# Gradient flow on extensive-rank positive semi-definite matrix denoising

Antoine Bodin and Nicolas Macris  School of Computer and Communication
Sciences
SMILS - IC - Ecole Polytechnique Fédérale de Lausanne
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

In this work, we present a new approach to analyze the gradient flow for a
positive semi-definite matrix denoising problem in an extensive-rank and high-
dimensional regime. We use recent linear pencil techniques of random matrix
theory to derive fixed point equations which track the complete time evolution
of the matrix-mean-square-error of the problem. The predictions of the
resulting fixed point equations are validated by numerical experiments. In
this short note we briefly illustrate a few predictions of our formalism by
way of examples, and in particular we uncover continuous phase transitions in
the extensive-rank and high-dimensional regime, which connect to the classical
phase transitions of the low-rank problem in the appropriate limit. The
formalism has much wider applicability than shown in this communication.

###### Index Terms:

Random Matrix Theory, Linear Pencils, Gradient Flow, Matrix Denoising,
Extensive-Rank, Phase transitions

##  I Introduction

Matrix denoising and factorization play a crucial role in a variety of data
science tasks such as matrix sensing, phase retrieval or synchronisation, or
matrix completion. The problem consists in reducing the amount of noise or
irrelevant information present in a dataset, allowing for more accurate
analysis and interpretation of the data, as well as better computational
efficiency and modeling by way of dimensionality reduction. The literature on
the subject is immense and we refer to [1, 2] for recent overviews of
applications and theory in various settings and formulations.

In this contribution we focus on the study of gradient-flow for the following
statistical formulation for positive definite matrix denoising. We consider a
”ground truth” signal $X^{*}\in\mathbb{R}^{n\times d}$ with randomly sampled
independent entries $X^{*}_{ij}\sim\mathcal{N}(0,\frac{1}{n})$ where the
dimensions $n,d$ are such that $\phi=\frac{d}{n}$ is fixed. Then we define the
corrupted data matrix $Y\in\mathbb{R}^{n\times n}$

$Y=X^{*}X^{*T}+\frac{1}{\sqrt{\lambda}}\xi$ (1)

where $\xi$ is an additive symmetric random noise with
$\xi_{ij}=\xi_{ji}\sim\mathcal{N}(0,\frac{1}{n})$ and $\lambda$ is
(proportional to) the signal-to-noise ratio. The objective is to estimate the
ground truth positive semi-definite matrix $X^{*}X^{*T}$ from the corrupted
data matrix $Y$ with a matrix $XX^{T}$ such that $X\in\mathbb{R}^{n\times m}$
where $m$ is set from the fixed ratio $\psi=\frac{m}{n}$. Note that we allow
$d$ and $m$ to be different. The estimator studied in this contribution is
given by the gradient flow $X(t)$ ($t$ is time) for an objective function with
regularization parameter $\mu$, defined as

$\mathcal{H}(X)=\frac{1}{4d}\left\lVert
Y-XX^{T}\right\rVert_{F}^{2}+\frac{\mu}{2d}\left\lVert X\right\rVert_{F}^{2}$
(2)

where $\left\lVert\cdot\right\rVert_{F}$ is the Frobenius norm. The
initialization of gradient flow is $X(0)=X_{0}\in\mathbb{R}^{n\times m}$
random with i.i.d matrix elements from $\mathcal{N}(0,\frac{1}{n})$. As a
measure of performance we adopt the expected matrix-mean-square-error

$\mathbb{E}\mathcal{E}=\frac{1}{d}\mathbb{E}\left\lVert
X^{*}X^{*T}-XX^{T}\right\rVert_{F}^{2}$ (3)

where the expectation is over $\xi$, $X^{*}$, $X_{0}$. Note that the objective
function and performance measure are not the same and can be thought of as
”training” and ”generalization” errors in the language of machine learning.

Summary of main contributions:

  * •

We derive a set of analytical fixed point equations whose solutions allow to
compute the full performance curve $t\to\mathbb{E}\mathcal{E}_{t}$ for the
extensive-rank and high-dimensional regime where $m,d,n$ all tend to infinity
while $\phi,\psi$ are kept fixed (results 1 and 2 in Sec. II). Continuous time
average behaviour of gradient flow is a proxy for the usual discrete gradient
descent algorithm, and has the advantage that it is more amenable to
analytical study. The numerical experiments confirm that (a) $\mathcal{E}_{t}$
concentrates over its expectation; (b) theoretical predictions of gradient
flow agree with gradient descent. See Fig. 1.

  * •

We further push the analysis of these equations in the time limit $t=+\infty$
and display specific examples where a critical value $\lambda_{c}$ can be
calculated such that: (a) for $\lambda\leq\lambda_{c}$ the performance error
of gradient flow is no better than the one of the null-estimator $X=0$; (b)
for $\lambda>\lambda_{c}$ better estimation is possible; (c) the phase
transition between the two regimes is a continuous type phase transition.
These results are displayed on Fig. 2.

  * •

We analyze the limit $\phi=\psi\to 0$ (after $n,m,d$ have been sent to
infinity) and derive a connection with the low-rank setting. It turns out that
the matrix-mean-square-error curve (when $t\to+\infty$) tends to the one of
the rank-one problem and the phase transition reduces to the well known BBP
transition at $\lambda_{c}=1$.

We use tools based on modern results in random matrix theory. Central to our
derivations, is the formalism of _linear-pencils_ , that initially appeared in
[3, 4] and has been further improved recently in the context of neural
networks [5, 6, 7]. In particular we make use of extensions provided in [7] to
derive closed-form expressions of non-trivial averages over $\xi$, $X^{*}$,
$X_{0}$ appearing in traces of complicated ”rational” expressions of these
random matrices. Although these techniques have not yet always been rigorously
proven they have been used successfully in different applications, and the
predictions are confirmed by numerical experiments. In addition, we use
holomorphic functional calculus for matrices [8].

Brief review of literature: The full time-evolution of gradient flow for the
rank-one problem (the so-called spiked Wigner model with $d=m=1$) has been
solved and rigorously analyzed in much the same spirit as the present work in
[9] with the difference that the spike is constrained to lie on a sphere all
along the evolution. For the present extensive-rank setting rigorous or even
analytical results on the whole time-evolution are scarce. Closely connected
to our work is the recent paper [10]. An essential difference however is that
in [10] the initialization $X(0)=X_{0}$ is taken to have eigenvectors aligned
with those of $Y$ (this pre-processing can be implemented empirically in
practice). Moreover the authors do not carry out the random matrix averages
fully analytically. Gradient flow has been studied in a variety of settings
more or less related to the present one, see [11, 12, 13, 14, 15, 16].

Bayesian approaches are quite well understood for the low-rank problem (mainly
rank-one). This context is quite different from the present one. To begin with
it is not dynamical. One studies the Minimum-Mean-Square-Estimator (MMSE)
computed as the conditional expectation of the signal with respect to the
Bayesian posterior probability distribution [17, 18, 19, 20, 21, 22, 23, 24,
25]. Bayesian-optimal as well as mismatched estimation settings have been well
studied with rigorous results on the mutual information, the MMSE, the cross-
entropy, and the problem displays a rich phenomenology of first and higher
order phase transitions depending on the nature of the priors. Related
dynamics of the Approximate Message Passing (AMP) algorithms is also well
understood for these problems [26, 27, 28] . The realm of extensive-rank
within such Bayesian and AMP approaches is quite open and very timely [29, 30,
31, 32, 33].

Finally other types of non-dynamical approach belong to the class of spectral
methods like Principal Component Analysis (PCA). The low rank case is covered
by [34, 35, 36]. For the extensive-rank setting the results are scarce and
little is known except for ensembles of rotation invariant signals for which
an interesting class of Rotation Invariant Estimators (RIE) has been proposed
[37].

##  II Results

###  II-A Preliminaries

We simplify the notations by introducing the variables $Z=XX^{T}$ and
$Z^{*}=X^{*}X^{*T}$ and the _order parameters_ $p$ and $q$ such that
$\mathcal{E}=r-2q+p$ with:

$q=\frac{1}{d}\text{Tr}\left[Z^{*}Z\right]\quad\quad
p=\frac{1}{d}\text{Tr}\left[Z^{2}\right]\quad\quad
r=\frac{1}{d}\text{Tr}\left[(Z^{*})^{2}\right]$ (4)

In the rank-one setting, $p$ can be seen as a norm of the estimator while $q$
represents the angle with the ground-truth. We consider the gradient flow

$\frac{\mathrm{d}X_{t}}{\mathrm{d}t}=-\phi\nabla\mathcal{H}(X_{t})$ (5)

and track the evolution of the matrix mean-square error $\mathcal{E}_{t}$
through the quantities $q_{t}$ and $p_{t}$. The factor $\phi$ amounts to a
rescaling of time which leads to more convenient expressions. With the
additional notation $H=Y-\mu I_{n}$, expanding the gradient provides:
$\frac{\mathrm{d}X_{t}}{\mathrm{d}t}=(H-Z_{t})X_{t}$, which in turns provides
the matrix Riccati differential equation:

$\frac{\mathrm{d}Z_{t}}{\mathrm{d}t}=HZ_{t}+Z_{t}H-2Z_{t}^{2}$ (6)

A general solution of this matrix differential equation is (see e.g., [10]):

$Z_{t}=e^{tH}X_{0}\left(I_{m}+2X_{0}^{T}\int_{0}^{t}e^{2sH}\mathrm{d}sX_{0}\right)^{-1}X_{0}^{T}e^{tH}$
(7)

This formula is valid regardless of the dimensions $n,m,d$. In particular,
when $m=d=1$ this is the solution of the rank-1 gradient flow. In the high-
rank case, it is not straightforward a priori how to track the evolution of
the matrix $Z_{t}$ as firstly the rank of $X_{0}X_{0}^{T}$ and $X^{*}X^{*T}$
(or $Y$ or $H$) are not necessarily equal when $d\neq m$, and secondly because
the eigenvectors of the two matrices are not aligned at the initialization.

In the following, we will consider the high-dimensional limit $n,m,d\to\infty$
with $d/n$ and $m/n$ fixed and make the following assumptions:

  * •

The limits of traces $p_{t}=\frac{1}{d}{\rm Tr}[Z_{t}^{2}]$,
$q_{t}=\frac{1}{d}{\rm Tr}[Z^{*}Z_{t}]$ (and $\mathcal{E}_{t}$) concentrate on
their expectation, as well as related traces used in the _linear-pencils_
method in Sec. III.

  * •

We assume that $H$ has a limiting spectral distribution whose support can be
enlaced in a finite contour $\Gamma\subset\mathbb{C}$.

To keep notations lighter we shall abusively denote by $p_{t}$, $q_{t}$,
$\mathcal{E}_{t}$ their limiting deterministic values.

###  II-B Main results

The MSE $\mathcal{E}_{t}$ of the problem is completely given by $q_{t}$,
$p_{t}$ and the constant $r$ which in the high-dimensional limit is found to
be $r=1+\phi$ from the second moment of the Marchenko-Pastur law [38]. The
main contribution of this paper is the self-consistent set of equations that
fully track $q_{t}$ and $p_{t}$:

(Result 1) In the high dimensional limit, the overlap $q_{t}$ evolves
according the integral:

$q_{t}=\int_{\mathbb{R}}\frac{z\rho_{Q}(z)\mathrm{d}z}{1-e^{-2tz}+z\tilde{q}_{t}e^{-2tz}}$
(8)

with the auxiliary function $\tilde{q}_{t}$ solution of the fixed-point
equation:

$\psi\tilde{q}_{t}=1+\int_{\mathbb{R}}\frac{(1-e^{-2tz})\rho_{P}(z)\mathrm{d}z}{\frac{1}{\tilde{q}_{t}}(1-e^{-2tz})+ze^{-2tz}}$
(9)

and $\rho_{P},\rho_{Q}$ are given by their inverse Stieltjes transforms
$P(z),Q(z)$. These are the analytic solutions of the degree 3 polynomials such
that $-zP(z)\to 1$ when $|z|\to\infty$ and $-zQ(z)\to 1$ when $|z|\to\infty$
where:

$P^{3}+P^{2}\left(\lambda(\mu+z)+1\right)+P\lambda\left(\mu+z-\phi+1\right)+\lambda=0$
$Q^{3}\phi+Q^{2}\left(\mu+z-2\phi-1-\frac{1}{\lambda}\right)-Q\left(\mu+z-\phi-2\right)=1$
(10)

(Result 2) In the high-dimensional limit, the eigenvalue distribution of
$Z_{t}$ is found by the inverse Stieltjes-Transform of $h_{t}(z)$ where:

$\displaystyle h_{t}(z)$ $\displaystyle=\frac{-1}{2\pi
i}\oint_{\Gamma}-\frac{(1+e^{-2tx}(\frac{x}{\tilde{h}_{t}(z)}-1))P(x)\mathrm{d}x}{x+z+ze^{-2tx}(\frac{x}{\tilde{h}_{t}(z)}-1)}$
(11) $\displaystyle\tilde{h}_{t}(z)$
$\displaystyle=1+\frac{1}{\psi}\frac{-1}{2\pi
i}\oint_{\Gamma}-\frac{(x+z-ze^{2tx})P(x)\mathrm{d}x}{x+z+ze^{-2tx}(\frac{x}{\tilde{h}_{t}(z)}-1)}$
(12)

in particular, we find:

$p_{t}=-\frac{1}{2\phi}\left.\frac{\partial^{(2)}}{\partial
z^{2}}\left(\frac{1}{z}h_{t}\left(\frac{1}{z}\right)\right)\right\rvert_{z=0}$
(13)

Note that a similar system of equations as (8) can be derived by calculating
the first and second derivatives in $z$ as given by (13) and using the
integrands in (11). However the resulting formulas are too cumbersome to be
presented here.

###  II-C Discussions and experiments

Figure 1 provides an example of the calculation of $q_{t}$ through time
compared with experimental runs: we see a good agreement between the curves
and the prediction.

Figure 1: Comparison of $q_{t}$ evolution with $10$ runs of a gradient descent
with $n=100$, $m=25$, $d=75$ and $\lambda=10^{4}$ and $\mu=0$.

Asymptotic Limit $t\to\infty$: An interesting question is to study the
asymptotics of $q_{t}$ when $t\to\infty$. We take the ansatz that
$q_{t}\sim\gamma e^{2\alpha t}$ in this limit with $\alpha>0$ and $\gamma>0$
another constant and plug this in equation (9):

$\displaystyle\psi$
$\displaystyle=\frac{1}{\tilde{q}_{t}}+\int_{\mathbb{R}}\frac{(1-e^{-2tz})\rho_{P}(z)\mathrm{d}z}{1-e^{-2tz}+z\tilde{q}_{t}e^{-2tz}}$
(14) $\displaystyle\simeq\frac{e^{-2\alpha
t}}{\gamma}+\int_{\mathbb{R}}\frac{(1-e^{-2tz})\rho_{P}(z)\mathrm{d}z}{1-e^{-2tz}+z\gamma
e^{-2(z-\alpha)t}}$ (15)
$\displaystyle\simeq\int_{\alpha}^{\infty}\rho_{P}(z)\mathrm{d}z=1-F_{P}(\alpha)$
(16)

With $F_{P}$ the CDF of $P$. Such a solution exists when we can find $\alpha$
such that $F_{P}(\alpha)=1-\psi$, effectively selecting the proportion $\psi$
of the eigenvalues of $H$ in the interval $(\alpha,+\infty)$. Due to the
assumption $\alpha>0$, a further condition for the existence of such an
$\alpha$ is $F_{P}(0)<1-\psi\leq 1$ or: $0\leq\psi<1-F_{P}(0)\leq 1$. This
implies that the ansatz is valid in the _under-parameterized_ regime $(m<n)$.
The asymptotic limit is thus given by
$q_{\infty}=\lim_{t\to\infty}q_{t}=\int_{\alpha}^{\infty}z\rho_{Q}(z)\mathrm{d}z$.
Note that the alternative ansatz that $\tilde{q}_{t}$ converges towards a
finite limit leads to a similar solution as but with $\alpha=0$.

A similar line of reasoning lead us to consider the term
$p_{\infty}=\frac{1}{\phi}\int_{\alpha}^{\infty}z^{2}\rho_{P}(z)\mathrm{d}z$
and thus a asymptotic mean square error:

$\mathcal{E}_{\infty}=r-\int_{\alpha}^{\infty}\left(2z\rho_{Q}(z)-\frac{1}{\phi}z^{2}\rho_{P}(z)\right)\mathrm{d}z$
(17)

As an example, for $\phi=\psi=1$ and $\mu=\frac{1}{\lambda}$, and $\alpha=0$
we expect from formula (17) that $\mathcal{E}_{\infty}=r$ when the support of
$\rho_{P}$ and $\rho_{Q}$ is located below $0$. This can be found by studying
the discriminant $\Delta_{P}(\lambda,z)$ of the polynomial solved by $P$:
because it is a order $3$ polynomial with coefficients in $\mathbb{R}$ when
$z\in\mathbb{R}$, either the solutions are all real $(\Delta_{P}>0)$ implying
$\rho_{P}(z)=0$, or one is real and two are complex conjugate $(\Delta_{P}=0)$
implying $\rho_{P}(z)>0$. At a specific $\lambda$, the support of $\rho_{P}$
is located below $0$ and touches $z=0$. This $\lambda_{c}$ is solution of
$\Delta_{P}(\lambda_{c},0)=0$ which provides the solution
$\lambda_{c}=\frac{4}{27}$. The whole error curve at $t=+\infty$ is shown in
Figure 2.

Figure 2: Experimental and theoretical $\mathcal{E}_{\infty}$ with
$\phi=\psi=1,\mu=\frac{1}{\lambda}$

The choice $\mu=\frac{1}{\lambda}$ is natural from a Bayesian point-of-view
because it would correspond to the situation where the statistician matches
its prior to the ground-truth when $\psi=\phi$.

Low rank limit when $\phi=\psi\to 0$: We bring to the reader’s attention that
the objective function $\mathcal{H}$ when $d=1$ and $n\to\infty$ with
$\mu=\frac{1}{\lambda}$ corresponds precisely to the spiked-Wigner problem.
This suggest to look at the limit $\phi=\psi\to 0$. In this situation, we
expect $\alpha$ should be close to the maximum eigenvalue of the bulk of
$\rho_{Q}$. We make the following observation in Figure 3: as $\phi$
decreases, $\rho_{P}$ in blue has two bulks of eigenvalues, one of which
disappears as $\phi$ grows. On the other hand, $\rho_{Q}$ in orange displays
also two bulks at the same locations but the second bulk develops a mass as
$\phi\to 0$. Therefore, we expect that $\alpha$ adjusts itself to the maximum
eigenvalue of the first bulk of $\rho_{P}$. Furthermore, interestingly we see
that these two bulks are getting closer when $\lambda$ is closer to $1$ as
seen in Figure 4.

Figure 3: Bulk of eigenvalues for $\lambda=5$ and different values of $\phi$
Figure 4: Bulk of eigenvalues for $\lambda=2$ and different values of $\phi$

With these observations, we expect that $Q(z)$ has a pole $z=1$ in the limit
$\phi\to 0$. Let’s consider a polynomial equation of $\hat{Q}$ solving the
reduced polynomial equation of $Q$ with $\phi=0$:

$\hat{Q}^{2}(z-1)-\hat{Q}(z-2+\frac{1}{\lambda})-1=0.$ (18)

In order to find a potential pole, we consider
$\mathcal{Q}(z)=(1-z)\hat{Q}(z)$ and check for potential limits of
$\mathcal{Q}$ when $z\to 1$. First of all, injecting $\mathcal{Q}$ in the
former polynomial equation, we find:

$\left(\mathcal{Q}^{2}\lambda+\mathcal{Q}(\lambda z-2\lambda+1)-\lambda
z+\lambda\right)/(z-1)=0.$ (19)

Therefore, on the upper-complex plane we find the numerator equals $0$, and by
analytic continuation, the limit $z\to 1$ follows:
$\mathcal{Q}(1)(\mathcal{Q}(1)\lambda-\lambda+1)=0$ so
$\mathcal{Q}(1)\in\\{0,1-\frac{1}{\lambda}\\}$. It is interesting to notice
the connection with the usual Bayesian overlap of the spiked Wigner model -
since $\mathcal{Q}(1)$ represents the squared overlap $q_{\infty}$ in the
limit $\phi\to 0$. Pushing further this analysis for $P(z)$ allows to
eventually get $p_{\infty}$ and $\mathcal{E}_{\infty}$ in the limit $\phi\to
0$ and check the connection with the Bayesian MMSE of the spiked Wigner model.

##  III Sketch of Proof

Our method relies on considering the interaction of the random matrices
$X_{0},X^{*},\xi$. We treat each term $q_{t}$ and $p_{t}$ separately with the
linear-pencil technique. In both cases, we first factor out the $X_{0}$
matrix, then decouple the time dependency from the remaining random matrix
expressions, and finally factor-out $X^{*},\xi$.

Our results are derived in the limit $n,m,d\to+\infty$. For a sequence of
matrices $A_{N}\in\mathbb{R}^{N\times N}$ we use the notation
$\text{Tr}_{N}\left[A_{N}\right]=\lim_{N\to\infty}\frac{1}{N}\text{Tr}\left[A_{N}\right]$.
As stated in Sec. II we assume that the limiting traces involved in the linear
pencil method concentrate.

###  III-A Tracking the angle $q_{t}$

The term $q_{t}=\text{Tr}_{d}\left[Z^{*}Z_{t}\right]$ can be completely
recovered from a sub-block of the following linear-pencil $M_{q}$:

$\displaystyle M_{q}=\left(\begin{array}[]{c|||c||c|ccc|c}0&I_{d}&0&0&0&0&0\\\
\hline\cr\hline\cr\hline\cr I_{d}&0&0&0&0&0&W_{t}\\\ \hline\cr\hline\cr
0&0&0&X_{0}&0&0&I_{n}\\\ \hline\cr 0&0&X_{0}^{T}&I_{m}&0&X_{0}^{T}&0\\\
0&0&0&0&L_{t}&I_{n}&0\\\ 0&0&0&X_{0}&I_{n}&0&0\\\ \hline\cr
0&W_{t}^{T}&I_{n}&0&0&0&0\end{array}\right)$ (27)

Where $W_{t}=X^{*T}e^{tH}$ and $L_{t}=2\int_{0}^{t}e^{2sH}\mathrm{d}s$. A
recursive application of the Schur-complement to compute $M_{q}^{-1}$ shows
that the block $(M_{q}^{-1})^{(1,1)}$ is the random matrix $X^{*T}Z_{t}X^{*}$.
So in fact: $q_{t}=\text{Tr}_{d}\left[(M_{q}^{-1})^{(1,1)}\right]$.

The random matrices $X_{0},X^{*},\xi$ are all independent and $X_{0}$ is not
part of the terms $W_{t},L_{t}$. Therefore, we can apply the linear-pencil
theory on $M_{q}$ over the random-matrix $X_{0}$ while considering the other
random matrices as fixed. To this end, we note the constant part
$C_{q}=\mathbb{E}_{X_{0}}[M_{q}]$, and consider matrix of sub-traces
$g\in\mathbb{R}^{7\times 7}$ such that for squared-blocks $ij$,
$g_{ij}=\text{Tr}_{N_{i}}\left[(M_{q}^{-1})^{(i,j)}\right]$ where $N_{i}$ is
the size of the block $ij$ in $M_{q}^{-1}$. Then we apply the fixed-point
equation described in Appendix D of [7] with
$g_{ij}=\frac{1}{N_{i}}\text{Tr}\left[((C_{q}-\eta(g)\otimes
I)^{-1})^{(ij)}\right]$ where $\eta(g)$ is the matrix defined by:

$\displaystyle\eta(g)=\left(\begin{array}[]{ccccccc}0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\\\ 0&0&\psi g_{44}&0&0&\psi g_{44}&0\\\
0&0&0&g_{33}+g_{36}+&0&0&0\\\ &&&g_{63}+g_{66}&&&\\\ 0&0&0&0&0&0&0\\\ 0&0&\psi
g_{44}&0&0&\psi g_{44}&0\\\ 0&0&0&0&0&0&0\end{array}\right)$ (36)

Further inversion of $C_{q}-\eta(g)\otimes I$ leads to:

$\displaystyle g_{11}$ $\displaystyle=\text{Tr}_{d}\left[g_{44}\psi
W_{t}(g_{44}\psi L_{t}+I_{n})^{-1}W_{t}^{T}\right]$ (37) $\displaystyle
g_{44}$ $\displaystyle=\frac{1}{1-g_{66}}$ (38) $\displaystyle g_{66}$
$\displaystyle=-\text{Tr}_{n}\left[L_{t}(g_{44}\psi L_{t}+I_{n})^{-1}\right]$
(39)

Let $\Gamma\subset\mathbb{C}$ be a contour enclosing the eigenvalues of $H$,
we use the fact that for any functional $f$ which applies on the eigenvalues
of a matrix we have $f(H)=\frac{-1}{2\pi
i}\oint_{\Gamma}f(z)(H-zI_{n})^{-1}\mathrm{d}z$ to obtain:

$g_{11}=\frac{-1}{2\pi i}\oint_{\Gamma}\frac{g_{44}\psi
e^{2zt}}{1+g_{44}\psi\int_{0}^{t}2e^{2sz}\mathrm{d}s}\text{Tr}_{d}\left[(H-zI_{n})^{-1}Z^{*}\right]\mathrm{d}z$

which leads with $Q(z)=\text{Tr}_{d}\left[X^{*T}(H-zI_{n})^{-1}X^{*}\right]$
to:

$\displaystyle g_{11}=\frac{-1}{2\pi i}\oint_{\Gamma}\frac{g_{44}\psi z}{\psi
g_{44}(1-e^{-2tz})+ze^{-2tz}}Q(z)\mathrm{d}z$ (40)

Similarly with $P(z)=\text{Tr}_{n}\left[(H-zI_{n})^{-1}\right]$

$g_{66}\psi=\frac{-1}{2\pi i}\oint_{\Gamma}\frac{1-e^{-2tz}}{\psi
g_{44}(1-e^{-2tz})+ze^{-2tz}}P(z)\mathrm{d}z$ (41)

We find the equations from the main results with $\tilde{q}_{t}=\frac{1}{\psi
g_{44}}$.

###  III-B Tracking the norm $p_{t}$

The term $p_{t}=\text{Tr}_{d}\left[Z_{t}^{2}\right]$ can also be recovered
from a similar calculation but would lead to design a much larger linear-
pencil. Another method is to track directly the eigenvalues of $Z_{t}$ with
the trace of the resolvent:
$h_{11}=\text{Tr}_{n}\left[(Z_{t}-zI_{n})^{-1}\right]$ with $h$ the solution
of the fixed point equation (Appendix D in [7]) stemming from the following
linear-pencil:

$M_{p}=\left(\begin{array}[]{c||c|ccc|c}-zI_{n}&0&0&0&0&e^{tH}\\\
\hline\cr\hline\cr 0&0&X_{0}&0&0&I_{n}\\\ \hline\cr
0&X_{0}^{T}&I_{m}&0&X_{0}^{T}&0\\\ 0&0&0&L_{t}&I_{n}&0\\\
0&0&X_{0}&I_{n}&0&0\\\ \hline\cr e^{tH}&I_{n}&0&0&0&0\end{array}\right)$ (42)

Which yields the set of equations:

$\displaystyle h_{11}$
$\displaystyle=-\text{Tr}_{n}\left[\left(L_{t}+\frac{1}{h_{33}}I_{n}\right)\left(e^{2tH}+zL_{t}+\frac{z}{h_{33}}I_{n}\right)^{-1}\right]$
$\displaystyle h_{33}$
$\displaystyle=1-\frac{1}{\psi}\text{Tr}_{n}\left[\left(zL_{t}+e^{2tH}\right)\left(e^{2tH}+zL_{t}+\frac{z}{h_{33}}I_{n}\right)^{-1}\right]$

Using the contour integration technique, we obtain:

$h_{11}=\frac{-1}{2\pi
i}\oint_{\Gamma}-\frac{\frac{1}{h_{33}}+\int_{0}^{t}2e^{2sx}\mathrm{d}s}{\frac{z}{h_{33}}+e^{2tx}+z\int_{0}^{t}2e^{2sx}\mathrm{d}s}P(x)\mathrm{d}x$
(43)

which is reduced to:

$h_{11}(z)=\frac{-1}{2\pi
i}\oint_{\Gamma}-\frac{1+e^{-2tx}(\frac{x}{h_{33}}-1)}{x+z+ze^{-2tx}(\frac{x}{h_{33}}-1)}P(x)\mathrm{d}x$
(44)

Similarly for $h_{33}$:

$h_{33}(z)=1+\frac{1}{\psi}\frac{-1}{2\pi
i}\oint_{\Gamma}-\frac{(x+z-ze^{2tx})P(x)\mathrm{d}x}{x+z+ze^{-2tx}(\frac{x}{h_{33}}-1)}$
(45)

Two possible ways to retrieve $p_{t}$ from $h_{11}$ and $h_{33}$: either with
$\phi p_{t}=\frac{-1}{2\pi i}\oint_{\Gamma}z^{2}h_{11}(z)\mathrm{d}z$, or
$\phi p_{t}=-\frac{1}{2}\frac{\partial^{(2)}}{\partial
z^{2}}\left(\frac{1}{z}h_{11}(\frac{1}{z})\right)\rvert_{z=0}$. In both cases,
there is an additional level of complexity in terms of calculation as it
either requires a double-contour integration, or computing derivative and
second derivative of the given functions yielding further new equations.

###  III-C Quantities $Q(z),P(z)$

There remains to calculate the terms $Q(z),P(z)$ which depends only on the
random matrices $X^{*},\xi$ and can be done altogether with the linear-pencil:

$M_{z}=\left(\begin{matrix}I_{n}&X^{*}&0&0\\\ 0&I_{d}&X^{*T}&0\\\
0&0&(z+\mu)I_{n}-\frac{1}{\sqrt{\lambda}}\xi&X^{*}\\\
0&0&X^{*T}&I_{d}\end{matrix}\right)$ (46)

Using the kernel $K=(H-zI_{n})^{-1}$, we can calculate the inverse:

$M_{z}^{-1}=\left(\begin{matrix}I_{n}&-X^{*}&-Z^{*}K&Z^{*}KX^{*}\\\
0&I_{d}&X^{*T}K&-X^{*T}KX^{*}\\\ 0&0&-K&KX^{*}\\\
0&0&-X^{*T}K&I_{d}-X^{*T}KX^{*}\end{matrix}\right)$ (47)

So that $Q(z)=-f_{13}$ and $P(z)=f_{33}$ where we $f$ is the analog of $g$ and
$h$ with the former linear-pencils. In particular we expect the following
structure:

$f=\left(\begin{matrix}1&0&-\phi Q(z)&0\\\ 0&1&0&-Q(z)\\\ 0&0&-P(z)&0\\\
0&0&0&1-Q(z)\end{matrix}\right)$ (48)

We can further compute the fixed point equation with:

$\eta(f)=\left(\begin{matrix}0&0&f_{22}\phi+f_{24}\phi&0\\\
0&f_{31}&0&f_{33}\\\ 0&0&\frac{f_{33}}{\lambda}+f_{42}\phi+f_{44}\phi&0\\\
0&f_{31}&0&f_{33}\end{matrix}\right)$ (49)

After some algebraic reductions, we obtain the degree 3 polynomials given in
equation (10). In general, these equations have multiple solutions but only
one corresponds to the analytic solution associated to the appropriate trace
of resolvent.

##  IV Conclusion

Our work primarily shows how we can take advantage of random matrix techniques
to derive fixed-point equations solving the time evolution of the matrix-mean-
square-error in the high-dimensional limit. Although we choose a specific data
model, as future considerations, the matrix $H$ can be generalized to other
structures for which the same methods would apply. In particular, if only the
noise structure changes, then only $\rho_{Q}$ and $\rho_{P}$ are changed. We
will come back to these issues in a more extensive and detailed contribution.

Acknowledgments The work of A. B is supported by Swiss National Fondation
Grant no 200020 182517. We also acknowledge instructive discussions with
Farzad Pourkamali and Jean Barbier.

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These authors contributed equally These authors contributed equally

# Locally Phase-Engineered MoTe2 for Near-Infrared Photodetectors

Jan Hidding  Cédric A. Cordero-Silis  Zernike Institute for Advanced
Materials, University of Groningen, 9747 AG Groningen, The Netherlands  Daniel
Vaquero  Nanotechnology Group, USAL—Nanolab, Universidad de Salamanca, E-37008
Salamanca, Spain  Konstantinos P. Rompotis  Zernike Institute for Advanced
Materials, University of Groningen, 9747 AG Groningen, The Netherlands  Jorge
Quereda  Departamento de Física de Materiales, GISC, Universidad Complutense
de Madrid, E-28040 Madrid, Spain  Marcos H. D. Guimarães<EMAIL_ADDRESS>Zernike Institute for Advanced Materials, University of Groningen, 9747 AG
Groningen, The Netherlands

###### Abstract

Transition metal dichalcogenides (TMDs) are ideal systems for two-dimensional
(2D) optoelectronic applications, owing to their strong light-matter
interaction and various band gap energies. New techniques to modify the
crystallographic phase of TMDs have recently been discovered, allowing the
creation of lateral heterostructures and the design of all-2D circuitry. Thus
far, the potential benefits of phase-engineered TMD devices for optoelectronic
applications are still largely unexplored. The dominant mechanisms involved in
the photocurrent generation in these systems remain unclear, hindering further
development of new all-2D optoelectronic devices. Here, we fabricate locally
phase-engineered MoTe2 optoelectronic devices, creating a metal (1T’)
semiconductor (2H) lateral junction and unveil the main mechanisms at play for
photocurrent generation. We find that the photocurrent originates from the
1T’–2H junction, with a maximum at the 2H MoTe2 side of the junction. This
observation, together with the non-linear IV-curve, indicates that the
photovoltaic effect plays a major role on the photon-to-charge current
conversion in these systems. Additionally, the 1T’–2H MoTe2 heterojunction
device exhibits a fast optoelectronic response over a wavelength range of 700
nm to 1100 nm, with a rise and fall times of 113 µs and 110 µs, two orders of
magnitude faster when compared to a directly contacted 2H MoTe2 device. These
results show the potential of local phase-engineering for all-2D
optoelectronic circuitry.

KEYWORDS: Scanning Photocurrent, Transition Metal Dichalcogenides, Crystal
Phase-Engineering

###### keywords:

Scanning Photocurrent, Transition Metal Dichalcogenides, Crystal Phase-
Engineering

##  1 Introduction

The large family of two-dimensional (2D) materials called the transition metal
dichalcogenides (TMDs) have gained much attention in the last two decades due
to their versatility, mechanical strength, atomically flat interfaces, and
strong absorption at the monolayer limit, making them promising candidates for
future (opto)electronic and (opto)spintronic applications 1. The most commonly
studied crystal structure of the TMD family is the hexagonal (2H) phase, for
which most TMDs are semiconducting, and possess a thickness-dependent band gap
2, 3. Apart from the 2H phase, however, TMDs can present a multitude of
different crystallographic phases, such as the semiconducting 3R phase, or the
semi-metallic 1T, 1T’ and 1Td phases, which possess different symmetries and
(opto)electronic properties. To benefit from these different properties in a
single device, researchers have recently focused on gaining control of the
crystallographic phase of TMDs, allowing them to transform the phase of single
TMD crystals at will.4, 5 This new and emerging field is now referred to as
the field of phase-engineering and opens the door to creating on-chip 2D
circuitry with 2D metals and semiconductors.

In literature, multiple methods are used to induce a 2H to 1T’ phase
transformation for different TMDs, such as crystal deformation 6, 7, 8, 9,
electrostatic doping 10, chemical doping 11, 12, laser heating 13, etc. In
particular, MoTe2 gained much attention as the energy barrier between the 2H
and 1T’ phase is the smallest ($\sim$40 meV) 14. It was shown electrically
that the Schottky barrier, present when directly contacting the 2H TMD with
metallic contacts, is significantly reduced when contacting a 2H TMD via a
phase transformed 1T’ region 11, 13, 15, 16, with these two-dimensional
lateral junctions, approaching the quantum limit for the contact resistance
17. The possibility of fabricating high-quality contacts to a 2D semiconductor
is essential to increase the speed of optoelectronic devices based on these
materials.

Apart from electrical characterization, only a few reports explored the
benefits of local phase engineering on the optoelectronic performance of TMD
devices 18, 19, 20. Lin et al. report an increased responsivity for 2H MoTe2
devices using 1T’ interlayer contacts 19. However, no scanning photocurrent
measurements are performed, which makes it difficult to disentangle the
possible microscopic mechanism involved in the photocurrent generation to
either the photovoltaic effect (PVE), due to the build in electric field at
the Schottky barriers, or the photothermal effect (PTE), due to the different
Seebeck coefficients of the 2H and 1T’ region 21, 22, 1.

Here, we perform scanning photocurrent measurements on 1T’–2H MoTe2
heterojunction devices, which allow us to spatially resolve the areas involved
in the photocurrent generation, giving insights on the underlying mechanisms
involved. First, we phase-transform the sides of an exfoliated 2H MoTe2
crystal to a 1T’ phase using local heating by laser irradiation, which allows
us to contact the 2H MoTe2 via the semi-metallic 1T’ regions. We find a clear
non-linear behavior for the 1T’–contacted 2H region, indicative of a Schottky
barrier between the 1T’ and 2H region. Additionally, using the scanning
photocurrent measurements, we clearly observe that the photocurrents are
generated at the 1T’–2H junction rather than at the Ti/Au electrodes or the
1T’ region. More specifically, we find that the peak of the photocurrent is
generated at the 2H side of the junction, which suggests that the observed
photocurrents in the 1T’–2H junction can be attributed to the PVE rather than
the PTE. Lastly, we characterize the optoelectronic performance of the MoTe2
photodetector by performing time-resolved, and laser power dependent
photocurrent measurements. We find fast rise and fall times of 113 µs and 110
µs, respectively, over a broad spectral range of 700 nm to 1100 nm. By
comparing our 1T’–2H MoTe2 photodetector to a 2H MoTe2 diode where the
electrodes are directly deposited on the 2H MoTe2 crystal, we are able to show
that the temporal response of 1T’–contacted 2H MoTe2 is two orders of
magnitude faster. This indicates that phase-engineering can be considered
another tool for improving the performance of TMD-based optoelectronic
devices.

##  2 Results and Discussion

Figure 1: (a) Optical micrograph of a phase-changed MoTe2 device, where the
phase changed regions are outlined with the white dashed line while the bright
green part is the unaltered 2H MoTe2 region. (b) The Raman spectra obtained
before (green) and after (purple) the phase transformation, which clearly
indicate a successful phase transformation. The spectra before the phase
change is multiplied by 3 for clarity. (c) The $I_{ds}-V_{ds}$ measurements,
as indicated in (a), with $V_{g}$ ranging from 0 V to 50 V, taken at 78 K. The
non-linear IV characteristics show the Schottky behavior. The IV measurement
for the two 1T’ regions are depicted in the inset, which clearly show Ohmic
behavior. (d) The transfer curve measured with a $V_{ds}$ of 3 V, taken at 78
K, shows clear n-type behavior.

###  2.1 Raman Spectroscopy

The device used to perform the optoelectronic measurements is depicted in Fig.
1(a). The green region is the untreated 2H MoTe2 while the dark green regions,
indicated by the dashed white line, were irradiated with a laser to induce the
phase transformation from 2H to 1T’ (details of the device fabrication can be
found in the Methods section). To confirm the phase transformation, we
performed Raman spectroscopy measurements as depicted in Fig. 1(b). Before
laser irradiation, we observe the in-plane $E_{2g}$ mode at 235 cm-1 and an
out-of-plane $A_{g}$ mode near 174 cm-1, indicative of the 2H MoTe2 phase.
After laser irradiation, these peaks disappear, and we observe two new peaks
at 124 cm-1 and 138 cm-1, corresponding to the $A_{g}$ mode of 1T’ MoTe2. This
significant change in the Raman spectrum indicates the successful phase
transformation of the irradiated regions.13, 23

###  2.2 Electrical Characterization

After fabricating electrical contacts to the phase-changed region, we
electrically characterize the device in a cryostat at 78 K. We swept the
drain-source voltage ($V_{ds}$) and measured the drain-source current
($I_{ds}$) for the different regions (both 1T’ and the 1T’–2H junction). In
Fig. 1(c), the 2-probe $I_{ds}$-$V_{ds}$ measurements are shown for the
1T’–2H–1T’ junction at different gate voltages, ranging from 0 V to 50 V. We
observe a clear non-linear behavior for the $I_{ds}$ as function of the
$V_{ds}$, indicative of a Schottky barrier present in our device, which could
either be between the Ti/Au contact and the 1T’–MoTe2, or the 1T’–2H junction.
To determine this, we performed the $I_{ds}$-$V_{ds}$ measurement on the phase
transformed 1T’ region only and observe a clear linear behavior showing Ohmic
contact between the Ti/Au contacts and 1T’ region, depicted in the inset of
Fig. 1(c). Therefore, we expect the non-linear behavior observed in Fig. 1(c)
to originate from a Schottky barrier between the 1T’–2H junction. For the 1T’
regions, we find a 2-probe resistance of 8 k$\Omega$ and 18 k$\Omega$, which
again indicate the successful transformation from the semiconducting 2H MoTe2
phase to the semi-metallic 1T’ phase.

Figure 2: (a) Reflectivity map of the scanning photocurrent measurement of the
device depicted in Fig. 1(a) with the corresponding photocurrent map in (b),
(c), and (d), taken at RT. The white outlines indicate the position of the
flake and Ti/Au contacts, while the white dashed lines indicates the 1T’–2H
junctions for clarity. The photocurrent maps are obtained with $\lambda=700$
nm, $P=1$ µW, and a $V_{ds}$ of (b) -2 V, (c) 0 V, and (d) 2 V. From the
photocurrent maps, we can clearly see that the induced photocurrent originates
from the 1T’–2H junction rather than from the Ti/Au contacts.

###  2.3 Optoelectrical Characterization

To observe the optoelectronic response of the 1T’–2H–1T’ sample, we perform
both scanning photocurrent measurements and time-resolved photocurrent
measurements, as described in the methods section. First, the scanning
photocurrent measurements are presented, which give more insight on the origin
of the photocurrent, after which the time-resolved photocurrent measurements
are discussed.

The scanning photocurrent measurements enable us to spatially identify where
the photocurrent is generated, and thus allows us to check whether the
photocurrent originates from the Ti/Au contacts or from the MoTe2 flake
itself. When performing the scanning photocurrent measurements, the laser beam
is focused and scanned across the sample in a raster-like fashion while
recording both the reflection and the generated photocurrent. Figure 2(a)
shows the recorded reflection map with an illumination wavelength of 700 nm, a
power of 1 µW, and a full width half maximum (FHWM) spot size of $0.70\pm
0.02$ µm (see supporting information). The contours of the flake and the Ti/Au
contacts are clearly visible and highlighted with the white outlines for
clarity. Note that the different phases of the MoTe2 can also be clearly
distinguished, and their junctions are highlighted with the white dashed line.
The corresponding photocurrent map with a $V_{ds}$ of -2 V, 0 V, and 2 V are
depicted in Fig. 2(b), (c), and (d), respectively. We clearly observe that the
photocurrent originates locally from the 1T’–2H junction, rather than the
Ti/Au contacts, which is the case when directly contacting the 2H MoTe2. This,
again, confirms a low Schottky barrier between the Ti/Au and 1T’ MoTe2,
indicating a successful phase transformation.

Figure 3: (a) A schematic of the photo-thermoelectric (PTE) and photovoltaic
effect (PVE). For the PTE, the laser locally heats the device which creates a
temperature gradient, which via the Seebeck effect causes an induced
photocurrent ($I_{PTE}$). For the PVE, the localized electric field at the
Schottky barrier causes a separation of the photo-induced carriers, resulting
in $I_{PVE}$. Note that the two effect produce a photocurrent with the same
direction, and that the induced photocurrent is opposite on both junctions.
(b) A line trace of the photocurrent mapping, indicated in black in Fig. 2(c),
showing the reflection (white) and negative (purple) and positive (green)
photocurrent peaks along the line trace. Both the maximum and minimum
photocurrent are obtained in the 2H region, as expected for a localized
electric field from a Schottky barrier. The phases are indicated by the purple
(1T’) and green (2H) background. Figure 4: (a) Time-resolved photocurrent,
taken at RT, where the photocurrent (purple) in the device is plotted together
with the chopper signal (grey) versus time show the fast response of our MoTe2
photodetector. The dashed grey rectangles indicate the region used to
determine the rise and fall time, as depicted in the inset of (b). (b) The
extracted rise (purple) and fall times (green) indicate no wavelength
dependence on the fast response for wavelengths ranging from 700 to 1100 nm.
The inset shows the rise (purple) and fall (green) curves of the photocurrent
from which the rise and fall times are extracted. (c) The power dependent
measurements for different $V_{ds}$, ranging from -2 V to 2 V, with a maximum
responsivity of $4.5\times 10^{-8}$ A/µW. Here, the responsivity ($R$) of the
device is plotted as function of the laser excitation power ($P$) and fitted
at high laser excitation power to a power law: $R\propto P^{\alpha-1}$. The
measured $R$ for 700 nm and 1068 nm are indicated by the filled circles and
unfilled squares, respectively. (d) The extracted index of the power law
($\alpha$) from the fitting in (c) versus $V_{ds}$ for the wavelengths 700 nm
(purple) and 1068 nm (green).

We observe photocurrents with opposite signs at the two 1T’-2H junctions for
$V_{ds}$ = 0 V. This is in line with the expected behavior for two possible
mechanisms: photovoltaic effect (PVE) due to a Schottky barrier at the 1T’–2H
junction, or the photo-thermoelectric effect (PTE) due to a different Seebeck
coefficient of the two MoTe2 phases. The two mechanisms are schematically
depicted in Fig. 3. While both the PVE and the PTE can give rise to
photocurrent in the 2H-1T’ interface, we conclude that the PTE contribution
should be negligible. For the PTE, local heating due to the laser irradiation
causes a temperature gradient ($\Delta T$) which is converted into a voltage
difference ($V_{PTE}$) due to a difference in the Seebeck coefficient between
the 1T’ ($S_{\mathrm{1T^{\prime}}}$) and 2H ($S_{\mathrm{2H}}$) phase 1, 24.
By using the maximum generated photocurrent in our measurements, we calculate
a range of unrealistic temperature gradients above $7000$ K, indicating that
the PTE cannot, solely, explain the observed photocurrent. A more detailed
explanation can be found in the supporting information.

On the other hand, for the PVE driven photocurrent, the localized electric
field at the 1T’–2H interface causes the photo-induced electron-hole pairs to
separate, resulting in a photocurrent 24. For our n-type MoTe2, the band
alignment is depicted in Fig. 3(a). The electric field from the Schottky
barrier is positioned in the 2H-region. By taking a line scan of the
reflection map and photocurrent map, indicated by the red line in Fig. 2(c),
we can more accurately determine the position of the photocurrent peak with
respect to the 1T’–2H junction. Here, we find that the peak of the
photocurrent arises in the 2H region rather than at the 1T’–2H junction, which
is in line with the expectation for the PVE 22. Additionally, we can estimate
the depletion region due to the Schottky barrier by using the following
equation:

$W=\sqrt{\frac{2\epsilon_{0}\epsilon_{r}\phi_{bi}}{eN_{d}}},$ (1)

where $W$ is the width of the Schottky barrier, $e$ is the electron charge,
$\epsilon_{0}$ is the permittivity of free space. By assuming a donor density
$N_{d}$ of $10^{11}$ cm2 25, a barrier height $\phi_{bi}$ of 60 meV 19 and a
relative $\epsilon_{r}$ of 12, we estimate $W$ to be $\sim 2$ µm, which
corresponds well to the FWHM of $1.5\pm 0.2$ µm we find by fitting the
positive photocurrent peak with a Gaussian.

Buscema et al. observe a similar photocurrent sign in their scanning
photocurrent measurements on an n-type MoS2 photodiode 26. However, they
attribute the induced photocurrent to the PTE, as they observe a clear
photocurrent generation in the center of their Ti/Au contacts and see a linear
$I_{ds}$-$V_{ds}$ behavior with no indication of a Schottky barrier. In
contrast, for our 1T’-2H MoTe2 junctions, we observe a clear non-linear IV-
curve, indicating that the Schottky barrier plays a more important role in our
devices. Furthermore, they see a pronounced photocurrent even when exciting
below the bandgap of MoS2. Unfortunately, our optoelectronic setup only allows
for excitation up to 1100 nm, which is still above the bandgap of MoTe2
($\sim$1.1 eV $\propto$ $\sim$1127 nm for bulk) 27, 28. Therefore, we suggest
further research to be performed on below band gap excitation to determine to
what extent the PTE is contributing to the observed photocurrent.

To characterize the optoelectronic performance of our MoTe2 photodetector, we
perform time-resolved and power-dependent photocurrent measurements, depicted
in Fig. 4. By measuring the induced photocurrent versus time, using a chopper
to modulate the light on and off (see Fig. 4(a)), we are able to extract the
rise ($\tau_{r}$) and fall time ($\tau_{f}$)) of the device, which are defined
as the time required for the photocurrent to increase from 10% to 90%, and
decrease from 90% to 10% of the maximum photocurrent, respectively. By
performing these measurements over a range of different excitation
wavelengths, we find that we get a short rise and fall time of $\sim$113 µs
and $\sim$110 µs, respectively, independent of the wavelength as depicted in
Fig. 4(b). These response times correspond to a 3 dB frequency of
$0.35/\tau_{r}=3$ kHz 29, which are close to the performance of
graphene/MoTe2/graphene photodetectors 30. In contrast, when directly
contacting the 2H MoTe2 with Ti/Au electrodes, we find a much slower response,
as shown in Fig. S2(c) in the supporting information. Here, we observe a
waveform similar to a “sawtooth” response, indicative of capacitive behavior,
already at 20 Hz. This is a result of the highly resistive contact with the 2H
region of the device, leading to photogating and slow charging of the device.
This shows that using the 1T’ regions to contact the 2H MoTe2 increases the
temporal response of our MoTe2 photodetector by more than two orders of
magnitude.

The response dynamics displayed at the 1T’–2H junction are fast compared to
other TMD-based photodetectors 31, 29. More specifically, compared to other
MoTe2 based photodetectors, they are one order of magnitude faster than the
report of Huang et al. on 2H MoTe2 32, and similar to the ones found by Lin et
al. in 1T’–contacted 2H MoTe2 19. On other TMD-based devices, a variety of
different response dynamics are reported, with the fastest responses reported
for deep UV and mid-IR detectors on graphene/MoTe2/black phosphorus devices,
which reach bandwidths of 2.1 MHz 33.

To determine the responsivity of our devices, we vary the excitation power at
a fixed excitation wavelength (700 nm and 1068 nm). From these power dependent
measurements, we are able to determine the responsivity by: $R=I_{PC}/P$,
where $I_{PC}$ is the induced photocurrent and $P$ is the power of the laser
1, 24, and find a maximum $R$ of $4.5\times 10^{-8}$ A/µW with a wavelength of
700 nm and a 2 V bias. The value we find is comparable to other reports on
TMD-based photodetectors, ranging approximately from $7.25\times 10^{-11}$
A/µW to $9.708\times 10^{-3}$ A/µW 31, 34, 32, 35. Given that there is
commonly a trade-off between fast response and high responsivity in these
devices 29, it is not unexpected that the large responsivity of our devices
falls within the middle or lower half of the reported range. Additionally, the
responsivity of our device is measured with a focused laser spot rather than
illuminating the entire device. This could lead to an underestimation of the
generated photocurrent as only a small fraction of the photodetector area is
used to generate the photocurrent.

Figure 4(c) clearly shows a decrease of responsivity with incident excitation
power for $P>1.9$ µW, which is commonly observed in TMD photodetectors 36, 37.
It can be associated with a reduced number of photogenerated carriers
available for extraction under high photon flux due to the saturation of
recombination/trap states that influence the lifetime of the generated
carriers 38. The responsivity versus laser power can be expressed by a power
law $R\propto P^{\alpha-1}_{d}$ for $P>1.9$ µW, where $P$ is the laser power,
and $\alpha$ is the index of the power law 32, 39. From the fit, we are able
to extract $\alpha$ for the two different wavelengths at different $V_{ds}$ as
shown in Fig. 4(d). The deviation from the ideal slope of $\alpha=1$, where
the responsivity does not dependent on the laser power, can be attributed to
complex processes in the carrier generation, trapping, and electron-hole
recombination in the MoTe2 40, 41. For the PTE, a value of $\alpha\sim 0.8$ is
expected, while we find a value of $\alpha\sim 0.25$, which indicates again
that the PTE is not primarily responsible for the generated photocurrent in
our device 21.

##  3 Conclusion

In conclusion, our results indicate that contacting the 2H region of MoTe2 via
a phase-transformed 1T’ region is beneficial for the temporal optoelectronic
response of MoTe2 based photodetectors. Our scanning photocurrent measurements
and non-linear IV curves, clearly show that the origin of the photocurrent in
our devices can be ascribed to the Schottky barrier between the 1T’–2H
junction, rather than the photo-thermoelectric effect or Schottky barriers at
the Ti/Au electrode-TMD interface. Contacting the MoTe2 via the phase-
transformed 1T’ region, therefore, allows one to study the intrinsic
properties of the TMD rather than the electrode-TMD interactions, beneficial
for fundamental research. Additionally, an increase of 2 orders of magnitude
in the optoelectronic temporal response is observed when contacting the 2H
MoTe2 via the 1T’ regions. This shows that tailoring the crystallographic
phase of TMDs locally, altering their optoelectronic response at will, can
have a profitable effect on the optoelectronic operation. Our results, in
combination with the wide variety of phase-engineering techniques and
different TMDs available, could lead to a further improved performance of TMD-
based optoelectronic devices, leading to more sensitive, faster and flexible
photodetectors.

##  4 Methods

###  4.1 Device fabrication

The 2H MoTe2 flakes are obtained by mechanical exfoliation (bulk crystal
supplied by HQ Graphene) and transferred onto a Si/SiO2 (285 nm) substrate in
a nitrogen environment. Using an optical microscope, the MoTe2 flakes are
selected based on their size, thickness, and homogeneous surface. Next, the
Raman spectra are obtained with an inVia Raman Renishaw microscope using a
linearly polarized laser in backscattering geometry. The excitation wavelength
and grating used were $\lambda=532$ nm and $2400$ l/mm respectively. The laser
power was $\sim 100$ $\upmu\mathrm{W}$ with a diffraction-limited spot of
$\sim 1$ µm. Using the same system, the 2H-1T’ phase transformation is
performed by selectively illuminating parts of the MoTe2 flake with the 532 nm
laser beam in a raster-like fashion, using steps of 500 nm and 0.1 s
illumination. We find that a laser power $\geq 3.25$ $\mathrm{mW}$ (laser spot
size around 500 nm) is needed to initiate the phase transformation. Finally,
using standard lithography techniques, the Ti/Au (5 nm/55 nm) contacts are
fabricated on top of the flake by means of electron beam lithography and
electron beam evaporation.

###  4.2 Optoelectronic measurements

The electrical characterization (i.e. IV-sweeps, transfer curves) is performed
using Keithleys 2400 and 2450 source measure units at 78 K. For the
optoelectronic measurements, a supercontinuum white light laser (NKT Photonics
SuperK EXTREME) is used as illumination source, and the measurements are taken
at room temperature. The induced photocurrent is measured in a short circuit
configuration using a Stanford Research Systems SR830 lock-in amplifier, which
is referenced to the frequency of the optical chopper. The photocurrents are
either measured directly by the lock-in amplifier, or converted to a voltage
using a home build current pre-amplifier, which is subsequently measured by
the lock-in amplifier. The time-resolved photoresponse of the device, depicted
in Fig. 3(a), is measured using a chopper and an oscilloscope (Keysight
DSOX1204A) at room temperature.

##  5 Acknowledgements

The authors acknowledge Prof. M. A. Loi and E. K. Tekelenburg for their help
with the Raman measurements and thank J. G. Holstein, H. Adema, H. de Vries,
A. Joshua, and F. H. van der Velde for their technical support. Sample
fabrication was performed using NanoLabNL facilities. This work was supported
by the Dutch Research Council (NWO—STU.019.014), the European Union (ERC,
2D-OPTOSPIN, 101076932), the “Materials for the Quantum Age” (QuMat) program
(Registration No. 024.005.006) which is part of the Gravitation program
financed by the Dutch Ministry of Education, Culture and Science (OCW), the
Zernike Institute for Advanced Materials, and the innovation program under
grant agreement No. 881603 (Graphene Flagship).

##  6 Author contributions

J.H. and C.A.C.S. fabricated the samples, and together with D.V. performed
both the electrical and optical measurements under the supervision of
M.H.D.G.. K.R. joined to perform measurements and sample characterization on
additional samples also under the supervision of M.H.D.G.. J.H. performed the
data analysis and, together with C.A.C.S and M.H.D.G., wrote the paper with
comments from all authors.

##  7 Supporting Information

The Supporting Information contains: details over the Photothermoelectric
temperature gradient calculation; the laser spot size calculation and data for
700 and 1064 nm for the reported measurements; equation and additional details
for charge carrier mobility and further details of another phase-engineered
device, including responsivity and transfer curves.

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