Current address: ] Centre for Materials Science and Nanotechnology, University
of Oslo, Oslo, 0318, Norway.

# The diamond (111) surface reconstruction and epitaxial graphene interface

B. P. Reed  Department of Physics, Aberystwyth University, Aberystwyth, SY23
3BZ, United Kingdom.  Centre for Doctoral Training in Diamond Science and
Technology, University of Warwick, Coventry, CV4 7AL, United Kingdom.
National Physical Laboratory, Teddington, TW11 0LW, United Kingdom.  M. E.
Bathen  Advanced Power Semiconductor Laboratory, ETH Zurich, Physikstrasse 3,
Zurich, 8092, Switzerland.  Centre for Materials Science and Nanotechnology,
University of Oslo, Oslo, 0318, Norway.  J. W. R. Ash  Department of Physics,
Aberystwyth University, Aberystwyth, SY23 3BZ, United Kingdom.  Centre for
Doctoral Training in Diamond Science and Technology, University of Warwick,
Coventry, CV4 7AL, United Kingdom.  C. J. Meara  Centre for Doctoral Training
in Diamond Science and Technology, University of Warwick, Coventry, CV4 7AL,
United Kingdom.  School of Electrical and Electronic Engineering, Newcastle
University, Newcastle-upon-Tyne, NE1 7RU, United Kingdom.  A. A. Zakharov  Max
IV Laboratory, Lund University, Lund, 221 00, Sweden.  J. P. Goss  School of
Electrical and Electronic Engineering, Newcastle University, Newcastle-upon-
Tyne, NE1 7RU, United Kingdom.  J. W. Wells  Centre for Materials Science and
Nanotechnology, University of Oslo, Oslo, 0318, Norway.  D. A. Evans
Department of Physics, Aberystwyth University, Aberystwyth, SY23 3BZ, United
Kingdom.  S. P. Cooil<EMAIL_ADDRESS>[  Department of Physics, Aberystwyth
University, Aberystwyth, SY23 3BZ, United Kingdom.  Centre for Materials
Science and Nanotechnology, University of Oslo, Oslo, 0318, Norway.

###### Abstract

The evolution of the diamond (111) surface as it undergoes reconstruction and
subsequent graphene formation is investigated with angle-resolved
photoemission spectroscopy, low energy electron diffraction, and complementary
density functional theory calculations. The process is examined starting at
the C$(111)\text{-}(2\times 1)$ surface reconstruction that occurs following
detachment of the surface adatoms at
$920\text{\,}\mathrm{\SIUnitSymbolCelsius}$, and continues through to the
liberation of the reconstructed surface atoms into a free-standing monolayer
of epitaxial graphene at temperatures above
$1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Our results show that the
C$(111)\text{-}(2\times 1)$ surface is metallic as it has electronic states
that intersect the Fermi-level. This is in strong agreement with a
symmetrically $\pi\text{-bonded}$ chain model and should contribute to
resolving the controversies that exist in the literature surrounding the
electronic nature of this surface. The graphene formed at higher temperatures
exists above a newly formed C$(111)\text{-}(2\times 1)$ surface and appears to
have little substrate interaction as the Dirac-point is observed at the Fermi-
level. Finally, we demonstrate that it is possible to hydrogen terminate the
underlying diamond surface by means of plasma processing without removing the
graphene layer, forming a graphene-semiconductor interface. This could have
particular relevance for doping the graphene formed on the diamond (111)
surface via tuneable substrate interactions as a result of changing the
terminating species at the diamond-graphene interface by plasma processing.

Suggested keywords

††preprint: PRB

##  I Introduction

A rejuvenated interest in diamond as an electronic material has occurred over
the last couple of decades as its importance for emerging state-of-the-art
technologies continues to become apparent. For example, the nitrogen vacancy
(NV) centres in diamond are one of the prime candidates for qubit storage in
quantum computers [1]. Within the field of diamond electronics, the majority
of studies have focused on the (100) orientation as it is the preferred growth
direction for synthesized material by low pressure chemical vapour deposition
(CVD) [2] and, until recently, atomically flat surfaces have been easier to
prepare [3]. However, the (111) surface has many desirable properties compared
to the other low index surfaces: a higher Hall electron mobility in phosphor
doped thin-films [4], a higher temperature superconducting transition
suggested to result from a non-uniform lattice expansion at high boron
concentrations [5, 6], and a higher sheet carrier density (hole mobility) for
the hole accumulation layer on hydrogen-terminated surfaces [7]; Finally, the
average alignment of NV centres can be controlled during CVD growth on (111)
substrates to a single crystallographic direction, increasing the homogeneity
of the NV centre and maximizing the collection efficiency of NV emission in
bulk single crystal samples. This is ideal for quantum information and sensing
applications, naturally securing (111) orientated diamond’s future in the
development of quantum devices [8, 9].

Despite these attractive attributes, fundamental studies of the diamond (111)
surface remain incomplete. For example, with respect to the reconstructed
C$(111)\text{-}(2\times 1)$ surface, controversy surrounding the electronic
nature of the surface has arisen by trying to match the published experimental
angle-resolved photoemission spectroscopy (ARPES) results to density
functional theory (DFT) calculations. In order to accomodate the
semiconducting nature of the surface states observed experimentally [10, 11,
12, 13], structural models suggesting dimerized or buckled $\pi\text{-bonded}$
surface chains have been presented [14]. The resulting electronic structure
from such models exhibits a pronounced bandgap opening in the surface state
that is well matched to the experimental results. However, the most
energetically favourable structural models suggest that symmetric
$\pi\text{-bonded}$ surface chains with little-to-no dimerization should occur
[15, 16, 17], and that the resulting electronic structure should feature a
metallic surface state that intersects the Fermi-level ($E_{\text{F}}$). The
most recent measurements of the C$(111)\text{-}(2\times 1)$ surface structure
using low energy electron diffraction (LEED) do suggest a lack of
dimerization, indicating that a surface with symmetric $\pi\text{-bonded}$
chains is accurate [18]. However, measurements of the electronic structure
that exhibit the expected metallic surface state indicative of this structural
model have not been presented to date. In this article we aim to provide clear
evidence of the metallic nature of the surface states by use of synchrotron-
based ARPES studies compared to complementary DFT calculations, along with
LEED and core-level X-ray photoelectron spectroscopy (XPS) measurements.

The growing demand for miniaturisation has also led to increased interest in
the nanoscale properties of diamond [19] and the production of diamond-
graphene ($\text{sp}^{3}$ – $\text{sp}^{2}$) interfaces, investigated both
experimentally [20, 21, 22, 23, 24, 25, 26, 27, 28, 29] and through DFT
studies [30, 31, 32, 33]. There now exists a multitude of graphene growth
systems with the ability to produce high quality epitaxial graphene. Of
relevance to this work are those methods for which the supply of carbon lies
within the substrate. These primarily include high temperature treatment of
SiC and the segregation of carbon from bulk single crystal metals (e.g. Ni,
Ru, Ir, and Pt). Direct measurements of the electronic structure of graphene
prepared by each method was necessary to assess its suitability for future
electronic applications. For example, the quality and electronic properties of
graphene grown on SiC is highly dependent on the chosen growth face [34, 35]
whilst doping [36], the opening of a band-gap at the Dirac-point [37], and
mini-gaps elsewhere in the electronic structure [38] have been shown to arise
from a variety of graphene substrate interactions. However for graphene grown
directly on the diamond (111) surface, we find no direct measurement of the
electronic structure in the literature. To this end, we will proceed to follow
the C$(111)\text{-}(2\times 1)$ system through to the detachment of the
reconstructed surface atoms, and their promotion to a freestanding graphene
layer in order to investigate the resulting electronic structure.

Figure 1: Structure models, surface quality, and schematics detailing the
nomenclature of the BZs used throughout the article. (a)–(c) The optimized
lowest energy configuration positions calculated by DFT. Namely, the
C$(1\times 1)\text{-}(111)$:H surface, the C$(111)\text{-}(2\times 1)$
symmetric $\pi\text{-bonded}$ chain reconstructed surface, and the graphitized
C$(111)\text{-}(2\times 1)$ surface respectively. The surface primitive unit
cells are indicated by the black solid lines on each plan view. (d) DFT
calculated electronic structure of the symmetrically $\pi\text{-bonded}$
C$(111)\text{-}(2\times 1)$ surface, with the surface state shown in black.
(e)–(f) Schematics of the reciprocal space patterns for the $(1\times 1)$ and
$(2\times 1)$ lattice respectively, along with the conventional notation used
for the high symmetry points of the BZs. The black circles represent the
reciprocal lattice originating from the diamond $(1\times 1)$ surface and the
grey dashed hexagon is the surface BZ with its irreducible part shaded in
grey. In (f), the red coloured circles represent the new $(2\times 1)$ surface
reciprocal lattice that arises following reconstruction and the red dashed
rectangle is the new surface BZ with its irreducible part shaded in red. (g)
Surface BZ for the three-domain C$(111)\text{-}(2\times 1)$ surface along with
the notation used when high symmetry points overlap as a result of the
spatially averaging nature of the photoemission techniques used. Each colour
(red, blue, and green) represents one orientation of the $(2\times 1)$ BZ. (h)
Multiple adjacent BZs with solid-coloured lines that represent the almost flat
band that traverses the short edge of the rectangular $(2\times 1)$ zone in
red, green, and blue for their respective rotations. Averaging over space
would lead to a repeating hexagram represented by all colours, at binding
energies close to $E_{\text{F}}$. The location of the high symmetry points at
the corner of graphene’s hexagonal BZ, $\overline{\text{K}}_{\text{G}}$, is
also shown relative to the $(2\times 1)$ BZs.

##  II Experiment and Calculations

###  II.1 Experimental Details

Two single crystal diamond (111) samples with a low miscut angle to the (111)
plane were prepared by mechanical polishing by Element Six (Harwell, UK). The
miscut angle was found to be $\pm$$None$ by Laue measurements. The root-mean-
square roughness of the diamond surfaces, determined by contact mode atomic
force microscopy, was found to be $0.16\text{\,}\mathrm{nm}$ and
$0.20\text{\,}\mathrm{nm}$ along and perpendicular to the polishing lines
respectively. A natural type IIb sample with a boron concentration of
${10}^{15}\text{\,}{\mathrm{cm}}^{-3}$ was used for the ARPES and ultra-violet
photoelectron spectroscopy (UPS) measurements, and a synthetic high-pressure
high-temperature (HPHT) type Ia sample was used for XPS measurements. Samples
were initially hydrogen terminated in a custom low-power
($150\text{\,}\mathrm{W}$) capillary discharge plasma system with open
geometry. H-termination was confirmed by the measurement of a sharp C$(1\times
1)$ LEED pattern, [39] as well as an increase in the secondary electron tail
measured by UPS, which is characteristic of a negative electron affinity (NEA)
of this surface. [40]

ARPES measurements were performed at beamline SGM3 of the ASTRID synchrotron
radiation facility (Aarhus, Denmark). Typically for ARPES measurements, low
photon energies $<$ $50\text{\,}\mathrm{eV}$ are preferred as this provides a
higher photoionization cross section and an improved energy and momentum
resolution [41]. However, diamond is a peculiar case where there are a lack of
free electron like final states (FEFS) available when using low excitation
energies, therefore necessitating the use of higher photon energies [12].
Several unsuitable energy ranges have been detailed [19], particularly the use
of photon energies typically available in home laboratories, such as the He(I)
and He(II) emission lines at $21.2\text{\,}\mathrm{eV}$ and
$40.08\text{\,}\mathrm{eV}$ respectively. Our ARPES measurements are recorded
from a plane close to the centre of a 3D Brillouin zone (BZ) for ease of
comparison to the DFT calculations. To achieve this, measurements of the
valence band maximum (VBM) were recorded whilst varying the photon energy
until the bulk $\sigma\text{-bands}$ had their maximum closest to
$E{{}_{\text{F}}}$, corresponding to $h\nu=$ $125\text{\,}\mathrm{eV}$.
Binding energies are referenced relative to $E{{}_{\text{F}}}$ measured on an
Ar+ sputter-cleaned Cu crystal.

Figure 2: $\mu$LEED, ARPES, and DFT results from the C$(111)\text{-}(2\times
1)$ surface reconstruction following in vacuo annealing at
$920\text{\,}\mathrm{\SIUnitSymbolCelsius}$. (a) $\mu$LEED pattern acquired at
a kinetic energy of $64\text{\,}\mathrm{eV}$, spots from all three rotational
domains are observed with comparable intensity. (b) Schematic diagram of the
$\overline{\Gamma}\text{-}\overline{\text{K}}_{\text{I,II}}$ direction through
the surface BZs used for the DFT and ARPES datasets presented in panels (d)
and (e), respectively. (c) Constant energy surface from the ARPES dataset at
$E_{\text{B}}$ = $0.1\text{\,}\mathrm{eV}$. The white arrow indicates the
direction of the slice used to produce the $E$ vs $k_{\parallel}$ data shown
in (e). (d) DFT results of the occupied valence band structure, the
$\pi\text{-band}$ originating from the reconstruction is shown in magenta and
the bulk $\sigma\text{-bands}$ in blue. (e) Photoemission intensity on the
left-hand side and the simulated intensity on the right. Overlaid on the image
is the $\pi\text{-band}$ that results from the 64 atom DFT supercell
calculation in magenta, along with the bulk $\sigma\text{-bands}$ from the
simple 2-atom unit cell DFT calculation in blue. The white vertical lines are
the boundaries of the integrated area used for the EDCs at
$\overline{\text{K}}_{\text{I,II}}$ and $\overline{\Gamma}$ shown in panels
(f) and (g), respectively.

$\mu$LEED measurements were performed at beamline I311 at the MaxLab
synchrotron (Lund, Sweden) using an Elmitec III SPELEEM photoemission and low
energy electron microscope. Atomic hydrogen was provided by a Tectra atomic
hydrogen source within the UHV system at a pressure of
$5\text{\times}{10}^{-9}\text{\,}\mathrm{mbar}$. XPS and UPS measurements were
carried out in the Aberystwyth University laboratory’s real-time electron
emission spectroscopy (REES) system equipped with a Specs PHOIBOS 100 electron
analyzer, 2D CCD detector, and non-monochromated dual anode X-ray source. The
temperatures reported from this system are measured by a type-K thermocouple,
for which the readings have been calibrated by means of in-situ Raman
spectroscopy measurements as detailed in references [42, 43], see
supplementary information Fig. S1 for more details.

###  II.2 Computational Details

The electron dispersions, total cohesion energies, and geometrical structures
were calculated using DFT employing the generalised gradient approximation
(GGA) implemented in the Perdew-Burke-Ernzerhof (PBE) exchange correlation
functional [44]. This was done using the Aimpro [45] and Vasp [46] packages to
ascertain if any differences manifest when including Van der Waals (VdW)
interactions, as this has been suggested to be necessary for the graphene on
diamond system [30, 31, 33].

For both codes, the surface was modeled by a supercell consisting of a slab
with two surfaces and a vacuum region of
$30\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. The diamond slab was comprised of
an orthorhombic unit cell of 60 $\text{sp}^{3}\text{-bonded}$ carbon atoms,
with the top and bottom faces terminated with 4 $\text{sp}^{2}\text{-bonded}$
C$(111)\text{-}(2\times 1)$ symmetric $\pi\text{-bonded}$ chain atoms. A
graphene monolayer was then placed on either side of the slab by the addition
of $8~{}\text{sp}^{2}\text{-bonded}$ carbon atoms. This structure was repeated
ad infinitum in the $xy$-plane, and alternating layers of diamond and vacuum
were then produced via repetition in the surface normal direction. The slab
thickness of 32 atomic layers proved to be thick enough to prevent any
electric field effects between opposite sides of the slab. The slab was
carefully optimized allowing the atoms to adjust their positions until the
maximum force on any atom was less than
$0.05\text{\,}\mathrm{eV}\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$. The
electronic structures were sampled using a Monkhorst-Pack scheme [47] with a
$8\times 4\times 1$ sampling grid. In the slab calculations performed, the
number of bulk bands observed depends on the slab thickness, and would
eventually approach a continuum for a very thick slab. [48]. The photoemission
measurements however probe a small value of $k_{\perp}$ resulting in far fewer
bands being observed in the measurement. In order to better compare the bulk
bands of the DFT calculations and our measurements, we have also calculated
the bulk band structure using a simple 2-atom unit cell repeated ad infinitum
in $x$,$y$ and $z$. The bare $\sigma\text{-bands}$ were then extracted at
$k_{\perp}=$
$0\text{\,}\mathrm{eV}\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ and
result in two bare bands representing the heavy and light hole bands.
Throughout this paper, when a comparison between the DFT and the ARPES
measurements is required, a constant Lorentzian broadening in momentum space
has been applied to the DFT-calculated bare bands. This is achieved by use of
a spectral function in which electron-defect interactions that limit the mean-
free path of the carriers are assumed to be the dominant broadening mechanism.
The comparatively low temperatures used during our ARPES measurements relative
to diamond’s much higher phonon temperature $(T_{\text{sample}}\ll
T_{\text{phonon}})$ means that other quasiparticle interactions such as
electron-phonon coupling, or electron-electron scattering are not assumed to
strongly contribute to the measured line width. This broadening is achieved by
setting the imaginary part of the self-energy
$2\Sigma^{{}^{\prime\prime}}\cong\Gamma_{\text{e-df}}$ to a constant, which in
turn, via the Kramers-Kronig relation requires the real part of the self
energy term $\Sigma^{{}^{\prime}}$ to have no particular structure. Here,
$\Gamma_{\text{e-df}}$ is the inverse lifetime of the core-hole due to
electron-defect scattering. The spectral function broadened bare bands have
then been convolved by our experimental resolutions and multiplied by a
normalized Fermi-function cut off at $E_{\text{B}}=$ $0\text{\,}\mathrm{eV}$,
and a width appropriate for the $\sim$$300\text{\,}\mathrm{K}$ measurements.

##  III Results and discussion

The structural models relating to the evolution of the three thermodynamically
stable stages of the diamond (111) surface whilst heating up to
$1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$ are shown in Fig. 1(a)–(c). These
models were constructed following optimisation of the atomic positions
obtained from DFT calculations. The monohydride terminated
C$(111)\text{-}(1\times 1)$:H surface is shown in Fig: 1(a) and is the
starting point for our experiments. The first double layer of the $(1\times
1)$ surface is a corrugated hexagonal lattice with alternating surface and
subsurface atoms, giving a rhombohedral primitive unit cell with lattice
vector $a_{0}=$ $2.52\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ as shown in Fig:
1(a). The resulting reciprocal space lattice is hexagonal as shown Fig: 1(e)
along with the surface $(1\times 1)$ BZ for which the calculated reciprocal
space distances for $\overline{\Gamma}\text{-}\overline{\text{K}}$ and
$\overline{\Gamma}\text{-}\overline{\text{M}}$ are
$1.66\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ and
$1.44\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ respectively.

The desorption of H adatoms from the diamond’s surface results in unstable
dangling bonds that cause a complex surface reconstruction as initially
suggested by Pandey for the Si(111) surface [49]. The surface atoms from the
upper and lower layer in the corrugated surface form symmetrically
$\pi\text{-bonded}$ chains along the surface as shown in Fig. 1(b). This
results in a real-space doubling of the surface periodicity in one surface
mesh direction whilst maintaining bulk periodicity in the other, leading to a
new rectangular $(2\times 1)$ surface unit cell and BZ shown in Fig: 1(b) and
(f). The calculated reciprocal distances of the high symmetry points from the
zone center of the new $(2\times 1)$ BZ are
$\overline{\Gamma}\text{-}\overline{\text{K}}=$
$1.44\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$,
$\overline{\Gamma}\text{-}\overline{\text{J}}=$
$1.25\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ and
$\overline{\Gamma}\text{-}\overline{\text{J'}}=$
$0.72\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$. As a result of the
three-fold symmetry of the hexagonal (111) surface, three rotational domains
of the $(2\times 1)$ reconstruction are possible, and due to the spatially
averaging nature of the photoemission technique, all three will be captured
concurrently. As in the previous experimental reports on this surface [11], we
will adopt a subscript of roman numerals when discussing the high symmetry
points of the three $(2\times 1)$ BZs, i.e. I, II, and III for the
$[\bar{1}01]$, $[\bar{1}10]$, and $[01\bar{1}]$ directions shown in red,
green, and blue respectively in Fig: 1(g). The labeling of the high symmetry
points for the $(1\times 1)$ BZ of diamond will also be dropped as the only
states close to $E_{\text{F}}$ are at the zone center.

Finally, at temperatures above $1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$
the surface reconstruction detaches forming a graphene layer above the diamond
surface. This graphene layer exists above a newly formed
C$(111)\text{-}(2\times 1)$ as shown in Fig. 1(c) and results in a new
hexagonal $(1\times 1)$ surface BZ of graphene. In order to differentiate
between the notation of the $(2\times 1)$ surface BZ and that of the graphene,
a subscript ‘G’ will be adopted when discussing its high symmetry points, with
the exception of $\overline{\Gamma}$ as the centre of all the BZs discussed
are congruent and will therefore not be presented with any subscript.

###  III.1 The C$(111)\text{-}(2\times 1)$ surface reconstruction

The $\mu$LEED pattern shown in Fig. 2(a) was acquired after annealing the
diamond (111) surface to $920\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and is as
expected when averaging three rotational domains of the
C$(111)\text{-}(2\times 1)$ reconstruction. ARPES measurements performed after
the same temperature treatment, along with electron dispersions extracted from
our DFT calculations for the C$(111)\text{-}(2\times 1)$ surface, are
presented in Fig. 2. Pandey’s original symmetrically $\pi\text{-bonded}$ chain
calculations [14] and ours shown in Fig. 1(d) indicate that the surface state
is weakly dispersing with energy (almost flat), along the short edge of the
rectangular $(2\times 1)$ BZ , i.e. along
$\overline{\text{J}}_{\text{I}}\text{-}\overline{\text{K}}_{\text{I}}$ in Fig.
1(f). This almost flat band should give rise to a constant energy surface at
energies close to $E_{\text{F}}$ represented by a single colour in Fig. 1(h)
e.g. blue, green or red. Whilst averaging all rotational domains will lead to
a repeating hexagram pattern represented by all these colours. The hexagram
constant energy surface is reproduced experimentally in Fig. 2(c). The white
arrow along $\overline{\Gamma}\text{-}\overline{\text{K}}_{\text{I,II}}$
indicates the direction through momentum space used for extracting the $E$ vs
$k_{\parallel}$ data in Fig. 2(d) and (e) from the DFT calculations and ARPES
dataset respectively. The direction is also shown schematically in Fig. 2(b),
and reaches an equivalent $\overline{\text{K}}_{\text{I,II}}$-point as
discussed in the work by Graupner _et al._ [11].

The DFT calculations presented in Fig. 2(d) agree well with earlier modelling
studies of the C$(111)\text{-}(2\times 1)$ surface by several groups that
predict symmetrically $\pi\text{-bonded}$ surface chains with states that
intersect $E_{\text{F}}$ [10, 15, 16, 17, 50, 51, 52]. For comparison with our
collected ARPES data, the $\pi\text{-band}$ shown as a solid magenta line in
Fig. 2(d) is directly overlaid on the ARPES dataset shown in Fig. 2(e),
whereas the solid blue lines of the $\sigma\text{-bands}$ overlaid on the
ARPES data are taken from the simple 2-atom bulk band calculation. It was
necessary to perform a rigid energy shift of the $\sigma\text{-bands}$ so that
their maxima lay $0.66\text{\,}\mathrm{eV}$ below $E_{\text{F}}$, matching the
position of the VBM of the C$(111)\text{-}(1\times 1)$:H surface extracted
from an energy distribution curve (EDC) at $\overline{\Gamma}$ (see
supplementary Fig. S2). These three bands have then been employed to generate
the simulated ARPES dataset presented on the right of Fig. 2(e).

There is a strong agreement between the experimental and theoretical data,
with the most obvious difference being the binding energy of the heavy hole
$\sigma\text{-band}$ minima at $\overline{\text{K}}_{\text{I,II}}$. However,
this is likely a result of underestimating the electron-electron correlation
in the DFT calculations performed due to the self-interaction error of the PBE
functional. EDCs taken at $\overline{\Gamma}$ and
$\overline{\text{K}}_{\text{I,II}}$, averaged from a region
$0.1\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ wide, are shown in Fig.
2(f) and (g) respectively. The EDC of the band maxima at $\overline{\Gamma}$
is fitted with two asymmetric-Voigt profiles separated by
$0.33\text{\,}\mathrm{eV}$, which is very similar to the energy separation
found between the maxima of the $\pi$ and $\sigma$ bare-bands at
$\overline{\Gamma}$ after compensating for the boron doping level in the
sample by performing the rigid energy shift of the $\sigma\text{-bands}$ from
our bulk diamond calculation. The asymmetry parameter is equal for both peaks
however the FWHM of the $\pi\text{-band}$ is almost double that of the
$\sigma\text{-band}$. An increase of the FWHM like this could result from an
increased density of defects present within the Pandey chains. These defects
most likely result from polishing lines or step edges and would strongly
enhance the broadening caused by impurity scattering for the $\pi$ states.
This is also not surprising given that the bulk states originate from a
crystal of exceptional quality whilst the surface has been subjected to
mechanical polishing. The EDC taken at $\overline{\text{K}}_{\text{I,II}}$
shows that the state is indeed metallic as its leading edge is best fit with a
Fermi-function at $0\text{\,}\mathrm{eV}$ and a width representative of a
sample temperature of $300\text{\,}\mathrm{K}$.

On the other hand, our ARPES measurements do not compare well with the
experimental results presented by Graupner _et al._ and Himpsel _et al._ [11,
13]. In their works, an energy gap of at least $0.5\text{\,}\mathrm{eV}$
between the maximum of the surface state at
$\overline{\text{K}}_{\text{I,II}}$ and $E_{\text{F}}$ is observed that led to
the promotion of a dimerized-chain model by Pandey [14]. We can only
speculative suggestions as to why these measurements show surface states with
such an energy gap. For example, a wider energy gap could result from a larger
miscut angle of their sample, therefore producing narrower (111) terraces and
more step edges. Rougher diamond surfaces could cause buckling of the surface
reconstructions, or a misalignment of the sample during the photoemission
studies could be responsible. This does not however appear to be the case as,
by taking cuts through our data sets at various rotations around
$\overline{\Gamma}$, a gap of $0.5\text{\,}\mathrm{eV}$ between the surface
state and $E_{\text{F}}$ could not be reproduced. This is likely due to the
almost flat nature of the states along the
$\overline{\text{J}}\text{-}\overline{\text{K}}_{\text{I}}$ direction of the
$(2\times 1)$ rectangular surface BZ.

Figure 3: ARPES measurements and DFT calculations of graphene formed above the
C:2$\times 1$(111) surface. (a) Constant energy surface at $E_{\text{B}}=$
$0.1\text{\,}\mathrm{eV}$ using $h\nu=$ $125\text{\,}\mathrm{eV}$. The
graphene $\overline{\text{K}}_{\text{G}}$-point is circled in a magenta dotted
line, and the white dotted lines are to guide the eye towards the hexagram
features already discussed for the C$(111)\text{-}(2\times 1)$ surface. (b)
Schematic diagram showing the
$\overline{\text{K}}_{\text{G}}\text{-}\overline{\text{M}}_{\text{G}}\text{-}\overline{\text{K}}\textsubscript{\hbox
to0.0pt{G\hss}}^{\prime}~{}\text{-}\overline{\Gamma}\hbox
to0.0pt{${}^{\prime}$\hss}$ direction along the edge of graphene’s hexagonal
BZ. (c) Photoemission intensity, DFT calculated bare bands and simulated
intensity. The ARPES dataset on the left hand side was acquired with $h\nu=$
$40.8\text{\,}\mathrm{eV}$. Overlaid on the image are the DFT calculated
$\pi\text{-band}$ as the dashed magenta line, and the same band after
performing a rigid shift and stretch as a solid magenta line. The solid line
is used for creating the simulated ARPES intensity on the right. (d) EDC taken
at the $\overline{\text{K}}_{\text{G}}$ point with a width of
$0.1\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ integrated between the
vertical dashed white lines in (c). (e) Constant energy surface at
$E_{\text{F}}$, using $h\nu=$ $40.8\text{\,}\mathrm{eV}$, the white arrow
indicates the slice taken to extract the $E$ vs $k_{\parallel}$ datasets shown
in (c). (f) The graphene atoms’ cohesion energy as a function of distance from
the C$(111)\text{-}(2\times 1)$ surface both with and without VdW
interactions.

###  III.2 Epitaxial graphene formation above the C$(111)\text{-}(2\times 1)$
surface reconstruction

At temperatures greater than $1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$, the
surface reconstructed atoms are liberated into a monolayer of freestanding
graphene as shown in Fig. 1(c). New electron states coexisting with those
already presented for the C$(111)\text{-}(2\times 1)$ surface are observed in
the ARPES datasets shown in Fig. 3. One of these features is shown in the
constant energy surface shown in Fig. 3(a), located close to the corner of the
hexagonal diamond ($1\times 1$) BZ, where no states from the diamond
C$(111)\text{-}(2\times 1)$ electronic structure are expected. For increasing
binding energy, the state displays the characteristic isoenergetic triangular
modulation of the photoemission intensity, becoming more pronounced at
energies away from $E_{\text{F}}$ [53] as shown in Fig. S3(a), providing
evidence that it originates from a graphene film. The feature labeled
$\overline{\text{K}}_{\text{G}}$ appears at a distance of $k_{\parallel}=$
$1.61\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ from $\overline{\Gamma}$
which is close to the $\overline{\Gamma}\text{-}\overline{\text{K}}$ distance
of the hexagonal diamond ($1\times 1$) zone and in good agreement with the
$\overline{\Gamma}\text{-}\overline{\text{K}}_{\text{G}}$ distance of
freestanding graphene at $1.70\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$
[54]. This is a good indication that the graphene is commensurate with the
diamond surface. To decouple the graphene features from that of the diamond
surface states, lower photon energies were used, making use of diamond’s
tendency to have weaker photoemission signal at low photon energies. The
photoemission intensity along the
$\overline{\text{K}}_{\text{G}}\text{-}\overline{\text{M}}_{\text{G}}\text{-}\overline{\text{K}}\textsubscript{\hbox
to0.0pt{G\hss}}^{\prime}$ edge of the graphene BZ, (shown schematically in
Fig. 3(b)), is presented in the left panel of Fig. 3(c).

Using the structural model presented in Fig. 1(c) the electronic dispersion
for the graphene on diamond system was calculated and found to be in good
agreement with our experimental data, shown as the dashed magenta line
overlaid on the ARPES dataset in Fig. 3(c). The main difference between the
calculated electron dispersion and the experimental results appears to be the
energy at which the band maximum and minimum are observed at the
$\overline{\text{K}}_{\text{G}}$ and $\overline{\text{M}}_{\text{G}}$ points
respectively. For the calculation to better match our experimental data, it
was necessary to perform a linear stretching of the calculated energy axis by
a factor of 1.2 and shift the energy of the band maximum at
$\overline{\text{K}}_{\text{G}}$ from $0.22\text{\,}\mathrm{eV}$ to
$E_{\text{F}}$. The need to stretch the energy axis can be understood in terms
of the commonly observed underestimation of electronic energies for GGA level
exchange correlation functionals. This is shown as the solid magenta line in
Fig. 3(c) and is the bare band used to simulate the ARPES data on the right-
hand side of Fig. 3(c). The direction of the cut taken through the ARPES data
unfortunately cuts through areas of weak photoemission intensity as a result
of the triagonal warping mentioned earlier. However, the adjusted bare-band
calculation matches well with the measured intensity as band progresses
towards the next zone centre $\overline{\Gamma}\hbox
to0.0pt{${}^{\prime}$\hss}$, as shown in second differential image Fig. S3(b).
The graphene $\overline{\text{K}}_{\text{G}}$-points are observed at
$k_{\parallel}=$ $0.81\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ (where
$k_{\parallel}=$ $0\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ is
reference to the saddle point of the band at the
$\overline{\text{M}}_{\text{G}}$-point), giving a
$\overline{\text{K}}_{\text{G}}\text{-}\overline{\text{K}}\textsubscript{\hbox
to0.0pt{G\hss}}^{\prime}$ distance of
$1.62\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ which is slightly smaller
than that expected for freestanding graphene at
$1.70\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$ but within the accuracy
of the $k$-warping performed in the analysis due to large emission angles. The
continued presence of the $(2\times 1)$ electronic states along with the
features now present from the graphene provides strong evidence that epitaxial
graphene is formed on the diamond (111) surface in co existence with the
C$(111)\text{-}(2\times 1)$ surface reconstruction. This high degree of
epitaxial registry is further confirmed by the lack of any states related to
rotational domains in our photoemission datasets. The constant energy surface
taken at $E_{\text{F}}$ shown in Fig. 3(e) shows only two
$\overline{\text{K}}_{\text{G}}$-points along the edge of graphene’s BZ and no
other points that would indicate rotational domains seen on some metal [55,
56] and semiconducting substrates [57, 58]. However, small scale rotational
disorder such as that observed for epitaxial graphene on SiC(0001) [59] cannot
be ruled out.

The $\overline{\text{K}}_{\text{G}}$-point intersects the Fermi level in the
ARPES data, as seen in the high contrast inset in Fig. 3(c) and the EDC in
Fig. 3(d) taken at $k_{\parallel}=$
$0.81\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$. The intrinsic nature of
the graphene formed on this surface is, as far as we are aware, contrary to
all calculations performed in the literature and in our own DFT calculations,
which show an n-type doping for graphene [33]. In pristine, undoped, and
unbuckled graphene, the binding energy of the graphene $\pi\text{-band}$ at
the $\overline{\text{M}}_{\text{G}}$-point should be
$\approx$$3\text{\,}\mathrm{eV}$. However, in both the ARPES and the DFT
modelling of graphene on the diamond C$(111)\text{-}(2\times 1)$ surface
presented here, the energy of the band at the
$\overline{\text{M}}_{\text{G}}$-point is observed closer to $E_{\text{F}}$,
i.e. $E_{\text{B}}=$ $2.65\text{\,}\mathrm{eV}$. This up-shift of the
$\pi\text{-band}$ at the $\overline{\text{M}}_{\text{G}}\text{-point}$ (and by
extension, a flattening of the dispersion along the
$\overline{\text{K}}_{\text{G}}\text{-}\overline{\text{M}}_{\text{G}}\text{-}\overline{\text{K}}\textsubscript{\hbox
to0.0pt{G\hss}}^{\prime}$ edge of graphene’s BZ) can potentially be attributed
to an increased third-nearest atomic neighbour coupling due to either uniaxial
pressure, shear, or strain forces, doping, or more likely in this situation,
an increased graphene-substrate interaction [60, 61].

Following optimisation of the atomic positions and the inclusion of VdW
interactions in our DFT calculations, the graphene sheet moved to a stable
distance of $3.29\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ away from the
C$(111)\text{-}(2\times 1)$ surface to avoid buckling as shown in Fig. 1(c),
which is only slightly smaller than the graphite interlayer distance of
$3.35\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. This structural model
contradicts previous experimental results that suggest an unreconstructed
C$(111)\text{-}(1\times 1)$ surface existing below the graphene formed at
$1250\text{\,}\mathrm{K}$ [21]. We found that the inclusion of VdW
interactions had little bearing on the electronic dispersions calculated using
Aimpro or Vasp, as there were no discernible differences between them. However
as shown in Fig. 3(f), the VdW interactions did play a role in determining the
optimized graphene distance above the C$(111)\text{-}(2\times 1)$ surface. The
Aimpro calculations showed no obvious minima in cohesion energy
($E_{\text{Cohesion}}$) as a function of distance away from the diamond
surface.

To investigate if the graphene layer will remain intact following exposure of
the substrate to atmosphere, and coinciding deconstruction of the diamond
C$(111)\text{-}(2\times 1)$ bonds, a study of the surface properties was
conducted using XPS, UPS and LEED. XPS measurements of the C 1s core level
acquired after reconstruction at $930\text{\,}\mathrm{\SIUnitSymbolCelsius}$
and graphitization at $1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$ are
presented in Fig. 4(a) and (b) respectively. Corresponding widescan spectra
are shown in Fig. S4 in the supplementary information. A type Ia substrate is
chosen here as the bulk C 1s core level for the type IIb sample has a very
similar $E_{\text{B}}$ to that of graphene and the C$(111)\text{-}(2\times 1)$
surface at $\approx$$284\text{\,}\mathrm{eV}$. The use of an insulating
diamond, together with a supply of charge compensating secondary electrons
from the unmonochromated twin-anode x-ray source, allows for a clear
distinction to be made between the conductive $\text{sp}^{2}\text{-bonded}$
surface components and the bulk $\text{sp}^{3}\text{-bonded}$ diamond
substrate.

Figure 4: XPS and UPS measurements of the diamond surface following high
temperature annealing and atomic hydrogen treatment. (a) and (b) Core level
intensity for C 1s spectra collected at a pass energy of
$20\text{\,}\mathrm{eV}$ and $h\nu=$ $1253.6\text{\,}\mathrm{eV}$ after
$950\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$ respectively on the type Ia
diamond substrate. (c) and (d) UPS spectra at various stages of preparation of
the type IIb diamond substrate collected at a pass energy of
$5\text{\,}\mathrm{eV}$ and $h\nu=$ $21.2\text{\,}\mathrm{eV}$. (c) SE cut off
and shape of the SE tail and (d) intensity of the integrated valence band. The
regions labelled I and II are related to bulk $\sigma\text{-states}$ where
region III relates to $\pi\text{-states}$. The inset panel in (d) shows a
$\mu$LEED pattern of the surface following hydrogen termination of the
graphene on the C$(111)\text{-}(2\times 1)$ diamond system.

Both the reconstructed diamond surface and the graphitized diamond surface
show two components: the larger component at higher binding energy is
naturally attributed to the $\text{sp}^{3}\text{-bonded}$ carbon of the
diamond substrate shown in blue, and the smaller component at lower binding
energy shown in green is consistent with $\text{sp}^{2}\text{-bonded}$ carbon
species (i.e. at $E_{\text{B}}=$ $284\text{\,}\mathrm{eV}$). The
$\text{sp}^{2}\text{-bonded}$ carbon component only emerges when the samples
are subjected to temperatures greater than
$930\text{\,}\mathrm{\SIUnitSymbolCelsius}$. After graphitization, the
$\text{sp}^{2}$ component has a small degree of asymmetry, implying that there
has been an increase in the density of states at $E_{\text{F}}$. This
component doubled in intensity after the sample is heated to
$1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$, from $5.4\text{\,}\mathrm{\char
37\relax}$ to $10.6\text{\,}\mathrm{\char 37\relax}$ of the total C 1s area.
This means that this component must account for two structures of
$\text{sp}^{2}$ carbon: The C$(111)\text{-}(2\times 1)$ reconstruction, and
the graphene layer. If the intensity of the $\text{sp}^{2}$ peak in Fig. 4(a)
represents a single layer of $\text{sp}^{2}$ carbon due to the reconstruction,
then it follows that a doubling of this intensity in Fig. 4(b) represents a
contribution from the graphene sheet, as two layers of reconstruction is not
possible and sub-surface reordering has not been demonstrated within the
literature.

Following annealing to $1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$ a new
feature is also observed in the UPS spectra in region III of Fig. 4(d) at
$\approx$$2\text{\,}\mathrm{eV}$. This feature is attributed to the
$\pi\text{-band}$s of the ($2\times 1$) reconstruction and graphene as seen in
the ARPES datasets. It is well known that hydrogen induces a deconstruction of
the C$(111)\text{-}(2\times 1)$ surface [10, 50, 62]. However, following
hydrogen treatment in our in-situ plasma system, the state at
$\approx$$2\text{\,}\mathrm{eV}$ is still evident, and appears to have
increased in intensity, despite the fact that the secondary electron (SE) tail
shown in Fig. 4(c) has also dramatically increased in intensity, a typical
characteristic of the H-terminated diamond (111) surface’s NEA [63]. $\mu$LEED
measurements performed after similar in-situ treatment reveal that the surface
is no longer reconstructed as the loss of second-order spots is evident when
compared to Fig. 2(f). A feature at $\approx$$2\text{\,}\mathrm{eV}$ is also
apparent in the density of states calculations for graphene adhered to a
H-terminated diamond (111) surface by Zhao $et~{}al.$ [31]. We believe that
this is strong evidence that the graphene grown in this manner survives the
gentle ($150\text{\,}\mathrm{W}$) hydrogen plasma processing, which would
allow for various diamond surface terminations to be achieved under the
graphene, doping the graphene film via charge-transfer either n- or p-type
depending on the terminating species [20, 31, 32, 33].

##  IV Conclusions

In summary, we have investigated the occupied electronic structure of the
C$(111)\text{-}(2\times 1)$ surface, experimentally using ARPES, and by DFT
calculations using the Aimpro and Vasp codes. Our results show that the
C$(111)\text{-}(2\times 1)$ surface exhibits a metallic surface state that
intersects $E_{\text{F}}$ at the $\overline{\text{K}}_{\text{I,II}}$-point.
This observation is in good agreement with DFT modelling both in this work and
the wider literature, in particular with Pandey’s original prediction of
symmetric $\pi\text{-bonded}$ surface chains, but contrary to previous ARPES
studies and dimerized surface chain models. The experimental observations are
attributed to a combination of the exceptional surface quality of the diamond
samples used in this study, their low miscut angle, and possibly to the higher
resolution of the ARPES instrumentation used. In conjunction with relatively
recent structural investigations that confirm the symmetric $\pi\text{-chain}$
model, we can now confidently say that a consistent and robust model of the
C$(111)\text{-}(2\times 1)$ surface exists that reconciles both its structural
_and_ electronic properties.

New states observed in the ARPES Fermi surface measurements after annealing
the C$(111)\text{-}(2\times 1)$ surface to
$1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in vacuo, are attributed to the
liberation of the surface atoms to a monolayer of graphene that coexists above
a newly reconstructed C$(111)\text{-}(2\times 1)$ surface. The graphene
appears to be undoped with its $\overline{\text{K}}_{\text{G}}$-point
intersecting $E_{\text{F}}$ within experimental resolution. This contrasts
with all calculations suggesting there should be a n-type doping of at least
$100\text{\,}\mathrm{meV}$ for graphene on the C$(111)\text{-}(2\times 1)$
surface. An increased graphene-substrate interaction is also observed,
evidenced by a flattening of the $\pi\text{-band}$ dispersion between the
$\overline{\text{K}}_{\text{G}}$ and $\overline{\text{M}}_{\text{G}}$ points.
Deconstruction of the diamond surface to a H-terminated
C$(111)\text{-}(1\times 1)$ structure after gentle plasma treatment does not
affect the graphene, providing the option for device manufacturers to tailor
the doping of the graphene sheet by altering the termination of the diamond
surface bonds beneath the graphene.

###### Acknowledgements.

We would like to acknowledge the financial support by: the EPSRC CDT in
Diamond Science and Technology; The European Regional Development Fund through
the Solar Photovoltaic Academic Research Consortium (SPARC II) operated by the
Welsh Government, the UK research council EPSRC grant #EP/G068216/1; The
Research Council of Norway through the Centres of Excellence scheme, project
#262633 “QuSpin” and FriPro ToppForsk-program #251131 FUNDAMeNT; the National
Infrastructure for High Performance Computing and Data Storage in Norway
UNINETT Sigma2; and the National Measurement System programme of the UK
Department of Business, Energy and Industrial Strategy. We would also like to
acknowledge the technical support from Dr Marco Bianchi and insightful
discussions held with Prof. Philip Hofmann at the ISA synchrotron in Aarhus,
and Dr Federico Mazzola for the many fruitful discussions and helping
implement the code used to simulate the ARPES results from DFT calculations.
MEB would like to acknowledge an ETH Zurich Postdoctoral Fellowship program.

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# Exponential lower bound for the eigenvalues of the time-frequency
localization operator before the plunge region

Aleksei Kulikov  Tel Aviv University, School of Mathematical Sciences, Tel
Aviv, 69978, Israel,
<EMAIL_ADDRESS>
###### Abstract.

We prove that the eigenvalues $\lambda_{n}(c)$ of the time-frequency
localization operator satisfy $\lambda_{n}(c)>1-\delta^{c}$ for
$n=[(1-\varepsilon)c]$, where $\delta=\delta(\varepsilon)<1$ and
$\varepsilon>0$ is arbitrary, improving on the result of Bonami, Jaming and
Karoui, who proved it for $\varepsilon\geq 0.42$. The proof is based on the
properties of the Bargmann transform.

Key words and phrases: prolate spheroidal wave functions, Bargmann–Segal–Fock
space, Hermite functions

##  1\. Introduction

For a measurable set $\Omega\subset\mathbb{R}$ we define the projection
$P_{\Omega}:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ as
$P_{\Omega}f=f\chi_{\Omega}$ and the Fourier projection
$Q_{\Omega}:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ as
$Q_{\Omega}=\mathcal{F}^{-1}P_{\Omega}\mathcal{F}$, where $\mathcal{F}$ is the
Fourier transform

$\mathcal{F}(f)(\xi)=\int_{\mathbb{R}}f(x)e^{-2\pi ix\xi}dx.$

For a pair of sets $T,\Omega\subset\mathbb{R}$ we define the time-frequency
localization operator $S_{T,\Omega}$, associated with them, as
$S_{T,\Omega}=P_{T}Q_{\Omega}P_{T}$. It is easy to check that $S_{T,\Omega}$
is a bounded self-adjoint non-negative definite operator.

If the measures of $T$ and $\Omega$ are finite then $S_{T,\Omega}$ is a
Hilbert–Schmidt operator with the Hilbert–Schmidt norm equal to $|T||\Omega|$
(see [6, proof of Theorem 2.3.1]). In particular, in this case $S_{T,\Omega}$
is a compact operator and as such it has a sequence of eigenvalues
$1>\lambda_{1}(T,\Omega)\geq\lambda_{2}(T,\Omega)\geq\ldots>0$. The first
eigenvalue $\lambda_{1}(T,\Omega)$ is equal to the norm of $S_{T,\Omega}$ and
in particular it is always at most $1$, but it can be shown that it is always
strictly less than $1$ (see [6, Theorem 2.3.3]). A highly non-trivial result
of Nazarov [11, Theorem II] shows that there exist absolute constants $c,C>0$
such that we always have $\lambda_{1}(T,\Omega)\leq 1-ce^{-C|T||\Omega|}$
regardless of the geometry of sets $T$ and $\Omega$.

The famous Donoho–Stark conjecture [4] says that if $T$ is an interval then
among the sets $\Omega$ of fixed measure the maximum of
$\lambda_{1}(T,\Omega)$ is achieved when both $T$ and $\Omega$ are intervals.
In this case, it can be seen by dilation that the eigenvalues depend only on
the product of lengths of the intervals $c=|T||\Omega|$, so we have a sequence
$1>\lambda_{1}(c)>\lambda_{2}(c)>\ldots>0$. The distribution of these
eigenvalues is the main subject of this paper.

It turns out, as was discovered by Slepian [14] and rigorously proved by
Landau and Widom [10], that the eigenvalues exhibit a phase transition around
the point $n_{0}=c$: if $c-n\gtrsim\log c$ then $\lambda_{n}(c)\approx 1$, if
$n-c\gtrsim\log c$ then $\lambda_{n}(c)\approx 0$, and only in the plunge
region $|c-n|\lesssim\log c$ the eigenvalues $\lambda_{n}(c)$ have
intermediate value. Specifically, they proved the following theorem.

######  Theorem 1.1.

For a fixed $b\in\mathbb{R}$ we have

(1.1) $\lim_{c\to\infty}\lambda_{n(c,b)}(c)=(1+e^{b})^{-1},$

where $n(c,b)=[c+\frac{1}{\pi^{2}}b\log(c)]$ and $[t]$ denotes the integer
part of $t$.

Their proof had no uniformity in $b$ and so in recent years there were some
results on getting similar results and bounding the size of the plunge region
for varying values of $b$, in particular by Israel [7], culminating in the
following result of Karnik, Romberg and Davenport [8] (note that its
formulation is slightly different since we use a different normalization of
the Fourier transform).

######  Theorem 1.2 ([8, Theorem 3]).

For all $c>0$ and $0<\varepsilon<\frac{1}{2}$ we have

(1.2)
$\left|\\{n:\varepsilon<\lambda_{n}<1-\varepsilon\\}\right|\leq\frac{2}{\pi^{2}}\log(50c+25)\log\left(\frac{5}{\varepsilon(1-\varepsilon)}\right)+7.$

For $c\to\infty,\varepsilon\to 0$ their bound has the form
$\log(c)\log(\varepsilon^{-1})(\frac{2}{\pi^{2}}+o(1))$, coinciding with the
bound from Theorem 1.1 asymptotically while being completely explicit and
uniform in all parameters.

There is also quite a few results establishing asymptotic and non-asymptotic
decay bounds on the eigenvalues $\lambda_{n}(c)$ when $n>c$. Widom [16] showed
that for fixed $c$ the eigenvalues decay like
$\lambda_{n}(c)\sim\left(\frac{e\pi c}{(8n+4)}\right)^{2n+1}$, and the works
of Osipov [12], Bonami and Karoui [3] and Bonami, Jaming and Karoui [2]
established uniform in $c$ upper bounds on $\lambda_{n}(c)$, in particular [3,
Theorem 1] essentially says that after the plunge region (say, for
$n>(1+\varepsilon)c$) the eigenvalues start with an exponential decay and then
catch up to the super-exponential decay similar to the Widom’s result.

On the other hand, not much is known about how close are the eigenvalues
$\lambda_{n}(c)$ to $1$ when $n$ is significantly less than $c$. Apart from
the general bounds on the plunge region like (1.2), we are aware of only two
results dealing with this regime: the result of Fuchs [5] who showed that for
fixed $n$ the eigenvalues $\lambda_{n}(c)$ satisfy

$1-\lambda_{n}(c)\sim\frac{\sqrt{\pi}}{2}\frac{8^{n}}{(n-1)!}\left(\frac{\pi
c}{2}\right)^{n-1/2}e^{-\pi c},c\to\infty,$

in particular showing that $\lambda_{n}(c)$ are exponentially close to $1$,
and the following result of Bonami, Jaming and Karoui [2](we present it in a
slightly weaker form to highlight the important parts).

######  Theorem 1.3.

For $0\leq n\leq c$ and $c>100$ we have

(1.3) $1-\frac{(\pi c)^{n}}{n!}e^{-\frac{\pi}{2}c}\leq\lambda_{n}(c)<1.$

Note that this lower bound is meaningful only if $\frac{(\pi
c)^{n}}{n!}e^{-\frac{\pi}{2}c}\leq 1$, which asymptotically means $n\leq
0.58c$, and for such values of $n$ we have an exponential lower bound
$\lambda_{n}(c)\geq 1-\gamma^{c}$ for some $\gamma<1$.

The goal of the present work is to push this estimate all the way to the
plunge region and obtain the following result.

######  Theorem 1.4.

For any $\varepsilon>0$ there exists constant $0<\delta=\delta(\varepsilon)<1$
such that for large enough $c$ we have

$\lambda_{n}(c)\geq 1-\delta^{c},$

where $n=[(1-\varepsilon)c]$.

Note that exponential lower bound is the best we can achieve since this is the
best possible bound already for the $\lambda_{1}(c)$.

The proof of Theorem 1.3 is based on applying the min-max principle to the
operator $S_{T,\Omega}$ and picking a suitable subspace of
$L^{2}(\mathbb{R})$. The subspace that they chose is generated by the first
$n$ Hermite functions. However, there are two reasons why they only prove the
result for $\varepsilon\geq 0.42$. The first one is that their elementary
real–analytic estimates for the tails of the Hermite functions are not the
strongest possible. The second one is that even with the best possible bounds
on the Hermite functions we can not prove the theorem in full generality – we
must consider more general subspaces generated by the time-frequency shifts of
Hermite functions and the best we can get from the Hermite functions alone is
$\varepsilon>1-\frac{\pi}{4}=0.21$.

Our approach is to also apply the min-max principle, but translate the problem
to the realm of complex analysis by means of the Bargmann transform and work
with the functions in the Bargmann–Segal–Fock space. Although the projection
$P_{\Omega}$ becomes not the most pleasant operator in this setting, since we
only care about the lower bounds we can approximate it by some crude estimates
which are enough to get Theorem 1.4. Additionally, since the time-frequency
shifts of the Hermite functions are no longer orthogonal we also need to upper
bound their inner products to show that they are almost orthogonal.

It turns out that in the Bargmann–Segal–Fock space the $k$’th Hermite function
lives essentially on the disk of area $k$ centred at the origin, and, if we
let $T=\Omega=[-\frac{\sqrt{c}}{2},\frac{\sqrt{c}}{2}]$, to have good bounds
on the operator $S_{T,\Omega}$ we need this disk to lie within the square
$T\times\Omega$. This gives us the aforementioned value of
$\varepsilon>1-\frac{\pi}{4}$. When we apply the time-frequency shift we shift
this disk on the complex plane and the Hermite functions corresponding to
different disks have small inner product if these disks do not intersect. This
idea naturally leads us to the following purely geometric lemma.

######  Lemma 1.5.

For any $\varepsilon>0$ there exists a finite union of pairwise disjoint
closed disks
$D_{1},D_{2},\ldots,D_{N}\subset(-\frac{1}{2},\frac{1}{2})\times(-\frac{1}{2},\frac{1}{2})$
such that the area of their union is at least $1-\varepsilon$.

Note that since the disks we consider are closed and the square we consider is
open, there exists $\gamma=\gamma(\varepsilon)>0$ such that all disks are at
least $\gamma$ away from the boundaries of the square and from each other.

Harder version of this lemma was given as a problem at the first USSR
mathematical olympiad for students [1]. To keep the article self-contained we
present its proof at the end of the text.

It is worth noting that our interest in the lower bounds for the eigenvalues
of the time-frequency operator came from our study of the Fourier
interpolation formulas [9], such as the Radchenko–Viazovska formula [13]. In
particular, Theorem 1.4 allows us to prove the main result of [9], although
with a weaker error term $o(4WT)$ instead of $O(\log^{2}(4WT))$. We also find
it interesting that the formula in [13] came from the celebrated work of
Viazovska on sphere packings [15] and our proof is based on a circle packing
of the square from Lemma 1.5, although it is crucial that we allow disks of
varying radii.

The structure of this paper is as follows. In Section 2 we recall the
definitions and basic properties of the Bargmann transform and the
Bargmann–Segal–Fock space. In Section 3 we construct an almost orthogonal
system of functions with strong time-frequency localization properties and in
Section 4 using simple functional analysis we prove Theorem 1.4. Finally, in
the last section we prove Lemma 1.5.

##  2\. Bargmann transform

Let $f$ be a function in $L^{2}(\mathbb{R})$. For $z\in\mathbb{C}$ we define
the Bargmann transform of $f$ at $z$ as

(2.1) $\mathcal{B}f(z)=2^{1/4}\int_{\mathbb{R}}f(t)e^{2\pi tz-\pi
t^{2}-\frac{\pi}{2}z^{2}}dt.$

The function $\mathcal{B}f$ turns out to be an entire function belonging to
the Bargmann–Segal–Fock space $\mathcal{F}$ of entire functions for which the
norm

$||F||_{\mathcal{F}}^{2}=\int_{\mathbb{C}}|F(z)|^{2}e^{-\pi|z|^{2}}dz$

is finite. Moreover,
$||f||_{L^{2}(\mathbb{R})}=||\mathcal{B}f||_{\mathcal{F}}$ and the Bargmann
transform is a bijection between $L^{2}(\mathbb{R})$ and $\mathcal{F}$. For
the proofs of these facts see [6, Section 3.4].

Since the space $L^{2}(\mathbb{R})$ has a rich group of isometries, coming
from translations and modulations, they should have a counterpart on the
Bargmann transform side. Indeed, for $w\in\mathbb{C}$ and $F\in\mathcal{F}$
consider

$T_{w}F(z)=F(z-w)e^{\pi z\bar{w}-\frac{\pi}{2}|w|^{2}}.$

Clearly, $F_{w}$ is still an entire function and direct computation shows that
$||T_{w}F||_{\mathcal{F}}=||F||_{\mathcal{F}}$, in particular if
$F\in\mathcal{F}$ then $T_{w}F\in\mathcal{F}$.

As usual for the spaces of analytic functions, point evaluations are
continuous in $\mathcal{F}$. Specifically, for all $F\in\mathcal{F}$ and
$z\in\mathbb{C}$ we have $|F(z)|\leq
e^{\frac{\pi}{2}|z|^{2}}||F||_{\mathcal{F}}$, which can be verified from (2.1)
and the Cauchy–Schwarz inequality.

The space $\mathcal{F}$ contains an orthonormal basis
$\\{\sqrt{\frac{\pi^{k}}{k!}}z^{k}\\}_{k\in\mathbb{N}_{0}}$. While it is not
technically necessary for our proof, it is important to mention that these
functions are exactly the Bargmann transform of the Hermite functions
$h_{k}(x)$. Thus, this basis is the Bargmann transform of the basis used by
Bonami, Jaming and Karoui. We will consider more generally $T_{w}$ applied to
this basis for various values of $w$.

Last fact that we need is that the Fourier transform corresponds to rotation
by $90$ degrees in the complex plane, that is
$(\mathcal{B}\mathcal{F}f)(z)=\mathcal{B}f(iz)$. This can be either verified
using the aforementioned fact about the Hermite functions, or derived directly
from (2.1) by using the fact that the Fourier transform is an isometry and the
Fourier transform of a Gaussian is a Gaussian.

##  3\. Construction of the system

Since eigenvalues $\lambda_{n}(I,J)$ for the intervals $I,J$ depend only on
the product $|I||J|$, we will assume without loss of generality that
$I=J=[-\frac{\sqrt{c}}{2},\frac{\sqrt{c}}{2}]$. Let us fix an $\varepsilon>0$
in the Theorem 1.4 and consider the set of disks $D_{1},D_{2},\ldots,D_{N}$
from Lemma 1.5 corresponding to this $\varepsilon$. By $w_{m}$ and $r_{m}$ we
denote the center and radius of $D_{m}$, respectively.

We consider the following set of functions from $\mathcal{F}$

(3.1) $\mathcal{U}=\\{T_{\sqrt{c}w_{m}}\frac{z^{k}}{\sqrt{k!}}\mid 1\leq m\leq
N,0\leq k\leq c\pi r_{m}^{2}\\}$

and put $\mathcal{V}=\mathcal{B}^{-1}\mathcal{U}$ (the set $\mathcal{V}$
consists of some time-frequency shifts of Hermite functions). Note that all
functions in $\mathcal{V}$ have $L^{2}(\mathbb{R})$ norm equal to $1$ and that
$\mathcal{V}$ contains at least $(1-\varepsilon)c$ elements. The goal of this
section is to show that the functions from $\mathcal{V}$ are almost orthogonal
and have very strong concentration on the interval $I$ and Fourier
concentration on the interval $J$. Specifically, we will prove the following
proposition.

######  Proposition 3.1.

There exists $\alpha=\alpha(\varepsilon)<1$ such that for big enough $c$ the
following conditions hold.

  1. (i)

For all $f,g\in\mathcal{V},f\neq g$ we have $|\langle
f,g\rangle|\leq\alpha^{c}$.

  2. (ii)

For all $f\in\mathcal{V}$ we have
$||(\operatorname{Id}-P_{I})f||\leq\alpha^{c}$ and
$||(\operatorname{Id}-Q_{J})f||\leq\alpha^{c}$.

###### Proof.

We begin with proving $(i)$. Since Bargmann transform is an isometry, it is
enough to bound $|\langle\mathcal{B}f,\mathcal{B}g\rangle|$. Set
$F=\mathcal{B}f,G=\mathcal{B}g$. If $F$ and $G$ correspond to the same disk
$D_{m}$ then they are orthogonal since the monomials $z^{k}$ are pairwise
orthogonal and $T_{\sqrt{c}w_{m}}$ is an isometry of the space $\mathcal{F}$.
Thus, we can assume that $F$ corresponds to the disk $D_{n}$ and $G$
corresponds to the disk $D_{m}$, $m\neq n$. Put
$v_{n}=\sqrt{c}w_{n},v_{m}=\sqrt{c}w_{m}$ and assume that
$F=T_{v_{n}}\frac{z^{k}}{\sqrt{k!}},G=T_{v_{m}}\frac{z^{l}}{\sqrt{l!}}$. We
are going to estimate $|\langle F,G\rangle|$ directly. We have

$\displaystyle|\langle
F,G\rangle|\leq\int_{\mathbb{C}}|F(z)||G(z)|e^{-\pi|z|^{2}}dz\leq$
$\displaystyle\int_{|z-v_{n}|>\sqrt{c}(r_{n}+\frac{\gamma}{2})}|F(z)||G(z)|e^{-\pi|z|^{2}}dz+\int_{|z-v_{l}|>\sqrt{c}(r_{m}+\frac{\gamma}{2})}|F(z)||G(z)|e^{-\pi|z|^{2}}dz,$

where in the second inequality we used that $D_{n}$ and $D_{m}$ are at least
$\gamma$ apart. We will estimate only the first of these two integrals since
the estimate for the other one is similar.

Since $||G||_{\mathcal{F}}=1$, we have $|G(z)|e^{-\frac{\pi}{2}|z|^{2}}\leq
1$. Therefore

$\int_{|z-v_{n}|>\sqrt{c}(r_{n}+\frac{\gamma}{2})}|F(z)||G(z)|e^{-\pi|z|^{2}}dz\leq\int_{|z-v_{n}|>\sqrt{c}(r_{n}+\frac{\gamma}{2})}|F(z)|e^{-\frac{\pi}{2}|z|^{2}}dz.$

We will take this integral in polar coordinates with
$|z-v_{n}|=r,\arg(z-v_{n})=\theta$. This way we get

$2\pi\sqrt{\frac{\pi^{k}}{k!}}\int_{\sqrt{c}(r_{n}+\frac{\gamma}{2})}^{\infty}r^{k+1}e^{-\frac{\pi}{2}r^{2}}dr.$

Note that to justify this formula we can either do a direct computation or
first apply $T_{v_{n}}^{-1}$ to the whole thing since it is an isometry of
$\mathcal{F}$. Next, we do the change of variables $s=r^{2}$ and get

$\pi\sqrt{\frac{\pi^{k}}{k!}}\int_{c(r_{n}+\frac{\gamma}{2})^{2}}^{\infty}s^{\frac{k}{2}}e^{-\frac{\pi}{2}s}ds.$

Observe that the expression in the integral is decreasing in $s$. Indeed,
differentiating it we can see that it is decreasing if $k<\pi s$ and since
$k\leq c\pi r_{n}^{2}$, we can see that it is decreasing. Therefore, the
integral is at most

$\sum_{p=0}^{\infty}\left(p+c\left(r_{n}+\frac{\gamma}{2}\right)^{2}\right)^{\frac{k}{2}}e^{-\frac{\pi}{2}(p+c(r_{n}+\frac{\gamma}{2})^{2})}.$

Let us estimate the ratio of two consecutive elements of this sum, setting
$p+c(r_{n}+\frac{\gamma}{2})^{2}=s$ for brevity. We have

$\left(\frac{s+1}{s}\right)^{\frac{k}{2}}e^{-\frac{\pi}{2}}\leq
e^{\frac{k}{2s}-\frac{\pi}{2}}\leq e^{\frac{c\pi
r_{n}^{2}}{2c(r_{n}+\frac{\gamma}{2})^{2}}-\frac{\pi}{2}}=e^{\frac{\pi}{2}\left(\frac{1}{(1+\frac{\gamma}{2r_{n}})^{2}}-1\right)},$

where in the first inequality we used the well-known fact that
$(1+\frac{1}{s})^{s}\leq e$. Therefore, the ratio of any two consecutive terms
is at most some $\beta=\beta(\varepsilon)<1$. Hence, the whole sum is at most
the first term multiplied by some constant depending only on $\varepsilon$.
Thus, the integral is at most

$C_{\varepsilon}\pi\sqrt{\frac{\pi^{k}}{k!}}\left(c\left(r_{n}+\frac{\gamma}{2}\right)^{2}\right)^{\frac{k}{2}}e^{-\frac{\pi}{2}c(r_{n}+\frac{\gamma}{2})^{2}}.$

The last ingredient that we need is the classical inequality
$k!\geq\left(\frac{k}{e}\right)^{k}$. Collecting everything, we get the upper
bound

$C_{\varepsilon}\pi\left(\frac{ce\pi\left(r_{n}+\frac{\gamma}{2}\right)^{2}}{k}\right)^{\frac{k}{2}}e^{-\frac{\pi}{2}c(r_{n}+\frac{\gamma}{2})^{2}}.$

Differentiating this quantity with respect to $k$ we can see that it is
increasing for $k<c\pi\left(r_{n}+\frac{\gamma}{2}\right)^{2}$, in particular
we can without loss of generality assume that $k=c\pi r_{n}^{2}$. Substituting
this value we get

$C_{\varepsilon}\pi\left(e\left(1+\frac{\gamma}{2r_{n}}\right)^{2}\right)^{\frac{c\pi
r_{n}^{2}}{2}}e^{-\frac{\pi}{2}c(r_{n}+\frac{\gamma}{2})^{2}}=C_{\varepsilon}\pi\left(e\left(1+\frac{\gamma}{2r_{n}}\right)^{2}e^{-(1+\frac{\gamma}{2r_{n}})^{2}}\right)^{\frac{c\pi
r_{n}^{2}}{2}}.$

Let $u=\left(1+\frac{\gamma}{2r_{n}}\right)^{2}-1>0$. Then the quantity in the
brackets is $(1+u)e^{-u}=\nu<1$, therefore the whole expression is

$C_{\varepsilon}\pi\nu^{c\frac{\pi r_{n}^{2}}{2}},$

which is exponentially small in $c$, as required. Doing the same for the
second integral we get $C_{\varepsilon}^{\prime}\pi\nu^{\prime c\frac{\pi
r_{m}^{2}}{2}}$. Let $\nu_{0}$ be the maximum of all $\nu$ over all pairs
$(n,m)$, $c_{\varepsilon}$ be the maximum of $C_{\varepsilon}$ over all pairs
$(n,m)$ and $r$ be the minimum of $r_{n}$. Then we get the upper bound

$2c_{\varepsilon}\pi\nu_{0}^{c\frac{\pi r^{2}}{2}}.$

Taking $\nu_{0}^{\frac{\pi r^{2}}{2}}<\alpha<1$ and taking $c$ big enough we
can have this quantity less than $\alpha^{c}$, as required.

Now, we turn to proving $(ii)$. We will only bound
$||(\operatorname{Id}-P_{I})f||$ since the Fourier transform corresponds to
the 90 degrees rotation of the complex plane, and when we rotate the square
with center at the origin we get back the same square (in fact, with slight
tweaking to the proof of Lemma 1.5 we can achieve that set $\mathcal{V}$ is
invariant under the Fourier transform). We have

$||(\operatorname{Id}-P_{I})f||^{2}=\int_{-\infty}^{-\frac{\sqrt{c}}{2}}|f(t)|^{2}dy+\int_{\frac{\sqrt{c}}{2}}^{\infty}|f(t)|^{2}dy.$

We will only bound the second term, the other one being similar. To estimate
it, we use duality:

$\int_{\frac{\sqrt{c}}{2}}^{\infty}|f(t)|^{2}dy=\sup_{\operatorname{supp}g\subset[\frac{\sqrt{c}}{2},+\infty),||g||_{2}=1}|\langle
f,g\rangle|^{2}.$

Since Bargmann transform is an isometry, this inner product is equal to
$\langle\mathcal{B}f,\mathcal{B}g\rangle$. Recall that $\mathcal{B}f$ is just
a $T_{w}$ shift of some normalized monomial. Specifically, let $f$ correspond
to the disk $D_{n}$, with center $w_{n}$ and radius $r_{n}$, so that
$\mathcal{B}f=F=T_{v_{n}}\frac{z^{k}}{\sqrt{k!}}$, where
$v_{n}=\sqrt{c}w_{n}$. We want to obtain an upper bound for

$\int_{\mathbb{C}}|F(z)||G(z)|e^{-\pi|z|^{2}}dz,$

where $G=\mathcal{B}g$. We will again split this integral into two integrals

$\int_{|z-v_{n}|\geq\sqrt{c}(r_{n}+\frac{\gamma}{2})}|F(z)||G(z)|e^{-\pi|z|^{2}}dz+\int_{|z-v_{n}|<\sqrt{c}(r_{n}+\frac{\gamma}{2})}|F(z)||G(z)|e^{-\pi|z|^{2}}dz.$

For the first integral we bound $|G(z)|e^{-\frac{\pi}{2}|z|^{2}}\leq 1$ and
proceed exactly like in the part $(i)$. For the second integral, we in turn
use an estimate $|F(z)|e^{-\frac{\pi}{2}|z|^{2}}\leq 1$ and so we need to
obtain a stronger pointwise bound for $|G(z)|$ when
$|z-v_{n}|\leq\sqrt{c}(r_{n}+\frac{\gamma}{2})$. We will do this by applying
the Cauchy–Schwarz inequality to the definition of the Bargmann transform
(2.1). Note that this will be the only place where we use an explicit formula
for the Bargmann transform. We have

$e^{-\frac{\pi}{2}|z|^{2}}\mathcal{B}g(z)=e^{-\frac{\pi}{2}|z|^{2}}2^{1/4}\int_{\frac{\sqrt{c}}{2}}^{\infty}g(t)e^{2\pi
tz-\pi t^{2}-\frac{\pi}{2}z^{2}}dt.$

Applying the Cauchy–Schwarz inequality we get

$\left|e^{-\frac{\pi}{2}|z|^{2}}\mathcal{B}g(z)\right|^{2}\leq
e^{-\pi|z|^{2}}2^{1/2}\int_{\frac{\sqrt{c}}{2}}^{\infty}e^{\operatorname{Re}(4\pi
tz-2\pi t^{2}-\pi z^{2})}dt.$

This quantity can be seen to be the tail of the Gaussian distribution. This
can be checked directly, but to guess this we can use the following heuristic:
if the lower limit was $-\infty$ then this would have been just $1$ from the
pointwise bound in the Bargmann–Segal–Fock space, and by applying the shift
$T_{z}^{-1}$ we will get the same result as if we were integrating from
$\frac{\sqrt{c}}{2}-\operatorname{Re}z$. So, we have the bound

$\left|e^{-\frac{\pi}{2}|z|^{2}}\mathcal{B}g(z)\right|^{2}\leq\sqrt{2}\int_{\frac{\sqrt{c}}{2}-\operatorname{Re}z}^{\infty}e^{-2\pi
t^{2}}dt.$

Observe that if $|z-v_{n}|\leq\sqrt{c}(r_{n}+\frac{\gamma}{2})$ then
$\operatorname{Re}z\leq\frac{\sqrt{c}}{2}(1-\frac{\gamma}{2})$, because disk
$D_{n}$ is at least $\gamma$ away from the boundary of the square
$(-\frac{1}{2},\frac{1}{2})\times(-\frac{1}{2},\frac{1}{2})$. Therefore, for
such $z$ we have

$\left|e^{-\frac{\pi}{2}|z|^{2}}\mathcal{B}g(z)\right|^{2}\leq\sqrt{2}\int_{\frac{\gamma\sqrt{c}}{2}}^{\infty}e^{-2\pi
t^{2}}dt.$

As in the part $(i)$, we observe that the function $e^{-2\pi t^{2}}$ is
decreasing and $e^{-2\pi(t+1)^{2}+2\pi t^{2}}\leq e^{-2\pi}<1$ for $t\geq 0$,
therefore the integral is at most $C_{\varepsilon}$ multiplied by
$e^{-2\pi\left(\frac{\gamma\sqrt{c}}{2}\right)^{2}}$ for some
$C_{\varepsilon}>0$. That is, we have

$\left|e^{-\frac{\pi}{2}|z|^{2}}\mathcal{B}g(z)\right|\leq\sqrt{C_{\varepsilon}\sqrt{2}}e^{-\frac{\pi\gamma^{2}}{4}c}.$

When we integrate this bound over the disk
$|z-v_{n}|\leq\sqrt{c}(r_{n}+\frac{\gamma}{2})$ we multiply by the area of
this disk. Since it is contained in the square
$(-\frac{\sqrt{c}}{2},\frac{\sqrt{c}}{2})\times(-\frac{\sqrt{c}}{2},\frac{\sqrt{c}}{2})$,
its area is at most $c$. Thus, the integral is at most

$c\sqrt{C_{\varepsilon}\sqrt{2}}e^{-\frac{\pi\gamma^{2}}{4}c}.$

This is exponentially decreasing in $c$, thus we proved that for some
$\alpha<1$ we have $|\langle F,G\rangle|\leq\alpha^{c}$ if $c$ is big enough,
as required. ∎

##  4\. Proof of Theorem 1.4

Let us take a subset $\mathcal{V}_{0}\subset\mathcal{V}$ of size
$n=[(1-\varepsilon)c]$ and let $V\subset L^{2}(\mathbb{R})$ be the subspace
generated by $\mathcal{V}_{0}$. We will apply min-max principle to $V$ to
deduce the lower bound for $\lambda_{n}(c)$ (in particular, we will show that
functions in $\mathcal{V}_{0}$ are linearly independent so that the dimension
of $V$ is indeed $[(1-\varepsilon)c]$).

We start with estimating $P_{I}Q_{J}P_{I}f$ for $f\in\mathcal{V}$. We have

$||(\operatorname{Id}-Q_{J}P_{I})f||=||(\operatorname{Id}-Q_{J})f+Q_{J}(\operatorname{Id}-P_{I})f||\leq||(\operatorname{Id}-Q_{J})f||+||(\operatorname{Id}-P_{I})f||\leq
2\alpha^{c},$

where we used $||Q_{J}||\leq 1$ since $Q_{J}$ is a projection. Doing this one
more time we get

$||(\operatorname{Id}-P_{I}Q_{J}P_{I})f||=||P_{I}(\operatorname{Id}-Q_{J}P_{I})f+(\operatorname{Id}-P_{I})f||\leq||(\operatorname{Id}-Q_{J}P_{I})f||+||(\operatorname{Id}-P_{I})f||\leq
3\alpha^{c}.$

Let $\mathcal{V}_{0}=\\{f_{1},f_{2},\ldots,f_{n}\\}$ and consider
$f(t)=\sum_{k=1}^{n}a_{k}f_{k}(t)$ where not all $a_{k}$ are equal to zero. We
are interested in the quantity $\frac{||P_{I}Q_{J}P_{I}f||}{||f||}$. We have

$||f||^{2}=\sum_{k=1}^{n}|a_{k}|^{2}+\sum_{1\leq k,l\leq n,k\neq
l}a_{k}\bar{a_{l}}\langle f_{k},f_{l}\rangle.$

From the Proposition 3.1 we know that $|\langle
f_{k},f_{l}\rangle|\leq\alpha^{c}$. Therefore, we can crudely bound

$||f||^{2}\geq\sum_{k=1}^{n}|a_{k}|^{2}-\left(\sum_{k=1}^{n}|a_{k}|\right)^{2}\alpha^{c}.$

Now, we can show that $\mathcal{V}_{0}$ is linearly independent. By the
inequality between the arithmetic mean and the quadratic mean we have
$\sum_{k=1}^{n}a_{k}^{2}\geq\frac{1}{n}\left(\sum_{k=1}^{n}|a_{k}|\right)^{2}$,
therefore, if $\alpha^{c}<\frac{1}{2c}$, we have

(4.1)
$||f||^{2}\geq(1-n\alpha^{c})\sum_{k=1}^{n}|a_{k}|^{2}\geq\frac{1}{2}\sum_{k=1}^{n}|a_{k}|^{2},$

in particular it is non-zero (here we used a crude bound $n\leq c$).

Similarly, we can estimate $||P_{I}Q_{J}P_{I}f||$. We have

$||(\operatorname{Id}-P_{I}Q_{J}P_{I})f||=||\sum_{k=1}^{n}(\operatorname{Id}-P_{I}Q_{J}P_{I})a_{k}f_{k}||\leq\sum_{k=1}^{n}|a_{k}|||(\operatorname{Id}-P_{I}Q_{J}P_{I})f_{k}||\leq
3\alpha^{c}\sum_{k=1}^{n}|a_{k}|.$

By the triangle inequality this implies

$||f||-3\alpha^{c}\sum_{k=1}^{n}|a_{k}|\leq||P_{I}Q_{J}P_{I}f||.$

Dividing this by $||f||$ we get

$\frac{||P_{I}Q_{J}P_{I}f||}{||f||}\geq
1-\frac{3\alpha^{c}\sum_{k=1}^{n}|a_{k}|}{||f||}.$

Using the estimate (4.1) and the inequality between the arithmetic mean and
the quadratic mean this is at least $1-3\sqrt{2c}\alpha^{c}$. Choosing
$\alpha<\delta<1$ and $c$ big enough this is at least $1-\delta^{c}$. By the
min-max principle this means that $\lambda_{n}(c)\geq 1-\delta^{c}$, as
required.

##  5\. Proof of Lemma 1.5

We will prove by induction on $n$ that there exists a finite set of disjoint
closed disks in $R=(-\frac{1}{2},\frac{1}{2})\times(-\frac{1}{2},\frac{1}{2})$
with total measure at least $1-2^{-n}$. For $n=1$ we can take disk with center
at the origin and radius $0.49$, its area is $\pi 0.49^{2}>\frac{1}{2}$.

Assume that we already constructed disks $D_{1},\ldots,D_{m}$ with total
measure at least $1-2^{-n}$. We want to add some disks so that the total
measure becomes at least $1-2^{-n-1}$.

Let us pick a very large integer $N$ to be determined later and slice $R$ into
the squares of side length $\frac{1}{N}$. We will split the squares into three
groups: squares in the group $A$ are those which lie entirely inside one of
the disks, squares in the group $B$ are those which do not intersect any of
the disks, and squares in the group $C$ are those which intersect the boundary
of one of the disks.

We claim that there exists a constant $c$, independent of $N$ (but depending
on the already chosen disks $D_{1},\ldots,D_{m}$) such that the number of
squares in $C$ is at most $cN$. Indeed, let us consider any square $H$ from
the group $C$ and assume that it intersects the boundary of $D_{l}$. Since the
diameter of $H$ is $\frac{\sqrt{2}}{N}$, square $H$ lies entirely in the
$\frac{\sqrt{2}}{N}$–vicinity of the circumferences of $D_{l}$. Therefore, the
total measure of disks in $C$ is not greater than the total measure of the
union of $\frac{\sqrt{2}}{N}$–vicinities of the circumferences of the disks
$D_{1},\ldots,D_{m}$. Let the radius of $D_{l}$ be $r_{l}$. Then the measure
of its $\frac{\sqrt{2}}{N}$–vicinity is

$\pi\left(r_{l}+\frac{\sqrt{2}}{N}\right)^{2}-\pi\left(r_{l}-\frac{\sqrt{2}}{N}\right)^{2}=\frac{\pi
4\sqrt{2}r_{l}}{N},$

assuming that $N$ is chosen so big that $\frac{\sqrt{2}}{N}$ is smaller than
the radius of every disk $D_{l}$. Therefore, the measure of the unions of
these vicinities is at most

$\frac{\pi 4\sqrt{2}\sum_{l=1}^{m}r_{l}}{N}.$

Since each square $H$ from our decomposition has area $\frac{1}{N^{2}}$, there
can be at most $\pi 4\sqrt{2}\sum_{l=1}^{m}r_{l}N$ squares in the set $C$, so
$c=\pi 4\sqrt{2}\sum_{l=1}^{m}r_{l}$ works.

We have $|A|+|B|+|C|=N^{2}$. Let $\mu$ be the measure of the union of the
already chosen disks $D_{1},\ldots,D_{m}$. By the induction hypothesis
$\mu\geq 1-2^{-n}$. For each square in $B$ we will consider the disk with the
center at the center of $B$ and radius $\frac{0.49}{N}$. Clearly, these disks
are pairwise disjoint, they are disjoint from the already chosen disks
$D_{1},\ldots,D_{m}$ and they lie inside of $R$ (since $0.49<0.50$, they don’t
even touch each other or the boundary of $R$, as required). Their total
measure is $\frac{|B|0.49^{2}\pi}{N^{2}}\geq\frac{3|B|}{4N^{2}}$. Adding them
to our set, the total measure we get is at least $\frac{3|B|}{4N^{2}}+\mu$. It
remains to show that this is at least $1-2^{-n-1}$.

Since all squares from $A$ lie inside of the disks $D_{1},\ldots,D_{m}$, we
have $\frac{|A|}{N^{2}}\leq\mu$. Subtracting this inequality from $1$, we get

$1-\mu\leq\frac{|B|+|C|}{N^{2}}.$

Subtracting further $\frac{3|B|}{4N^{2}}$ from both sides, we obtain

(5.1) $1-\mu-\frac{3|B|}{4N^{2}}\leq\frac{|B|+4|C|}{4N^{2}}.$

Since all the squares from $B$ do not intersect any of the disks
$D_{1},\ldots,D_{m}$, their total measure is at most $1-\mu$, that is
$\frac{|B|}{N^{2}}\leq 1-\mu\leq 2^{-n}$. Plugging this into (5.1) and
recalling that $|C|\leq cN$ we get

$1-\mu-\frac{3|B|}{4N^{2}}\leq 2^{-n-2}+\frac{c}{N}.$

Choosing $N$ so that $\frac{c}{N}<2^{-n-2}$ we get the desired result.

### Acknowledgments

I would like to thank Fabio Nicola, Kristian Seip and Mikhail Sodin for
helpful discussions. This work was supported by BSF Grant 2020019, ISF Grant
1288/21, and by The Raymond and Beverly Sackler Post-Doctoral Scholarship.

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