Neural networks trained with stochastic gradient descent (SGD) starting from different random initialisations typically find functionally very similar solutions, raising the question of whether there are meaningful differences between different SGD solutions.
Entezari et al. recently conjectured that despite different initialisations, the solutions found by SGD lie in the same loss valley after taking into account the permutation invariance of neural networks.
Concretely, they hypothesise that any two solutions found by SGD can be permuted such that the linear interpolation between their parameters forms a path without significant increases in loss.
Here, we use a simple but powerful algorithm to find such permutations that allows us to obtain direct empirical evidence that the hypothesis is true in fully connected networks. Strikingly, we find that two networks already live in the same loss valley at the time of initialisation and averaging their random, but suitably permuted initialisation performs significantly above chance.
In contrast, for convolutional architectures, our evidence suggests that the hypothesis does not hold.
Especially in a large learning rate regime, SGD seems to discover diverse modes.

§ INTRODUCTION

Neural networks trained from different initialisations typically have similar decision boundaries [Somepalli et al., 2022], their predictions agree on a large fraction of data [Meding et al., 2021] and even their internal representations are similar [Li et al., 2015, Kornblith et al., 2019]. This raises the question if  all trained instances of neural networks are created equal.
Entezari et al., 2021 recently proposed one way to formalise this idea and hypothesised that any two solutions found by stochastic gradient descent (SGD) live in the same loss valley when accounting for the permutation invariance of neural networks. More precisely, they suggest the following hypothesis.

[All SGD solutions are created equal, Entezari et al., 2021]

For two solutions $\theta_1, \theta_2$ obtained by SGD, there is a permuted instance of the first solution $\pi(\theta_1)$ so that there is no (significant) increase in loss along the linear interpolation between $\pi(\theta_1)$ and $\theta_2$.

If Hypothesis <ref> is true, this would greatly improve our understanding of the loss landscape of neural networks, likely impact ensemble learning [Garipov et al., 2018] as well as network sparsification [Frankle et al., 2020] and it would potentially allow to improve optimisation and generalisation [Izmailov et al., 2018] in neural networks.
Here, we investigate this hypothesis by explicitly constructing permutations with the desired properties, leading to a range of novel findings that elucidate the structure of neural loss landscapes.

* We find strong, direct evidence in support of the hypothesis for fully connected networks of moderate and large width on standard datasets.

* For fully connected networks, we find that an even stronger version of the hypothesis is true: The networks already live in the same loss valley at initialisation.
In particular, averaging the random initialisations $\pi(\theta_1^{\text{init}}), \theta_2^{\text{init}}$ using the permutation revealed through training, gives a network performing significantly above chance.

* For convolutional neural networks (CNNs), our evidence suggests that there is a notable loss barrier between solutions also after accounting for permutation invariance. In particular, and interestingly, in the large learning rate regime, SGD seems to explore a genuinely diverse set of solutions.

We discuss related work on permutation invariance and mode connectivity in Appendix <ref>.

Training procedure and accuracy heat map of plane spanned by three networks $\theta_1, \theta_2, \pi(\theta_1)$, for trained networks (top) and at initialisation (bottom). After training two independently initialised networks, performance decreases when the trained weights are interpolated. This barrier vanishes after applying the permutation found by matching activations. Strikingly, when applying the permutation to the untrained initialisations, interpolating between the two results yields significantly above chance performance.

§ BACKGROUND

We would like to evaluate whether two sets of parameters $\theta_1$, $\theta_2$ lie in the same loss valley after accounting for permutation invariance. To this end, we seek a permutation $\pi^*$ of the parameters $\theta_1$ that minimizes the loss barrier between the two sets of parameters after applying the permutation, i.e. we would like to solve

\begin{gather}
\min_\pi \, \mathrm{Barrier}(\pi(\theta_1), \theta_2)
\\
\mathrm{Barrier}(\theta_1, \theta_2) \coloneqq \max_{\lambda\in[0,1]} \ell\bigl(\lambda \theta_1 + (1-\lambda \theta_2)\bigr) - \Bigl( \lambda\ell(\theta_1) + (1-\lambda)\ell(\theta_2) \Bigr)
\end{gather}

While this problem is computationally intractable, a simple heuristic allows us to find good permutations in practice.
We relax the problem by considering the neuron activations of each layer separately. For each layer, we then find a permutation that makes the resulting neuron activations as similar as possible.

Formally, given a set of $d$ data points, the neuron activations of the networks parameterized by $\theta_1$ and $\theta_2$ are given by the matrices $A, B \in \mathbb{R}^{d\times n}$, where $n$ denotes the number of neurons in the hidden layer (or the number of channels for CNNs).

We define a similarity function $k(\cdot, \cdot)$ which measures for any pair of neurons $i, j$ how similar their activation vectors $A_i$, $B_j$ are and specify our surrogate objective as finding the permutation $\pi$ which maximizes the summed similarity between all matched neuron activations $s(\pi)$

\begin{align}
s(\pi) \coloneqq \sum_{i=1}^n k(A_{\pi(i)}, B_i)
\end{align}

Maximizing $s(\pi)$ corresponds to finding a maximum matching in a bipartite graph and can be solved efficiently using the Hungarian algorithm.

We explored different variants of this idea which all performed similarly. Unless noted otherwise we use the cosine similarity as similarity function $k$ and consider the neuron activations after the non-linearity.

Finding permutations based on this idea has already been explored in [Li et al., 2015, Collier and Dalalyan, 2016] and re-described e.g. in [Yurochkin et al., 2019, Tatro et al., 2020, Singh and Jaggi, 2020].

§ RESULTS

Loss barrier reduces after permutation and depends on network depth and width.
All networks are fully connected and trained on Fashion MNIST for five epochs with AdamW. First, we plot the performance of such a network (“Single Net”, black line). Then, we plot the performance when the parameters of two trained networks are averaged, with (“Perm Avg.”, blue) or without (“Vanilla Avg.”, red) being permuted. Thus, the loss barrier is the difference between the “Single Net” line (black) and the “Perm Avg” line (blue). Shaded region show twice the sem (approx. 95% confidence interval) over 6 seeds.

§.§ Fully connected networks

For the first set of experiments, we will focus on fully-connected networks with varying width and depth trained on MNIST [LeCun, 1998], Fashion MNIST [Xiao et al., 2017] or CIFAR 10 [Krizhevsky, 2009]. In the main paper we will show results on Fashion MNIST; results on other datasets are qualitatively similar and can be found in Appendix <ref>.

The training and evaluation procedure is simple and outlined in Figure <ref>(a). We train two networks independently, for 5 epochs each with AdamW [Kingma and Ba, 2014, Loshchilov and Hutter, 2018] with PyTorch [Paszke et al., 2019] default settings. We then compute the permutation making the networks' activations as similar as possible. Subsequently, we evaluate networks along the linear interpolation between the two networks, either using their unpermuted or permuted versions.

Trained networks lie in the same loss valley after permutation. Figure <ref> show the results of these experiments. The loss barrier between solutions decreases dramatically after accounting for permutation invariance. This decrease confirms that permutation invariance is important for understanding mode connectivity.
Furthermore, by sufficiently increasing the width of the networks the loss barrier is eliminated completely by accounting for permutation invariance. Thus, in wide networks, this directly confirms the hypothesis of Entezari et al., 2021, that there is no significant loss barrier between SGD solutions after taking permutation invariance into account.

However, in narrower networks, we did not find permutations that eliminate the loss barrier suggesting that the two solutions are genuinely different even when accounting for permutation invariance.
That said, we cannot exclude the possibility that our permutations are suboptimal, and that “better” permutations would eliminate the loss barrier.
Indeed, a thorough falsification of the hypothesis in narrower networks would require an exhaustive (and therefore infeasible) search over the space of permutations.

Permutations found after training project random initialisations to same loss valley.
Left: Network with two hidden layers of 2048 neurons each, trained on Fashion MNIST. Left- and rightmost point correspond to two random initialisations. The average of the two permutations performs significantly above chance, demonstrating that, after the permutation, the initialisations lie in the same loss valley.
Mid: Networks with one hidden layer and different widths. For $n>2$ networks, all networks are permuted to be similar to the first network, and then averaged with equal weights. Averaging increasing numbers of random initialisations gradually improves performance.
Right: Same architecture as left panel, but permutations are computed and applied at different points at training.
We match, permute and average the activations after each parameter update (“Current Perm”, dashed blue).
Additionally we apply the permutations found after $0$, $50$ respectively $100$ parameter updates at different times in training (“Perm@x”, shades of pink). Permutations that reveal that initialisations live in the same loss valley can also be found without training the networks, see Appendix <ref>.

Averaged, random initialisations perform above chance after permutations.
We have seen that trained networks usually live in the same, or at least similar, loss valleys when accounting for permutation invariance. Next, we ask how much training is required until the networks enter a shared loss valley.

To investigate this, we conduct experiments as outlined in Figure <ref>(b), i.e. we take two random initialisations, train them with independently shuffled data, determine a permutation that makes the trained networks similar and then apply this permutation to the (untrained) initialisation.
Figure <ref>(left) shows the interpolation between those permuted initialisations and shows a rather surprising discovery: Already the initialisations live in a shared loss valley (up to a permutation), and simply averaging the permuted initialisations gives significantly above chance performance.
Figure <ref>(mid) shows that performance of these random initialisations can be further improved by averaging more than two of them.
Figure <ref>(right) also shows that comparatively little training is required to reveal the shared loss valley between random initialisations.

We also note that it is possible to use the training data to find permutations that put initialisations into the same loss valley without actually training the networks, see Appendix <ref>.

Overall, these findings give a significantly improved understanding of the loss landscape of fully-connected networks. They reveal that, up to permutation, fully-connected feedforward networks live in the same loss valley at initialisation. Moreover, even as the networks start specialising during training, the loss barrier separating any two SGD solutions remains small.

Diversity of SGD solutions depends on training regime for CNNs & random initialisations reside in different loss valleys.
Left & Mid: Two identical CNNs but different initialisations (Net 1 and Net 2) trained with different learning rates. With a smaller learning rate (left), the loss barrier between different solutions is very small after accounting for permutation invariance; whereas with a larger learning rate (mid) the loss barrier increases notably. This suggests that SGD explores more diverse solutions with larger learning rates. Analogous behaviour can be observed in other settings, e.g. for CIFAR 10 and AdamW, see Appendix <ref>.
Right: Different permutations applied across different times, analogous to Figure <ref> (right). This plot also shows that for CNNs the average of permuted initialisations does not perform notably above chance.

§.§ Convolutional Networks

Next, we investigate simple CNNs purely composed of convolutional layers with no skip connections and a linear readout layer (see Appendix <ref> for network and training details).

We again find that accounting for permutation invariance significantly reduces the loss barrier between SGD solutions, in a wide range of settings and as exemplified in Figure <ref>; however a notable barrier remained in most of our settings, including wide networks. This is also consistent with results from more complex architectures, see Fig. 3 in [Tatro et al., 2020].

In addition, we found that the loss barrier in CNNs is significantly more sensitive to training details, like the learning rate.[As a side effect, it is harder to compare the loss barrier between architectures of different width and depth, since for each architecture the choice of training setup has a non-negligible effect.]
One example of this behaviour is shown in Figure <ref> (left&mid), where with a larger learning rate the barrier between solutions increases. We emphasise that this behaviour can be observed across different convolutional architectures, datasets and also optimizers (AdamW as well as SGD+momentum). In all these settings, the loss barrier increases for large learning rates, which are at the edge of training stability (occasionally, runs with these high learning rates fail to converge). These results suggest that for CNNs trained with large learning rates, SGD explores diverse loss valleys, which differ also after accounting for permutation invariance.[
We did not find similar behaviour for fully connected networks, despite looking for it.]

Moreover, we note that for CNNs, different initialisations do not appear to come from a shared loss valley, see Figure <ref> (right).

Summary. These results reveal two interesting differences between fully-connected and convolutional architectures. Firstly, in fully-connected networks already the initialisations live in a shared loss valley, while this does not seem to be the case for CNNs. Secondly, CNNs exhibit a richer training behaviour and in particular for large learning rates, close to training instability, they appear to explore genuinely different loss valleys, while solutions found by SGD in fully connected network are more similar.

§ ACKNOWLEDGEMENTS

Simon Schug was supported by grant no. PZ00P3_186027 from the Swiss National Science Foundation.
Robert Meier was supported by grant no. CRSII5 173721 of the Swiss National Science Foundation.
Johannes von Oswald was funded by the Swiss Data Science Center (J.v.O. P18-03).
Nicolas Zucchet and Yassir Akram were supported by ETH Research Grant no.  ETH-23 21-1.

§ RELATED WORK

The question how similar different neural networks are has inspired research from a variety of angles.
For example, Li et al., 2015 train independent copies of CNNs and investigate how well individual neurons and groups of neurons from one network correspond to groups of different neurons; part of their methodology also relies on computing matchings analogously to our approach.

Kornblith et al., 2019 investigates similarity of layer representations drawing on work from neuroscience (e.g. [Kriegeskorte et al., 2008]) and machine learning [Laakso and Cottrell, 2000]. Their similarity measure is theoretically well-motivated and suggests that different networks learn considerably more similar representations than suggested by similarity measures of prior work [Raghu et al., 2017, Wang et al., 2018, Morcos et al., 2018].

Mehrer et al., 2020 also study network similarity with methods from neuroscience and additionally conduct insightful experiments on how inter- or intra- class variance contribute to differences between networks.

All [Li et al., 2015, Kornblith et al., 2019, Mehrer et al., 2020] find that representations agree very strongly in early layers and decay to some extent for deeper layers. This is consistent with our findings in Figure <ref> suggesting that deeper networks are separated by higher loss barriers.

In addition, our work is closely connected to mode connectivity. It was first shown theoretically that any two modes are connected by a (potentially complex, non-linear) path of low loss in networks with one hidden layer [Freeman and Bruna, 2017]. This was confirmed in a range of empirical settings [Draxler et al., 2018, Garipov et al., 2018].

Tatro et al., 2020 show that these paths can be simplified or their loss barrier lowered when taking permutation invariance into account. They find permutations similarly as us. Thus, their work is overall similar in spirit to ours, with the difference that we investigate linear paths between modes and how architecture and training choices affect mode connectivity.

Theoretically, [Brea et al., 2019, Simsek et al., 2021]  investigate the connectivity between permutations of a given mode. We note that this is different from our objective to determine whether independently obtained modes are (approximately) permutations of one another.

Additionally, there is work developing network-/kernel-representations which are invariant to permutations and rotations [Aitchison et al., 2021].

Linear mode connectivity has also been investigated in different contexts, e.g. focusing on training trajectories of neural networks [Goodfellow et al., 2014], relating it to flatness and generalisation [Izmailov et al., 2018], and connecting it to sparsity and the lottery ticket hypothesis [Frankle and Carbin, 2018, Frankle et al., 2020].

§ FINDING GOOD PERMUTATIONS AT INITIALISATION WITHOUT TRAINING

In the main paper we described how a permutation obtained by training networks and matching their features leads to averaged initialisations that perform above chance.

Here, we briefly note that such permutations can also be obtained without training the networks and describe how this can be done.

As a first observation note that for our desired permutation $\pi$, the loss decreases when we start at $\theta_2$ and move towards $\pi(\theta_1)$, i.e. in the direction of $v_\pi=\pi(\theta_1) - \theta_2$. This suggests that the direction $v_\pi$ should be (negatively) aligned with the gradient at $\theta_2$. Thus, we can try to find permutations that minimize the dot product $v_\pi \cdot \frac{\partial L(\theta)}{\partial \theta}|_{\theta=\theta_2}$. This can be formalised as a minimum matching problem again and thus be solved with the Hungarian algorithm.

The idea described above does not yet give good permutations and we need two additional ideas.

* First, and most importantly, we replace the vanilla gradient $\frac{\partial L(\theta)}{\partial \theta}|_{\theta=\theta_2}$ by the update direction of a more powerful optimizer. We used FOOF [Benzing, 2022], which was inspired by obtaining a better understanding of KFAC [Martens and Grosse, 2015, Desjardins et al., 2015].

* Second, we symmetrise the loss. I.e. (using the vanilla gradient rather than more advanced optimizers for notational simplicity) we minimize

\begin{align}
\min_\pi\;
(\pi(\theta_1)-\theta_2) \cdot \frac{\partial L(\theta)}{\partial \theta}|_{\theta=\theta_2}
(\pi^{-1}(\theta_2)-\theta_1) \cdot \frac{\partial L(\theta)}{\partial \theta}|_{\theta=\theta_1}
\end{align}

We note that this gives permutations with very similar performance (at initialisation) as the permutations described in the main text.

Moreover, in the case of CNNs this algorithm also does not find permutations which put different initialisations in the same loss valley, further supporting our finding from the main paper that CNN initialisations live in distinct loss valleys.

§ FINDING PERMUTATIONS FOR SECOND-ORDER TAYLOR EXPANSIONS AROUND THE LOSS

The algorithm described in the preceding section (almost) amounts to finding a permutation that minimizes a first-order Taylor expansion of the loss around $\theta_2$ evaluated at $\frac12\pi(\theta_1) + \frac12\theta_2$.[We choose the midpoint between $\pi(\theta_1)$ and $\theta_2$ here since empirically, this is a very good approximation of the point along the linear interpolation that maximizes the loss barrier.]
Thus or otherwise motivated, one might be tempted to find permutations that minimize higher-order Taylor expansions of the loss around $\theta_2$ evaluated at $\frac12\theta_2+\frac12\pi(\theta_1)$ both for initialisations or trained networks $\theta_2$.

If one uses KFAC [Martens and Grosse, 2015] to obtain a computationally tractable approximation of the second-order expansion of the loss around $\theta_2$, and then computes the permutations sequentially layer-by-layer, the minimization problem to find good permutations turns out to be equivalent to the quadratic assignment problem (and thus NP hard). We experimented with convex relaxations of this problem over the Birkhoff polytope but failed to find better permutations than with the simpler method described in the main paper.

§ VARIANTS OF FEATURE MATCHING AND INTERPOLATION

We experimented with different similarity functions $k(\cdot,\cdot)$ for measuring the similarity between neuron activations. In particular, we investigated negative $\ell_2$-distance and cosine similarity. We also experimented with extracting neuron activations both before and after the non-linearity. Broadly speaking, all variants performed very similarly.

We also experimented with combining the permutations with learned / non-learned “small” rotations. This often decreased the loss barrier slightly, but did not affect results qualitatively.

Moreover, we tried interpolating between each layers' weights using a rotation, rather than a linear path, but found that this significantly increased the loss barrier.

Finally, we also tried matching weights instead of neuron activations. This also performed worse, i.e. the loss barrier increased.

See also Section <ref> for a different approach to find permutations that is based on minimizing higher-order approximations of the loss along the interpolation between $\pi(\theta_1)$ and $\theta_2$.

§ RECAP OF PERMUTATION INVARIANCE

It is well known that neural networks are permutation invariant. To make this term precise, consider a fully connected network consisting of a sequence of weight matrices $W_1, \ldots, W_\ell$, where $W_i$ has dimensions $n_{i-1} \times n_i$, and a pointwise non-linearity $\sigma$ (for simplicity of notation, we assume that each layer is followed by the same non-linearity, but this assumption can easily be dropped). For a given input $X$, the output of the $i$-th layer $A_{i}$ is defined iteratively as

\begin{align}
A_0 &= X \\
A_i &= \sigma(W_{i} A_{i-1}) \quad\text{for } 1\leq i \leq \ell
\end{align}

and the output of the neural network is simply the output of the last layer $A_\ell$.

If we are given a sequence of permutation matrices $P_0, \ldots, P_{\ell}$, where $P_i$ has dimensions $n_i \times n_i$ and where the first and the last permutation matrix $P_0, P_\ell$ are the identity, then the permuted neural network consisting of the weight matrices

$$\tilde{W}_i = P_i^T W_i P_{i-1}$$
computes precisely the same output as the network consisting of weight matrices $W_i$ (since non-linearity and permutation commute and since $P_i^T P_i = \mathrm{Id}$). Thus the input-output function specified by the network is invariant to permutations as described above.

While we used fully connected networks above, permutation invariance straightforwardly applies for example to convolutional networks, where channels can be permuted.

§ EXPERIMENTAL DETAILS

The term “SGD” in the paper usually refers to SGD with the usual 0.9 heavy-ball momentum, but we obtained qualitatively similar results for SGD without momentum.

We used a batch size of 100 in all experiments, but we found similar results across a range of batch sizes.

The permutations are usually computed using the activations of 10000 images, and we verified that the quality of permutations saturates already for smaller numbers of images.

We used Kaiming-He init and ReLU nonlinearities for intermediate layers. The final layer is followed by a softmax, in combination with the cross-entropy loss.

§.§ Fully Connected Networks

Networks presented here are trained for 5 epochs with AdamW PyTorch [Paszke et al., 2019] default settings, except Figure <ref> mid-panel, where permutations were obtained by training for one epoch. We note that for narrow networks, permutations that obtain even better performance when used to average initialisations can be obtained by training shorter.

§.§ CNNs

All CNN architectures use a sequence of convolutional filters and 2x2 average pooling layers, based on standard CNN design practices. These layers are finally followed by a global average pooling layer and one fully connected layer.

Based on this general design, we trained two kinds of CNN architectures,
one of type

$$k - k - \text{avg} - 2k - 2k$$

where “$\text{avg}$” denotes a 2x2 average pooling layer and the remaining numbers denote the number of (hidden) channels in the convolutional layers; and another one of type:

$$k - \text{avg} - 2k - \text{avg} - 4k.$$

We used $k=32, 64, 128, 256$

The CNN in Figure <ref> is of type $128-128-\text{avg}-256-256$, but we observed similar behaviour across different optimizers, architectures and datasets.  One more example for CIFAR10, AdamW and type $64-64-\text{avg}-128-128$ is shown in Figure <ref>.

§ ADDITIONAL FIGURES

Corresponding Train Loss Heat Map to the right part of Figure <ref>.

Analogous to Figure <ref>, but on CIFAR 10.

Loss increases at very high width, because in this case lower learning rates would be optimal, but we stick to AdamW default settings.

Analogous to Figure <ref>, but on MNIST.

Loss increases at very high width, because in this case lower learning rates would be optimal, but we stick to AdamW default settings.

Analogous to Figure <ref>, but on CIFAR 10.

Analogous to Figure <ref>, but on MNIST.

Analogous to Figure <ref>, but on CIFAR10 with AdamW.

Additional results on Fashion MNIST when varying the initialization scheme and loss function. Top row: The effect of vanilla and permuted averaging, at initialization and after training, for two different initialization schemes. We test Kaiming-He (as in the main text) [He et al., 2015], Xavier (not shown) and Orthogonal  [29] initialization and find that the mode connectivity remains linear for the different initializations. Behaviour with Xavier initialization is very similar to the Orthogonal one and therefore omitted. Bottom row: When turning to the mean squared error (MSE) loss instead of the cross-entropy loss, we observe worse performance for vanilla averaging before and after training, for the two initializations, while the networks stay linearly connected after permuting.

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# Quasiperiodic gallium adlayer on i-Al-Pd-Mn

Pramod Bhakuni1, Marian Krajčí2,∗, Sudipta Roy Barman1,† 1UGC-DAE Consortium
for Scientific Research, Khandwa Road, Indore - 452001, Madhya Pradesh, India
2Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, SK-84511
Bratislava, Slovak Republic

###### Abstract

Using scanning tunneling microscopy (STM), low energy electron diffraction
(LEED), and density functional theory (DFT), we demonstrate the formation of
quasicrystalline gallium adlayer on icosahedral ($i$)-Al-Pd-Mn. Quasiperiodic
motifs are evident in the STM topography images, including the Ga white flower
(GaWF) and $\tau$ inflated GaWF ($\tau$-GaWF), where $\tau$ is the golden
mean. A larger and more complicated ring motif is also identified, comprised
of a bright center and an outer ring of pentagons. The fast Fourier transform
of the STM images exhibits distinct quasiperiodic spots, thereby establishing
quasiperiodicity on a length scale of $\sim$350 nm. Based on our DFT
calculations, the preferred adsorption sites of Ga on $i$-Al-Pd-Mn are
determined to be either the Mn atoms at the center of the Penrose P1 tile or
the vertices of the P1 tile containing Pd atoms at the center of a cluster of
5 Al atoms (5-Al). The GaWF motif is modeled by an inner 6 atom Ga cluster
(6-Ga) around the central Mn atom and an outer ring of 5 Ga atoms adsorbed at
the centers of the 5-Al clusters, both having pentagonal symmetry. The
$\tau$-GaWF motif is modeled by the 6-Ga arranged on the $\tau$-P1 tiling,
while the ring motif is modeled by Ga atoms adsorbed at the center of 5-Al
clusters above a Pd atom. The side lengths and diameters of the GaWF,
$\tau$-GaWF, and the ring motifs are $\tau$ scaled and show excellent
agreement with the DFT-based models. An additional indication of the
quasiperiodic characteristics of the Ga monolayer is the 5-fold LEED patterns
that were detected throughout the entire range of beam energy that was
measured.

##  I Introduction

Quasicrystals are a distinct type of material that display aperiodic order
with rotational symmetries, including 5-fold, 8-fold, 10-fold, and 12-fold,
which are forbidden in crystals with translational order [1, 2]. The
quasicrystalline phase was first identified in a binary alloy of Al6Mn with
icosahedral symmetry [3], and since then it has been observed in a variety of
systems, which include ternary and binary intermetallic compounds [4],
nanoparticle superlattices [5], colloidal systems [6], perovskite barium
titanate on Pt(111) [7], molecular assemblies [8, 9], twisted bilayer graphene
[10, 11, 12], chalcogenides [13], elemental films [14, 15, 16, 17], and even
in naturally occurring minerals [18, 19]. Inorganic quasicrystals exhibit
unusual properties such as low thermal and electrical conductivity, low
specific heat, low frictional coefficient, and large hardness [20, 21, 22, 23,
24]. Their stability has been related to the pseudogap i.e., a reduced density
of states (DOS) around the Fermi level [25, 26, 27, 28]. The possibility of
higher order topological states occurring in quasiperiodic systems has been
predicted by theory [29, 30, 31]. Our recent research utilizing hard x-ray
photoemission spectroscopy and density functional theory (DFT) has established
the occurrence of Anderson localization in icosahedral ($i$)-Al-Pd-Re [32].

Figure 1: (a) STM topography image of 0.1 ML Ga/$i$-Al-Pd-Mn ($I_{T}$ = 0.9 nA
, $U_{T}$ = -0.7 V) showing the Ga white flower (GaWF) motifs encircled by
white circles and ovals, dashed circles show the incomplete motifs. The color
scale representing the height is shown on the left, zero corresponds to the
bearing height. (b) A height profile across the GaWF along the orange dashed
line in panel a. (c) Zoomed view of a GaWF motif with a white dashed pentagon
overlaid on it. (d) The atomic structure of the surface plane (2.3
nm$\times$2.4 nm) of the 5-fold $i$-Al-Pd-Mn derived from the 2/1 approximant
to the bulk. The white, magenta, and red colored circles represent the Al, Pd,
and Mn atoms, respectively. The Penrose P1 tiling is shown by black lines. (e)
An 11 Ga atom DFT based model of the GaWF centered on the Mn site of the
5-fold $i$-Al-Pd-Mn. The green colored circles represent the Ga atoms.

Since more than two decades, researchers have studied elemental
quasicrystalline films to investigate the impact of aperiodic order on the
physical and electronic properties that are independent of the chemical
complexity of the ternary quasicrystals. But to date, only a few elements have
demonstrated quasiperiodicity [14, 33, 34, 35, 36, 37, 38, 39, 40, 15, 16,
17]. Franke et al. discovered that Bi and Sb are quasiperiodic on $i$-Al-Pd-Mn
and decagonal Al-Ni-Co quasicrystalline substrates up to 1 monolayer (ML)
coverage using low energy electron diffraction (LEED) and He atom scattering
[14]. Shukla et al. [39] showed that Na forms a regular 5-fold quasiperiodic
bilayer on $i$-Al-Pd-Mn. Pb/$i$-Ag-In-Yb was reported to exhibit 5-fold growth
isostructural with the $i$-Ag-In-Yb substrate to a maximum height of 0.7 nm
[15]. Recent research utilizing scanning tunneling microscopy (STM), LEED, and
DFT has demonstrated that a 4 nm thick Sn adlayer deposited on the fivefold
surface of an $i$-Al-Pd-Mn substrate exhibits long range quasiperiodic order
as a metastable realization of an elemental, clathrate family quasicrystal
[16]. Long range decagonal clathrate quasiperiodic ordering of Sn thin films
on $d$-Al-Ni-Co has also been observed up to 0.9 nm, with partial retention of
decagonal structural correlations up to 10 nm film thickness [17].

Gallium is an interesting element due to its relatively low melting point (303
K) and its complex solid-state structure. The latter produces a notable dip at
the Fermi level ($E_{F}$), which is believed to be the consequence of partial
covalent bonding [41, 42]. Recently, gallanene i.e., a honeycomb monolayer of
Ga that is analogous to graphene, has been reported [43, 44, 45, 46, 47].
However, Ga deposition on single crystalline metal surfaces has been scarcely
studied [48], and there has been no study on any quasicrystalline substrates
to date. Ga could be a possible candidate since its surface energy (0.55 J/m2)
[49] is less than that reported for the i-Al-Pd-Mn surface (0.82 J/m2) [50],
indicating the possibility of layered growth. In this work, we use STM, LEED,
and density functional theory (DFT) to reveal the occurrence of long range
quasiperiodicity in a gallium monolayer deposited at room temperature (RT) on
$i$-Al-Pd-Mn. The quasiperiodic motifs such as Ga white flower (GaWF), $\tau$
scaled GaWF ($\tau$-GaWF), and the ring observed from STM have been modeled by
our DFT calculations. The fast Fourier transform of the large area STM images
exhibits distinct quasiperiodic spots.

##  II Methods

The STM measurements were carried out using a variable temperature STM from
Scientaomicron GmbH at a base pressure of 5$\times$10-11 mbar. All the STM
measurements were performed at RT in the constant current mode using a
tungsten tip that was prepared by sputtering and the voltage pulse method. The
tip was biased, and the sample was kept at the ground potential. STM images
were recorded for varying bias voltages. The Fast Fourier transform (FFT) was
averaged for 2-4 STM images, as in our previous work [17]. The STM images are
shown after low-pass Fourier transform filtering (Fig. S1 of the Supplementary
material (SM) [51]). The zero in the z-scale of the STM image corresponds to
the most frequently occurring height (bearing height) based on the height
histogram. In a height profile, the difference of the average $z$ corrugation
on the Ga adlayer and the substrate provides the height of the former [52].
The average height for a particular deposition is determined by fitting a
Gaussian curve to the height distribution obtained from more than 50 height
profiles derived from various parts of the STM images. LEED and AES were
performed using a four-grid rear view optics and a hemispherical retarding
field analyzer, respectively. The I-V curves were extracted using the EasyLeed
software [53].

The 5-fold surface of the monocrystalline i-Al-Pd-Mn was polished using a 0.25
$\mu$m diamond paste before inserting it into the ultra-high vacuum chamber.
The polished i-Al-Pd-Mn surface was treated in-situ by repeated cycles of Ar+
ion sputtering at 1-2 keV for 30-60 min and annealing to 970 K for 2-2.5 hr to
produce an atomically clean surface with a composition similar to the bulk
[54, 55]. Gallium of 99.99$\%$ purity was evaporated using a water-cooled
Knudsen cell [56] operating at 1050 K at a pressure better than 2$\times$10-10
mbar. The substrate surface was freshly prepared for each deposition.

The DFT calculations to probe the interaction of Ga atoms with the $i$-Al-Pd-
Mn surface were performed using the method of the Vienna Ab initio Simulation
Package (VASP) [57, 58, 59]. The models of the surface have been derived from
the Katz-Gratias-Boudard model [60] of bulk $i$-Al-Pd-Mn. The atomic structure
of the 5-fold surface is derived from the icosahedral approximants by cleaving
at a plane perpendicular to one of the 5-fold axes. Details of construction of
the surface models can be found in our previous papers [61, 62, 40, 63]. The
adsorption energy ${E_{A}[\rm Ga]}$ of Ga adatom is defined with respect to
the energy in the crystalline structure as $E_{A}[\rm Ga]$ = $E_{tot}[\rm
Ga]$$-$$E_{surf}$+$N_{A}$$\times$$\mu[\rm Ga]$, where $E_{tot}$ is the energy
of the $i$-Al-Pd-Mn surface model [60, 40] with the adsorbed adatoms.
$E_{surf}$ is the energy of the model $i$-Al-Pd-Mn surface representing the
atomic structure of the clean quasicrystalline 5-fold surface. ${N_{A}}$ is
the number of adatoms in the cluster on the surface. $\mu[\rm Ga]$ is the
chemical potential of Ga in the crystalline structure, $\mu$[Ga] =
$-E_{cryst}$[Ga.oP8]= 2.934 eV. The adsorption energy per atom $E_{a}$ is
$E_{a}$[Ga] = $E_{A}$[Ga]/$N_{A}$.

##  III Results and discussion

###  III.1 Nucleation of Ga on $i$-Al-Pd-Mn

A STM topography image for a deposition time ($t_{d}$) of 2 min results in a
coverage of 0.1 ML Ga on $i$-Al-Pd-Mn. The coverage is determined by the
fraction of the total area covered by the Ga clusters that are identified by
the bright regions in Fig. 1(a). The clusters have pentagonal symmetry that
resembles flowers with five petals and are highlighted by white circles and
ovals. We refer to these as Ga white flowers (GaWFs). The height of the GaWFs
is determined by subtracting the average $z$ corrugation of the substrate from
that of the GaWF, as illustrated in Fig. 1(b). In Fig. S2(a) of the SM [51], a
distribution of the heights obtained from several height profiles is fitted
with a Gaussian function. The position of its maximum gives the average height
of the GaWF to be 0.18$\pm$0.03 nm. A zoomed view of a GaWF motif is shown in
Fig. 1(c). The length of the sides of the pentagon (white dashed lines)
joining its petals has been determined for several motifs. Its average length
turns out to be 0.79$\pm$0.1 nm, and the corresponding distribution is shown
in Fig. S2(b) of SM [51].

To understand the origin of the GaWF, we have performed DFT calculation for
the adsorption of Ga atoms on the 5-fold $i$-Al-Pd-Mn surface that was modeled
as the 2/1 approximant [Fig. 1(d)]. The 5-fold $i$-Al-Pd-Mn surface is
superposed with the Penrose P1 tiling (black lines) [61]. The P1 tiling has an
edge length ($a_{0}$) of 0.776 nm. The Mn atoms (red circles) appear at the
centers of the pseudo-Mackay clusters (pMC). The vertices of the P1 pentagon
are occupied by 5 Pd atoms (magenta circles). This figure shows that the
surface Mn atoms are in the center of the P1 tiling, around which the well
known white flower motifs of the substrate form [64, 65].

Quasicrystalline surfaces offer a greater number of inequivalent adsorption
sites compared to ordinary crystals, allowing for the formation of various
adatom configurations. On the 5-fold $i$-Al-Pd-Mn surface, there are two kinds
of regular sites that preferably adsorb Ga atoms. The first site is the center
of the surface 5-Al cluster that has a Pd atom. These Pd atoms can be 0.048 nm
or 0.126 nm below the surface plane. Therefore, there are 2 kinds of the Pd
sites and only one can adsorb the adatoms. The Ga atoms are adsorbed
preferably only at the centers of the 5-Al cluster with the deeper Pd atom.
These deeper Pd atoms are the centers of the Bergman clusters [60]. In Fig.
1(d), these sites are at the vertices of the P1 tiles and are also marked by
white dots in Fig. 4(f) and Fig. S3(b) of SM. The binding energy of a single
Ga atom in such sites is $-$1.357 eV (see Table 1). The second preferable
adsorption site for a Ga adatom is the central Mn atom at the surface. The
main reason for the higher reactivity of these Mn atoms is their low
coordination: In the bulk pMC cluster, in the first shell around the Mn atom,
there are only 7 nearest neighbors [60, 66]. The binding energy of Ga atoms at
the surface Mn atoms is between $-$1.2 eV and $-$1.4 eV.

In previous studies [40, 66, 63, 55, 16, 37], it was observed that the adatoms
(Bi, Pb, and Sn) form WF clusters around the surface Mn atoms. Our DFT
calculations show that Ga atoms adsorbed on the 5-fold $i$-Al-Pd-Mn surface
also form clusters similar to the previously observed WFs, but the internal
structure is distinct because of the notably different electronic structure of
Ga [41, 42]. In the previous studies [40, 66, 63, 55, 16, 37], the adatoms
form the WF cluster, in which the outer and inner rings are oriented in the
same direction and are therefore interconnected to form a starfish-like
configuration. The GaWF cluster also consists of the outer ring of five Ga
atoms, each adsorbed at the vertices of the P1 pentagon that are the centers
of the reactive 5-Al clusters [Fig. 1(e)] that is the same as in the previous
WFs. However, in contrast, the central pentagon of Ga adatoms has an opposite
orientation compared to the pentagon of the atoms in the outer ring. Moreover,
to stabilize this central pentagon, there must be an additional Ga atom  at
the center, making it a 6 atom Ga cluster (6-Ga), while retaining the
pentagonal symmetry. The energies calculated for different Ga clusters are
discussed later and compared in Table 1. The opposite orientation of the
pentagonal rings reflects the lower attractive interaction of adatoms between
the outer ring and central 6-Ga cluster. In the case of the Ga adatoms, the
outer and inner rings can also exist independently.  The formation of the
central 6-Ga cluster is slightly (-0.16 eV/Ga atom) preferred over the
formation of both rings simultaneously. The central Ga atom in the 6-Ga
cluster is elevated by 0.1 nm compared to the five Ga neighbors. Therefore,
the center of the GaWF cluster may appear brighter in the STM images than the
outer ring. Also, on the 6-Ga cluster at the center, additional Ga atoms can
be adsorbed, which can lead to a brighter center and an irregular shape. We
also note that, due to stochastic reasons, some GaWF motifs have a dark
center, which could be caused by the absence of the 6-Ga cluster at the
center.

It should be noted that the edge length of the P1 tiling $a_{0}$ (= 0.776 nm)
is in excellent agreement with the side length of the GaWF from STM
(0.79$\pm$0.1 nm). Additionally, the typical distance between two neighboring
Mn atoms on which the 6-Ga clusters are centered is 1.255 nm (=
$\tau$$\times$$a_{0}$, where $\tau$= 1.618). This distance also agrees well
with the distance between the nearest neighbor GaWFs, as shown by the white
ovals in Fig. 1(a), where the distance between their centers is $\sim$1.25 nm.
This confirms that the 6-Ga clusters nucleate on the Mn atoms i.e., on the
white flower motifs of the substrate.

Table 1:  Adsorption energies per atom ${E_{a}}$ [eV] of Ga clusters
consisting from ${N_{A}}$ adatoms on the 5-fold $i$-Al-Pd-Mn surface.
| Single Ga | Outer 5-Ga | WF cluster | Inner 6-Ga
---|---|---|---|---
${N_{A}}$ | 1 | 5 | 11 | 6
${E_{a}}$ |  $-$1.357 |  $-$0.451 |  $-$0.354 |  $-$0.515

A comparison of the adsorption energies of Ga adatoms on the 5-fold $i$-Al-Pd-
Mn surface are presented in Table 1. As mentioned above, the adsorption energy
of $-$1.357 eV corresponds to a single Ga adatom adsorbing at the reactive
center of the 5-Al surface cluster above the Pd atom. The adsorption energy
per atom ($E_{a}$) of the pentagonal 5-Ga outer ring of adatoms adsorbed in
the five reactive 5-Al surface sites is $-$0.451 eV. The GaWF cluster with an
$\rm{E_{a}}$ of $-$0.354 eV consists of the outer 5-Ga ring and the inner 6-Ga
cluster. The $E_{a}$ of the central 6-Ga cluster formed alone around the
surface Mn atom is the strongest ($-$0.515 eV). The adsorption of adatoms in
the clusters is weaker compared to the adsorption of a single adatom because,
in the cluster, in addition to bonding of adatoms to the surface, there are
also interactions between the adatoms.

For comparison, in Discussion I of the SM, we also present the adsorption
energies of the previously studied Sn white flower (SnWF) cluster [16] on the
same 5-fold $i$-Al-Pd-Mn surface. ${E_{a}}$ of the complete SnWF cluster is
$-$0.795 eV (see Table S1 of the SM), which is significantly larger than
${E_{a}}$ = $-$0.354 eV of the GaWF cluster. This could be a possible reason
for the somewhat worse quality of the GaWF motifs [Fig. 1(a)] compared to the
SnWF motifs on $i$-Al-Pd-Mn [see Fig. 10(a) and Fig. S8 of Ref. 16]. Another
possible reason is that the STM in the present work has been performed at room
temperature, which is close to the melting point of Ga, resulting in enhanced
thermal effects.

The 5-fold surface of $i$-Al-Pd-Mn is primarily composed of Al atoms, with the
top two atomic planes of the 5/3-approximant consisting of 75.9% Al, 19.6% Pd,
and 4.5% Mn [66]. So, it can be interesting to compare the adsorption of Ga
adatoms on the crystalline Al(111) surface with the results in Table I for the
quasicrystalline surface. Discussion II of SM [51] shows that the reactivity
of the adsorption sites on $i$-Al-Pd-Mn is significantly higher compared to
Al(111).

Figure 2: (a) STM topography image of 0.4 ML Ga/$i$-Al-Pd-Mn ($I_{T}$ = 0.5
nA, $U_{T}$ = 1.0 V), where white, yellow, and red circles highlight the Ga
white flower (GaWF), $\tau$-GaWF, and ring motifs, respectively. (b) Fast
Fourier transform (FFT) calculated for panel a by selecting only the Ga
islands and masking the substrate using the height threshold procedure, the
spots are numbered as 1-10. (c) STM topography image of 0.7 ML Ga ($I_{T}$=
0.7 nA, $U_{T}$ = 0.9 V), the motifs are highlighted as in panel a. (d) FFT of
the Ga islands in panel c after masking the substrate with the spots numbered.
(e) The height profile along the red and orange dashed lines in panel c.

###  III.2 Submonolayer Ga on $i$-Al-Pd-Mn

In Fig. 2(a), the deposition of Ga for $t_{d}$= 4 minutes results in the
formation of condensed islands with a lateral size of $\sim$20 nm. The
coverage turns out to be 0.4 ML. The height profiles show that their height is
similar to the GaWFs, indicating that islands constitute monolayer Ga (the
distribution is shown in Fig. S2(c) [51]). The formation of such islands – in
contrast to a dispersed phase reported for alkali metals on $i$-Al-Pd-Mn [36]
– shows that the interaction between the adatoms is considerable.

Figure 3: (a) STM topography image of 1.2 ML Ga/i-Al-Pd-Mn ($I_{T}$= 0.8 nA,
$U_{T}$ = 0.5 V) showing formation of a uniform Ga monolayer. The motifs of
the monolayer are highlighted by circles of similar color, as shown in Fig. 2.
The motifs in the bright condensed islands that represent Ga bilayer regions
are shown by dashed circles. FFT of (b) the monolayer region by masking the
bilayer regions and (c) the whole image in panel a. (d) The height profile
along the red and yellow dashed lines in panel a. (e) Height histogram of
panel a. The double-sided black arrows in panels d and e indicate the
thickness of the second Ga layer. (f) Auger electron spectra as a function of
deposition time ($t_{d}$) show an increase in the Ga MNN signal (red arrow)
and a decrease in the Al LMM (blue arrow) signal. (i) A large area (250
nm$\times$250 nm) STM topography image of 1.2 ML Ga/i-Al-Pd-Mn ($I_{T}$= 0.6
nA, $U_{T}$ = $-$1.9 V) and its (j) FFT. The inset of panel i shows the height
profile along the white dashed line. FFT of the different regions of panel i,
as indicated by the (g) black, (h) green, and (k) gray squares.

It is interesting to note here that besides the GaWF motifs (white circles),
larger sized white flower motifs [yellow circles in Fig. 2(a)] are observed.
We refer to these as $\tau$-inflated GaWF or $\tau$-GaWF as their size scales
with $\tau$ compared to the GaWF motif. In addition, red circles highlight a
larger, more complicated motif with a bright center and an outer ring of
pentagons. We refer to it as the “ring” motif. In subsection III.4, we
elaborate on the origin of these motifs based on our DFT calculation.

To establish the quasiperiodic nature of the Ga adlayer at 0.4 ML coverage, in
Fig. 2(b) we show the FFT of the Ga island regions of the STM image by masking
the substrate using the height threshold procedure. The quasiperiodic nature
of the islands is confirmed by the 10-fold FFT spots, with the angle subtended
by two adjacent spots at the center being 36${}^{\circ}\pm$ 2∘. In Fig. S4(a)
of SM [51], the presence of all the spots is confirmed by the intensity
profiles that show a peak that represents each spot.

The STM topography image in Fig. 2(c) shows that for 0.7 ML Ga coverage
($t_{d}$= 8 min), the condensed islands grow larger in size and merge
together. The $\tau$-inflated GaWF motifs (yellow circles) are more prevalent
at this coverage. Additionally, their orientation is found to vary, as
indicated by the yellow dashed pentagons that are rotated by 36∘.

The FFT in Fig. 2(d) [see also Fig. S4(b) of SM [51]] for the Ga adlayer
region displays 10-fold spots with the angle subtended at the center by the
adjacent spots being 36∘$\pm$2∘. Thus, the FFT shows that the quasiperiodicity
is retained up to this coverage. The height profiles taken along the red and
orange dashed lines shown in Fig. 2(c) are plotted in Fig. 2(e). Considering
several such height profiles, the average height turns out to be 0.18$\pm$0.03
nm, in agreement with that determined for 0.1-0.4 ML. Note that the thickness
of the Ga monolayer is somewhat smaller compared to other quasiperiodic layers
of elemental metals on $i$-Al-Pd-Mn, such as Sn (0.2 nm) [16] and Pb (0.23 nm)
[38, 40]. A possible reason is the smaller size of the Ga atom (empirical
atomic diameter being 0.26 nm [67]) compared to Sn (0.29 nm) and Pb (0.36 nm).

###  III.3 Quasiperiodicity of monolayer gallium

Figure 3(a) illustrates the STM topography image for $t_{d}$= 16 minutes, in
which the substrate is uniformly covered by the Ga monolayer. Note that the
characteristic quasiperiodic motifs such as $\tau$-GaWF and ring – similar to
those observed in Fig. 2 – are also observed on the Ga monolayer, these are
highlighted by similar colored circles in Fig. 3(a).

Curiously, in this figure, isolated bright condensed islands are observed. In
Fig. 3(d), the height profiles along the red and yellow dashed lines in Fig.
3(a) show that the average height of these islands relative to the monolayer
is 0.19$\pm$0.02 nm. This is nearly similar to the monolayer’s height
discussed in the previous section, and thus the bright islands can be related
to Ga bilayer regions. This is further supported by the height histogram in
Fig. 3(e), which displays two peaks corresponding to the mono- and bilayer,
the separation between which yields a similar height of the latter (horizontal
double-sided arrow). From the area under the two peaks of the height
histogram, we find that the bilayer forms over an area that is $\approx$20% of
the total area. Thus, the coverage for this deposition turns out to be 1.2 ML.

Fig. 3(b) displays the FFT of only the monolayer (i.e., excluding the bilayer
regions), while Fig. 3(c) displays the FFT of the entire image. 10-fold spots
are observed in both. Notably, from a comparison of their intensity profiles
in Fig. S5 of SM [51], it is apparent that the intensity of the spots is
larger with the inclusion of the bilayer regions, indicating an improvement in
the quality of the FFT. In addition, close scrutiny reveals the characteristic
quasiperiodic motifs such as $\tau$-GaWF and an incomplete ring on the bilayer
islands [dashed circles in Fig. 3(a)], indicating the possible existence of
quasiperiodic structural correlations in some regions of the bilayer.

In Fig. 3(f), the Ga MNN Auger electron spectroscopy signal at 55 eV increases
with a concomitant decrease of the substrate Al LMM signal at 68 eV as a
function of $t_{d}$. This shows that the Ga atoms are deposited incrementally
on the $i$-Al-Pd-Mn surface. No change in the position of either of the peaks
is observed over the whole $t_{d}$ range (see the vertical dashed arrows),
indicating the absence of any surface alloying or chemical bonding with the
substrate.

A STM topography image in Fig. 3(i) for 1.2 monolayer coverage ($t_{d}$= 16
min) that spans a larger length scale (250nm$\times$250 nm) shows three
adjacent terraces of the substrate. The bottom terrace is marked as t1 (blue),
while the other terraces that exhibit step heights of 0.4$\pm$0.02 nm (= $L$)
and 1.05$\pm$0.02 nm (= $2L+S$, $L$ and $S$ are the basic heights related by
$L=\tau\times S$) are referred to as t2 (light blue ) and t3 (whitish blue),
respectively; see the inset for the height profile along the white dashed
line. $L$ and $(2L+S)$ are the characteristic step heights of the substrate
[68, 69, 70]; their presence indicates that these terraces are uniformly
covered by the Ga monolayer (the small bright isolated islands are the bilayer
regions). The quasiperiodicity of Ga on these terraces is established by the
presence of sharp 10-fold spots in the FFT, as shown in Fig. 3(g) for an
overlapping area of t2 and t3 terraces (black square). In Fig. 3(h), FFT is
shown for an overlapping area of all the three terraces (green square), while
Fig. 3(k) shows it for t1 and t2 (gray square). The whole image also shows FFT
spots [Fig. 3(j)]. Thus, the Ga monolayer is quasiperiodic over t1, t2, and t3
terraces, whose combined area is 0.0525 $\mu$m2. The largest length scale over
which quasiperiodicity is observed here is $\sim$350 nm, which is the diagonal
of the image in Fig. 3(i). The signature of long range quasiperiodicity of the
Ga monolayer is thus obtained over a range larger than the previous reports in
the literature for other elements [15, 38, 16, 17, 37, 34].

Figure 4: Zoomed STM images of the $\tau$-GaWF motif with a dark center
observed for the Ga (a) monolayer and (b) bilayer. (c) A $\tau$-GaWF motif
with a bright center. The color scale representing the height in nm is shown
on the left side of panels a-c and f, zero corresponds to the bearing height.
(d) The atomic structure of the surface plane of 5-fold $i$-Al-Pd-Mn derived
from the 5/3 approximant (5.32 nm$\times$ 6.25 nm), the thick blue (black)
lines show the $\tau$-P1 (P1) tiling. The white, magenta, and red colored
circles represent the Al, Pd, and Mn atoms, respectively. (e) Same as panel d
with DFT based 30-Ga atom (green circles) model representing the $\tau$-GaWF
motif on the $i$-Al-Pd-Mn surface. (f) A model for the ring motif, the white
dots show all possible adsorption sites on the Pd atoms at the center of the
5-Al cluster. The blue arrow shows a vertex of the outer pentagon that is not
a favorable site. The black lines show the P1 tiling that is shifted compared
to panel d. (g) The green circles show the adsorbed Ga atoms of the model
connected by green dashed lines overlaid on the zoomed ring motif from STM.
Both panels f and g have same length scale.

###  III.4  DFT based models for the $\tau$-GaWF and ring motifs

The STM images in Figs. 2-3 show that the $\tau$-GaWF motif with a dark center
is most frequently observed, whose zoomed views are shown in Figs. 4(a,b).
Some of these motifs are also observed with a bright center [Figs. 4(c)]. To
show that these motifs are $\tau$ inflated compared to the GaWF motifs, we
determine the edge length of the pentagon shown by yellow dashed lines in
Figs. 4(a-c). Its average value is 1.26$\pm$0.1 nm, the corresponding
distribution is shown in Fig. S6 of SM [51]. The experimental ratio of the
side length of $\tau$-GaWF and the GaWF motifs is $\frac{1.26}{0.79}$= 1.59,
which is close to $\tau$ (= 1.618). Thus, STM shows that the $\tau$-GaWF motif
is indeed $\tau$ inflated with respect to the GaWF motif.

The size of this motif can be approximately estimated by the diameter of a
circle circumscribing it drawn such that the $\tau$-GaWF motif is contained in
it. The distribution of the diameter, considering several such motifs, is
plotted as a histogram in Fig. S7(a) [51]. The average diameter of the
$\tau$-GaWF motif turns out to be 3.2$\pm$0.2 nm. This is approximately $\tau$
scaled with respect to the diameter of the GaWF (2$\pm$0.1 nm), thus
justifying the name “$\tau$-GaWF”.

Fig. 4(d) shows the atomic structure of the surface plane of the 5-fold
$i$-Al-Pd-Mn derived from the 5/3 approximant. To model larger sized motifs by
DFT, the $\tau$-P1 tiling is more appropriate (blue lines), where the size of
the tiles is $\tau$ times larger with the edge length $a_{1}$= 1.255 nm
(=$\tau$$\times$$a_{0}$, $a_{0}$= 0.776 nm) compared to the P1 tiling (black
lines). Here, all the surface Mn atoms are at the vertices of the $\tau$-P1
tiling, but not all vertices of the $\tau$-P1 tiling are occupied by Mn atoms.
In the central P1 pentagon, all 5 vertices are occupied by Mn atoms. At the
center of this $\tau$-P1 pentagon, a dip can be observed. In previous studies,
such a surface dip was also characterized also by a charge density minimum
(see e.g., Fig. 3 in Ref. 63).

If around the Mn atoms of the $\tau$-P1 pentagon the 6-Ga clusters grow, a new
Ga superstructure can be observed that would represent $\tau$-GaWF, as shown
in Fig. 4(e). This model is supported by the excellent agreement of $a_{1}$
estimated from the side length of the $\tau$-GaWF motifs from STM
(1.26$\pm$0.1 nm) with the theoretical value (1.255 nm). Also, this model
accurately reflects the distinct petals in the $\tau$-GaWF motif observed from
STM. On $i$-Al-Pd-Mn surfaces, the dips could be filled by Ga adatoms, and the
dark center of the $\tau$-GaWF could disappear. This could explain the bright
centered $\tau$-GaWF motif in Figs. 2(c). On the $i$-Al-Pd-Mn surface,
$\tau$-P1 pentagons with Mn atoms at all 5 vertices exist in varying
orientations. This explains why $\tau$-GaWF clusters could be rotated relative
to one another, as seen in Figs. 4(a,b) and Fig. 2(c).

Figure 5: LEED patterns at a beam energy ($E_{p}$) of 81 eV for (a) 0, (b)
0.7, and (c) 1.2 ML Ga/i-Al-Pd-Mn in an inverted gray scale, the coverage is
indicated at the bottom right corner of each panel.

A zoomed view of the ring motif (red circles in Figs. 2\- 3), which comprises
of a nearly circular arrangement of outer pentagons, is shown in Fig. 4(g).
The center of the ring motif is bright and is surrounded by a relatively
darker region. An estimate of the average diameter of this motif is
5.2$\pm$0.4 nm, indicating that it is approximately $\tau$ inflated with
respect to the $\tau$-GaWF motif (3.2 nm). The distribution of the diameters
is shown in Fig. S7 [51].

In Fig. 4(f), we present a DFT based model of the ring motif that consists of
an outer ring of pentagons formed by Ga atoms that occupy the favorable Pd
sites at the middle of the 5-Al cluster that occur at the vertices of the
pentagons of the P1 tiling shown by black lines. All possible favorable sites
for Ga adsorption are shown by white dots in Fig. 4(f), while the green
circles in Fig. S3(c) of SM represent a complete ring motif. The bright center
of the ring motif also arises from a Ga adatom adsorbed at the center Pd atom
surrounded by the 5-Al cluster. This is the common vertex of three inner P1
pentagons, however two vertices of each of these pentagons are not
energetically favorable. To be noted is that 6-Ga clusters on the Mn atoms at
the center of the alternate pentagons could form a GaWF, but it is somewhat
energetically less favored compared to only the outer 5-Ga considered in the
model (Table 1). A satisfactory agreement can be noticed in both the atomic
positions and motif size when this model is superimposed on the ring motif
depicted in Fig. 4(g) subsequent to a 120∘ rotation. It is worth noting that
the majority of the vertices comprising the outer ring of 10 pentagons are
energetically favorable (as indicated by white dots). However, there is one
unfavorable vertex, denoted by a blue arrow in Figs. 4(f,g), which lacks both
the Pd atom and the 5-Al cluster. Additionally, a neighboring vertex of the
ring motif in Fig. 4(g) is unoccupied for stochastic reasons.

###  III.5 Low energy electron diffraction study of Ga/i-Al-Pd-Mn

Figure 5(a) displays the LEED pattern of $i$-Al-Pd-Mn that has different sets
of sharp 5-fold spots in agreement with the literature [71, 39, 16]. The most
intense inner and outer sets of spots are indicated by blue and black arrows,
respectively. These spots with similar relative intensities between the 5-fold
sets are clearly visible for 1.2 ML coverage in Fig. 5(c)], where the surface
is uniformly covered by Ga, as shown by STM in Fig. 3, also see Fig. S8 for
other $E_{p}$ values. The spots are also visible for the submonolayer coverage
of 0.7 ML [Figs. 5(b)]. The intensity profiles of the inner and outer sets of
spots for 1.2 ML LEED pattern in Fig. S9(a) of SM [51] show that the angular
position (separation between spots being 36∘$\pm$1∘) and the width of the
spots remain nearly unchanged with respect to the substrate. The profiles
along three radial directions in Figs. S9(b-d) show similar $k_{\parallel}$
for both sets of spots, indicating an unchanged quasi-lattice parameter of the
Ga adlayer compared to the substrate. There is no evidence of splitting or
broadening of the spots indicating the absence of surface defects or steps of
multiple heights [72]. It may be noted that rotational epitaxy instead of
quasiperiodicity has been reported for several metals deposited on $i$-Al-Pd-
Mn such as Al [73], Ag [74], Fe [75], and Ni [76]. In these cases, the adlayer
manifests as five crystalline domains that mirror the symmetry of the
substrate, and these domains add extra spots to the LEED pattern. Such extra
spots are absent for the Ga monolayer for the whole $E_{p}$ range, as shown by
a series of patterns from 30 eV $<$$E_{p}$$<$ 180 eV at a step of 2 eV in
video files named “Ga0.7ML” and “Ga1.2ML” for 0.7 and 1.2 ML, respectively of
the SM [51]. This rules out the possibility of the formation of Ga domains
with a particular orientational relationship with the substrate. Thus, our
electron diffraction study shows that the Ga monolayer exhibits quasiperiodic
order within the coherence length scale of the LEED optics, which is 10-20 nm
[77]. There is however a decrease in the intensity of the Ga adlayer spots,
indicating an increase in disorder and roughness. This statement is supported
by the fact that the roughness of the monolayer (0.05 nm), as measured by STM,
is larger compared to the substrate (0.02 nm).

Figure 6: LEED patterns of (a-c) i-Al-Pd-Mn, and (d-f) 1.2 ML Ga/i-Al-Pd-Mn
for three different Ep values are shown at the bottom right corner in eV. The
differences are highlighted by blue and red lines. Experimental I-V curves for
0.7 and 1.2 ML Ga/$i$-Al-Pd-Mn compared with i-Al-Pd-Mn (0 ML) for different
spots (g) (10000), (h) (11000), (i) (000$\overline{1}$0), and (j)
(00$\overline{1}$$\overline{1}$0). These spots are marked in Fig. 5(a) by
green, blue, yellow, and red circles, respectively. The insets compare the 1.2
ML I-V with the substrate after normalizing the most intense peak to the same
height.

Although the LEED patterns of the Ga adlayer resemble the substrate in Fig. 5,
for some $E_{p}$, the relative intensities of the two sets of 5 fold spots are
different [Figs. 6(a-f)]. For example, at $E_{p}$= 55 eV in Fig. 6(d), the
outer ring has nearly equal intensities of the two sets (thus appearing like a
decagon, red lines), while for the substrate in Fig. 6(a) one set is more
intense compared to the other (thus appearing as two 36∘ rotated pentagons,
shown by solid and dashed red lines). This is also observed at 86 eV.
Moreover, at 55 eV, the inner pentagon highlighted by blue lines is 36∘
rotated between the two. At 63 eV, the inner (outer) set of spots of the Ga
layer portrays a blue (solid and dashed red) pentagon, while for the
substrate, the spots are of equal intensity.

To understand these differences, the intensity ($I$) of a LEED spot as a
function of $E_{p}$ – called I-V curve [72, 78] – has been shown in Figs.
6(g-j). The I-V curves of the substrate for different spots, such as (10000),
(11000), (000$\overline{1}$0), and (00$\overline{1}$$\overline{1}$0) are in
agreement with the earlier reports [79, 16]. Although a comparison with the
substrate shows a fair qualitative closeness in the positions of the main
peaks, there are perceptible differences in particular for
(000$\overline{1}$0) and (11000) spots, which are apparent in the insets where
their most intense peak has been normalized to the same height. The difference
in the shapes of the I-V curves could arise from the structural difference
caused by preferred adsorption positions of Ga on $i$-Al-Pd-Mn as shown by DFT
as well as non-structural parameters such as change in the surface potential
[72]. The differences in the LEED patterns between the Ga adlayer and the
substrate in Figs. 6(a-f) can be explained by the difference in the shapes of
the I-V curves [dashed lines in Figs. 6(g-j)].

##  IV Conclusions

In conclusion, our present study utilizing scanning tunneling microscopy
(STM), low energy electron diffraction (LEED), and density functional theory
(DFT) demonstrates that Ga monolayer on $i$-Al-Pd-Mn exhibits quasiperiodic
order at room temperature. The FFTs of the STM images exhibit the
characteristic quasiperiodic spots, while the real space images reveal motifs
such as Ga white flower (GaWF) and $\tau$-inflated GaWF ($\tau$-GaWF), where
$\tau$ is the golden mean. A larger sized ring motif made up of pentagons in a
circular congregation with a bright center is also observed. STM shows that
the quasiperiodic order persists over a length scale of $\sim$350 nm. The
quasiperiodic nature of the Ga monolayer is further supported by the 5-fold
LEED patterns observed across the entire range of the beam energy. Auger
electron spectroscopy provides additional evidence that the Ga adlayer does
not exhibit any surface alloying or chemical reactions with the substrate.

Our DFT calculations indicate that Ga prefers to adsorb on $i$-Al-Pd-Mn at two
specific sites: (i) the vertices of the Penrose P1 tile located at the center
of a cluster consisting of 5 Al atoms that has a Pd atom at the center but
positioned below the surface plane, and (ii) the Mn atom positioned at the
center of the P1 tile. The GaWF motif is represented by a pentagonal outer
ring comprising 5 Ga atoms adsorbed at the center of 5-Al clusters and a
pentagonal inner cluster consisting of 6 Ga atoms (6-Ga) surrounding the
central Mn atom. The 6-Ga clusters arranged on the $\tau$-P1 tiling serve as a
model for the $\tau$-GaWF motif, while the ring motif is represented by Ga
atoms that adsorb at the reactive centers of 5-Al clusters above the marked Pd
sites. Excellent agreement is observed between the side length and the
diameter of the motifs between the experiment and the DFT-based models. Our
research reveals a previously unknown characteristic of gallium, i.e., its
quasicrystallinity. It is intriguing that Ga exhibits quasiperiodic order
despite the substrate temperature being near its melting point; this
characteristic sets it apart from other systems.

##  V Acknowledgments

M.K. is thankful for the support from the Slovak Grant Agency VEGA (No.
2/0144/21) and APVV (No. 19-0369). We are grateful to M. Mihalkovič and K.
Pussi for insightful discussions. We thank D. L. Schlagel and T. A. Lograsso
for providing us with the $i$-Al-Pd-Mn substrate. Special appreciation is
extended to V. K. Singh for his suggestions regarding the STM measurement.

e-mail addresses:
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
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