# Head-Related Transfer Function Interpolation from Spatially Sparse
Measurements Using Autoencoder with Source Position Conditioning

###### Abstract

We propose a method of head-related transfer function (HRTF) interpolation
from sparsely measured HRTFs using an autoencoder with source position
conditioning. The proposed method is drawn from an analogy between an HRTF
interpolation method based on regularized linear regression (RLR) and an
autoencoder. Through this analogy, we found the key feature of the RLR-based
method that HRTFs are decomposed into source-position-dependent and source-
position-independent factors. On the basis of this finding, we design the
encoder and decoder so that their weights and biases are generated from source
positions. Furthermore, we introduce an aggregation module that reduces the
dependence of latent variables on source position for obtaining a source-
position-independent representation of each subject. Numerical experiments
show that the proposed method can work well for unseen subjects and achieve an
interpolation performance with only one-eighth measurements comparable to that
of the RLR-based method.

Index Terms—  head-related transfer functions, deep neural networks,
autoencoder, hypernetworks, spatial audio

Fig. 1:  (a) Overview of the proposed model. (b) Structure of the encoder. The
weight/bias generators output the weights and biases for the linear layers
enclosed by dashed lines. (c) Structure of the decoder. (d) Structure of the
weight/bias generators.

##  1 Introduction

Recently, demand for high-quality spatial audio technology has increased with
the spread of virtual/augmented reality technology. One of the spatial audio
methods is the binaural method, which is based on listening with headphones.
Binaural signals are synthesized by convolving the head-related transfer
function (HRTF), which represents the transfer characteristics from the sound
source to both ears, into a sound source signal. HRTFs depend on the anatomy
of a subject (e.g., pinna shape), which vary greatly from subject to subject.
Hence, using somone else’s HRTF leads to poor localization of a sound image
[1]. Therefore, the ideal binaural synthesis requires the subject’s own HRTFs.
However, measuring HRTFs takes a long time because HRTFs are measured by
sequentially recording impulse responses from hundreds of source positions in
a dense grid on a sphere. This measurement typically takes 60 to 90 minutes
per subject [2]. Long-time measurements are burdensome for subjects, making it
difficult to measure their own HRTFs nonrigorously.

One way to reduce this long measurement time is HRTF interpolation, which
refers to spatial upsampling from sparsely measured HRTFs to dense HRTFs. The
fewer the number of observation positions, i.e., source positions, the shorter
the measurement time. Typical HRTF interpolation methods use bases in the
spatial domain, such as spherical harmonic functions [3, 4, 5], spherical
wavefunctions [6], and spatial principal components [7]. As shown in Sect. 3,
these methods amount to solving regularized linear regression (RLR) problems.
Thus, when the measured HRTFs are spatially very sparse, we are forced to
solve underdetermined problems or to use a small number of bases, which
degrades interpolation accuracy, as we will show later in Sect. 5. Therefore,
these methods require somewhat dense HRTFs for high-precision interpolation
and thus do not contribute to a significant reduction in measurement time.

One approach to overcome this difficulty is a training-based approach. The use
of deep neural networks (DNNs) has shown promising results for HRTF-related
tasks such as real-time binaural synthesis [8, 9], HRTF range extrapolation
[10], and HRTF regression from anthropometric features [11, 12, 13, 14, 15] or
ear images [16]. An HRTF interpolation method using a shallow neural network
(NN) has been proposed [17]. The network estimates an HRTF magnitude at a
target source position from an HRTF magnitude measured at the nearest source
position. These successes led us to use a DNN.

We propose a method of interpolating dense HRTF magnitudes from sparse
measurements based on an autoencoder conditioned on source positions. The
proposed method is drawn from our finding that the RLR-based method can be
interpreted as an autoencoder, which we will show in Sect. 4. This finding
reveals that the key feature of the RLR-based method is to convert measured
HRTFs into a subject representation independent of source positions by using
source-position-dependent basis expansions. Inspired by this feature, we
design our network architecture such that only the encoder and decoder, not
the latent variables, depend on source positions. We also construct a loss
function based on cosine distances between the latent variables of each
subject at different source positions. The proposed model architecture and
loss function promote the latent variables to capture the HRTF individuality
of each subject.

##  2 Problem statement

As in [6], we define HRTF as an acoustic transfer function from the sound
source to both ears. HRTFs for each ear depend on the physical features of the
subject and the position of the sound source. Let
$p_{s,b,\text{ch},l}\in\mathbb{C}$ denote the HRTF for subject
$s\in\\{1,\ldots,S\\}$, source position
$b\in\\{1,\ldots,B\\}\eqqcolon\mathcal{B}$, channel
$\text{ch}\in\\{\text{left},\text{right}\\}$, and frequency bin
$l\in\\{1,\ldots,L\\}$, where $S$ is the number of subjects, $B$ is the number
of source positions, and $L$ is the number of frequency bins. The HRTF
interpolation problem of our interest is formulated as the problem of
estimating spatially dense HRTFs $\\{p_{s,b,\text{ch},l}\\}_{b\in\mathcal{B}}$
from spatially sparse HRTFs
$\\{p_{s,b,\text{ch},l}\\}_{b\in\mathcal{B}^{\prime}}$, where
$\mathcal{B}^{\prime}\subset\mathcal{B}$. Let
$B^{\prime}\coloneqq|\mathcal{B}^{\prime}|$. To distinguish
$b\in\mathcal{B}^{\prime}$ from $b\in\mathcal{B}$, we call the former
measurement position and the latter target position.

To describe sound source positions, we use three-dimensional Cartesian
$\boldsymbol{x}_{b}\coloneqq(x_{b},y_{b},z_{b})$ or spherical coordinates
$(r_{b},\theta_{b},\phi_{b})$ with the origin fixed at the center of the
subject’s head, where $r_{b}$, $\theta_{b}$, and $\phi_{b}$ denote the radius,
azimuth angle, and zenith angle, respectively. The $+x$ and $+z$ directions of
the Cartesian coordinate system are set to the direction of the subject’s view
and the top of the head, respectively. Cartesian coordinates
$\boldsymbol{x}_{b}$ and spherical coordinates $(r_{b},\theta_{b},\phi_{b})$
are related by
$\boldsymbol{x}_{b}=(r_{b}\sin\theta_{b}\cos\phi_{b},r_{b}\sin\theta_{b}\sin\phi_{b},r_{b}\cos\theta_{b})$.

##  3 Regularized-linear-regression-based method

We briefly review the HRTF interpolation method based on spherical
wavefunction expansions proposed in [6]. For simplicity, we drop the
subscripts $s$ and ch from $p_{s,b,\text{ch},l}$; thus, we use $p_{b,l}$ in
the following. On the basis of the reciprocity [18], an HRTF can be regarded
as an acoustic transfer function from a sound source placed at the ear to a
microphone placed at an original source position. Therefore, the HRTFs can be
regarded as the exterior sound field satisfying the homogeneous Helmholtz
equation. Such a sound field is approximately represented by the spherical
wavefunction expansion as

$p_{b,l}\simeq\sum_{n=0}^{N}\sum_{m=-n}^{n}C_{n,m,l}h_{n}^{(1)}(k_{l}r_{b})Y_{n}^{m}(\theta_{b},\phi_{b}),$
(1)

where $h_{n}^{(1)}(\cdot)$ is the $n$th-order spherical Hankel function of the
first kind , $Y_{n}^{m}(\cdot)$ is the spherical harmonic function of order
$n$ and degree $m$, and $(r_{b},\theta_{b},\phi_{b})$ is the original source
position in the spherical coordinates [19]. $k_{l}=2\pi f_{l}/v$ is the $l$th
wave number, where $f_{l}$ and $v$ are the $l$th frequency and speed of sound
in the air, respectively. $\Phi_{n,m,b,l}\coloneqq
h_{n}^{(1)}\left(k_{l}r_{b}\right)Y_{n}^{m}(\theta_{b},\phi_{b})$ and
$C_{n,m,l}$ are the spherical wavefunction and its expansion coefficient,
respectively. $N$ is the truncation order and both sides of (1) are equal as
$N\to\infty$.

This interpolation method consists of two steps: In the first step, the
expansion coefficients $C_{n,m,l}$ are estimated, and in the second step, HRTF
$p_{b,l}$ is obtained by substituting the expansion coefficients $C_{n,m,l}$
and the target position $(r_{b},\theta_{b},\phi_{b})$ into (1). The estimation
of $C_{n,m,l}$ from HRTFs at $\mathcal{B}^{\prime}$ is as follows. We can
write (1) in a matrix–vector product form:

$\boldsymbol{p}_{l}\simeq\boldsymbol{\Phi}_{l}\boldsymbol{c}_{l},$ (2)

where $\boldsymbol{p}_{l}\in\mathbb{C}^{B^{\prime}}$ and
$\boldsymbol{c}_{l}\in\mathbb{C}^{(N+1)^{2}}$ are vectors consisting of
$p_{b,l}$ and $C_{n,m,l}$, respectively, and
$\boldsymbol{\Phi}_{l}\in\mathbb{C}^{B^{\prime}\times(N+1)^{2}}$ is a matrix
consisting of $\Phi_{n,m,b,l}$. For an overdetermined case, i.e.,
$B^{\prime}>(N+1)^{2}$, the estimation problem of $\boldsymbol{c}_{l}$ amounts
to the following RLR problem:

$\mathop{\rm
minimize}\limits_{\boldsymbol{c}_{l}\in\mathbb{C}^{(N+1)^{2}}}\quad\mathcal{L}_{\text{RLR}}=\left\|\boldsymbol{p}_{l}-\boldsymbol{\Phi}_{l}\boldsymbol{c}_{l}\right\|_{2}^{2}+\lambda\left\|\boldsymbol{D}^{1/2}\boldsymbol{c}_{l}\right\|_{2}^{2},$
(3)

where $\lambda>0$ is a regularization parameter and
$\boldsymbol{D}\in\mathbb{R}^{(N+1)^{2}\times(N+1)^{2}}$ is a diagonal matrix
whose diagonal components are $1+n(n+1)$. The minimizer of
$\mathcal{L}_{\text{RLR}}$ is given as

$\hat{\boldsymbol{c}}_{l}=\left(\boldsymbol{\Phi}_{l}^{\mathsf{H}}\boldsymbol{\Phi}_{l}+\lambda\boldsymbol{D}\right)^{-1}\boldsymbol{\Phi}_{l}^{\mathsf{H}}\boldsymbol{p}_{l}.$
(4)

Although only overdetermined cases are discussed in [6], (4) is a reasonable
solution for a balanced or an underdetermined case, i.e.,
$B^{\prime}\leq(N+1)^{2}$.

##  4 Proposed method

###  4.1 Motivation and strategy

The RLR-based method described in Sect. 3 works well when a sufficient amount
of measured HRTFs of each subject are available. However, when measurement
positions are spatially sparse, the RLR-based method can be less accurate. In
fact, as we will show in Sect. 5, the estimated HRTFs from a small number of
measured HRTFs differed from the ground truths in notches and peaks. Although
decreasing the truncation order $N$ stabilizes the estimation performance, it
restricts the expressive power of the spherical wavefunction expansion, which
makes it difficult to represent the fine structure of HRTFs.

To overcome this problem, we focus on our finding that the RLR-based method
can be reinterpreted from a DNN perspective. As described in Sect. 3, for
given $\mathcal{B}^{\prime}$ with $\mathcal{B}$, the RLR-based method consists
of two linear transformations given by (4) and (2). Hence, we can interpret
the RLR-based method as a linear autoencoder, where the encoder is a linear
layer with weights of
$\left(\boldsymbol{\Phi}_{l}^{\mathsf{H}}\boldsymbol{\Phi}_{l}+\lambda\boldsymbol{D}\right)^{-1}\boldsymbol{\Phi}_{l}^{\mathsf{H}}$,
the decoder is a linear layer with weights of $\boldsymbol{\Phi}_{l}$, and the
latent variables are the expansion coefficients. This analogy reveals the key
feature of the RLR-based method. That is, it decomposes HRTFs into source-
position-dependent and source-position-independent factors, i.e., the
spherical wavefunction expansion and expansion coefficients, respectively.

On the basis of this key feature, we propose an HRTF interpolation method
using an autoencoder whose encoder and decoder are conditioned on source
positions (see Fig. 1(a)). To obtain a source-position-independent
representation, we introduce an aggregation module between the encoder and
decoder, which aggregates latent variables of measurement points. The proposed
autoencoder operates the estimation in the HRTF magnitude domain, similarly to
DNN-based methods for HRTF-related tasks [13, 15]. To obtain the head-related
impulse responses (HRIRs) of the estimated HRTF magnitudes, we can use a
method of assigning a phase to the HRTF magnitude. For example, a method based
on minimum phase restoration provides the phase similar to that of the
original HRTFs in sound image localization [20].

###  4.2 Model

Encoder: The encoder converts the logarithmic magnitudes of measured HRTFs
$\left\\{\log_{10}\left|\boldsymbol{p}_{s,b,:,:}\right|\right\\}_{b\in\mathcal{B}^{\prime}}$
into the latent variables
$\\{\boldsymbol{z}_{s,b}\\}_{b\in\mathcal{B}^{\prime}}$, referring to the
measurement positions $\mathcal{B}^{\prime}$ . Here, $|\cdot|$ denotes an
element-wise absolute value. The measured HRTF is processed independently for
each measurement position, which enables us to use the proposed method for
various numbers of measurement positions without retraining.

The encoder consists of linear layers conditioned on $\mathcal{B}^{\prime}$ ,
layer normalization (LN) layers, and rectified linear unit (ReLU) layers. Fig.
1(b) shows the encoder architecture, where Linear, LN, and ReLU denote the
linear layer, LN layer, and ReLU nonlinearity, respectively. The two values in
the parenthesis after Linear denote input and output feature sizes,
respectively, and the value in the parenthesis after LN denotes a feature
size. The dimension of latent variables was set to 64. For the conditioning on
$\mathcal{B}^{\prime}$ , we use the hypernetwork idea [21]. Unlike usual
layers, it generates weights and biases of the linear layers from auxiliary
information by a DNN, which we call a weight/bias generator. Fig. 1(d) shows
the architecture of the weight/bias generator, where $d^{\textsf{(in)}}$ and
$d^{\textsf{(out)}}$ denote the input and output feature sizes of the
corresponding linear layer, respectively. As an auxiliary information, we can
use measurement positions in Cartesian coordinates.

Aggregation Module: The aggregation module yields a source-position-
independent representation $\bar{\boldsymbol{z}}_{s}$, which we call a
prototype. First, this module normalizes $\boldsymbol{z}_{s,b}$ so that its
$\ell_{2}$ norm is one:

$\boldsymbol{z}_{s,b}\leftarrow\boldsymbol{z}_{s,b}/\|\boldsymbol{z}_{s,b}\|_{2}.$
(5)

Second, to reduce the dependence of $\boldsymbol{z}_{s,b}$ on the source
position, the module averages $\boldsymbol{z}_{s,b}$ as

$\bar{\boldsymbol{z}}_{s}=\frac{1}{B^{\prime}}\sum_{b\in\mathcal{B}^{\prime}}\boldsymbol{z}_{s,b}.$
(6)

Note that the averaging operation is inspired by a few-shot learning method
presented in [22]. Finally, the prototype is normalized as

$\bar{\boldsymbol{z}}_{s}\leftarrow\bar{\boldsymbol{z}}_{s}/\|\bar{\boldsymbol{z}}_{s}\|_{2}.$
(7)

Since the prototype no longer depends on $b$, we can use it as a
representation of a subject.

Decoder: The decoder converts the prototype into a logarithmic magnitude of
the HRTFs $\log_{10}\left|\boldsymbol{\hat{p}}_{s,b,:,:}\right|$ at each
target position $b\in\mathcal{B}$. Fig. 1(c) shows the architecture of the
decoder. Similarly to the encoder, it consists of two linear layers whose
weights and biases are generated from the weight/bias generators, LN layers,
and ReLU nonlinearities. The inputs of the weight/bias generators are target
positions in Cartesian coordinates.

###  4.3 Loss function

To train the proposed model, we use a loss function $\mathcal{L}$ defined as

$\mathcal{L}=\text{LSD}+\alpha~{}\text{CosDist},$ (8)

where $\alpha\geq 0$ is a hyperparameter that controls the importance of the
second term. The first term on the right-hand side of (8) is a reconstruction
error of HRTFs. The second term is expected to make the latent variable
$\boldsymbol{z}_{s,b}$ less dependent on the source positions
$\boldsymbol{x}_{b}$ and instead better represent the characteristics of each
subject $s$. The first term is the log-spectral distortion (LSD) of HRTFs and
is defined as

$\text{LSD}\coloneqq\frac{1}{2SB}\sum_{s,b,\text{ch}}\sqrt{\frac{1}{L}\sum_{l}\left(20\log_{10}\frac{\left|\hat{p}_{s,b,\text{ch},l}\right|}{\left|p_{s,b,\text{ch},l}\right|}\right)^{2}},$
(9)

where $\hat{p}_{s,b,\text{ch},l}$ and $p_{s,b,\text{ch},l}$ denote an
estimated and a true element of HRTFs, respectively. The second term on the
right-hand side of (8) is a measure of nonuniformity of the latent variables
$\boldsymbol{z}_{s,b}$ for each subject $s$ and is given as

$\text{CosDist}\coloneqq\sqrt{\frac{1}{SB^{\prime}}\sum_{s,b}\left(1-\frac{\boldsymbol{z}_{s,b}^{\mathsf{T}}\bar{\boldsymbol{z}}_{s}}{\|\boldsymbol{z}_{s,b}\|_{2}\|\bar{\boldsymbol{z}}_{s}\|_{2}}\right)^{2}}.$
(10)

##  5 Experimental evaluation

###  5.1 Experiment condition

####  5.1.1 Preparation of training, validation, and test data

We conducted numerical experiments of HRTF interpolation to quantitatively and
qualitatively evaluate the effectiveness of the proposed method. To generate
training, validation, and test data, we used the HUTUBS dataset [23, 24]. It
includes HRIRs of 94 subjects, excluding duplicates. The HRIRs of each subject
were measured at 440 measurement positions on a sphere of radius
$r=$1.47\text{\,}\mathrm{m}$$. We used 77, 10, and 7 subjects to generate
training, validation, and test data, respectively. The HRTFs were obtained
from the HRIRs as follows. First, the HRIRs were resampled at
$32\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ and the filter length was set to
$256$ by zero padding. Second, FFT at $256$ points was performed, and after
taking the complex conjugate, only the positive frequency bins were extracted.
Finally, HRTFs at $128$ frequency bins
$$125\text{\,}\mathrm{H}\mathrm{z}$,$250\text{\,}\mathrm{H}\mathrm{z}$,\ldots,$16\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$
were obtained.

####  5.1.2 Compared methods

We conducted a preliminary experiment and compared an HRTF interpolation
method based on a shallow NN [17] with the RLR-based method. We found that the
RLR-based method greatly outperformed the shallow-NN-based method in LSD, and
we chose the RLR-based method for comparison. The expansion coefficients were
obtained up to the order of
$\min\\{\left\lceil\mathrm{e}k_{l}R/2\right\rceil,N\\}$ for each of the cases
where $B^{\prime}=9,16,\ldots,196$ for the subjects in the test data. Here,
$\mathrm{e}$ is Napier’s constant and $R=$0.45\text{\,}\mathrm{m}$$. The
$B^{\prime}$ measurement positions were obtained by sampling the nearest
neighbor points of the points contained in the spherical $t$-design [25] with
$t=\sqrt{B^{\prime}}-1$ from the total 440 measurement positions. The HRTFs at
440 target positions were obtained from the estimated expansion coefficients.
As an evaluation metric, LSD in (9) was calculated. For the regularization
parameter $\lambda$ and the maximum truncation order $N$, the four settings
shown in Table 1 were used.

Table 1: Setting of the RLR-based method [6] used in the experiment. Two regularization parameters and two maximum truncation orders were used. U and B in the labels indicate that the system was underdetermined and balanced, respectively. Label | $\lambda$ | $N$
---|---|---
RLR, RP6, U | $10^{-6}$ | $19$
RLR, RP6, B | $10^{-6}$ | $\sqrt{B^{\prime}}-1$
RLR, RP7, U | $10^{-7}$ | $19$
RLR, RP7, B | $10^{-7}$ | $\sqrt{B^{\prime}}-1$

For the proposed method, the logarithmic magnitudes of measured HRTFs were
standardized to have zero mean and a unit variance before fed into the
encoder. The reverse operation of the above standardization was applied to the
decoder outputs. The mean and variance were determined using the training
data.

The proposed model was trained with an Adam optimizer for 1000 epochs. The
learning rate was set at $10^{-3}$. The LSD for the validation data was
calculated for each epoch, and the model with the lowest validation LSD was
finally selected. During training and validation, $B^{\prime}$ and $B$ were
set to 440. For the test data, we varied $B^{\prime}$ from $9$ to $196$. The
hyperparameter $\alpha$ was set to one.

###  5.2 Results and discussion

Fig. 2:  LSDs of the proposed and RLR-based methods at various numbers of
measurement positions. Proposed denotes the proposed method and the remaining
labels are summarized in Table 1.  Fig. 3:  Example of magnitude frequency
responses of ground truth and estimated HRTFs with $B^{\prime}=9$. Ground
Truth and Proposed denote the ground truth and the proposed method,
respectively, and RLR, RP6, B is defined in Table 1.

Fig. 2 shows the LSD of HRTFs at 440 target positions obtained by the proposed
and RLR-based methods. We can see that the LSDs of the RLR-based methods
increased as the number of measurement positions $B^{\prime}$ decreased, i.e.,
the accuracy decreased, regardless of whether the system was underdetermined
or balanced. On the other hand, the LSDs of the proposed method were almost
similar regardless of $B^{\prime}$. Furthermore, the proposed method can
interpolate HRTFs at only $B^{\prime}=25$ with almost the same accuracy as the
RLR-based method at $B^{\prime}=196$. Note that the proposed method does not
focus on the situation where HRTFs at a large number of measurement positions
are known, although it is expected that the accuracy of the RLR-based method
will exceed that of the proposed method when the number of measurement
positions $B^{\prime}$ increases to more than $196$.

Fig. 3 shows magnitude frequency responses of HRTFs estimated by the RLR-based
method and the proposed method for $B^{\prime}=9$. The estimated HRTF by the
RLR-based method had a greatly different spectral shape from the ground truth.
For example, although the ground truth had the peak in the frequency band
between 10 to $12\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, the estimate of the
RLR-based method did not have such a peak. In contrast, it can be confirmed
that the HRTF estimated by the proposed method appropriately captured the
peaks and notches of the ground truth, even though the HRTF is from a subject
not included in the training data. This indicates that the proposed method can
interpolate HRTFs from a small number of measurement positions more accurately
than the RLR-based method.

##  6 Conclusion

We proposed a method of HRTF interpolation from sparsely measured HRTFs using
an autoencoder with source position conditioning. The proposed architecture
was designed from an analogy between the RLR-based method and an autoencoder
architecture. Inspired by this analogy, we made the layers of the encoder and
decoder dependent on source positions. To further reduce this dependence, we
introduced a loss function based on cosine distances between the latent
variables at different measurement positions. Numerical experiments showed
that the proposed method can work well for subjects not included in the
training data and achieve an interpolation performance with only one-eighth
measurements comparable to that of the RLR-based method.

##  7 Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP22H03608 and JST FOREST
Program (Grant Number JPMJFR216M, Japan).

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# Observation of an Abrupt 3D-2D Morphological Transition in Thin Al Layers
Grown by MBE on InGaAs surface

A. Elbaroudy  corresponding author<EMAIL_ADDRESS>Department of
Electrical and Computer Engineering, University of Waterloo, Waterloo N2L 3G1,
Canada  Waterloo Institute for Nanotechnology, University of Waterloo,
Waterloo N2L 3G1, Canada  B. Khromets  Department of Electrical and Computer
Engineering, University of Waterloo, Waterloo N2L 3G1, Canada  Institute for
Quantum Computing, University of Waterloo, Waterloo N2L 3G1, Canada
Department of Chemistry, University of Waterloo, Waterloo N2L 3G1, Canada  F.
Sfigakis  Institute for Quantum Computing, University of Waterloo, Waterloo
N2L 3G1, Canada  Northern Quantum Lights inc., Waterloo N2B 1N5, Canada
Department of Chemistry, University of Waterloo, Waterloo N2L 3G1, Canada  E.
Bergeron  Institute for Quantum Computing, University of Waterloo, Waterloo
N2L 3G1, Canada  Department of Chemistry, University of Waterloo, Waterloo N2L
3G1, Canada  Y. Shi  Department of Electrical and Computer Engineering,
University of Waterloo, Waterloo N2L 3G1, Canada  Department of Physics,
University of Waterloo, Waterloo N2L 3G1, Canada  Waterloo Institute for
Nanotechnology, University of Waterloo, Waterloo N2L 3G1, Canada  M.C.A. Tam
Department of Electrical and Computer Engineering, University of Waterloo,
Waterloo N2L 3G1, Canada  T. Blaikie  Department of Electrical and Computer
Engineering, University of Waterloo, Waterloo N2L 3G1, Canada  Waterloo
Institute for Nanotechnology, University of Waterloo, Waterloo N2L 3G1, Canada
George Nichols  Institute for Quantum Computing, University of Waterloo,
Waterloo N2L 3G1, Canada  Department of Physics, University of Waterloo,
Waterloo N2L 3G1, Canada  J. Baugh  Institute for Quantum Computing,
University of Waterloo, Waterloo N2L 3G1, Canada  Department of Physics,
University of Waterloo, Waterloo N2L 3G1, Canada  Northern Quantum Lights
inc., Waterloo N2B 1N5, Canada  Department of Chemistry, University of
Waterloo, Waterloo N2L 3G1, Canada  Waterloo Institute for Nanotechnology,
University of Waterloo, Waterloo N2L 3G1, Canada  Z. R. Wasilewski
<EMAIL_ADDRESS>Institute for Quantum Computing, University of
Waterloo, Waterloo N2L 3G1, Canada  Department of Physics, University of
Waterloo, Waterloo N2L 3G1, Canada  Northern Quantum Lights inc., Waterloo N2B
1N5, Canada  Department of Electrical and Computer Engineering, University of
Waterloo, Waterloo N2L 3G1, Canada  Waterloo Institute for Nanotechnology,
University of Waterloo, Waterloo N2L 3G1, Canada

###### Abstract

Among superconductor/semiconductor hybrid structures, in-situ aluminum (Al)
grown on InGaAs/InAs is widely pursued for the experimental realization of
Majorana Zero Mode quasiparticles. This is due to the high carrier mobility,
low effective mass, and large Landé g-factor of InAs, coupled with the
relatively high value of the in-plane critical magnetic field in thin Al
films. However, growing a thin, continuous Al layer using the Molecular Beam
Epitaxy (MBE) is challenging due to aluminum’s high surface mobility and
tendency for 3D nucleation on semiconductor surfaces. A study of epitaxial Al
thin film growth on In0.75Ga0.25As with MBE is presented, focusing on the
effects of the Al growth rate and substrate temperature on the nucleation of
Al layers. We find that for low deposition rates, 0.1 Å/s and 0.5 Å/s, the
growth continues in 3D mode during the deposition of the nominal 100 Å of Al,
resulting in isolated Al islands. However, for growth rates of 1.5 Å/s and
above, the 3D growth mode quickly transitions into island coalescence, leading
to a uniform 2D Al layer. Moreover, this transition is very abrupt, happening
over an Al flux increase of less than 1%. We discuss the growth mechanisms
explaining these observations. The results give new insights into the kinetics
of Al deposition and show that with sufficiently high Al flux, a 2D growth on
substrates at close to room temperature can be achieved already within the
first few Al monolayers. This eliminates the need for complex cryogenic
substrate cooling and paves the way for the development of high-quality
superconductor-semiconductor interfaces in standard MBE systems.

††preprint: AIP/123-QED

##  I Introduction

The combination of superconductor (SP) and semiconductor (SE) heterostructures
has garnered significant attention in the field of condensed matter physics.
In particular, these structures are crucial to the progression of topological
quantum computation through braiding-based techniquesSarma, Freedman, and
Nayak (2015); Aasen  _et al._ (2016) and transmon qubitsWang  _et al._ (2022).
The pursuit of topological qubits within SP/SE material systems relies on two
primary methods: one-dimensional self-assembled nanowiresKrogstrup  _et al._
(2015) and gated two-dimensional electron gas (2DEG) systemsShabani  _et al._
(2016). For the latter approach, Al deposited in situ on a quantum well
surface made of InAs/InGaAs is the most promising platform. Al thin films
($\sim$10 nm thick) are especially well suited for the superconductive layer
due to their high values of superconducting coherence length, which is the
measure of the size of a Cooper pair (distance between the two electrons), and
relatively high value of the in-plane critical magnetic fieldShabani  _et al._
(2016); Tedrow and Meservey (1982). InAs is a preferred material choice for
the quantum well due to its large Landé g-factor, strong spin-orbit
interaction, and small effective mass, meeting the conditions for the
formation of Majorana zero modes (MZMs)Lutchyn  _et al._ (2018); Thomas  _et
al._ (2018). Majorana quasiparticles offer a promising platform for robust
practical qubits due to their non-Abelian statistics and potential for
topological protectionLutchyn  _et al._ (2018). Self-assembled nanowires have
been successfully used in experimental studiesMourik  _et al._ (2012);
Kürtössy  _et al._ (2022), yet scaling up the nanowires to large device arrays
for the Majorana quasiparticle braiding or fusion applications would be very
challenging. On the other hand, top-down fabrication techniques for 2DEG
system design are more promising for practical applications and large-scale
productionCheah  _et al._ (2023). Sub-gap states are typical in the materials
with ex-situ Al depositionWang  _et al._ (2015). Specifically, defects due to
impurities at the metal-semiconductor interfaces enhance the density of sub-
gap states, resulting in undesirable dissipative effects. These sub-gap states
reduce the qubit coherence time by randomly absorbing energy and adding noise
to the stored phase of a qubit stateFrolov, Manfra, and Sau (2020). Therefore,
an atomically clean metal-semiconductor interface is essential for the
development of topological qubits. In-situ Al deposition in the MBE system,
following the growth of the InAs quantum well, yields an improved SP/SE
interface of the hybrid heterostructure materials system, resulting in a hard
superconducting gap with no sub-gap statesCheah  _et al._ (2023); Wang  _et
al._ (2015). Aside from purity, essential to viable devices are very thin,
flat, and continuous Al layers. These have been consistently demonstrated only
in systems capable of Al deposition at cryogenic temperaturesAghaee  _et al._
(2023), while broadly accessible MBE growth at temperatures closer to 300 K
typically results in a distinct 3D island morphology. Several fundamental
studies have focused on the in-situ epitaxial deposition of Al on a variety of
semiconductor surfaces. This includes Al on GaAsPilkington and Missous (1999);
Tournet  _et al._ (2016), InGaAsShabani  _et al._ (2016); Pilkington and
Missous (1999), and more recently, 2DEG systems utilizing near-surface InAs
quantum wellsCheah  _et al._ (2023).

In the work done by PilkingtonPilkington and Missous (1999), a consistent
trend was observed for growing 1200 Å thick epitaxial Al on arsenide
semiconductors at room temperature with growth rates of 1 Å/min and 20 Å/min.
Initially, an adlayer formed on the semiconductor surface, followed by 3D
nucleation in three orientations: (1 0 0), (1 1 0), and (1 1 0)R. The islands
that formed on these nucleation centers continued to grow as more Al was
deposited. At layer thicknesses above about 400 Å, the (1 0 0) orientation
began to dominate, transitioning the growth into a partial 2D mode. This
process was assessed by Reflection High-Energy Electron Diffraction (RHEED)
analysis and Atomic Force Microscopy (AFM) imaging. The latter revealed that
the final surface was not that of a continuous flat layer but had a morphology
characterized by hillock-valley structures, regardless of the initial buffer
layer compositions. A more recent studySarney  _et al._ (2018) reported that
when depositing Al at thicknesses of 35.9 Å and 71.8 Å on InAs, at
temperatures below 0 °C and with a growth rate of about 0.09 Å/s, Al tends to
form 3D islands rather than a 2D smooth layer. It was observed that on InAs,
the height of these islands was reduced when grown on thicker AlAs
intermediate layers, suggesting the potential for achieving a flat Al film by
adjusting this layer’s thickness. The study found that Al films grown on both
AlAs and In0.91Ga0.09As did not exhibit laminar growth either, forming 3D
islands instead. Additionally, Ga-rich surfaces showed significantly less
reactivity with Al compared to InAs or AlAs surfaces. Recently, Cheah et
al.Cheah  _et al._ (2023) investigated the growth of 120 Å of Al on a 2DEG
system consisting of an InAs quantum well with a 13.4 nm top barrier of
In0.75Ga0.25As, using an Al growth rate of 1 Å/s and a nominal substrate
temperature of approximately -30 °C. Intermediate layers of 2 and 5 monolayers
of GaAs were used to prevent indium diffusion into Al and to improve chemical
stability during device fabrication. This approach was found to enhance
mobility and the strength of the induced proximity effect without affecting
the electron mobility or superconducting properties of the system.

Despite these different studies, a comprehensive analysis of growth parameters
influencing the growth mode of Al, particularly the effects of the Al growth
rate and substrate temperature during deposition, has not been reported yet.
In the present work, we study the quality of epitaxial Al thin films ($\sim$10
nm) grown by MBE directly on In0.75Ga0.25As with no intermediate layers, at
temperatures above zero °C. We discuss the influence of Al growth rates and
substrate temperature on the Al nucleation path and final layer morphology.
Distinct Al islands were observed at relatively low Al deposition rates. Even
though 3D nucleation is observed for all cases, it transitions quickly to 2D
growth mode for the fastest deposition rates, resulting in near-atomically
flat final Al films, as evidenced by Scanning Electron Microscope (SEM) and
AFM measurements. Moreover, we present measurements of the critical in-plane
magnetic field for the superconductor to normal transition for the grown Al
layers. This work sheds new light on the growth mechanisms of Al thin films,
carrying implications for the advancement of topological superconducting
qubits.

##  II Experimental

###  II.1 Molecular Beam Epitaxy Growth

In this study, we present experimental results based on two sets of samples
grown during two different MBE campaigns, A and B. The nominal structures of
all samples were the same and are shown in Figure 1. Also, all the growth
steps were the same up to and including the top In0.75Ga0.25As layer. The key
difference between the samples was in the deposition rate of Al, as detailed
in Table 1. Also, for the samples grown during Campaign A, the main
manipulator shutter was not available. As detailed later, this resulted in a
higher initial substrate temperature during Al deposition.

Figure 1: Material structures of all samples grown in this study.

All the samples were grown using a Veeco GEN10 MBE system. The structures were
grown on quarters of 3” semi-insulating, Fe-doped, (001) InP epi-ready
substrates. The wafers were In-free mounted, sandwiched between two Mo plates
mounted in a wafer holder designed for 3” substrates, all supplied by Veeco.
Initially, the substrates were outgassed at 200 °C for two hours in the
system’s load lock. This was followed by a transfer to the preparation module
for an additional one hour of outgassing at 400 °C. Subsequently, the
substrates were moved to the growth module to begin the epitaxial growth
process. The native oxide was removed by elevating the substrate temperature
to 535 °C under As4 overpressure with substrate rotation at 20 rpm. The
wafer’s temperature was measured using band edge thermometry (BET)Johnson
(1995). Following this, the substrate temperature was lowered to 470 °C for
the growth of a nominally lattice-matched 100-nm-thick In0.53Ga0.47As buffer
layer, followed by a 6 nm thick layer of In0.75Ga0.25As. The latter
constitutes a typical top barrier for a shallow InAs QW used to fabricate
structures supporting MZMs (Figure 1). The RHEED diffraction patterns along
the (110) and (-110) substrate azimuths were continuously monitored on a
rotating wafer using a kSA 400 RHEED acquisition system triggered at
appropriate azimuths. After the growth of the In0.75Ga0.25As layer, the
resulting films exhibited a consistent and well-ordered (2x4) reconstruction,
indicative of high-quality 2D growth and precise surface stoichiometry
control. Upon completing the growth of the top In0.75Ga0.25As layer, the
substrate’s temperature was lowered to 400 °C, at which point the As flux was
interrupted by closing the As cracker valve and the As cell shutter. For
series B, to reduce surface contamination and prevent substrate heating by the
idling effusion cells, the main manipulator shutter located beneath the
substrate was closed. The wafer’s temperature was maintained at 400 °C for a
duration of at least four hours. This step was introduced to eliminate
condensation of As on the substrate at lower temperatures and to ensure that
the subsequent growth of epitaxial Al was taking place on a possibly clean
(2×4) In0.75Ga0.25As surface. Once the background pressure in the growth
module decreased to below 1.2E-10 Torr, the manipulator heater power was
reduced to zero, and the wafers were allowed to cool down overnight in the
growth module prior to the Al deposition. Helped by a liquid nitrogen-cooled
shroud environment, the substrates reached temperatures below 13 °C by the
following morning for the series B samples. Because of the lack of protection
from the main shutter, the lowest temperature achieved by samples from series
A was between 20 and 30 °C. For the deposition of the Al layer, both Al cells
installed in the reactor were used simultaneously, and Al deposition was done
on stationary substrates. For each growth rate used, the Al cells were ramped
up from their idle temperature of 750 °C to the target temperatures shortly
before the Al deposition to minimize the thermal radiation load in the
reactor. For samples in series B, the main shutter was opened throughout the
duration of Al deposition only. After deposition, the temperatures of the Al
cells were quickly ramped down back to idling.

Table 1: List of all samples reported in this study. Sample ID | Al growth rate (Å/s) | Al thickness (Å)
---|---|---
A1 | 0.1 | -
A2 | 0.5 | -
A3 | 2.0 |  $\sim$ 100
B1 | 0.1 | -
B2 | 0.5 | -
B3 | 1.5 |  $\sim$ 85
B4 | 2.0 |  $\sim$ 100
B5 | 3.0 |  $\sim$ 100

###  II.2 Substrate Temperature Monitoring

The temperature of the substrates was monitored throughout the entire growth
process using a BET technique utilizing two array spectrometers, NIR for the
900-1700 nm spectral range and Vis-IR for the 400-1050 nm spectral range. This
allowed us to track InP absorption edge location, and thus the substrate
temperature, through the entire range of temperatures accessible in our
system. The NIR spectrometer was used for oxide desorption and InGaAs layers
deposition, while the Vis-IR spectrometer was used to measure the substrate’s
temperature in the 0-150 °C range. Since the manipulator heater was turned off
during the Al deposition, the radiation needed to track the substrate
absorption edge was supplied by a halogen lamp located at the top of the
manipulator. The light was guided to the back of the substrate by a quartz
light pipe installed in the manipulator. Dedicated tests performed prior to
the experiments revealed that the halogen power level used did not cause
significant local heating of the samples prepared for Al deposition. The raw
transmission spectra were subsequently used to determine the band edge
position. During the growth process, the radiation emitted from the Al cells
led to a gradual increase in the substrate temperature. Since the thermal
radiation power emitted from the Al cell is expected to be proportional to the
fourth power of the Al melt temperature, while the Al flux (thus the growth
rate) increases exponentially with increasing the melt temperature, the total
thermal radiation dose absorbed by the wafer during Al deposition was expected
to be lowest for the fastest Al deposition rates. Figure 2a-c compares the
evolution of substrate temperature over time during Al deposition for the
three samples of series A. The manipulator thermocouple readings, Tthr, are
shown in addition to the actual wafer temperature, TBET, measured with BET.
The thermocouple junction is located in the space between the heater element
and the wafer and is not in contact with the wafer.

Figure 2: Substrate temperatures during the growth of Al layers for series A
samples: (a) A1, (b) A2, and (c) A3. The shaded regions within the temperature
data indicate the periods of opening the Al cells’ shutters, marking the
duration of Al growth. Figure (d) shows the relationship between the radiative
energy output from the Al cells as a function of the progression of TBET for
all samples in series A.

As expected, the layer deposited at the slowest growth rate – sample A1 –
received the highest thermal dose and heated up by more than 90 °C during the
Al deposition cycle. Remarkably, the thermocouple readings, Tthr, increased by
less than 10 °C during the same period. This emphasizes the importance of
monitoring the wafer temperature directly. The substrate temperature increases
recorded during the deposition process for all reported here samples are
listed in Table 2. Figure 2d shows the evolution of TBET for all three samples
as a function of the cumulative energy output D(t) from the unit surface area
of the molten aluminum in the Al cells. Here, we used the approximation:

$\displaystyle D(t)=\frac{1}{2}\sigma\left(T_{1}^{4}+T_{1}^{4}\right)$ (1)

where $\sigma=5.67\times 10^{-8}\text{ W}\cdot\text{m}^{-2}\cdot\text{K}^{-4}$
is the Stefan-Boltzmann constant for black body radiation, and we assume the
emissivity of the cell to be close to one. The T1 and T2 are the absolute Al
melt temperatures for the two aluminum cells used. Without heat losses from
the wafers, we would expect all the points in Figure 2d to fall on the same
curve and wafer temperature to remain constant after Al deposition. The
approximately exponential dependencies seen for the wafer heating and cooling
phases in Figure 2a-c are the direct measure of the heat loss channels
present. Although we do expect some radiative heat loss by the wafers, most
likely, the main heat loss is through the wafer’s direct contact with the
holder and through it to the rest of the fairly massive manipulator, whose
temperature is not altered much during the Al deposition cycles. The relative
displacement of the traces in Figure 2d may also be caused by the deviation of
the effective cell orifice radiation power from the assumed T4. Both Al cells
used here are dual filament cells equipped with 40 cc conical PBN crucibles.
Only bottom filaments are used, though, and the crucible has a strong
temperature gradient from the melt to the top of the cell. Also, this gradient
is expected to change significantly between the lowest and the highest Al melt
temperatures used in the reported here experiments. None of these effects are
reflected in the equation used.

Table 2: Substrate Temperature Measurements for All Samples During Al Growth: This table presents ’T${}_{\text{BET}}$ Start’, which indicates the substrate temperature at the beginning of Al growth. ’$\Delta T_{\text{BET}}$’ and ’$\Delta T_{\text{thr}}$’ represent the differences in substrate temperature between the start and end of the growth period, measured by BET and thermocouple, respectively. Sample ID | T${}_{\text{BET}}$ Start (°C) |  $\Delta\text{T}_{\text{BET}}$ (°C) |  $\Delta\text{T}_{\text{thr}}$ (°C)
---|---|---|---
A1 | 25.3 | 90.6 | 10.0
A2 | 26.3 | 51.2 | 0.94
A3 | 29.7 | 23.6 | 0.4
B1 | 8.0 | 103.3 | 13.6
B2 | 13.0 | 88.1 | 3.3
B3 | 12.1 | 49.6 | 0.6
B4 | 12.8 | 39.4 | 0.8
B5 | 11.3 | 38.6 | 0.6

Still, additional complications exist. The wafer heating is affected not only
by the radiation absorbed by the wafer itself but also by the heating rate of
the holders. The latter will depend on the materials deposited on them in the
past. We see evidence that such quarter wafers are not uniformly heated by
radiation from Al cells. Such temperature gradients most likely result from
the variation in wafer-holder contact along the wafer periphery. One other
element that can affect the heating rate of the substrate exposed to a fixed
thermal radiation load is the light absorption in the small bandgap InGaAs
epitaxial layers, and also in the interfacial InAs layer. The latter is
typically formed on InP substrates during oxide desorption under As
overpressure and is due to the displacement of phosphorus by arsenic. From the
fitting of a dynamical scattering theory to XRD measurements, we find the
thicknesses of such InAs layers varying between about 0.5 to 1.5 ML. Thus, any
variation in the InGaAs compositions or InAs interfacial layer thickness may
also influence the observed heating rate and the maximum temperature achieved
during deposition. Notably, even though the slowest deposition rate leads to
the largest total absorbed radiative power and results in the largest
temperature increase by the wafer, the rate of temperature rise increases with
increasing Al growth rate. For example, over the first 150 seconds, the
temperature of sample A2 (0.5 Å/s) rose by over 40 °C, while an increase of
only about 25 °C was observed for sample A1 (0.1 Å/s). We will come back to
this observation in the discussion section.

##  III RESULTS AND DISCUSSION

###  III.1 Scanning Electron Microscope (SEM)

The surface morphology of all the series B samples was tested using a JEOL
JSM-7200F SEM. All scans were conducted at the same magnification for
comparison purposes (Figure 3). The sample grown at 0.1 Å/s shows small 3D
islands of Al with small separations, while the sample grown at 0.5 Å/s shows
a larger radius of 3D islands of Al with larger separations, as depicted in
Figure 3(a) and (b), respectively. With increasing the growth rate to 2 Å/s, a
continuous Al layer with occasional very small voids was observed, as seen in
Figure 3(c). Finally, at the highest tested growth rate of 3 Å/s, the
smoothest surface was achieved, characterized by an Al layer free of voids, as
seen in Figure 3(d).

Figure 3: High-resolution Scanning Electron Microscope images illustrating the
variations in surface morphology among series B samples (a) B1, (b) B2, (c)
B3, and (d) B4.

For the B3 sample, the Al cells’ temperatures ramped down during the
deposition, leading to a change in Al growth rate from 2 to 1.4 Å/s over 50
seconds and resulting in a deposition of approximately 8.5 nm thick Al film.
As in the other four series B samples, this layer was deposited on a
stationary substrate, which led to a variation of Al flux across the wafer.
This deposition resulted in a wafer covered only partially by a continuous
smooth Al layer. The remaining part showed rough morphology with distinct Al
islands. As shown in Figure 4, the 3D-2D transition region is abrupt,
happening across a fraction of a millimetre. Across this region, round 3D Al
clusters transition into elongated clusters, which then merge to form a
continuous Al layer. Remarkably, the change of Al flux across this transition
region is less than 1%. We do expect some temperature gradient across the
wafer. Still, the variation in the local wafer temperature across this region
is expected to be only a fraction of a degree throughout the entire deposition
process. This abruptness of the observed here 3D-2D morphological transition
for thin Al layers makes it akin to other phase transitions, such as melting
points of solids.

Figure 4: (a) B3 Sample Surface: Shows the rough and smooth areas,
highlighting the light spot analyzed in (b). Figure (b) Nomarski Microscopy
image illustrates the transition from continuous to discontinuous Al layers.

###  III.2 Atomic Force Microscopy (AFM)

To gain further insight into the series B samples’ morphologies, we performed
atomic force microscopy measurements. These were done using a Bruker Dimension
FastScan AFM system. Figure 5 illustrates the shift in nucleation behavior of
the Al thin layers, marking a transition from 3D Al islands formed at lower
growth rates, as shown in Figure 5(a) and (b), to a 2D laminar Al layer, shown
in Figure 5(c) and (d), with the increase in deposition rate from 0.1 Å/s to 3
Å/s. To better illustrate the quality of the continuous B4 and B4 layers, the
cross sections of these layers are also shown in panels (c) and (d). This was
derived from the surface morphology scan along the lines between the arrows.

Figure 5: Atomic Force Microscopy illustrating the variations in surface
morphology among samples (a) B1, (b) B2, (c) B4, and (d) B5.

###  III.3 Low temperature resistivity measurements with in-plane magnetic
field

Using standard AC lock-in techniques, the transition from superconducting to
normal resistance in our epitaxial Al films in series B samples was
characterized as a function of the magnetic field. These experiments were
conducted in a 3He/ 4He dilution refrigerator (model TLM from Oxford
Instruments) equipped with a 20 Tesla magnet and achieving a base temperature
of 12 mK. Four-terminal measurements (I+, I-, V+, V-) on cleaved small samples
(3 mm x 1 mm) allowed for the precise determination of the epitaxial Al
resistance, effectively eliminating the influence of contact resistance, line
resistances, and filters, among other factors. Figure 6 illustrates the
critical in-plane magnetic field measurements of B3, B4, and B5 samples which
showed a superconducting transition. In contrast, B1 and B2 samples failed to
conduct at all, due to their discontinuous layers of Al. Sample B3 has the
highest critical field of 1.1 T, in part because of having thinner, 8.5 nm, Al
layer. Aluminum, a type I superconductor, typically undergoes an abrupt
transition from superconducting to the normal state with an increasing
magnetic field. However, in all tested samples, a second transition for the
critical magnetic field is observed, which we attribute to small variations in
the Al thickness across the Al layer in each sample. If a film exhibits
interwoven patches having two distinct thicknesses interconnected in series
along the possible current paths, two transitions will be evident: the thicker
patches will transform to a normal state at a lower magnetic field compared to
the thinner ones. The resistivity should drop to zero only when there’s a
continuous superconducting path. The initial drop in resistances seen in
Figure 6, most likely, is the result of only the thinner patches becoming
superconducting, while the entire layer achieves superconductivity only at the
still lower magnetic field.

Notably, the out-of-plane critical field for thin Al films is very small
(about 10 mT) and does not depend on the layer thickness. Even though our
samples were intentionally mounted to have a magnetic field along the surface,
even small misalignment would introduce a perpendicular component, effectively
lowering the measured critical field. For example, if the in-plane critical
field of our Al film was 2 T, a misalignment angle as small as 2° would
introduce a perpendicular component of the magnetic field sufficient to reduce
the measured critical field to below 1 T. We note that absolute values of
critical fields alone cannot serve as quality indicators. Importantly, the
critical fields reported here for the continuous layers are sufficiently large
for pursuits of MZMs quasiparticles.

Figure 6: In-plane magnetic field measurements for B3, B4, and B5 samples. It
shows the relationship between critical magnetic fields and normalized 4-point
resistance values at 20 mK.

###  III.4 Discussion

Several comprehensive studies of the deposition of Al on GaAs surfacesTournet
_et al._ (2016); Donner  _et al._ (1989); Landgren, Ludeke, and Serrano
(1982); Massies and Linh (1982) reveal a complex behaviour of Al, strongly
dependent on the starting GaAs surface reconstruction. There is a consensus
that, for As-rich surface reconstruction, up to three monolayers of Al can be
deposited before the transition to 3D islanding is observed. This is likely
due to aluminum’s preference to form covalent bonds with surface arsenic. We
expect similar mechanisms to be active for the case of Al deposition on
In0.75Ga0.25As As-rich surface. Since Al-As bonds are stronger than Ga-As or
In-As, aluminum may engage not only the excess As on the surface but also
displace some of the surface Ga and In atoms from the existing bonds with As.
Since reported here Al depositions were performed on stationary wafers, we do
not have enough experimental data to speculate about the surface evolution
during the growth of initial Al layers on In0.75Ga0.25As. However, the
available RHEED observations are consistent with surface evolution resembling
that reported by TournetTournet  _et al._ (2016) for the Al deposition on GaAs
(2×4) surface with the transition to 3D growth at about 3 Al monolayers. At
that point, it becomes energetically favourable for arriving Al to grow
isolated Al islands rather than attach to the surface. Here, we propose a
simple model that qualitatively explains our key finding, namely 3D-2D
morphological transition at high Al growth rates: Once the growth enters the
stage when the attachment to islands becomes energetically favourable, one
expects the density of small “seed” islands to increase with decreasing Al
surface mobility. So, for a given flux of arriving Al atoms, the Al islands
density should increase with lowering the substrate temperature. On the other
hand, at a fixed temperature, the Al islands density should increase with
increasing Al flux, i.e. increasing the target growth rate. Indeed, a higher
Al arrival rate will increase the population of mobile Al on the surface, thus
promoting Al-Al encounters and leading to the formation of new seed islands.
The higher the seed island density, the more likely their coalescence and
formation of a continuous Al layer, i.e. 3D-2D morphological transition. For
the case of Al deposition in an MBE system, such as described here, we can fix
Al flux, but the substrate temperature rises throughout the deposition
process, and the rate of this temperature rise increases with the increase in
the nominal Al flux. It is reasonable to assume that once the Al seed islands
reach some critical size, their density will remain constant throughout the
remaining layer deposition. Indeed, since the temperature is continuously
increasing and the Al flux is constant, there will be no drive to form new
islands. The now more mobile Al will quickly attach to the existing islands,
whose density was established at lower temperatures when Al diffusivity was
lower.

However, the process is more complex. Indeed, based on the above
considerations alone, one would expect that with a given starting substrate
temperature, there will be a critical minimum Al flux above which very dense
seeding of Al islands would take place, leading in turn to their early
coalescence and subsequent 2D growth. Yet, the observation of larger islands
with smaller surface density for faster deposition rate, shown in Figure 3b,
signals the presence of other mechanisms. Indeed, at a growth rate of 0.1 Å/s,
a high density of small Al islands is observed (Figure 3a), while at a 0.5 Å/s
growth rate (Figure 3b), a smaller density of larger Al islands is seen. This
observation can be explained by the much faster temperature rise rate for the
0.5 Å/s growth rate than the 0.1 Å/s one. Thus, it is plausible that the Al
seed island population was established at a lower temperature for the latter,
resulting in higher final island density despite a considerably lower Al
arrival rate. As it is clear from the above, the path to forming a continuous,
very thin Al layer of uniform thickness is a complex one but depends primarily
on two parameters - the starting substrate temperature and the nominal Al
growth rate. Importantly, with a sufficiently fast Al deposition rate, it is
possible to grow 10 nm thick uniform Al layers showing a high critical
magnetic field necessary for establishing conditions for forming MZM
quasiparticles.

##  IV SUMMARY AND CONCLUSIONS

In this study, we successfully demonstrated the in-situ growth of continuous,
2D 10 nm thick Al films on In0.75Ga0.25As surface, starting at substrate
temperatures above approximately 10 °C. We investigated the impact of varying
Al growth rates on the formation of Al layers and explored how substrate
temperature influences the growth mode of Al. We show a significant variation
in substrate temperature evolution during growth, which strongly depends on
the Al growth rate. A high growth rate of 3 Å/s was shown to produce very
smooth, continuous Al layers with a critical in-plane magnetic field higher
than 0.63 T at 20 mK for 10 nm thick layers. In contrast, layers grown at
growth rates smaller than 1 Å/s showed 3D Al islands and were not conducting
during the low-temperature resistivity measurements. Interestingly, the
transition from 3D to 2D layer morphology is abrupt, taking place over an Al
flux change smaller than 1%. The results show the path for the growth of high-
quality superconducting thin Al layers in a standard configuration MBE system
without the need for cryogenic UHV chambers dedicated to in-situ Al
deposition.

###### Acknowledgements.

We acknowledge support from funding sources and institutions. This research
was made possible partly by the financial aid from the Canada First Research
Excellence Fund, specifically the Transformative Quantum Technologies (TQT)
program, and the Natural Sciences and Engineering Research Council (NSERC) of
Canada. The research was primarily conducted in the QNC-MBE lab. Additionally,
significant use was made of the University of Waterloo’s QNFCF Facility. The
realization of this infrastructure owes much to the substantial contributions
from CFREF-TQT, the Canada Foundation for Innovation (CFI), Innovation,
Science and Economic Development Canada (ISED), the Ontario Ministry of
Research and Innovation, as well as the generous support of Mike and Ophelia
Lazaridis. The assistance and resources provided by all these entities are
deeply appreciated and have been instrumental in the progression of this work.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

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# Nonplanar isoperimetric inequality for random groups

Tomasz Odrzygóźdź  Institute of Mathematics, Polish Academy of Science,
Warsaw, Śniadeckich 8<EMAIL_ADDRESS>
##  1\. Introduction

In [gro93] Gromov introduced the notion of a random finitely presented group
on $m\geq 2$ generators at density $d\in(0,1)$. The idea was to fix a set of
$m$ generators and consider presentations with $(2m-1)^{dl}$ relators, each of
which is a random reduced word of length $l$. Gromov investigated the
properties of random groups when $l$ goes to infinity. We say that a property
occurs in the Gromov density model with overwhelming probability (w.o.p.) if
the probability that a random group has this property converges to $1$ when
$l\rightarrow\infty$.

There are many important properties of this model: for densities
$>\frac{1}{2}$ a random group is trivial w.o.p. [inv, Theorem 11]; for
densities $<\frac{1}{2}$ a random group is, w.o.p., infinite, hyperbolic and
torsion-free [inv, Theorem 11]; for densities $<\frac{1}{5}$ a random group
does not have Property (T) with overwhelming probability [ow11, Corollary
7.5].

One of the basic tools to investigate the geometry of the Cayley complex of a
random group is the ’’isoperimetric inequality‘‘ proved in [inv]:

######  Theorem 1.1.

For any $\varepsilon>0$, in the Gromov random group model at density
$d<\frac{1}{2}$ with overwhelming probability all reduced van Kampen diagrams
associated to the group presentation satisfy

$|\partial D|\geq l(1-2d-\varepsilon)|D|,$

Here $\partial D$ denotes to set of boundary edges of diagram $D$ and $|D|$
denotes the number of faces of $D$.

One of the corollaries of this theorem is the fact that in the Gromov density
model for densities $<\frac{1}{2}$ a random group is w.o.p. hyperbolic.

The goal of this note is to generalize Theorem 1.1 to the class of non-planar
diagrams of bounded number of faces.

######  Definition 1.2.

Suppose $Y$ is a $2$–complex, not necessarily a disc diagram. The
_cancellation_ in $Y$ is

$\mathrm{Cancel}(Y)=\sum_{e\in Y^{(1)}}(\deg(e)-1).$

Let the _size_ $|Y|$ denote the number of $2$–cells of $Y$.

By $G$ we will denote the random group with the presentation
$\left<S|R\right>$ and by $X$ the Cayley complex of $G$ with respect to this
presentation.

######  Definition 1.3.

We say that $Y$ is _fulfilled_ by a set of relators $R$ if there is a
combinatorial map from $Y$ to the presentation complex $X/G$ that is locally
injective around edges (but not necessarily around vertices).

In particular, any subcomplex of the Cayley complex $X$ is fulfilled by $R$.
From the definition we see that every $2$-cell $f$ of $Y$ bears some relator
$r\in R$, which means that the edges of the boundary of $f$ are labeled with
the consecutive letters of $r$, there is one edge which corresponds to the
first letter of $r$ and there is an orientation of the face $f$, which
determines the direction of labeling edges with letters or $r$.

######  Definition 1.4.

Let $G=\left<S|R\right>$ be a finite presentation. Let $Y$ be a $2$–complex.
Suppose that for $L>0$ there are chosen $L$ embedded edge paths
$A_{1},A_{2},\dots,A_{L}\subset Y^{(1)}$ each of which is contained in the
boundary of some face $f\in Y^{(2)}$. The set of fixed edges is
$\mathrm{Fix}(Y):=\bigcup_{i=1}^{L}A_{i}$. Suppose that the edges belonging to
$\mathrm{Fix}(Y)$ are prescribed with the elements of $S$ or their inverses.
Such $Y$ we call a diagram with L fixed paths or shortly a diagram with L
fixed paths if the value of $L$ is irrelevant.

######  Definition 1.5.

Let $Y$ be a diagram with fixed paths. Suppose that the following additional
information is given:

  1. (1)

which faces of $Y$ bear the same relator

  2. (2)

for each face there is chosen an edge corresponding to the first letter of the
relator labeling this face

  3. (3)

the orientation of each face.

Such $Y$ we call an abstract diagram with fixed paths. We say that $Y$ is
fulfilled by a set of relators $R$ if it is fulfilled as $2$-complex, letters
on fixed edges agrees with this fulfilling (meaning that under the
combinatorial map given by the fulfilling a fixed edge prescribed by $s$ is
mapped onto the edge in $X/G$ also labeled by $s$) and this fulfilling agrees
with the additional information.

######  Definition 1.6.

A reduction pair is a pair of two adjacent faces in $2$-complex $Y$ that under
the combinatorial map from $Y$ to the presentation complex $X/G$ are mapped
onto the same $2$-cell.

Note that if abstract diagram with fixed paths $Y$ is fulfilled and the are no
proper powers in the set of relators $R$ fulfilling $Y$, then $Y$ has no
reduction pairs. For densities $d<\frac{1}{2}$ with overwhelming probability
there are no proper powers in the random set of relators.

We will prove the following

######  Theorem 1.7 (local version with fixed edges).

For each $K,L,\varepsilon>0$, w.o.p. there is no $2$-complex $Y$ with $|Y|\leq
K$ and with at most $L$ fixed paths fulfilling $R$ and satisfying

(1.1) $\mathrm{Cancel}(Y)+|\mathrm{Fix}(Y)|>(d+\varepsilon)|Y|l,$

where $l$ is the length of the relators in the presentation.

Our proof is only a slight modification of the Olliver‘s proof of 1.1. The
crucial point in our reasoning was to define of the $\mathrm{Cancel}(Y)$,
which was done thanks to Piotr Przytycki. We start with

######  Proposition 1.8.

Let $R$ be a random set of relators at density $d$ and at length $l$. Let $Y$
be a $2$-complex with fixed paths. Then either
$\mathrm{Cancel}(Y)+|\mathrm{Fix}(Y)|<(d+2\varepsilon)|Y|l$ or the probability
that there exists a tuple of relators in $R$ fulfilling $Y$ is less than
$(2m-1)^{-\varepsilon l}$.

To prove this proposition we need some more definitions. Let
$N:=|\mathrm{Fix}(Y)|$ and let $n$ be the number of distinct relators in $Y$.
For $1\leq i\leq n$ let $m_{i}$ be the number of times relator $i$ appears in
$Y$. Up to reordering the relators we can suppose that $m_{1}\geq
m_{2}\geq\dots\geq m_{n}$.

For $1\leq i_{1},i_{2}\leq n$ and $1\leq k_{1},k_{2}\leq l$ we say that
$(i_{1},k_{1})>(i_{2},k_{2})$ if $i_{1}>i_{2}$ or $i_{1}=i_{2}$ but
$k_{1}>k_{2}$. Suppose that for some $s\geq 2$ an edge $e$ is adjacent to
faces: $f_{1},f_{2},\dots,f_{s}$ bearing relators $i_{1},i_{2},\dots,i_{s}$
accordingly. Suppose moreover that for $1\leq j\leq s$ the edge $e$ is the
$k_{j}$–th edge of the face $f_{j}$. Since $Y$ is reduced, for every $1\leq
j,j^{\prime}\leq s,j\neq j^{\prime}$ holds:
$(i_{j},k_{j})>(i_{j^{\prime}},k_{j^{\prime}})$ or
$(i_{j},k_{j})<(i_{j^{\prime}},k_{j^{\prime}})$ (otherwise there will be a
reduction pair in $Y$). Therefore this relation defines linear lexicographical
order, so there is a minimal element $1\leq j_{min}\leq s$. If $e$ is not a
fixed edge we say that edge $e$ belongs to faces $f_{j}$ for $j\in\\{1,2,\dots
s\\}\setminus\\{j_{min}\\}$. If $e$ is a fixed edge we say that it belongs to
all faces to which it is adjacent.

Let $\delta(f)$ be the number of edges belonging to a face $f$. For $1\leq
i\leq n$ let

$\kappa_{i}=\max_{f\text{ face bearing relator i}}\delta(f)$

Note that

(1.2) $\mathrm{Cancel}(Y)+N=\sum_{f\in Y^{(2)}}\delta(f)\leq\sum_{1\leq i\leq
n}m_{i}\kappa_{i}$

######  Definition 1.9.

We say that $Y$ is _partially fulfilled_ by a set of relators $R$ if there is
a combinatorial map from a subcomplex $Y^{\prime}\subset Y$ to the
presentation complex $X/G$ that is locally injective around edges (but not
necessarily around vertices).

######  Lemma 1.10.

For $1\leq i\leq n$ let $p_{i}$ be the probability that $i$ randomly chosen
words $w_{1},w_{2},\dots,w_{i}$ partially fulfill $Y$ and let $p_{0}=1$. Then

(1.3) $\frac{p_{i}}{p_{i-1}}\leq(2m-1)^{-\kappa_{i}}.$

###### Proof.

Suppose that first $i-1$ words $w_{1},\dots,w_{i-1}$ partially fulfilling $Y$
are given. We will successively choose the letters of the word $w_{i}$ in a
way to fulfill the complex. Let $k\leq l$ and suppose that the first $k-1$
letters of $w_{i}$ are chosen. Let $f$ be the face realizing
$\delta(f)=\kappa_{i}$ and let $e$ be the $k$-th edge of the face $f$.

If $e$ belongs to $f$ this means that there is another face $f^{\prime}$
meeting $e$ which bears relator $i^{\prime}<i$ or bears $i$ too, but $e$
appears in $f^{\prime}$ as a $k^{\prime}<k$-th edge or that $e$ is a fixed
edge. In all these cases the letter on the edge $e$ is imposed by some letter
already chosen so drawing it at random has probability $\leq\frac{1}{(2m-1)}$.

Combining all these observations we get that the probability to choose at
random the correct word $w_{i}$ is at most $p_{i-1}(2m-1)^{-\kappa_{i}}$. ∎

Now we can provide the proof of Proposition 1.8.

###### Proof of Proposition 1.8.

For $1\leq i\leq n$ let $P_{i}$ be the probability that there exists an
$i$-tuple of words partially fulfilling $Y$ in the random set of relators $R$.
We trivially have:

(1.4) $P_{i}\leq|R|^{i}p_{i}=(2m-1)^{idl}p_{i}$

Combining equations (1.2) and (1.3) we get

$\mathrm{Cancel}(Y)+N\leq\sum_{i=1}^{n}m_{i}(\log_{2m-1}p_{i-1}-\log_{2m-1}p_{i})=$

$=\sum_{i=1}^{n-1}(m_{i+1}-m_{i})\log_{2m-1}p_{i}-m_{n}\log_{2m-1}p_{n}+m_{1}\log_{2m-1}p_{0}.$

Now $p_{0}=1$ so $\log_{2m-1}p_{0}=0$ and we have

$\mathrm{Cancel}(Y)+N\leq\sum_{i=1}^{n-1}(m_{i+1}-m_{i})\log_{2m-1}p_{i}-m_{n}\log_{2m-1}p_{n}$

Now from (1.4) and the fact that $m_{i+1}-m_{i}\leq 0$ we have

$\mathrm{Cancel}(Y)+N\leq\sum_{i=1}^{n-1}(m_{i+1}-m_{i})(\log_{2m-1}P_{i}-idl)-m_{n}\log_{2m-1}(P_{n}-ndl)$

Observe that
$\sum_{i=1}^{n-1}(m_{i}-m_{i+1})idl+m_{n}ndl=dl\sum_{i=1}^{n}m_{i}=dl|Y|$.
Hence

$\mathrm{Cancel}(Y)+N\leq
l|Y|d+\sum_{i=1}^{n-1}(m_{i+1}-m_{i})\log_{2m-1}P_{i}-m_{n}\log_{2m-1}P_{n}$

Setting $P=\min_{i}P_{i}$ and using the fact that $m_{i+1}-m_{i}\leq 0$ we get

$\mathrm{Cancel}(Y)+N\leq
l|Y|d+(\log_{2m-1}P)\sum_{i=1}^{N-1}(m_{i+1}-m_{i})-m_{N}\log_{2m-1}P=$
$=l|Y|d-m_{1}\log_{2m-1}P\leq|Y|(ld-\log_{2m-1}P),$

since $m_{1}\leq|Y|$. It is clear that a complex is fulfillable if it is
partially fulfillable for any $i\leq n$ and so:

$\mathrm{Probability(}\emph{Y is fullfillable by relators of R}\mathrm{)}\leq
P\leq(2m-1)^{\frac{|Y|ld-\mathrm{Cancel}(Y)-N}{|Y|}},$

which was to be proven. ∎

###### Proof of Theorem 1.7.

Let $C(K,L,l)$ be the number of abstract complexs with $L$ fixed paths and
having at most $K$ faces, each of which is an $l$-gon. It can be easily
checked that for fixed $K$, $C(K,L,l)$ grows polynomially with $l$. We know
from Proposition 1.8 that for any reduced abstract complex violating the
inequality $\mathrm{Cancel}(Y)+|\mathrm{Fix}(Y)|<(d+\varepsilon)|Y|l$ the
probability that it is fulfilled by a random set of relators is
$\leq(2m-1)^{\varepsilon l}$. So the probability that there exists a reduced
complex with at most $K$ faces, violating the inequality is $\leq
C(K,L,l)(2m-1)^{-\varepsilon l}$, so converges to $0$ when
$l\rightarrow\infty$. ∎

## References