# High mobility SiMOSFETs fabricated in a full 300mm CMOS process

Timothy N. Camenzind  These authors contributed equally to this work
Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel,
Switzerland  Asser Elsayed  These authors contributed equally to this work
Department of Physics and Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001
Leuven, Belgium  IMEC, Kapeldreef 75, B-3001 Leuven, Belgium  Fahd A.
Mohiyaddin  IMEC, Kapeldreef 75, B-3001 Leuven, Belgium  Ruoyu Li  IMEC,
Kapeldreef 75, B-3001 Leuven, Belgium  Stefan Kubicek  IMEC, Kapeldreef 75,
B-3001 Leuven, Belgium  Julien Jussot  IMEC, Kapeldreef 75, B-3001 Leuven,
Belgium  Pol Van Dorpe  Department of Physics and Astronomy, KU Leuven,
Celestijnenlaan 200D, B-3001 Leuven, Belgium  IMEC, Kapeldreef 75, B-3001
Leuven, Belgium  Bogdan Govoreanu  IMEC, Kapeldreef 75, B-3001 Leuven, Belgium
Iuliana Radu  IMEC, Kapeldreef 75, B-3001 Leuven, Belgium  Dominik M. Zumbühl
Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel,
Switzerland

###### Abstract

The quality of the semiconductor-barrier interface plays a pivotal role in the
demonstration of high quality reproducible quantum dots for quantum
information processing. In this work, we have measured SiMOSFET Hall bars on
undoped Si substrates in order to investigate the quality of the devices
fabricated in a full CMOS process. We report a record mobility of $17.5\times
10^{3}$ cm2/Vs with a sub-10 nm oxide thickness indicating a high quality
interface, suitable for future qubit applications. We also study the influence
of gate materials on the mobilities and discuss the underlying mechanisms,
giving insight into further material optimization for large scale quantum
processors.

The spin of an electron in Silicon has been considered as one of the most
promising candidates for large-scale quantum computers, due to its long
coherence time, compactness, potential to operate at relatively high
temperatures, and compatibility with CMOS technology for upscaling [1, 2, 3].
High fidelity single and two qubit operations have been demonstrated in
academic lab-based devices [4, 5]. The scale up process to multi-qubit arrays
will however entail high quality qubits in large numbers, necessitating a
transition to industrial fabrication techniques [6].

While spin qubit-devices can be fabricated using semiconductor manufacturing
techniques, a detailed and careful analysis on the impact of different
fabrication process steps on the performance of spin qubits is crucial.
Hallbar mobility is a widely used metric to characterize the MOS interface
quality, which can provide valuable feedback to fabrication process
optimization. In this work, we perform transport measurements on SiMOSFETs and
investigate the quantum Hall effect. The gate stack for SiMOSFET is the same
as that for qubit integration [7]. We extract key characteristics of the MOS
gate stack such as critical density and mobility as well as transport and
quantum lifetimes. Comparing the obtained results to a transport model, we
identify and quantify the leading scattering terms. Additionally, we explore
possible mobility limiting factors when changing the top gate material from
PolySi to titanium nitride (TiN).

The devices measured in this work are undoped inversion gated Hall bars
fabricated in a state-of-the art 300 mm fab line [7]. The starting substrate
consists of a (100) silicon wafer with a background doping <$5\times 10^{15}$
cm-3. The Hall bar top gates are defined by electron beam lithography and
subsequent dry etching process, and the fanout metal pins are defined by deep
UV lithography. After depositing the gate material, the whole wafer is covered
with a passivation layer. In order to measure the electronic properties of the
investigated samples, a positive voltage has been applied to the top gate to
form a 2-dimensional electron gas (2DEG) at the interface of Si and SiO2.
Throughout this work, we perform four-wire electronic transport measurements
using standard lock-in techniques. Both the longitudinal and transverse Hall
voltages were measured simultaneously with differential voltage amplifiers.
Additionally, the current through the sample was measured using an IV
transimpeadance amplifier. The measurements were carried out in a dilution
refrigerator with a base temperature of 20 mK with a 9 T magnet. Higher
temperatures were achieved by heating the mixing chamber, resulting in
electron temperatures between 100 mK and 4.2 K.

Fig.1 shows the density and mobility study performed on a sample with a PolySi
top gate at three different temperatures. First, the density of charge
carriers in the 2DEG was extracted from the transverse Hall effect in the
linear regime far from quantum oscillations, particularly at the lowest
temperature with magnetic field between $\pm$1 T. The density of charge
carriers in the 2DEG has an overall linear dependence on the applied top gate
voltage $V_{G}$ as expected from a plate-capacitor model. Assuming the
relative dielectric constant $\epsilon_{r}(\mathrm{SiO_{2}})=3.9$, we extract
the thickness of the oxide and obtain a value of 8.1 $\pm$ 0.1 nm. This in
good agreement with the oxide thickness of 8 nm defined during the fabrication
process. As one can see in the inset of Fig.1a, the extracted density deviates
from a linear regime at very low $V_{G}$ near the threshold voltage,
particularly at the lowest temperature. A temperature dependent dielectric
constant would result in a temperature dependent slope of the $n_{S}$
regardless of the $V_{G}$ [8], which is not observed in this study. Partial
localization of charge carriers in the disorder potential at low sheet
density, on the other hand, could explain this [9]. Here, the disorder
potential is mainly due to bulk charge scattering and scattering from oxide
charges. At these rather low densities close to the 2D Metal to Insulator
Transition (MIT), partial localization would effectively reduce the sheet
density of mobile carriers below a simple capacitor model, as observed, and
thus slightly enhance the mobility, which is inversely proportional to the
density.

The extracted densities, in combination with the sheet conductance from the
longitudinal voltage measurements, allow for the calculation of the mobility
$\mu$ at various temperatures. The resulting curves are shown in Fig.1b. We
measured a maximum mobility of $17.5\times 10^{3}$ cm2/Vs, among the highest
mobility reported for SiMOSFET devices with sub 10 nm oxide thickness [10, 11,
12]. We fit the mobility curve with the Kruithof-Klapwijk-Bakker (KKB) model
(Ref. [13]) which includes remote impurity scattering and surface roughness
terms [13, 14]. Solving the model self-consistently for three separate
temperatures (0.5, 0.9 and 4.2 K) yields the charged impurity center density
$N_{C}=$ $1.53\pm 0.03\times 10^{10}$ cm-2, the rms height of the amplitude of
the interface roughness $\Delta=$ 5.9 $\pm$ 0.51 Å and the correlation length
$L=$ 1.8 $\pm$ 0.35 nm. These numbers indicate a smooth and clean interface
which is critical for future qubit implementations.

As shown in Fig.1b, there is a slight deviation of the maximum mobility from
the theory curve at 0.9 K, and the theoretical curve does not agree with the
reported data for 0.1 K in the low density regime. This could be explained by
the assumption of a linear relationship between mobility and temperature in
the KKB theory. According to our fitting, this assumption seems to be only
valid above 1 K and can become a more complex function of temperature below 1
K if the low density regime becomes non-linear. This is due to the fact that
the slightly reduced density (compared to the capacitor model) results in a
slightly higher mobility, explaining the deviation of our data from the
theory. Furthermore, for sheet densities $n_{S}$ close to the depletion charge
density $n_{dep}$, the calculation of the finite extent of the wave function
in the KKB theory can become unreliable because of the assumed approximations.
The theoretically predicted curve can be recovered by using a linearly
extrapolated density during the calculation of the mobility. Further details
are discussed in Appendix A.

Another key parameter is the percolation density, which indicates the minimum
density required to form a conducting channel. Generally, the percolation
density describes the disorder in the channel at low densities, where quantum
devices typically operate. In Fig.1d, we employed a MIT model in order to fit
the density-dependent conductivity $\sigma_{xx}\sim(n-n_{p})^{p}$ [15], where
$n_{p}$ and $p$ are the percolation density and universal exponent,
respectively. However, this model is only valid in the low density regime and
deviates above $n=$$8\times 10^{11}$ cm-2. We fixed the exponent $p=1.31$, as
expected in a two-dimensional system, and allowed $n_{p}$ to vary [15]. Our
fits yield $n_{p}=$ $1.86\times 10^{11}$ cm-2 at $T=$ 4.2 K. As seen in the
inset of 1d the extracted $n_{p}$ decreases with decreasing temperature, which
is in agreement with previous studies [15], and is below $1\times 10^{11}$
cm-2 at the lowest temperature. Such a low percolation density is among the
lowest reported for similar devices[12, 16]. The small percolation density and
high peak mobility indicate low interface disorder of the MOS gate stack,
which is crucial for large scale quantum dot arrays.

Additionally, we investigate the temperature dependence of mobility at various
top gate voltages. For sheet densities less than $2\times 10^{12}$ cm-2 the
mobility is a function of temperature, with a linear dependence for
temperatures above 1 K (see Fig.1c). The temperature dependence of the
mobility is only visible for low electron densities because mobilities at
higher densities are limited by surface roughness, which is not temperature
dependent. Additionally, the peak mobility shifts to higher densities at
higher temperatures due to the activation of background charges limiting the
mobility at high temperatures.

In order to extract more information about the transport properties and the
dominating scattering mechanism, we investigated the transverse resistance Rxy
and longitudinal resistivity $\rho_{xx}$ in the quantum Hall regime. The
magnetic field $B$ was swept up to 8.5 T at a fixed sheet density of
$7.1\times 10^{11}$ cm-2 near the peak mobility. As shown in Fig.2a, we
observe Shubnikov-De Haas (SdH) oscillations starting at 0.8 T and begin to
resolve spin splitting at 3.2 T. The $\rho_{xx}$ shows well developed zeros
for magnetic fields larger than 3.8 T. In combination with the single
frequency of the SdH oscillations extracted below 2.5 T and the absence of
beatings strongly indicate single subband occupancy. We further observe a
four-fold degeneracy in the filling factor which changes to a two-fold
degeneracy as we increased the magnetic field (Fig.2b). The remaining two-fold
degeneracy is due to the two lowest lying valleys which are split off from the
remaining four valleys due to the z-confinement in Silicon (100) [17]. A
linear fit of the filling factor (Fig.2b) with respect to inverse magnetic
field yields a sheet density $7.1\times 10^{11}$ cm-2 in agreement with the
extracted Hall density, further confirming clean single subband transport in
the investigated samples.

Upon further investigations into the SdH oscillation amplitudes below 2.5 T,
we were able to extract an effective mass $m^{*}=0.21\pm 0.02\cdot m_{0}$ from
the temperature dependent data, which corresponds to the transverse mass
expected for Silicon samples with similar background doping [18]. Further
details are discussed in Appendix B. From fitting the SdH envelope as a
function of temperature and density, we can extract the quantum lifetime
$\tau_{q}$. By combining $\tau_{q}$, which includes all possible scattering
events, and the transport lifetime $\tau_{t}=e*\mu/m^{*}$, which includes only
the large angle scattering, we extract the so-called Dingle ratio
$\tau_{t}/\tau_{q}$. The Dingle ratio thus characterizes the dominant
scattering mechanism ranging from a ratio of $1$ for predominant large angle
scattering to values $>>1$ for dominant small angle scattering. Fig.2c shows
an increase of the Dingle ratio from $\approx 1$ to $\approx 2$ with
increasing temperature, indicating a shift in the type of scattering
dominating at different temperature ranges. We can see an inverted trend for
the Dingle ratio with respect to density. For a fixed temperature of 0.5 K, we
observe a decrease from a ratio of $\approx 1.5$ to $\approx 1$ by increasing
the density, again showing a slight shift towards large angle scattering. This
is true for all measured temperatures and densities. In all, the observed
Dingle ratio around 1 to 2 shows that short range / large angle scattering is
the dominant scattering mechanism in this system, where the scattering centers
are near or in the channel itself [18].

Furthermore, we investigate the effect of two different gate metals on the
mobility and percolation density. We chose PolySi and titanium nitride (TiN)
as top gate materials since both are widely applied gating materials for
integrated CMOS device fabrication. While PolySi has generally larger
mobilities, which indicates a small disorder potential and therefore a low
dot-to-dot variation, TiN exhibits low resistivity and is superconducting at
low temperatures, which is desirable for microwave qubit control and readout.
The density dependent mobilities at 4.2 K, for both TiN and PolySi are shown
in Fig.3a. There is a decrease by a factor of three in the maximum mobility
when changing the material from PolySi to TiN while all other parameters in
the fabrication remain the same. To investigate the underlying mechanism for
the mobility difference, we perform high magnetic field Hall measurements on
the TiN devices as well. Unlike PolySi which shows vanishing $\rho_{xx}$ in
Fig.2a, $\rho_{xx}$ of TiN does not have well developed zeros. This can be
attributed to the lower mobility and could be due to multiple subband
transport [19] (see Appendix C for further details). At high electron
concentrations and in the presence of compressive stress, multiple subbands
can be occupied and thus introduce inter-subband scattering at low
temperatures.

To explain the inter-subband scattering, we firstly consider the effect of
strain. On silicon (100) surfaces, materials used as top gates with certain
coefficient of thermal expansion (CTE) introduce compressive strain [20]. This
leads to a decrease in the energy splitting of two subsequent subbands [18,
21, 22]. Strain simulations, described in Appendix D, indicate a large
difference in strain between TiN and PolySi. However, the strain could only
decrease the TiN mobility by roughly 10 percent rather than the three fold
decrease observed [22]. This difference could be explained by the oxygen
scavenging of TiN on SiO2 [23]. When depositing TiN on a SiO2 surface, SiO2
gets reduced to SiOx and the free oxygen combines with TiN to form TiO2 and
titanium oxynitride (TiNxOy). This process creates charged oxygen vacancies in
the SiO2 layer and because the oxide is very thin in these samples, the charge
carriers can then scatter at these defects thus reducing the overall mobility.
Indeed, fits of the KKB model to the TiN data indicate a significantly higher
$N_{C}$ than PolySi consistent with the scavenging hypothesis. Those two
effects combined therefore could explain the difference in mobilities between
the two gate metals.

In summary, we have characterized SiMOSFET samples using Hall effect
measurements. We found a record mobility of $17.5\times 10^{3}$ cm2/Vs which
is the highest reported mobility to date for SiMOSFET samples with sub-10 nm
oxide thickness. Further, we have shown that the dominant mobility limiting
factor are charges near the conducting channel, which can be improved by
further reducing the impurity levels in the substrate. For example, an
epitaxial layer can reach a background doping level of roughly $1\times
10^{12}$ cm-3, which is three orders of magnitude lower than the samples
measured in this study. Additionally, we show that the choice of top gate
material directly affects the mobility and percolation density. Factors such
as strain and oxygen scavenging could explain the strong influence on both
parameters. This demonstrates that the choice of top gate material is of
paramount importance. The MOS interface has been widely identified as the
source for charge noise and disorder sites, which limits the qubit performance
and upscaling. To further improve the qubit device fabrication, more
systematical investigations are needed. The Hallbar mobility and the analysis
presented in this study could provide valuable insight into the gate stack
over a large device area and direct further fabrication optimizations, which
could pave the way for a large scale spin qubit processor integration with a
full CMOS process.

## Acknowledgements

This work was performed as part of imec’s Industrial Affiliation Program
(IIAP) on Quantum Computing. Additionally, this work was supported by the
Swiss Nanoscience Institute (SNI), the Swiss National Science Foundation
(SNF), the National Centers of Competence in Research (NCCR) SPIN and the EU
H2020 European Microkelvin Platform (EMP) grant No. 824109.

## Author contributions

T.N.C., A.E. and D.M.Z performed the experiments, analysed the measurements
and wrote the manuscript with input from all authors. B.G. designed the test
structures. S.K. and J.J. fabricated the devices. F.A.M. performed the strain
simulations. The work was completed with assistance from R.L., P.VD. and I.R..

## Data availability statement

The data supporting the plots of this paper are available at the Zenodo
repository at https://doi.org/10.5281/zenodo.xxxxx.

## Appendix

###  .1 A. Mobility as a function of linear density and MIT fitting range

During the calculation of the mobility at the lowest temperature, we observed
a peak mobility higher than the KKB model (see Fig.1b). We associated this
deviation to the non-linear density dependence in the small top gate voltage
regime at the lowest temperature as shown in the inset of Fig.1a. This
assumption is further supported because the theoretical curve can be recovered
using a linear extrapolated density when calculating the mobility (see Fig.4).
This behaviour can be explained within the KKB theory, where the depletion
charge density $n_{dep}$ is implicitly assumed to be significantly smaller
than the sheet density $n_{S}$ especially in the critical regime of maximum
mobility [13]. If the density becomes non-linear, this violation of the linear
density assumption has a significant impact when calculating the finite extent
of the wave function $b$ within the Coulomb scattering term
$\braket{}{U_{C}(q)}{{}^{2}}$ and the dielectric response function
$\varepsilon$. Further research in this regime is needed in order to capture
the behaviour found in this work, but it exceeds the scope of this work.
Nevertheless, we emphasize the mobility of $17.5\times 10^{3}$ cm2/Vs is the
properly measured mobility, and the slightly lower peak mobility of
$16.5\times 10^{3}$ cm2/Vs in Fig.4 is a modeled mobility assuming a linear
density model. The previous mobility record was $\approx$$10\times 10^{3}$
cm2/Vs for a comparable sample although the oxide was slightly thicker in that
work[12].

The metal to insulator transition model (MIT) we used in this work is only
valid at very low densities (below $1\times 10^{12}$ cm-2 and therefore is
rather sensitive to the fitting range. In order to find the appropriate
fitting range, we varied both the minimum and maximum value independently. As
shown in Fig.4, we determined that fitting the data between $2.7\times
10^{11}$ cm-2 and $6.0\times 10^{11}$ cm-2 yields the most reliable outcome
because variations in the fitting range do not significantly change the
result.

###  .2 B. Extraction of effective mass and quantum lifetime

In order validate our evaluation approach and our experiments, we extract the
effective mass from the temperature dependence of the SdH oscillations. For
that, we subtract a polynominal background from the $\rho_{xx}$ data and use
the following formula as described in [24, 25]

$\frac{\Delta\rho_{xx}(T)}{\Delta\rho_{xx}(T_{0})}=\frac{T}{T_{0}}\cdot\frac{\sinh\chi(T)}{\sinh\chi(T_{0})}$
(1)

with $\chi(T)=2\pi^{2}k_{B}T/\hbar\omega_{c}$ and $\omega_{c}=eB/m^{*}$. Here,
$T$ is the temperature, $T_{0}$ the lowest temperature, $k_{B}$ is the
Boltzmann constant, $\hbar$ the reduced Plank constant, $e$ the electric
charge, $B$ the magnetic field and $m^{*}$ the effective mass. Using the
effective mass as the only free fit parameter, we extracted $m^{*}=0.21\pm
0.02\cdot m_{0}$, which is close to the expected value of $m^{*}=0.19\cdot
m_{e}$.

Further, the envelope of the SdH oscillations allows for the extraction of the
quantum lifetime $\tau_{q}$. For that, we use the following formula

$ln\left(\frac{\Delta\rho_{xx}(B)\sinh\chi}{\chi}\right)=C-\frac{\pi
m^{*}}{eB\tau_{q}}$ (2)

Therefore, by analysing the slope of the magnetic field dependent logarithmic
term we can extract $\tau_{q}$ for all measured densities and temperatures. In
combination with the transport lifetime $\tau_{t}$ we can then further extract
the dingle ratio as described in the main text.

###  .3 C. TiN data at high fields

In comparison to the reported data in Fig.2a, TiN does not show well developed
zeroes in the longitudinal resistivity $\rho_{xx}$ at high magnetic fields at
a temperature of 500 mK (see Fig.7), as expected for a much lower mobility.
Further, the transversal Hall resistance $R_{xy}$ shows oscillating features
instead of the expected Hall plateaus. This could be due to multiple subbands
contributing to the overall transport or with the increased density of
background charges due to the above mentioned oxygen scavenging.

###  .4 D. Strain simulations

We quantify the stain induced in the 2DEG due to the different thermal
expansion coefficients of both gate materials. The strain was simulated using
a commercially available software [26]. The silicon was simulated
isotropically. The temperature dependence of the CTE for PolySi up to room
temperature is assumed to be linear because there is no reported data above
200 K [27]. Further, we neglect the intrinsic strain induced during the
fabrication process, which is beyond the scope of this work. Thermal
contraction is considered from room temperature down to the cryogenic
measurement temperatures. All simulations were performed in three dimensions
and the components of strain $\epsilon_{x},\epsilon_{y},\epsilon_{z}$ are
calculated 5 nm below the oxide due to the finite extension of the
wavefunction. As shown in Fig.7b and c, there is a large difference in strain
between TiN and PolySi. While TiN induces strain up to $2.4\times 10^{-4}$ ,
PolySi induces significantly less strain of approximately only $0.28\times
10^{-4}$ .

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Figure 1: Density and Mobility study (a) Linear relationship between 2DEG
sheet density $n_{S}$ and the top gate voltage $V_{G}$. Below a top gate
voltage of $\approx$ 0.4 V the extracted density deviates from the expected
linear behaviour as shown in the inset. (b) Channel mobility $\mu$ measured as
a function of $n_{S}$ and corresponding fit to a mobility model (black) which
includes scattering from charge impurities and surface roughness. The inset
shows a SEM image of a similar device as the one used during the experiments
with the same dimensions. (c) Channel mobility $\mu$ trends with respect to
temperature at various sheet densities. The symbols coincide with the ones
used in panel b. The dashed lines are linear fits to the extracted mobilities
and appear curved due to the logarithmic scale. (d) Longitudinal conductivity
$\sigma_{xx}$ in the low density range and fit to a percolation theory
(black). A linear trend with respect to temperature for $n_{p}$ is observed in
the inset. Figure 2: Extended Hall studies with transport properties (a)
Longitudinal resistivity $\rho_{xx}$ (blue) and transverse resistance Rxy
(black) at $n_{S}=$ $7.1\times 10^{11}$ cm-2 as a function of magnetic field
$B$. The arrow indicates the magnetic field spin is resolved. (b) Linear
relationship between filling factor $\nu$ (blue scatter) and inverse magnetic
field. The solid line is a fit from which the sheet density is calculated. (c)
and (d) Extracted dingle ratio $\tau_{t}/\tau_{q}$ as a function of
temperature $T$ and sheet density $n_{S}$, respectively. The dingle ratio is
slightly temperature and density dependent and is $\approx 1$, which indicates
that large angle scattering is most dominant scattering mechanism in this
sample. Figure 3: Gate material study (a) The channel mobility $\mu$ with
respect to sheet density $n_{S}$ for PolySi (green) and TiN (violet), both at
4.2 K. The mobility shows a decrease by a factor of three when changing the
top gate material from PolySi to TiN. (b) Longitudinal conductivity
$\sigma_{xx}$ with respect to sheet density. The fits to percolation theory
(black) show a significant difference between the PolySi and TiN percolation
densities. Figure 4: Mobility as a function of linear density and MIT fitting
range (a) When using a linearly extrapolated (instead of the carefully
extracted) density in order to calculate the mobility of our sample (blue),
the peak mobility becomes slightly smaller than previously demonstrated in
Fig.1b. Additionally, we can fit the KKB theory (black) self-consistently with
the other temperature data highlighting the importance of the density values.
(b) By varying $\Delta n_{s}$ for extracting the percolation density $n_{p}$
within the metal to insulator transition (MIT) model, we determine the optimal
fitting range. We chose $\Delta n_{s}=$$3.2\times 10^{11}$ cm-2 (indicated by
the red box) because the extracted $n_{p}$ does not change significantly while
changing $\Delta n_{s}$. Figure 5: SdH oscillations (a) We can extract the
effective mass by analysing the temperature dependent SdH oscillations at
various magnetic field values $B$. (b) The envelope of the SdH oscillations in
the fully spin and valley degenerate regime ($<$ 2.5 T) we can extract the
quantum lifetime $\tau_{q}$ for various densities. Curves are offset for
clarity. Figure 6: TiN data at high fields Longitudinal resistivity
$\rho_{xx}$ (blue) and transverse Hall resistance $R_{xy}$ (black) of the
sample measured with a TiN top gate at 500 mK. Both the not well developed
zeroes in $\rho_{xx}$ and the oscillating $R_{xy}$ indicate multiple subband
occupancy. Figure 7: Gate material strain simulations (a) Schematic of the
Hall bar used for the strain simulation. The strain in the channel was
calculated using sprocess in the region indicated by the dashed white line.
(b, c) The strain in the channel for the PolySi and TiN respectively. The
model uses literature values for the thermal expansion coefficient for both
materials.


# The Top Dimensional Singular Set $\text{sing}_{*}u$

Owen Drummond
Department of Mathematics, Rutgers University
<EMAIL_ADDRESS>

Advised by: Natasa Sesum

###### Abstract

In this survey, we examine the properties of the top dimensional singular set
$\text{sing}_{*}(u)$, including several theorems, geometric properties, and
relation to homogeneous degree zero minimizers.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Energy and $\epsilon$-regularity
    2. 2.2 Tangent Maps and sing(u)
  3. 3 Definition of $\text{sing}_{*}u$
  4. 4 H.D.Z.M.’s with $\dim S(\varphi)=n-3$
  5. 5 The Geometry of $\text{sing}_{*}(u)$

##  1 Introduction

In geometric analysis, energy minimizing maps have been a prominent area of
study for nearly 40 years, particularly in the 1980’s and 1990’s. One specific
area of interest within this topic is the structure of the singular set
$\text{sing}(u)$, where $u\in W^{1,2}(\Omega;N)$ is energy minimizing,
$\Omega\subset\mathbb{R}^{n}$, and N is a Riemannian target manifold
isometrically embedded into some Euclidean Space $\mathbb{R}^{p}$. For
understanding the behavior of energy minimizing maps near singular points, one
investigates the nature of tangent maps, that is, maps that the extrapolate
the local behavior of an energy minimizing map to a global scale by
convergence along a subsequence. Such convergence is guaranteed by several key
compactness results about the Sobolev Space $W^{1,2}$, yet much remains
unknown about tangent maps. The question of uniqueness of tangent maps is
classically quite a difficult problem, both in this context, and in the
geometric flows as well, namely mean curvature flow and Ricci flow.

In his notable 1996 work Theorems on Regularity and Singularity of Energy
Minimizing Maps[1], Leon Simon investigates this question of uniqueness from
the perspective of geometric analysis and measure theory. In Chapter 3 of his
work, which is concerned with the structure of $\text{sing}(u)$, Simon
introduces the Top Dimensional Singular Set $\text{sing}_{*}(u)$, which is
concerned with tangent maps of dimension $(n-3)$. This set is fundamental in
his discussion of the uniqueness of tangent maps under certain prescribed
conditions, such as in Section 3.8. This paper follows Simon’s book and will
serve as an exposition of this set $\text{sing}_{*}(u)$, including key
properties and theorems.

##  2 Preliminaries

One can reference my paper Geometric Analysis of Energy Minimizing Maps[2] to
study these preliminaries in greater detail. We will assume that the reader is
familiar with basic Real Analysis and PDE Theory. The general framework which
we will operate under remains the same: let $\Omega$ is an open subset of
$\mathbb{R}^{n}$, for $n\geq 2$, and that $N$ is a compact, $C^{\infty}$
Riemannian Manifold which is isometrically embedded into $\mathbb{R}^{p}$ for
some $p$. Assume $u:\Omega\rightarrow\mathbb{R}^{p}$ such that
$u(\Omega)\subset N$. For $\Omega$ and $N$ defined in this way, we denote the
Sobolev Space $W_{\text{loc}}^{1,2}(\Omega;N)$ to be the set of functions
$u\in W_{\text{loc}}^{1,2}(\Omega;\mathbb{R}^{p})$ with $u(x)\in N$ a.e.
$x\in\Omega$.

###  2.1 Energy and $\epsilon$-regularity

  * Definition2.1.1 (The Energy Functional).

The energy $\mathcal{E}_{B_{\rho}(y)}(u)$ for a function $u\in
W_{\text{loc}}^{1,2}(\Omega;N)$ in a ball $B_{\rho}(y)=\\{x:|x-y|<\rho\\}$
with $\bar{B}_{\rho}\subset\Omega$ is

$\mathcal{E}_{B_{\rho}(y)}(u)=\int_{B_{\rho}(y)}|Du|^{2}$

  * Definition2.1.2 (Energy Minimizing Maps).

A map $u\in W^{1,2}(B_{\rho}(y);N)$ is called energy minimizing if for each
ball $B_{\rho}(y)\subset\Omega$

$\mathcal{E}_{B_{\rho}(y)}(u)\leq\mathcal{E}_{B_{\rho}(y)}(w)$

for every $w\in W^{1,2}(B_{\rho}(y);N)$ with $w=u$ in a neighborhood of
$\partial B_{\rho}(y)$.

This gives way to the following fact: a map $\psi$ is harmonic if it is a
critical point of the energy functional. More specifically, let
$\bar{B}_{\rho}(y)\subset\Omega$ as before, and for any $\delta>0$, we have
the 1-parameter family $\\{\psi_{s}\\}_{s\in(-\delta,\delta)}$ such that
$\psi_{0}=\psi$, $D\psi_{s}\in L^{2}(\Omega)$, and $\psi_{s}\equiv\psi$ in a
neighborhood of $\delta B_{\rho}(y)$ for all $s\in(-\delta,\delta)$. Thus, any
such $\psi_{s}$ is harmonic if and only if

$\frac{d\mathcal{E}_{B_{\rho}(y)}(\psi_{s})}{ds}|_{s=0}=0$

this same equation leads to the Euler-Lagrange equations for the map, which
are harmonic map equations. As a consequence, harmonic maps are also local
minimizers of the energy functional, that is, in an $\epsilon$-neighborhood of
a map, no other map has lower energy than a harmonic map. However, there is no
such global condition for harmonic maps as energy minimizers.

Here is an important theorem of Schoen and Uhlenbeck [3] that bears on the
regularity of energy minimizing maps and will be employed several times
throughout this paper:

  * Theorem2.1.3 ($\epsilon$-Regularity Theorem).

Let $\Lambda>0$, $\theta\in(0,1)$. There exists
$\varepsilon=\varepsilon(n,\Lambda,N,\theta)>0$ such that if $u\in
W^{1,2}(\Omega;N)$ is energy minimizing on $B_{R}(x_{0})\subset\Omega$ and if

$R^{2-n}\int_{B_{R}(x_{0})}|Du|^{2}\leq\Lambda\quad\text{and}\quad
R^{2-n}\int_{B_{R}(x_{0})}|u-\lambda_{x_{0},R}|^{2}<\varepsilon^{2},$

then there holds $u\in C^{\infty}(B_{\theta R}(x_{0}))$, and for
$j=1,2,\ldots$ we have the estimates

$R^{j}\sup_{B_{\theta R}(x_{0})}|D^{j}u|\leq
C(R^{-n}\int_{B_{R}(x_{0})}|u-\lambda_{x_{0},R}|^{2})^{1/2},$

where $C$ depends only on $j$, $\Lambda$, $N$, $\theta$, and $n$.

Essentially, this theorem states that if the average deviation from an
approximate mean of an energy miniziming map is sufficiently small within some
ball $B_{R}(x_{0})$, then $u$ must be quite regular within a smaller ball
$B_{\theta R}(x_{0})$. This is foundational as it bridges the gap between
local behavior and global regularity, directly impacting our understanding of
singularities in geometric flows.

###  2.2 Tangent Maps and sing(u)

Next, we introduce the notion of tangent maps. In essence, these maps
extrapolate local behavior of some energy minimizing map to a global function.
Given $u:\Omega\rightarrow\mathbb{R}^{p}$ and $B_{\rho_{0}}(y)$ such that
$\bar{B}_{\rho_{0}}(y)\subset\Omega$, and for any $\rho>0$, consider the
scaling function $u_{y,\rho}$ given by

$u_{y,\rho}(x)=u(y+\rho x)$

Note that on $B_{\rho_{0}}(0)$, $u_{y,\rho}$ is well defined. For $\sigma>0$
and $\rho<\frac{\rho_{0}}{\sigma}$, after making a change of variables with
$\tilde{x}=y+\rho x$ in the energy integral for $u_{y,\rho}$ and noting that
$Du_{y,\rho}(x)=\rho(Du)(y+\rho x)$, we have

$\displaystyle\sigma^{2-n}\int_{B_{\sigma}(0)}|Du_{y,\rho}|^{2}=(\sigma\rho)^{2-n}\int_{B_{\sigma\rho}(y)}|Du|^{2}\leq\rho_{0}^{2-n}\int_{B_{\rho_{0}}(y)}|Du|^{2}$
(1)

by the Monotonicity Formula. Therefore, if $\rho_{j}\downarrow 0$, then
$\limsup_{j\rightarrow\infty}\int_{B_{\sigma}(0)}|Du_{y,\rho_{j}}|^{2}<\infty$
for all $\sigma>0$, and so by the compactness, there is a subsequence
$\rho_{j^{\prime}}$ such that $u_{y,\rho_{j^{\prime}}}\rightarrow\varphi$
locally in $\mathbb{R}^{n}$ w.r.t. the $W^{1,2}$-norm.

  * Definition2.2.1 (Tangent Map).

Any $\varphi$ obtained this way is called a tangent map of u at y. Moreover,
$\varphi:\mathbb{R}^{n}\rightarrow N$ is an energy minimizing map with
$\Omega=\mathbb{R}^{n}$.

Simon’s approach to understanding the properties of tangent maps is studying
degree zero minimizers, a broader class of functions which tangent maps
belong.

  * Definition2.2.2 (Degree Zero Minimizer).

A function $\varphi:\mathbb{R}^{n}\rightarrow N$ is a degree zero minimizer if
$\varphi(\lambda x)\equiv\varphi(x)$ for all $\lambda>0$, $x\in\mathbb{R}^{n}$

More on the properties of degree zero minimizers can be found in my previous
paper.

  * Definition2.2.3 (Regular and Singular Sets).

If $u\in W^{1,2}(\Omega:\mathbb{R}^{p})$, then

$\text{reg}(u):=\\{x\in\Omega:u\text{ is }C^{\infty}\text{ in a neighborhood
of }x\\}$

is the regular set of $u$, and

$\text{sing}(u):=\Omega\setminus\text{reg}(u)$

is the singular set of $u$.

The singular set $\text{sing}(u)$ is topic of great interest in the study of
energy minimizing maps due to its obscure geometric structure. Certain
techniques from geometric measure theory, primarily rectifiabilty and gap
measures, have proven to be particularly effective, and are explored in
Simon’s work.

##  3 Definition of $\text{sing}_{*}u$

As seen in my previous work (p. 13)[2], $\text{sing}(u)$ consists of only
isolated points, that is, $\mathcal{H}^{n-2}(\text{sing}(u))=0$. Thus,
$\text{sing}(u)$ is best understood in the context of codimension 3, that is,
dimension $n-3$. In 2015, Aaron Naber and Daniele Valtorta[4] have proven the
remarkable result that for a smooth, compact target manifold $N$,
$\mathcal{H}^{n-3}(\text{sing}(u))<\infty$ uniformly, and this result was
previously unknown for some time. This was proven using modern technique of
quantifiable rectifiability in geometric measure theory.

Here, we are concerned with target manifolds $N$ of dimension $n-3$. However,
this discussion is still meaningful in the case of dimension $m\leq n-4$ as
long as $N$ happens to be such that all homogeneous degree zero (tangent) maps
$\varphi\in W_{\text{loc}}^{1,2}(\mathbb{R}^{n};N)$ of $u$ satisfy $\dim
S(\varphi)\leq m$.

  * Definition3.1 (Top Dimensional Singular Set).

The top dimensional part $\text{sing}_{*}(u)$ of $\text{sing}(u)$ is the set
of points $y\in\text{sing}(u)$ such that some tangent map $\varphi$ of $u$ at
$y$ has $\dim S(\varphi)=n-3$.

from this we obtain the immediate result:

  * Theorem3.2.

For $\Omega\subset\mathbb{R}^{n}$ open and $u\in W^{1,2}(\Omega;N)$,

$\dim(\text{sing}(u)\setminus\text{sing}_{*}(u))\leq n-4$

###### Proof.

Given the fact that $\text{sing}(u)=\mathcal{S}_{n-3}$, for
$y\in\text{sing}(u)\setminus\text{sing}_{*}(u)$, any tangent map $\varphi$ of
$u$ at $y$ must satisfy

$\dim
S(\varphi)=\dim\\{y\in\mathbb{R}^{n}:\Theta_{\varphi}(y)=\Theta_{\varphi}(0)\\}=n-3$

Putting these two facts together, it must be the case that this same tangent
map $\varphi$ associated with $y\in\text{sing}(u)\setminus\text{sing}_{*}(u)$
must have $\dim S(\varphi)\leq n-4$, and hence

$\text{sing}(u)\setminus\text{sing}_{*}(u)\subset\mathcal{S}_{n-4}$

Now by Lemma 1 of Section 3.4 of Simon (p. 54)[1], which states that for each
$j=0,...,n-3$, $\dim\mathcal{S}_{j}\leq j$, he result follows, namely

$\dim(\text{sing}(u)\setminus\text{sing}_{*}(u))\leq\dim(\mathcal{S}_{n-4})\leq
n-4$

as desired.

∎

To continue our discussion of $\text{sing}_{*}(u)$, we will examine
homogeneous degree zero minimizers $\varphi:\mathbb{R}^{n}\rightarrow N$ with
$\dim S(\varphi)=n-3$.

##  4 H.D.Z.M.’s with $\dim S(\varphi)=n-3$

Let $\varphi:\mathbb{R}^{n}\rightarrow N$ be an arbitrary homogeneous degree
zero minimizer (H.D.Z.M.) with $\dim S(\varphi)=n-3$. Now modulo an orthogonal
transformation of $\mathbb{R}^{n}$ which takes $S(\varphi)$ to
$\\{0\\}\times\mathbb{R}^{n-3}$, and so we can write
$\varphi(x,y)\equiv\varphi_{0}(x)$, where $(x,y)\in\mathbb{R}^{n}$ with
$x\in\mathbb{R}^{3}$ and $y\in\mathbb{R}^{n-3}$, and where
$\varphi_{0}:\mathbb{R}^{3}\rightarrow N$ is a H.D.Z.M.

  * Theorem4.1.

$\varphi_{0}|_{S^{2}}:S^{2}\rightarrow N$ is a $C^{\infty}$ harmonic map.

###### Proof.

We claim that $\text{sing}\varphi_{0}=\\{0\\}$, which is sufficient to
establish the smoothness of $\varphi_{0}$. First, note that
$\text{sing}\varphi_{0}\supset\\{0\\}$, otherwise $\varphi_{0}$, and by
extension, $\varphi$ would be constant, yet $\dim S(\varphi)=n-3$, which is a
contradiction. Now suppose that $\xi\in\text{sing}\varphi_{0}$ and $\xi\neq
0$. By the homogeneity of $\varphi_{0}$, we have that

$\\{\lambda\xi:\lambda>0\\}\subset\text{sing}\varphi_{0}\implies\\{(\lambda\xi,y):\lambda>0,y\in\mathbb{R}^{n-3}\\}\subset\text{sing}\varphi$

Note that the set $\\{(\lambda\xi,y):\lambda>0,y\in\mathbb{R}^{n-3}\\}$ is a
half-plane of dimension (n-2), and hence

$\mathcal{H}^{n-2}(\text{sing}\varphi)\geq\mathcal{H}^{n-2}(\\{(\lambda\xi,y):\lambda>0,y\in\mathbb{R}^{n-3}\\})=\infty$

and thus $\mathcal{H}^{n-2}(\text{sing}\varphi)=\infty$. However, since
$\text{sing}(u)$ consists of only isolated points, for any energy minimizing
map $u\in W^{1,2}(\Omega;N)$, where $\Omega\subset\mathbb{R}^{n}$,
$\mathcal{H}^{n-2}(\text{sing}(u))=0$. Thus,
$\mathcal{H}^{n-2}(\text{sing}\varphi)=\infty$ is a contradiction, so $\xi=0$,
which establishes that $\text{sing}\varphi_{0}=\\{0\\}$. Therefore,
$\varphi_{0}|_{S^{2}}$ is away from the singularity at $0$, so the regularity
set is the entire domain $S^{2}$, so $\varphi_{0}|_{S^{2}}:S^{2}\rightarrow N$
is smooth and harmonic. ∎

Next is an important theorem related the boundedness of the $L^{2}$-norm of
the gradient of $\varphi$, with the general finiteness of the gradient of
$\varphi_{0}$ constructed in this way.

  * Theorem4.2.

If $\varphi^{(j)}$ is any sequence of H.D.Z.M.’s with
$\varphi^{(j)}(x,y)\equiv\varphi_{0}^{(j)}(x)$ for each $j\in[1,\infty)$, and
if
$\displaystyle\limsup_{j\to\infty}\int_{B_{1}(0)}|D\varphi^{(j)}|^{2}<\infty$,
then

$\displaystyle\limsup_{j\to\infty}[\sup_{S^{2}}|D^{\ell}\varphi_{0}^{(j)}|]<\infty\hskip
21.33955pt\forall\ell\geq 0$

###### Proof.

By compactness, there exists a subsequence of $\varphi^{(j)}$, say
$\varphi^{(j^{\prime})}$, such that
$\varphi^{(j^{\prime})}\rightarrow\varphi$, where $\varphi$ is a H.D.Z.M. with
$\varphi(x,y)=\varphi_{0}(x)$, and by the previous theorem,
$\varphi_{0}|_{S^{2}}\in C^{\infty}$. Thus, for
$z\in\mathbb{R}^{n}\setminus(\\{0\\}\times\mathbb{R}^{n-3})$, that is, a point
in $z=(z_{1},z_{2},...,z_{n})\in\mathbb{R}^{n}$ such that
$(z_{1},z_{2},z_{3})\neq(0,0,0)$, and a positive
$\sigma<\text{dist}(z,\\{0\\}\times\mathbb{R}^{n-3})$, we have

$|\varphi(x)-\varphi(z)|\leq
C\sigma<C\text{dist}(z,\\{0\\}\times\mathbb{R}^{n-3})=C\sqrt{z_{1}^{2}+z_{2}^{2}+z_{3}^{2}}$

by the smoothness of $\varphi$. This formulation admits a kind of Lipschitz
continuity. Hence,

$\int_{B_{\sigma}(z)}|\varphi^{(j^{\prime})}-\varphi(z)|^{2}\leq\int_{B_{\sigma}(z)}|\varphi^{(j^{\prime})}-\varphi|^{2}+C\sigma^{2}$

Now by the $\epsilon$-regularity theorem, and by letting $\sigma$ be
sufficiently small, and for $j^{\prime}$ sufficiently large, the convergence
of $\varphi^{(j^{\prime})}$ to $\varphi$ is with respect to the $C^{k}$ norm
for each $k$ on compact subsets
$K\subset\mathbb{R}^{n}\setminus(\\{0\\}\times\mathbb{R}^{n-3})$. In light of
the fact that the subsequence $\varphi^{(j^{\prime})}$ was arbitrary, it
follows that for all H.D.Z.M.’s $\varphi(x,y)\equiv\varphi_{0}(x)$, where
$x\in\mathbb{R}^{3},y\in\mathbb{R}^{n-3}$, with
$\int_{S^{2}}|D\varphi_{0}|^{2}\leq\Lambda$, we have that

$\sup_{S^{2}}|D^{\ell}\varphi_{0}|^{2}\leq C,\hskip 21.33955pt\ell=1,2,...$

and $C$ depends on the target manifold $N$, the constant $\Lambda$, and the
index $\ell$. ∎

This theorem will be a powerful resource for investigating the geometric
picture of $\text{sing}_{*}(u)$.

##  5 The Geometry of $\text{sing}_{*}(u)$

For the general framework, let $K\subset\Omega$, and $z\in\text{sing}_{*}u\cap
K$, and let $\varphi$ be a tangent map of $u$ at $z$ with $\dim
S(\varphi)=n-3$. Also, let $\varphi(x,y)=\varphi_{0}(x)$, where
$x\in\mathbb{R}^{3},y\in\mathbb{R}^{n-3}$ be as before. Leveraging the
properties of $\text{sing}_{*}(u)$, which includes the existence of a tangent
map $\varphi$ of $u$ at every point $z\in\text{sing}_{*}(u)$ that can
approximate $u$ in the $L^{2}$ over smaller and smaller balls around $z$, we
have that for $\varphi^{(z)}(x,y)\equiv\varphi((x,y)-z)$, there is a sequence
$\rho_{j}\downarrow 0$ such that

$\displaystyle\displaystyle\lim_{j\to\infty}\rho_{j}^{-n}\int_{B_{\rho_{j}}(z)}|u-\varphi^{(z)}|^{2}=0$
(2)

As a result, for $\rho=\rho_{j}$ with $j$ sufficiently large, we can control
the scaled $L^{2}$-norm given by

$\rho^{-n}\int_{B_{\rho}(z)}|u-\varphi^{(z)}|^{2}$

by making it as small as we please. From this we obtain the following theorem
which describes the geometric picture of $\text{sing}_{*}(u)$.

  * Theorem5.1.

For any H.D.Z.M.’s $\varphi:\mathbb{R}^{n}\rightarrow N$, and any ball
$B_{\rho_{0}}(z)$ with $\bar{B}_{\rho_{0}}(z)\subset\Omega$ we have the
estimate

$\textrm{sing}(u)\cap
B_{\rho/2}(z)\subset{x:\textrm{dist}(x,(z+\\{0\\}\times\mathbb{R}^{n-3}))<\delta(\rho)\rho}\hskip
14.22636pt\forall\rho\leq\rho_{0}$

where

$\delta(\rho)=C\left(\rho^{-n}\int_{B_{\rho}(z)}|u-\varphi^{(z)}|^{2}\right)^{\frac{1}{n}}$

and $C$ only depends on $n,N,\Lambda$, and $\Lambda$ is any upper bound for
$\rho_{0}^{2-n}\int_{B_{\rho_{0}}(z)}|Du|^{2}$.

Figure 1: The Geometric Picture of $\text{sing}_{*}u$

######  Remark 5.1.

In light of the fact that we can control the scaled $L^{2}$-norm as seen in
(2), one might suggest that $\text{sing}_{*}(u)$ in contained in a Lipschitz
manifold, if not a $C^{1}$ manifold of dimension $n-3$. However, given the
definition of $S(\varphi)$, $\delta(\rho)$ is small only when $\rho$ is
proportionally close to one of the $\rho_{j}$. Without further information, we
cannot say much about the geometric structure of $\text{sing}_{*}(u)$.

###### Proof.

Without loss of generality, assume $z=0$. Let $\rho<\rho_{0}$, and define
$w=(\xi,\eta)\in\text{sing}_{*}(u)\cap B_{\rho/2}(0)$. Choose
$\sigma=\beta_{0}|\xi|$, and $\beta_{0}\leq\frac{1}{2}$. By the
$\epsilon$-regularity theorem,
$\exists\epsilon_{0}=\epsilon_{0}(n,N,\Lambda)>0$ such that

$\displaystyle\epsilon_{0}\leq\sigma^{-n}\int_{B_{\sigma}(w))}|u-\varphi(w)|^{2}\leq
2\sigma^{-n}\int_{B_{\sigma}(w)}|u-\varphi|^{2}+2\sigma^{-n}\int_{B_{\sigma}(w)}|\varphi-\varphi(w)|^{2}$
(3)

which follows from triangle inequality. Now by Theorem 4.2, we have that

$|D\varphi_{0}(x)|\leq C|x|^{-1}$

and $C$ depends only on $N,\Lambda$, hence

$|\varphi(w)-\varphi(x)|\leq C|\xi|^{-1}\sigma\leq C\beta_{0}$

for $x\in B_{\sigma}(w)$. Now by (3):

$\epsilon_{0}\leq
2\beta_{0}^{-n}|\xi|^{-n}\int_{B_{\rho}(0)}|u-\varphi|^{2}+C\beta_{0}^{2}$

So by selecting $C\beta_{0}^{2}\leq\frac{1}{2}\epsilon_{0}$ and multiplying
through by $|\xi|^{n}$ on both sides, we have

$\displaystyle|\xi|^{n}\leq
C(\rho^{-n}\int_{B_{\rho}(0)}|u-\varphi|^{2})\rho^{n}$
$\displaystyle\implies|\xi|\leq
C(\rho^{-n}\int_{B_{\rho}(0)}|u-\varphi|^{2})^{\frac{1}{n}}$

where $C$ is some constant depending only on $n,N$, and $\Lambda$, which is
the inequality we desired to show. ∎

## References

  * [1] Leon Simon.  Theorems on Regularity and Singularity of Energy Minimizing Maps.  Birkhäuser Verlag, Basel, Boston, Berlin, 1996.
  * [2] Owen Drummond.  Geometric Analysis of Energy Minimizing Maps: Tangent Maps and Singularities.  2024\.  arXiv:2405.09661.  https://arxiv.org/abs/2405.09661.
  * [3] R. Schoen and K. Uhlenbeck.  A Regularity Theory for Harmonic Mappings.  Journal of Differential Geometry, 17:307–335, 1982.
  * [4] Aaron Naber and Daniele Valtorta.  Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps.  _Annals of Mathematics_ , 193(3):607–704, 2021.  https://arxiv.org/pdf/1504.02043