) | (b)
Figure 3: Learning performance during last set of experiments on the MNIST
Dataset where reply packets are dropped. Metrics were evaluated on a
validation set. Subfigure (a) shows the loss, and (b) shows the accuracy. The
metrics were averaged over all the nodes.

The results of these experiments are displayed in Figure 3. As we can see,
dropping packets did not seem to greatly effect the performance of the
algorithm. This combined with the results from the first experiment show that
this algorithm is fairly robust to intermittent communications failures. In
fact, sending the reply message does not seem to be necessary, at least for
these experiments. However we chose to keep it as an optional part of our
algorithm for the sake of the DAC algorithm that was the inspiration for this
algorithm.

###  3.4 Additional Experiments

Table 1: Training results on multiple datasets  |  | MNIST | Fashion MNIST | CIFAR10
---|---|---|---|---
Experiment |  | Loss | Acc. | Loss | Acc. | Loss | Acc.
Monolithic |  | 0.0905 | 0.9758 | 0.3528 | 0.8832 | 0.9415 | 0.6822
Update Rate | $M_{i}=$ |  |  |  |  |  |
| 0 | 0.2775 | 0.9425 | 0.6376 | 0.8416 | 3.3870 | 0.5395
| 1 | 0.0924 | 0.9720 | 0.3743 | 0.8685 | 1.1968 | 0.6411
| 2 | 0.0925 | 0.9727 | 0.3683 | 0.8687 | 1.0734 | 0.6492
| 5 | 0.0935 | 0.9728 | 0.3698 | 0.8706 | 1.0350 | 0.6416
Mixed Data | $r_{m}=$ |  |  |  |  |  |
| 10 | 0.3860 | 0.8853 | 0.8047 | 0.7571 | 2.4799 | 0.2925
| 20 | 0.2316 | 0.9298 | 0.5967 | 0.8008 | 1.9104 | 0.3852
| 30 | 0.1936 | 0.9411 | 0.5122 | 0.8243 | 1.6351 | 0.4768
| 40 | 0.1589 | 0.9526 | 0.4700 | 0.8386 | 1.3562 | 0.5549
| 50 | 0.1428 | 0.9563 | 0.4393 | 0.8465 | 1.3635 | 0.5567
| 60 | 0.1340 | 0.9597 | 0.4172 | 0.8537 | 1.1509 | 0.6080
| 70 | 0.1222 | 0.9637 | 0.4139 | 0.8547 | 1.1921 | 0.6095
| 80 | 0.1161 | 0.9644 | 0.3948 | 0.8602 | 1.1631 | 0.6252
| 90 | 0.1127 | 0.9670 | 0.3863 | 0.8640 | 1.2044 | 0.6280
Droprate | $dr=$ |  |  |  |  |  |
| 0% | 0.1012 | 0.9703 | 0.3735 | 0.8685 | 1.0633 | 0.6517
| 25% | 0.0987 | 0.9703 | 0.3662 | 0.8711 | 1.0213 | 0.6537
| 50% | 0.1027 | 0.9697 | 0.3719 | 0.8691 | 0.9947 | 0.6692
| 75% | 0.1027 | 0.9699 | 0.3707 | 0.8688 | 1.0194 | 0.6575
| 100% | 0.1057 | 0.9691 | 0.3812 | 0.8648 | 1.0960 | 0.6387

We also performed the previously mentioned experiments on two additional
datasets, the Fashion MNIST Dataset, and the CIFAR10 dataset. Table 1 shows
the metrics for the results of these experiments as well as the MNIST results.
For each dataset, the resulting neural networks were run on a test dataset
consisting of data points that the networks did not see during training or
validation, and the same test set was used for each experiment. The resulting
metrics are averaged over all the nodes that were used during training. We can
see the same patterns in the results with these new datasets that were
apparent in the MNIST experiments. In all cases, When $M_{i}=0$, the the
decentralized consensus method trained networks that perform poorly compared
to when $M_{i}>0$, where the trained networks were able to achieve comparable
results to the network produced by the monolithic method given enough time
(100 epochs in all cases). Further, increasing $M_{i}$, and adjusting the
dropout rate of return packets did not seem to have a noticeable effect on the
convergence and performance of the algorithm in any of our test cases.
Finally, when the data was divided into highly biased subsets, the
decentralized consensus method was still able to produce a network that is
able to recognize categories not prevalent in a given node’s local dataset.
The overall performance increased with the mixing rate in all cases.

##  4 Conclusion

In this paper we propose a Consensus Learning paradigm and an algorithm that
implements it. This algorithm takes inspiration from a Distributed Averaging
Consensus method to reach a consensus on the model being trained, while using
traditional training algorithms to train the model on local data. This
algorithm does so without significant requirements on the communication
network the nodes are operating on, and is robust to intermittent failure in
that network. We demonstrate this algorithm by using it to train a Deep Neural
Network model on the MNIST Dataset of handwritten numerals, as well as the
Fashion MNIST dataset, and the CIFAR10 dataset. We demonstrate the robustness
of this algorithm to failures in the communication network, as well as showing
its capacity to overcome bias in local datasets in all three cases.

## Broader Impacts

This work does not present any foreseeable societal consequence.

## Acknowledgments and Disclosure of Funding

The work on this project was funded by the Karles Fellowship at Naval Research
Laboratory under number 300000145350

## References

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  * [2] C. Noel and S. Osindero, “Dogwild!-distributed hogwild for cpu & gpu,” in _NIPS Workshop on Distributed Machine Learning and Matrix Computations_ , 2014, pp. 693–701.
  * [3] V. Mnih, A. P. Badia, M. Mirza, A. Graves, T. Lillicrap, T. Harley, D. Silver, and K. Kavukcuoglu, “Asynchronous methods for deep reinforcement learning,” in _International conference on machine learning_ , 2016, pp. 1928–1937.
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  *[ML]: Machine Learning
  *[A3C]: Asynchronous Advantage Actor Critic
  *[FL]: Federated Learning
  *[CL]: Consensus Learning
  *[DAC]: Distributed Averaging Consensus
  *[FIFO Buffer]: First In, First Out Buffer
  *[DNN]: Deep Neural Network
  *[MLP]: Multi-Layered Perceptron
  *[ReLU]: Rectified Linear Unit
  *[CNN]: Convolutional Neural Network


#  Shake-off in the 164Er neutrinoless double electronic capture and the dark
matter puzzle

F. F. Karpeshin111e-mail<EMAIL_ADDRESS>
Mendeleev All-Russian Research Institute of Metrology
190005 Saint-Petersburg, Russia

and
M. B. Trzhaskovskaya222Deceased
Petersburg Nuclear Physics Institute of the National Research Center
“Kurchatov Institute”
188300 Gatchina, Russia

###### Abstract

Traditionally double neutrinoless electronic capture is considered as a
resonance process. We have fulfilled shake-off probability calculations,
leading to ionization of the electron shell, in the case of 164Er. Allowance
for the shake-off removes the requirement of resonance, leading to a radical
increase of the capture rate. In the case of 164Er, the contribution of the
new mechanism increases the capture rate by a factor of 5.6 as compared to the
conventional resonance fluorescence mechanism. It also increases the
probability of electron capture from higher shells, which must be foreseen in
the experimental studies. Moreover, effect of the shake-off can expand the
list of candidate nuclei for experimental research. The influence of the
shake-off is also expected to manifest itself in the other $\beta$ processes
which are used in the studies of the neutrino nature, its mass and role in the
dark matter puzzle.

##  1 Introduction

The discovery of dark matter in the universe stimulates the development of
theories beyond the Standard Model. As a rule, they include violation of the
lepton quantum number, unless special restrictions are introduced. This
attracts great interest in the study of double beta processes, including the
$2e$ decay of a nucleus and the capture of two orbital electrons by it [1].
Within the framework of the Standard Model, the lepton quantum number is
conserved. This excludes double neutrinoless beta decay or $e$ capture. The
latter become possible only if neutrinos have mass and if neutrinos are
particles of the Majorana nature. However, the discovery of mass in neutrinos
and their oscillations has already marked the observation of processes beyond
the Standard Model. Thus, the search for neutrinoless binary processes should
answer the question about the Majorana nature of neutrinos. Of the two
neutrinoless processes, the $2e0\nu$ decay has the highest decay rate. The
$2e0\nu$ capture, although inferior to the $2e0\nu$ decay by several orders of
magnitude in probability, is more convenient from the viewpoint of detection.

The essential point is that neutrinoless $2e$ capture has traditionally been
considered as a resonance process, since not a single particle is emitted as a
result of nuclear transformation [2]. At the same time, the conservation law
requires the transfer of the part of the released energy to a third body. This
is the electron shell of the atom. The law of energy conservation is restored,
for example, due to the emission of a fluorescence quantum, the energy of
which includes the excessive $Q$ value. Another reason for the shift is
formation of the two vacancies in the places of the captured electrons and
thus inflated in comparison with a normal atom electron shell. Therefore, the
energy of these photons differs from ordinary fluorescence photons, as usually
fluorescence takes place in ionized atoms with a single hole in the inner
shell. The detection of such satellites in the fluorescence spectrum can serve
as an indicator of either neutrinoless or two-neutrino double electron capture
[3, 4]. Thus, the neutrinoless capture amplitude permanently includes the
radiative vertex, which retards the process by two orders of magnitude.
Therefore, the main criterion is focused on the study of nuclei with the small
$Q$ values. In Ref. [5], a list of three most suitable candidates for
experimental research was selected: 152Gd, 164Er, and 180W. In this paper, we
refine the question of the probability of the 164Er $\to^{164}$Dy process
based on the new shake-up mechanism, which was proposed in [6].

The shake-off mechanism does not require a resonance. For this reason, its
contribution decreases more slowly with increasing the resonance defect
$\Delta$ than the conventional resonant fluorescence mechanism. The
restoration of the energy conservation law occurs due to the ionization of the
electron shell. In this case, the excess energy is carried away by the ejected
electron. Estimates of the shake-off efficiency were made in [6] using the
example of the 152Gd $\to$ 152Sm decay, which has the smallest resonance
defect of $\Delta$ = 0.910 keV among the known candidates. Quantitatively, the
contribution of the new mechanism turned out to be at the level of 23% as
compared to the traditional mechanism. In this paper, we consider the 164Er
$\to$ 164Dy process, with a higher value of $\Delta$ = 6.82 keV. The results
confirm expectations. Accounting for shake-off reduces the expected half-life
of the process by almost 6 times. In the next section, we recall the basic
formulas. The calculation results are given in Sec. 3. Section 4 is devoted to
a discussion of the results obtained in this work.

##  2 Comparison of the two mechanisms of neutrinoless $2e$ capture: physical
principles and formulae for calculation

In the case of $2e0\nu$ capture, the atom remains generally neutral.
Therefore, the energy release is determined by the difference in the masses of
neutral atoms, the initial $M_{1}$ and the daughter one $M_{2}$ (we use the
relativistic system of units $\hbar=c=m_{e}=1$, with $m_{e}$ being the
electron mass, unless otherwise noted):

$Q=M_{1}-M_{2}\,.$ (1)

However, the process with the total energy release (1) could be realized only
with the capture of the outermost, valence, electrons. As a rule, the capture
of internal electrons is much more probable, as their density on the nucleus
is higher. Accordingly, the atom remains in an excited state with the energy
$E_{A}$ and two holes in the inflated electron shell [4]. Accordingly, instead
of (1), the effective energy release is realized

$Q_{\text{eff}}=M_{1}-M_{2}-E_{A}=Q-E_{A}\,.$ (2)

The process is energetically possible at $Q>0$, but $Q_{\text{eff}}$ can also
be negative: the excessive energy can be either added to or subtracted from
the energy of the satellite quantum. It is $Q_{\text{eff}}$ that acts as the
resonance defect $\Delta=|Q_{\text{eff}}|$. We write the formula for the
resonance mechanism in the pole approximation using the traditional model. The
formula is obtained (cf., for example, [5]) by multiplying the squared
amplitude of the capture itself, $\Gamma_{2e}$, which plays the role of the
formation of the doorway state, by the Breit—Wigner resonance factor

$\Gamma_{2e}^{(\gamma)}=\Gamma_{2e}B_{W}\,,$ (3)

where

$B_{W}=\frac{\Gamma/2\pi}{\Delta^{2}+(\Gamma/2)^{2}}\,.$ (4)

In Eq. (4), $\Gamma$ is the width of the inflated state of the daughter atom
with the two holes. In Ref. [5] it was taken as the total width of the both
holes. This is not correct: one must also add the width of the final state
[7]. As a result, the value of $\Gamma$ at least doubles [4]. However, the
decay of 164Er was not considered in Ref. [4]. For this reason, we make
comparison with Ref. [5], keeping their $\Gamma$ value. The shake-off
contribution is independent of the $\Gamma$ value.

Figure 1:  Typical dependence of the Breit—Wigner factor (4) (relativistic
units) on the resonance defect $\Delta$.

A typical value is $\Gamma\approx$ 30 eV. For illustration, the scale of the
variation of the $B_{W}$ factor is shown in fig. 1 versus the resonance
defect. It decreases by a factor of 2 at $\Delta$ = 1.5 keV. Only one
candidate is known with such a value: 152Gd, which has $\Delta$ = 0.91 keV. In
other cases under discussion, $\Delta$ is at least from a few keV to one or
two tens of keV, while the Breit—Wigner factor drops to six orders of
magnitude. Shake-off is energetically possible only for positive
$Q_{\text{eff}}>0$, and from the shells whose ionization potential $I_{i}$ in
the daughter atom (with the two vacancies in the electron shell) satisfies the
condition

$I_{i}<Q_{\text{eff}}\,.$ (5)

Then the energy of the shake-off electrons is defined by their difference as
follows:

$E_{sh}=Q_{\text{eff}}-I_{i}\,.$ (6)

The shake-off arises due to the very rapid, instantaneous change in the inner-
atomic electrostatic potential $V_{Z}(r)$ in the initial atom to the potential
in the daughter atom $V_{Z-2}(r)$, see Fig. 2.

Figure 2: Scheme of a representative shake-off process: the $L_{1}L_{1}$
electron capture creates two holes in the $2s$ shell. Changed instantaneously
inner-atomic electrostatic potential expels the 3$s$ electron into continuum
with momentum $p$.

Therefore, the electron wave functions of the initial and final atoms are non-
orthogonal, even with the same quantum numbers. To calculate the shake-off
probability, most important is the non-orthogonality of the one-electron wave
function $\phi_{i}(r)$ in the parent atom and $\psi_{f}(r)$ in the daughter
atom. Let’s denote the change in the potential $\Delta V(r)\equiv
V_{Z}(r)-V_{Z-2}(r)$. Thus, the wave function $\phi_{i}(r)$ belongs to the
parent neutral atom in the ground state, and $\psi_{f}(r)$ is calculated in
the field of the daughter atom with three vacancies: two in the places of the
captured electrons and one in the place of the emitted shake-off electron.
Then the shake-off amplitude reads as follows [8]:

$F_{\text{sh}}(\Delta)=\langle\psi_{f}|\phi_{i}\rangle\,.$ (7)

Similarly to (3), one can put down the full amplitude factorized as follows:

$F_{2e}^{\text{(sh)}}=F_{2e}F_{\text{sh}}(\Delta)\,.$ (8)

Comparing Eqs. (8) to (3), one arrives at the following expression for the
relative correction to the decay probability:

$G=\Gamma_{2e}^{(sh)}/\Gamma_{2e}^{(\gamma)}=\sum_{i}N_{i}|\langle\psi_{f}|\phi_{i}\rangle|^{2}/B_{W}\equiv\sum_{i}N_{i}|F_{\text{sh}}|^{2}/B_{W}\,,$
(9)

with $N_{j}$ being the occupation numbers.

Within the framework of the resonance fluorescence mechanism, the main
contribution comes from the capture of the two $L_{1}$ electrons in the 164Er
atoms. The capture of lower electrons is energetically forbidden, that of
higher ones is suppressed by the decrease in their density at the nucleus
together with a decreasing Breit—Wigner factor, brought about by an increase
in $Q_{\text{eff}}$. As for the nonresonance shake-off mechanism, a decrease
in the electron density at the nucleus, for example, in the capture from the
$M1$ shell, on the contrary, is partially compensated by an increase in
$Q_{\text{eff}}$, since the shake-off channel from the $L_{1}$ shell opens.
This leads to the fact that, as we shall see, the probability of capture from
higher shells gets even higher than the probability of the traditional
resonance mechanism. Let $ik$ capture occur from the higher $i$, $k$ shells.
Then the acceleration factor can be calculated in comparison with the most
probable resonance $L_{1}L_{1}$ capture by means of the formula

$G_{ik}=\frac{\rho_{i}(0)\rho_{k}(0)}{\rho_{L_{1}}^{\
2}(0)}\sum_{j}N_{j}|F_{\text{sh}}^{(j)}(|Q_{\text{eff}}^{(ik)}|)|^{2}/B_{W}\,.$
(10)

In Eq. (10), the summation is carried out over all shells $j$ where shake-off
is energetically allowed. $F_{\text{sh}}^{(j)}(|Q_{\text{eff}}^{(ik)}|)$ is
still the overlap integral of the wave functions of the electron in the
initial shell $j$ and the electron in the continuum, but calculated for the
actual energy release $Q_{\text{eff}}^{(ik)}$ corresponding to the $ik$
capture. In the case of the most probable $L_{1}L_{1}$ capture, the lowest
shell where the shake-off occurs is the $M$ shell. Alternatively, if the
capture of one of the electrons occurs from the $M$ shell, then the value of
$Q_{\text{eff}}^{(LM)}$ increases by the difference of the ionization
potentials of the $L$\- and $M$-shells. This automatically opens the shake-off
channel from the $L$-shell ($L_{1}$, $L_{2}$, $L_{3}$), which leads to a
stepwise increase of the shake probability.

##  3 Results of calculation

Calculations by means of Eqs. (9), (10) were performed using the RAINE
software package [9, 10]. The one-electron wave functions and their
eigenvalues were calculated by means of the self-consistent Dirac—Fock method.
In order to better understand the physics of the process, matrix elements (7)
were calculated for a number of hypothetical values of $\Delta$ from 0.05 to
20 keV for all electrons whose ionization potentials are less than the given
$\Delta$ value and which, therefore, contribute to the amplitude of the shake-
off mechanism of the process. The total widths of the electron hole states are
taken from [10].

The calculation results are shown in Figs. 2 – 5, as well as in tables 1, 2.
Our wavefunctions are normalized at unity for discrete states and the $\delta$
function on the energy scale — in the continuum. Therefore, the square of the
matrix element $F_{sh}(\Delta)$ acquires the dimension reciprocal of the
energy. The matrix elements are presented below in the relativistic system of
units.

Figure 3: Matrix elements $F_{sh}$ for the $2p_{1/2}$\- (solid curve) and
$2p_{3/2}$\- (dashed curve) subshells of 164Dy atoms against the resonance
defect $\Delta$.

The closer the shell is to the core, the greater its contribution to the
shake-off, if it is not energetically forbidden. This is illustrated in Fig.
2, which shows the $F_{sh}$ matrix elements for the $L_{2}$ and $L_{3}$
subshells. The curves start from different thresholds: 9.264 and 8.358 keV,
respectively. Both thresholds are higher than the effective energy release;
therefore, in this case, neither of the curves contributes to shake-off in the
most probable case of the $L_{1}L_{1}$ capture. Fig. 3 shows matrix elements
for the $2p_{1/2}$ – $5p_{1/2}$ shells. The matrix elements for the rest of
the shells are illustrated in fig. 5.

Figure 4: Matrix elements $F_{sh}$ for the $np_{1/2}$-subshells of 164Dy atom:
$n$=2 (solid line), $n$=3 (dashed line), $n$=4 (dash-dotted line) and $n$=5
(dotted line).

Figure 5:  Comparison of shell contributions to the shake amplitude as a
function of the orbital angular momentum. Matrix elements $F_{sh}$ for the
$2s$ subshell (dashed line), $2p_{1/2}$ subshell (solid curve), $3d_{3/2}$\-
(dash-dotted line), and $4f_{5/2}$\- (dotted line) subshells of 164Dy atom.

The total acceleration factor corresponding to the shake-off mechanism from
all the electrons, brought about by the $L_{1}L_{1}$ capture, relative to the
resonance mechanism is shown in Fig. 6. The probability of this process has a
pronounced stepwise character due to the fact that with increasing $Q$, deeper
and deeper shells are switched on, and the deeper the shell lies, the greater
its contribution at the threshold. As expected, the main contribution comes
from the $s$\- and $p$-electrons. It can be seen that at small $Q$, the
resonance mechanism dominates. At the actual value of $Q$ = 6.82 keV, the
contribution of the nonresonance mechanism is three times as high as that of
the traditional mechanism.

Figure 6: The gain of the shake-off mechanism $G$ (9) as compared to the
resonance-fluorescent mechanism in the probability of double neutrinoless
$L_{1}L_{1}$-capture in 164Er, depending on the resonance defect $\Delta$.

Shown in figs. 3 – 5 values can be used in order to estimate the shake-off
contribution in the cases of electron capture from the other, higher shells.
Using the total width of the $L_{1}$ hole in the Dy atom: $\Gamma_{L_{1}}$ =
4.3 eV [10], we obtain by means of formula (10) the total acceleration factors
for capture from the $L$, $M$, and $N$ shells. They are presented in Table 1.
As one can see from the Table, taking into account higher shells leads to an
additional increase of the capture rate by 1.8 times. And the total gain is
5.6 times. Supposing that it is only the shake-off mechanism which makes $e$
capture from higher orbits significant, one can estimate from Table 1 the
total fraction of these captures to be of about one third. One can easily
calculate the mean shake-off probability: 82% per capture.

Table 1: Partial gains $G_{ik}$ (10) from the shake-off mechanism calculated for $e$ capture in the various shells $ik$. $\rho(0)$ is the normalized at the $L_{1}L_{1}$ capture product of the densities [11] of the both captured electrons at the origin  Shell |  $\Delta$ (keV) | $\rho(0)$ | $G$
---|---|---|---
$LL$ | 6.82 | 1 | 2.81
$LM$ | 14 | 0.218 | 1.22
$MM$ | 21 | 0.048 | 0.20
$LN$ | 15.6 | 0.051 | 0.29
$MN$ | 22.6 | 0.011 | 0.05
NN | 24.3 | 0.003 | 0.01
Total: | — | — | 4.58

##  4 Discussion of the results and future prospects

1\. Allowance for shake-off processes in the neutrinoless double nuclear
$e$-capture shifts the theoretical estimates much closer to the experimental
possibilities. Its peculiarity is that shake-off renders obsolete the
requirement of resonance between the initial and final atoms. Therefore, it
can be expected that it will be more effective in the cases of nuclei with big
$Q_{\text{eff}}$ values, when traditional resonance fluorescent mechanism is
suppressed. Consideration of the shake-off mechanism significantly refines the
half-life estimate. The above calculations confirm this assumption: taking
into account the new mechanism increases the estimate of the double capture
probability by a factor of about 6 in the case of 164Er. Note that theoretical
half-life relative to the $2e0\nu$ capture scales with $(1/m_{\nu})^{2}$,
where $m_{\nu}$ is the effective neutrino mass. Taking into account a
traditional estimate of the half-life of this nucleus as $2\times 10^{30}$
years for $m_{\nu}$ = 1 eV [5], we obtain a refined estimate of the half-life,
taking into account the shake-off mechanism, as $T_{1/2}^{0\nu}\approx
3.6\times 10^{29}$ (1 eV/$m_{\nu})^{2}$ years. In other cases of heavier
nuclei with an effective energy release of $\gtrsim$10 keV, including 180W,
the gain achieves already a full order of magnitude. We summarize the expected
results for the half-lives of the above three candidates: 152Gd, 164Er and
180W in Table 2. In view of that the decay period of another candidate for
measuring the $2e0\nu$ capture of 152Gd remains four orders of magnitude
shorter, we can conclude that it remains a more likely candidate for setting
up an experiment than 164Er. At the same time, the expected lifetime of 180W
with respect to the $2e0\nu$ capture turns out to be only 4 times longer than
that of 152Gd. This can make it a better candidate, given that the abundance
of the 152Gd isotope in nature is only 0.2%.

Table 2: Resulting halflives of 152Gd, 164Er and 180W double neutrinoless $e$ capture, taking into account both mechanisms Nuclei |  152Gd$\to^{152}$Sm |  164Er $\to$ 164Dy |  180W $\to$ 180Hf
---|---|---|---
Decay channel | KL | LL | KK
$\Delta$ (keV) | 0.910 | 6.82 | 12.5
Resonance half-lives (years) |  1027 | $2\times 10^{30}$ | $3\times 10^{28}$
Shake-off half-lives (years) | $8\times 10^{26}$ | $3.6\times 10^{29}$ | $3\times 10^{27}$

2\. Shake-off leads to a radical change of the fluorescence spectrum. As
mentioned in the Introduction, the appearance of two satellites at each
fluorescence line comprises a characteristic feature of $2e$ capture. First,
the energies of the satellites receive an additional shift due to availability
of the third vacancy on the place of the shake-off electron. Second, in the
case of capture from higher orbits, new satellite fluorescence photons arise.
In the case of 164Er, in one third of the cases, the capture occurs from
higher orbits than $L_{1}$. Consequently, satellites fluorescence quanta
appear, which correspond to the transitions of electrons to the states of the
$M$\- and $N$-shells. This new phenomenon must be taken into account in
experiments. One can use it for the purpose of more reliable identification of
the process and its mechanism. A more detailed analysis of the appearing
satellite spectrum can be performed elsewhere.

3\. Allowance for shake-off is also important in investigation of $\beta$
decay and other $\beta$ processes. Such studies are carried out with the beta
decay of tritium [12] in the KATRIN experiment aimed at measuring the mass of
an electron antineutrino and searching for sterile neutrinos. Shake-off
modifies spectrum of emitted electrons near the upper bound — in the region
which is most sensitive to the experimental determination of the neutrino
mass. Restrictions on the mass of the Majorana neutrinos are also established
in the study of double beta decay [13, 14, 15] and Refs. cited therein. Shake-
off processes are of great importance for correct analysis of the experimental
lifetimes.

Similar measurements are also carried out by studying the $e$-capture at
163Ho, 159Dy. Preference is given to these sources because of the minimum $Q$
value. The lower the $Q$ value, the greater the number of events falls close
to the upper bound of the spectrum. In this case, the neutrino spectrum can be
measured by detecting the secondary processes accompanying formation of the
vacancy: fluorescence photons, Auger electrons by the calorimetric method. The
calorimetric spectrum of $e$-capture in 159Dy to the 363.5449 keV level of
159Tb, $Q$ = 1.14(19) keV was calculated in Ref. [16] with no allowance for
shake-off. The calculation was performed within the Vatai approximation, in
which the remaining electrons of the daughter atom inherit the quantum numbers
of the parent atom. A continuous calorimetric spectrum was supposed to be
created by the width of the formed vacancy. The main contribution near the
upper bound was obtained due to the $N_{1}$-capture with the formation of an
excited state corresponding to the configuration [Xe]$4s^{-1}4f^{10}6s^{2}$.
However, one can expect a significant contribution from the shake-off, for
example, with the formation of a daughter atom in a configuration
[Xe]$4s^{-1}5s^{-1}4f^{10}6s^{2}$. Then the continuity of the calorimetric
spectrum is provided by the escape of the $5s$ electron into the continuum.
Moreover, a significant contribution can also be expected from the electron
shake-off from the $N_{2}$ – $N_{5}$ and higher shells. As a result, the total
contribution of the shake-off can be comparable or greater than the
contribution from the resonance fluorescence mechanism.

Summing it up, one can conclude that the nonresonance shake-off mechanism of
double neutrinoless $e$-capture is an important example where this process,
being of great interest in itself, manifests itself surprisingly clearly.
Search for such surprising manifestations in other $\beta$ processes seems to
be a challenging task of the contemporary investigation.

One of the authors (FFK) would like to thank Yu. N. Novikov for initiating
discussions.

## References

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# Selective area epitaxy of GaAs films using patterned graphene on Ge

Zheng Hui Lim  Sebastian Manzo  Patrick J. Strohbeen  Vivek Saraswat  Michael
S. Arnold  Jason K. Kawasaki  Materials Science and Engineering, University of
Wisconsin-Madison, Madison, WI 53706

###### Abstract

We demonstrate selective area epitaxy of GaAs films using patterned graphene
masks on a Ge (001) substrate. The GaAs selectively grows on exposed regions
of the Ge substrate, for graphene spacings as large as 10 microns. The
selectivity is highly dependent on the growth temperature and annealing time,
which we explain in terms of temperature dependent sticking coefficients and
surface diffusion. The high nucleation selectivity over several microns sets
constraints on experimental realizations of remote epitaxy.

##  I Introduction

Selective area epitaxy is a promising strategy for synthesis of confined
quantum devices Gao et al. (2014); Aseev et al. (2018); Lee et al. (2019) and
for reducing dislocation densities in highly lattice mismatched systems Ujiie
and Nishinaga (1989); Fitzgerald and Chand (1991); Park et al. (2007); McMahon
et al. (2018). This growth mode is typically implemented by patterning
openings in an inert dielectric mask such as SiO2, on a crystalline substrate.
The openings serve as selective nucleation sites for epitaxial film growth,
while the mask defines the lateral dimensions and blocks dislocations from the
substrate or film/substrate interface. Continued epitaxial lateral overgrowth
can be used to produce coalesced planar films Nishinaga et al. (1988); Ujiie
and Nishinaga (1989); McMahon et al. (2018); Ironside et al. (2019).

Due to its chemical inertness, fast surface diffusion, and atomic thinness,
patterned graphene is an attractive alternative mask material. Early studies
demonstrated epitaxial lateral overgrowth of GaAs films on patterned
multilayer graphite on a Si substrate, by liquid phase epitaxy (LPE)
Zytkiewicz et al. (2001, 1998). More recently, monolayer graphene masks have
been used to selectively nucleate GaN films at 75 nm openings on a SiC
substrate Puybaret et al. (2016), for GaN grown by metalorganic chemical vapor
deposition (MOCVD). Open questions remain, however, about whether nucleation
selectivity can be achieved using graphene over larger length scales, whether
selectivity requires chemical precursors or can be achieved using physical
vapor deposition techniques like molecular-beam epitaxy (MBE), and how growth
kinetics affect the selectivity.

Here we demonstrate that monolayer graphene masks provide a highly selective
barrier for selective area epitaxy of MBE-grown GaAs films, using patterned
graphene stripes on a Ge (001) substrate. We demonstrate $>99\%$ nucleation
selectivity on the exposed Ge regions, for graphene stripe widths and spacings
as large as 10 microns. The nucleation selectivity depends strongly on Ga
surface diffusion and desorption, which we control experimentally by
increasing the growth temperature and periodic anneal time. The high
selectivity over a 10 micron scale also places constraints on experimental
realizations of “remote epitaxy” Kim et al. (2017), where the goal is to grow
epitaxial films via remote interactions through continuous graphene with no
intentional openings.

##  II Results and Discussion

Figure 1: Patterned graphene on Ge (001). (a) Scanning electron micrograph
(SEM) image of the graphene mask with 10 $\mu$m feature size under in-lens
secondary electron imaging mode. Inset shows the cartoon of graphene mask. (b)
AFM height (top) and phase (bottom) images of a graphene region. (c) AFM
images of an exposed Ge region.

To make the patterned graphene on Ge, we first grow continuous monolayer
graphene on a Ge (001) substrate by chemical vapor deposition using a methane
precursor, following Refs. Kiraly et al. (2015); Wang et al. (2013). We define
the stripes by UV photolithography (Shipley 1813 photoresist), etch the
exposed graphene regions using an oxygen plasma (50 W), and rinse the
remaining photoresist using sequential acetone and isopropanol baths. The
resulting patterned template consists of alternating 5-10 microns wide stripes
of graphene, separated by 5-10 microns of exposed Ge substrate (Fig. 1a). At
this stage, scanning electron microscopy (SEM) confirms that the pattern is
translated to the graphene with no obvious long-range tears (Fig. 1a).

Atomic force microscopy (AFM) images reveal a faceted morphology on both the
graphene regions (Fig. 1b) and exposed Ge regions (Fig. 1c). The faceting is
consistent with previous studies and is induced by graphene growth McElhinny
et al. (2016). On the Ge regions we also observe spherical particles that we
attribute to Ge that has been oxidized during the oxygen plasma etch. On
graphene regions we observe $\sim 10$ nm diameter pinholes. These pinholes
appear as bright spots in the AFM phase images, which we attribute to a
different elastic modulus of graphene versus the Ge substrate. The pinholes
are only faintly visible in the AFM height images. Based on these images we
estimate a pinhole concentration of $\sim 40$ $\mu m^{-2}$, which is an order
of magnitude smaller than the pinhole concentration for transferred graphene
on GaAs (001) after native oxide desorption Manzo et al. (2021). We attribute
these pinholes to the photolithography and etching processes, since no
pinholes were detected by AFM on the graphene/Ge samples before
photolithography and etching.

Figure 2: GaAs nucleation selectivity at the patterned openings. (a) Secondary
electron SEM image of a GaAs film nucleated on patterned graphene/Ge(001). The
stripe pattern consists of 10 microns graphene and 10 microns of exposed Ge.
(b) Left: Raman intensity map of the GaAs TO mode. Right: representative point
spectra on a GaAs covered region and a graphene region. (c) Raman intensity
map of the graphene 2D mode and corresponding spectra.

Prior to GaAs film growth, we etch the exposed Ge oxides in a deionized water
bath at 90∘C. We then dry samples with nitrogen and immediately load them into
a high vacuum loadlock ($p<10^{-8}$ Torr) and outgas them at $150^{\circ}$C
for at least 1 hour to remove organic residuals, followed by an anneal at
$650^{\circ}$C in the ultrahigh vacuum MBE chamber ($p<3\times 10^{-10}$ Torr)
to remove the remaining Ge-oxides. GaAs films are grown by MBE using elemental
Ga from a standard effusion cell and a mixture of As2/As4 from a thermal
cracker cell. Typical Ga fluxes were $2.74\times 10^{16}$ atoms/(cm2s),
calibrated by reflection high energy electron diffraction oscillations.

Fig. 2 shows scanning electron microsocopy (SEM) and Raman spectroscopy for a
GaAs film grown at $613^{\circ}$C by periodic supply of Ga and constant supply
of As, on patterned graphene / Ge (001). We observe high nucleation
selectivity for GaAs at exposed region of the Ge substrate, and negligible
parasitic GaAs nucleation on the graphene mask. In the SEM image (Fig. 2a),
the GaAs appears as coalesced islands with high secondary electron intensity
on the Ge regions. We attribute the island morphology to the faceted Ge (001)
surface, which is induced by graphene growth (Fig. 1c). Note that CVD graphene
on (111) and (110)-oriented Ge is known to produce smoother surfaces Kiraly et
al. (2015), free from the facetting of the (001) surface. We expect that GaAs
grown on these orientations may produce a smoother film surface morphology.

Raman spectroscopy maps (Horiba Labram, $\lambda=532$ nm) of the GaAs TO and
graphene 2D modes confirm that these islands are GaAs film, which grow
preferentially on the exposed Ge (Fig. 2b) and not on the graphene mask (Fig.
2c). Representative point spectra on the GaAs-covered Ge regions (blue curves)
and on the graphene regions (black curves) are shown on the right hand side of
Figs 2(b) and (c). Note that in this scattering geometry, only the GaAs LO
mode is symmetry allowed for for a perfect GaAs crystal – the GaAs TO mode is
not symmetry allowed Abstreiter et al. (1978). We attribute the presence of a
GaAs TO mode to strain or point defects.

Figure 3: Enhancing selectivity with growth temperature and periodic supply
epitaxy (PSE). (a) SEM images of GaAs grown by co-deposition, for increasing
growth temperature. (b) SEM images for samples grown by PSE. (c) Schematic of
co-deposition. The Ga shutter is kept open for the entire growth. (d)
Schematic of PSE. The Ga shutter is periodically shuttered open and closed.
Figure 4: Quantification of selectivity. (a) Percent area coverage of GaAs
film nucleated on exposed Ge regions ($A_{Ge}$) and on graphene regions
($A_{gr}$), for growth by co-deposition and by periodic supply epitaxy. The
percent area coverage was determined from the SEM images. (b) Nucleation
selectivity $S=A_{Ge}/(A_{Ge}+A_{gr})$ on the exposed Ge regions.

We find that the selectivity for GaAs growth on Ge rather than the graphene
mask is highly dependent on the growth temperature and annealing. Figure 3(a)
shows SEM images of GaAs films grown by codeposition of Ga and As, at varying
substrate temperature. All samples were exposed to a total flux of $4.56\times
10^{14}$ Ga atoms/cm2, which would correspond to a GaAs film thickness of 14
nm for a Ga sticking coefficient of 1. In these images the GaAs corresponds to
the light islands, the graphene corresponds to the dark stripes, and the Ge
corresponds to lighter stripes. At a growth temperature of $580^{\circ}$C, the
concentration of parasitic nucleated islands on graphene is $\sim 10$
$\mu$m-2, which is slightly less than the $\sim 40$ $\mu$m-2 concentration of
graphene pinholes before GaAs growth (Fig. 1b). We hypothesize that the
parasitic GaAs growth on graphene is caused by nucleation in the pinholes.
Similar nucleation at pinholes is observed for GaAs films grown on unpatterned
graphene/GaAs (001) Manzo et al. (2021). Further optimization is required to
reduce the graphene pinhole density created during lithography and etching.

With increasing growth temperature, the amount of GaAs nucleated on both
graphene and Ge regions decreases, i.e. the sticking coefficient decreases.
The rate of GaAs sticking on graphene decreases more rapidly than on Ge, such
that for a growth temperature of $613^{\circ}$C there is negligible GaAs
growth on graphene and highly preferred growth on Ge. We quantify these
effects in Fig. 4. Here, $A_{Ge}^{codep}$ is the area fraction of GaAs on the
exposed Ge regions and $A_{gr}^{codep}$ is the area fraction of GaAs on
graphene (Supplemental Information). We define the selectivity as
$S=A_{Ge}/(A_{gr}+A_{Ge})$. At growth temperatures greater than $610^{\circ}$C
we obtain a selectivity $S>99\%$.

We further enhance the nucleation selectivity by employing periodic supply
epitaxy (PSE). Here, we fix the As shutter open and periodically modulate the
Ga shutter between the open position (GaAs growth) and the closed position
(annealing, Fig. 3d). Fig. 3b shows SEM images of GaAs films grown by PSE,
where we let $t_{growth}=$ 1 minute and $t_{close}=$ 6 minutes, for a total of
15 cycles. We find that PSE decreases the total sticking coefficient (Fig. 4a)
and increases the selectivity for GaAs nucleation on Ge compared to on
graphene (Fig. 4b).

We understand effects of temperature and anneal time in terms of the
microscopic processes sketched in Fig. 4b insert. High selectivity results
from a combination of fast surface diffusion on graphene and high desorption
rate from graphene, compared to Ge which has a sticking coefficient closer to
1. Upon arrival at the surface, Ga adatoms diffuse over a characteristic
length $\lambda$. Ga adatoms will then either desorb into the vapor or attach
to the surface to form GaAs. With increasing temperature, the desorption rate
of Ga from graphene decreases dramatically, as observed by the factor of
$10^{3}$ drop in GaAs surface coverage on graphene ($A_{gr}$) from 560 to
625∘C (Fig. 4a). In contrast, for the same range of temperatures the GaAs
coverage on Ge ($A_{Ge}$) decreases by less than a factor of 10. Surface
diffusion also increases with temperature, resulting in an increased
probability for Ga to diffuse and attach to reactive areas of the exposed Ge
substrate. These combined factors explain the general observation that
increasing the temperature increases the selective growth on Ge. Increasing
the anneal time by periodic supply epitaxy plays a similar role as
temperature, by increasing the time for desorption and increasing the
diffusion length $\lambda$. These general trends are consistent with selective
area growth using conventional SiO2 masks Nishinaga et al. (1988).

The high selectivity and long diffusion length also set constraints for
experimental realizations of “remote epitaxy” on unpatterned graphene. “Remote
epitaxy” refers to the epitaxial growth of a film on a continuous graphene-
terminated substrate, with no intentional openings. In this growth mode,
epitaxial registry between film and substrate is thought to occur via “remote”
interactions that permeate through graphene. However, pinholes or openings in
the graphene can serve as alternate nucleation sites, followed by lateral
overgrowth Manzo et al. (2021) or growth by intercalation Briggs et al.
(2020). In the present work, our results show that GaAs can nucleate at
graphene openings over length scales as large as 10 microns. Experimental
realizations of remote epitaxy may require graphene defect control over a
length scale of several microns, in order to clearly distinguish a “remote”
mechanism from a selective area and lateral epitaxy mechanism.

##  III Conclusions

We demonstrated selective area epitaxy of GaAs using a patterned graphene mask
on Ge (001), over a length scale of 10 microns. The selectivity is comparable
to previous demonstrations of selective area epitaxy by MBE using SiO2 masks,
where smaller spacings of a 100 nm to 2 microns are more typical Ironside et
al. (2019); Lee et al. (2002); Bacchin and Nishinaga (2000). The main limiting
feature in the current work is the rough surface morphology of GaAs on Ge
(001), which we attribute to graphene-induced faceting of the Ge (001)
surface. We expect that growth on graphene/Ge in (111) or (110) orientation
may be a path for smoother films, since these orientations do not produce the
large graphene-induced faceting. Further improvements in lithography and
etching may also be required to reduce the graphene pinholes and enhance the
selectivity for growth at temperatures below $600^{\circ}$C.

##  IV Acknowledgments

This work was primarily supported by the Defense Advanced Research Projects
Administration (DARPA Grant number D19AP00088). Graphene synthesis and
characterization are supported by the U.S. Department of Energy, Office of
Science, Basic Energy Sciences, under award no. DE-SC0016007. GaAs film
characterization was supported by the NSF Division of Materials Research
through the University of Wisconsin Materials Research Science and Engineering
Center (Grant No. DMR-1720415) and the CAREER program (DMR-1752797). We
gratefully acknowledge the use of Raman and electron microscopy facilities
supported by the NSF through the University of Wisconsin Materials Research
Science and Engineering Center under Grant No. DMR-1720415.

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##  V Supporting Information

Figure S-1: Determination of surface coverage. (a) SEM image of a GaAs film
grown at $580^{\circ}$C. (b) Threshold mask image, used to determine the areal
coverage of GaAs film. (c) Histograms of the pixel intensity for the red and
blue boxed regions. Distinct peaks in the histograms correspond to Ge,
graphene, and GaAs. The threshold for the mask is shown as the dashed line:
pixels with intensity higher than this threshold correspond to GaAs. The
threshold is determined based on the mid-point between the FWHM of graphene
and GaAs peaks.