Present address: ]Center for Quantum Devices, Niels Bohr Institute, University
of Copenhagen, 2100 Copenhagen, Denmark

# Orbital-selective Band Hybridisation at the Charge Density Wave Transition
in Monolayer TiTe2

Tommaso Antonelli  SUPA, School of Physics and Astronomy, University of St
Andrews, St Andrews KY16 9SS, United Kingdom  Warda Rahim  Department of
Chemistry and Thomas Young Centre, University College London, 20 Gordon
Street, London, WC1H 0AJ, United Kingdom  Matthew D. Watson  SUPA, School of
Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United
Kingdom  Diamond Light Source, Harwell Campus, Didcot, OX11 0DE, United
Kingdom  Akhil Rajan  Oliver J. Clark  Alisa Danilenko  [  SUPA, School of
Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United
Kingdom  Kaycee Underwood  SUPA, School of Physics and Astronomy, University
of St Andrews, St Andrews KY16 9SS, United Kingdom  Igor Marković  Edgar
Abarca-Morales  SUPA, School of Physics and Astronomy, University of St
Andrews, St Andrews KY16 9SS, United Kingdom  Max Planck Institute for
Chemical Physics of Solids, Nöthnitzer Straße 40, 01187 Dresden, Germany  Seán
R. Kavanagh  Department of Chemistry and Thomas Young Centre, University
College London, 20 Gordon Street, London, WC1H 0AJ, United Kingdom  Department
of Materials and Thomas Young Centre, Imperial College London, Exhibition
Road, London SW7  P. Le Fèvre  F. Bertran  Synchrotron SOLEIL, CNRS-CEA,
L’Orme des Merisiers, Saint-Aubin-BP48, 91192 Gif-sur-Yvette, France  K.
Rossnagel  Institut für Experimentelle und Angewandte Physik, Christian-
Albrechts-Universität zu Kiel, 24098 Kiel, Germany  Ruprecht Haensel
Laboratory, Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany
David O. Scanlon  Department of Chemistry and Thomas Young Centre, University
College London, 20 Gordon Street, London, WC1H 0AJ, United Kingdom  Phil D. C.
King<EMAIL_ADDRESS>SUPA, School of Physics and Astronomy,
University of St Andrews, St Andrews KY16 9SS, United Kingdom

###### Abstract

An anomalous $(2\times 2)$ charge density wave (CDW) phase emerges in
monolayer 1T-TiTe2 which is absent for the bulk compound, and whose origin is
still poorly understood. Here, we investigate the electronic band structure
evolution across the CDW transition using temperature-dependent angle-resolved
photoemission spectroscopy. Our study reveals an orbital-selective band
hybridisation between the backfolded conduction and valence bands occurring at
the CDW phase transition, which in turn leads to a significant electronic
energy gain, underpinning the CDW transition. For the bulk compound, we show
how this energy gain is almost completely suppressed due to the three-
dimensionality of the electronic band structure, including via a
$k_{z}$-dependent band inversion which switches the orbital character of the
valence states. Our study thus sheds new light on how control of the
electronic dimensionalilty can be used to trigger the emergence of new
collective states in 2D materials.

## Introduction

Transition Metal Dichalcogenides (TMDs) offer a versatile platform to study
the interplay of different collective quantum states [1, 2, 3]. Excitingly,
the charge density wave (CDW) phases in these materials have been shown to
exhibit a strong dependence on material thickness down to the monolayer (ML)
limit, with the emergence of modified ordering wavevectors [4] and transition
temperatures [5], and evidence for significant phase competition [6]. A
particularly intriguing case is that of TiTe2. This is the sister compound of
the famous charge-density wave material TiSe2 [7, 8], in which an
unconventional $(2\times 2\times 2)$ CDW emerges from a narrow-gap
semiconducting normal state [9]. CDW order persists down to the monolayer
limit, with a $(2\times 2)$ ordering [10]. In TiTe2, a larger $p$-orbital
valence bandwidth (due to the more extended Te 5$p$ orbitals) and enhanced
$d$-$p$ charge transfer (promoted by the lower electronegativity of Te) cause
the Ti $d$-derived conduction band and Te $p$-derived valence band to
substantially overlap (Fig. 1(a-c)). This semi-metallic state persists in the
bulk to low temperatures, and no CDW transition occurs [11, 12, 13].

Figure 1: (a) Crystal structure of bulk 1T-TiTe2 and (b) its measured
electronic structure from ARPES ($h\nu=144$ eV, $T=20$ K) along the A-L
direction of the Brillouin zone. (c) Illustration of the bulk Fermi surface.
(d) Calculated phonon dispersion of bulk TiTe2, indicating stability of the 1T
crystal structure. (e) Crystal structure of ML-TiTe2, where the black arrows
indicate the expected direction of the atomic displacements of the periodic
lattice distortion associated with a CDW instability. (f) ARPES spectrum of
ML-TiTe2 measured along the $\Gamma$-M direction ($h\nu$ = 20 eV, $T=16$ K).
(g) Schematic Fermi surface of ML-TiTe2, with the three CDW wave vectors
$q_{1}$, $q_{2}$ and $q_{3}$, indicated. (h) Calculated phonon dispersions of
ML-TiTe2, indicating the instability of the 1T crystal structure. (i) Tracking
the potentials energy curve $\Delta U$ as a function of the corresponding
normal-mode coordinates $Q_{M}$, with finite but small energy gain for the
distortion; corresponding calculations for ML-TiSe2 (green) indicate a much
more pronounced instability at the same wavevector as for the Te case
(orange).

Surprisingly, however, signatures of a CDW phase have recently been observed
in monolayer TiTe2, although not in bilayer or thicker films [14]. This new
CDW is associated with a $(2\times 2)$ periodic lattice distortion, in
striking similarity to the CDW observed in ML-TiSe2 (Fig. 1(e-g)), despite the
significant band overlap and semimetallicity in the normal state of the Te-
based system. The origins of this instability, and why it is only stabilised
in the monolayer limit, have remained elusive to date. Here, we study the CDW
transition in ML-TiTe2 using temperature-dependent angle-resolved
photoemission spectroscopy (ARPES), hybrid density functional theory (DFT)
calculations, and minimal-model based approaches. Through this, we find
evidence of a marked but symmetry-selective band hybridisation between the
valence and conduction bands which ultimately provides the energetic incentive
to drive the CDW phase transition. For the bulk system, we demonstrate how
pronounced out-of-plane band dispersions lead to mismatched Fermi surfaces and
inverted valence band symmetries. As a result, the corresponding electronic
energy gain is highly suppressed, explaining the marked difference in the
ground state properties of single vs. multi-layer TiTe2.

## Results

### CDW state in ML-TiTe2

To confirm the previously reported CDW state in ML-TiTe2 as an intrinsic
instability of the perfect ML, we start with DFT calculations of the lattice
dynamics. The calculated phonon dispersions of bulk 1T-TiTe2 are shown in Fig.
1(d). These show an absence of any imaginary frequency modes: the 1T crystal
structure thus is predicted to remain stable down to 0 K. This is consistent
with the absence of any CDW transition in our measurements of the electronic
structure from bulk TiTe2 in Fig. 1(b) and with previous experiments [11, 12,
13]. In contrast, for the monolayer (Fig. 1(h)), our calculations indicate
that the 1T crystal structure is no longer stable. The imaginary frequency
mode at the M point indicates the tendency of the structure to undergo a
$(2\times 2)$ lattice distortion. This phonon mode has an $A_{u}$ irreducible
representation at M, consistent with the soft phonon mode thought to underpin
the triple-$q$ CDW instability in ML-TiSe2 [15]. Tracing the stability of this
mode (Fig. 1(i)), we find a classic anharmonic double-well potential,
confirming the intrinsic instability of ML-TiTe2 to undergoing a periodic
lattice distortion upon cooling.

The underlying lattice instability is qualitatively the same as that in the
sister compound ML-TiSe2 (Fig. 1(i)), but we find that the depth of the
potential well in the telluride is much smaller. This indicates that the
instability in ML-TiTe2 is intrinsically weaker than that of TiSe2, where
signatures of a strong-coupling CDW transition have been observed [16, 10].
Nonetheless, consistent with a recent work [14], we find that the new periodic
potential arising from the softening of the M-point phonon mode in ML-TiTe2 is
sufficiently strong to induce a ‘backfolded’ copy of the valence band
dispersions, evident at the original M point in our measurements of the low-
temperature electronic structure (Fig. 1(f)). The backfolded spectral weight
is weaker than the corresponding spectral signatures in ML-TiSe2 [16],
qualitatively consistent with a ‘weaker’ instability in the telluride as
discussed above. Nonetheless, its observation points to a clear electronic
component to the predicted periodic lattice distortion.

Figure 2: Temperature-dependent ARPES measurements of ML-TiTe2 ($h\nu=15$ eV)
along the $\Gamma$-M direction and centered at the (a) M and (c) $\Gamma$
points. (b) Intensity of the backfolded valence bands extracted from fits to
MDCs at $E-E_{\mathrm{F}}=-0.14$ eV (along the dashed line in (a)). (d)
temperature-dependent EDCs extracted at $k=0$ (dashed line in (c)). The inset
for the 14 K spectrum in (c) shows a second-derivative analysis of the
measured band dispersion.

To investigate this in detail, we study the temperature-dependent evolution of
the electronic structure of ML-TiTe2 in Fig. 2. With decreasing temperature,
the spectral weight of the characteristic backfolded valence bands at the
M-point gradually increases (Fig. 2(a)). This is evident in the spectral
weight extracted from fits to momentum distribution curves (MDCs) at $E-E_{F}$
= -0.14 eV (Fig. 2(b), see Supplemental Fig. S1 for the fitting details),
where a rapid increase of the backfolded weight is evident at a temperature
$T\approx 110$ K. This is similar, although slightly higher, than the value of
92 K reported by Chen et al. [14], and we assign this as the CDW transition
temperature in this system, $T_{\mathrm{CDW}}$. We note that this critical
temperature is substantially lower than $T_{\mathrm{CDW}}\approx 220$ K in ML-
TiSe2 [16]. This is qualitatively consistent with the calculated double-well
potentials shown in Fig. 1(i): the depths of these wells are related to the
thermal energy at which the distorted phase is no longer stable as compared to
the high-temperature structure, located at the saddle point of these wells.
The smaller depth of the potential well in the case of ML-TiTe2 than TiSe2,
therefore, indicates that the CDW ordering would be expected to onset at a
lower temperature in the former, as observed here.

The $(2\times 2)$ order predicted by our calculations is in agreement with a
superstructure modulation imaged by scanning tunnelling microscopy in Ref.
[14]. This naturally explains the valence band backfolding from the $\Gamma$
to the M-point of the Brillouin zone, although a concomitant backfolding of
the conduction band from M to $\Gamma$ has not been observed to date. Here, we
find the appearance of weak spectral weight in between the innermost valence
band in the ordered state (Fig. 2(c)). Its spectral weight grows with
deceasing temperature, as evident from a peak at $\approx 100$ meV binding
energy in energy distribution curves (EDCs) taken at the Brillouin zone centre
(Fig. 2(d)). A second-order derivative analysis (inset of Fig. 2(c)) indicates
how this additional spectral weight derives from an electron-like band,
suggesting its origin as due to a backfolding of the conduction bands in the
CDW phase of ML-TiTe2.

Figure 3: (a) ARPES spectra ($h\nu=20$ eV) measured around $\Gamma$ along the
$\Gamma$-M direction in the normal (left, $T=160$ K) and CDW (right, $T=16$ K)
state. (b,c) Intensities extracted from fits to MDCs in the vicinity of the
Fermi level from the dispersions in (a), shown for the (b) inner and (c) outer
valence bands. (d,e) Intensity difference between the MDC fits to the high and
low temperature ARPES spectra (black dots). The green lines are the
corresponding intensity difference extracted from the simulated spectra shown
in Fig. 5(c, e). The grey and green arrows indicate the energy where the
backfolded conduction and valence bands intersect (shown in the inset from a
measurement around the M point). Figure 4: (a,b) Tight-binding band structure
of ML-TiTe2 neglecting SOC, revealing the $p_{x}$ (a, red colouring) and
$p_{y}$ (b, green colouring) character along the M-$\Gamma$-K path. The inset
shows the evolution of the valence band orbital character around the Fermi
surface, indicating the radial and tangential orbital textures of the inner
and outer bands, respectively. (c) Overlap integrals
$\left\langle{}p_{x}|d_{x^{\prime}y^{\prime}}\right\rangle$ (red) and
$\left\langle{}p_{y}|d_{x^{\prime}y^{\prime}}\right\rangle$ (green) as a
function of the atom displacement caused by softening of an Au phonon mode at
M (see insets; only the radial part of the wavefunctions are shown). (d)
Schematic illustration of the two relevant coordinate systems, $xyz$ and
$x^{\prime}y^{\prime}z^{\prime}$, used to describe the orbital character in
the tight-binding calculation within the crystal and octahedral basis,
respectively. Below, a schematic of the 5 bands used in the minimal model is
shown.

### Orbital-selective band hybridisation

The second-derivative analysis also suggests the opening of spectral gaps in
the reconstructed electronic structure. To investigate this in detail, to
understand better how the electronic structure around the Fermi level evolves
across the CDW transition, we performed high-resolution measurements at the
$\Gamma$ point of the Brillouin zone. Here, the spectral weight is dominated
by the valence bands (Fig. 3(a)). Comparing the high- and low-temperature
spectra, a small spectral gap can be seen to open in the vicinity of the Fermi
level for the outer valence band, while no such gap is obvious for the inner
valence band. To probe this quantitatively, we show the intensities of
Lorentzian fits to MDCs of the outer and inner valence bands in Figs. 3(b,c),
respectively (see also Supplemental Fig. S2 for details of the fitting). The
intensities vary smoothly for the high-temperature measurements, with an
overall decrease in intensity approaching the Fermi level reflecting the
effects of transition matrix elements on the photoemission measurements, and
also the influence of spectral broadening and rotational disorder of the grown
films. This global intensity variation is still present at low temperature,
but with an additional modulation leading to dips in fitted intensity at
binding energies of $\approx 50$ meV and $\approx 100$ meV for the outer and
inner bands, respectively. This is clearly visible in normalised difference
plots of the fitted intensity at high and low temperatures (coloured arrows in
Fig. 3(d,e)). The suppressed spectral weight occurs at the energies at which
the original valence and conduction bands intersect when backfolded in the
low-temperature phase (inset of Fig. 3(b)). This suggests their origin is due
to a band hybridisation between the backfolded conduction and valence bands in
the CDW state, opening band gaps that have remained hidden in the spectral
broadening in measurements to date. Intriguingly, however, these gaps are
highly asymmetric, with a pronounced minimum evident for the outer band, while
only a small dip is observed at the band crossing energies for the inner
valence band.

We will show below that this reflects an orbital- and symmetry-selective band
hybridisation, providing key insight into the microscopic mechanisms of the
stabilisation of CDW order in ML-TiTe2, and its absence in the bulk. To
demonstrate this, we introduce a minimal 5-band model capturing the low-energy
electronic structure of ML-TiTe2 in the CDW state (see Methods Methods):

$\mathcal{H}=\begin{pmatrix}e_{1}&0&0&\Delta c(\theta)&\Delta s(\theta)\\\
0&e_{2}&0&\Delta c(\theta-\frac{2\pi}{3})&\Delta s(\theta-\frac{2\pi}{3})\\\
0&0&e_{3}&\Delta c(\theta-\frac{4\pi}{3})&\Delta s(\theta-\frac{4\pi}{3})\\\
\Delta c(\theta)&\Delta c(\theta-\frac{2\pi}{3})&\Delta
c(\theta-\frac{4\pi}{3})&h_{out}&i\lambda_{SO}\\\ \Delta s(\theta)&\Delta
s(\theta-\frac{2\pi}{3})&\Delta
s(\theta-\frac{4\pi}{3})&-i\lambda_{SO}&h_{in}\end{pmatrix}$ (1)

where $e_{i}$ {$i=1,2,3$} represent the normal state dispersion of the
elliptical electron pockets centered at neighbouring M points which are
backfolded to $\Gamma$ in the low-temperature phase (Fig. 1(g)), $h_{i}$
{$i=out,in$} are the two hole bands centered at $\Gamma$, spin-orbit coupling,
$\lambda_{SO}$, is included for the Te-derived hole bands, and $c(x)$ and
$s(x)$ represent $\cos(x)$ and $\sin(x)$, respectively.

Figure 5: (a) Fermi surface calculated within our minimal model for ML-TiTe2,
with three electron pockets from M backfolded to $\Gamma$, but without
hybridisation ($\Delta$ = 0 meV, top). As the hybridisation is turned on
($\Delta$ = 42 meV, bottom), the Fermi surface is partially gapped where
allowed by symmetry. (b,d) Bare bands calculated from the minimal model along
$\Gamma$-M for $\Delta$ = 0 meV and $\Delta$ = 42 meV, respectively, projected
onto the character of the original conduction and valence states (blue-yellow
colouring). (c,e) Comparison between the experimental data and simulated ARPES
spectra from the model calculation in (b) and (d), respectively. (f)
Calculated DOS from the model, indicating the effects of band hybridisation,
providing a significant electronic energy gain in the occupied states (shaded
region).

The key physics of the resulting band hybridisation between the backfolded
conduction and valence states is included via an interaction term $\Delta$,
whose strength is modulated by angle-dependent form factors, where $\theta$ is
the angle of the momentum vector within the 2D plane,
$\theta=\tan^{-1}(k_{y}/k_{x})$. These form factors derive from the particular
orbital textures of the valence states. Along $\Gamma$-M, our tight-binding
calculations, informed from our DFT-calculated electronic structure and
optimised to match the experimental normal-state dispersions (see Methods
Methods), indicate that the inner valence band is derived mainly from Te
5$p_{x}$ orbitals, while the outer valence band has a dominant $p_{y}$ orbital
character, where $x$ and $y$ are referenced to the global coordinate system
(Fig. 4(d)). Moving away from this direction, the orbital content rotates such
that, for the perpendicular $\Gamma$-K direction, the inner band is
predominantly of $p_{y}$ character, while the outer band is $p_{x}$-like. The
inner and outer valence bands thus host radial and tangential orbital textures
as shown in the inset of Fig. 4(a,b), much like those recently uncovered in,
e.g., topological states of Bi2Se3 [17][18] and Rashba states of BiTeI [19].
Here, we will show that these orbital textures lead to a strong momentum-
dependence of the allowed band hybridisation in the ordered state of TiTe2.

To demonstrate this, we consider the overlap of the $p$ states with one of the
three backfolded conduction band pockets, $e_{1}$, which has predominantly
$d_{x^{\prime}y^{\prime}}$ symmetry in the octahedral basis
({$x^{\prime},y^{\prime},z^{\prime}$}, Fig. 4(d)) [15]. The situation for the
other bands follows from the three-fold rotational symmetry of the system. To
understand the hybridisation, we consider two core symmetries of the normal-
state 1T crystal structure: a mirror plane, $m$, that is oriented along
$\Gamma$-M, and a $C_{2}$ rotational symmetry axis oriented along $\Gamma$-K.
The 3$d_{x^{\prime}y^{\prime}}$ orbital has even parity in both of these (see
inset in Fig. 4(c)), consistent with the $A_{g}$ irreducible representation of
the electron pocket at M [20]. In contrast, the state made from $p_{y}$
orbitals on the two chalcogen sites is even in $C_{2}$, but odd in $m$, while
the opposite is true for the $p_{x}$-derived bands (see insets in Fig. 4(c)).

Considering at the $\Gamma$ point, where both of these symmetries are present
for the high-temperature 1T crystal structure, the overlap integrals
$\left\langle{p_{x}|d_{x^{\prime}y^{\prime}}}\right\rangle$ and
$\left\langle{p_{y}|d_{x^{\prime}y^{\prime}}}\right\rangle$ are strictly zero,
as a result of the opposite parity of these states under the $C_{2}$ rotation
and the mirror symmetry, respectively. At the CDW transition, however, the
softening of three $A_{u}$ phonon modes (Fig. 1(h)) leads to a periodic
lattice distortion [15] which breaks the mirror symmetry, while preserving the
$C_{2}$ rotational symmetry (the corresponding atomic displacements arising
from the softening of one of the three phonon modes are indicated by the black
arrows in the insets of Fig. 4(c)). The hybridisation of the $p_{x}$\- and
$d$-derived states,
$\left\langle{p_{x}|d_{x^{\prime}y^{\prime}}}\right\rangle$, remains zero due
to the $C_{2}$ symmetry. However, the mirror-symmetry enforced constraint of a
lack of hybridisation between the $p_{y}$\- and $d$-derived states in the
normal state is relaxed in the distorted structure. This is evident from our
calculated overlap integrals in Fig. 4(c), where the hybridisation matrix
element between $p_{y}$ and $d_{x^{\prime}y^{\prime}}$ tesseral harmonics
increases linearly with the amplitude of the atomic displacements within the
periodic lattice distortion, while
$\left\langle{p_{x}|d_{x^{\prime}y^{\prime}}}\right\rangle$ remains zero
irrespective of the atomic displacement.

While these arguments are formally valid only at $\Gamma$ where both the
symmetries are present, they provide the basis for an understanding of the
band hybridisation throughout the Brillouin zone, where the $C_{2}$ (in both
the normal and low-temperature states) and $m$ (in the normal state)
symmetries are present along the entire $\Gamma$-K and $\Gamma$-M directions,
respectively. In particular, they indicate that new channels for band
hybridisation between the backfolded valence and conduction band pockets are
only opened up by the periodic lattice distortion where conduction bands of
$d_{x^{\prime}y^{\prime}}$ character intersect valence bands of
$p_{y}$-orbital character (and symmetry-equivalent versions of these for the
other conduction band pockets).

Combined with the momentum-dependent orbital textures of the valence bands
discussed above, this significantly constrains and modulates the hybridisation
in the 2D $k$-space, as captured in the minimal model of Equation 1, and shown
from our model calculations in Fig. 5(a,b,d). While the strict symmetry
protections outlined above are weakened by spin-orbit coupling on the Te site,
which mixes the $p_{x}$ and $p_{y}$ states, it is clear how the above symmetry
constraints lead to a strong asymmetry in the band gaps which open up between
crossings of, e.g., the $e_{1}$ and $h_{in}$ and $h_{out}$ bands for finite
values of hybridisation strength $\Delta$ (Fig. 5(b,d)). The overall
hybridisation scheme derived from our model is qualitatively consistent with
the results of recent DFT calculation of ML-TiTe2 in the $(2\times 2)$ phase
[21, 22]: although the energy gap is overestimated in such calculations, they
predict a weakly hybridised band laying in between the CDW gap as in our
model.

To validate our model and benchmark it against our experimental data, we
simulate the resulting ARPES spectra expected from these hybridised bands (as
described in Methods Methods). We find a good agreement between our simulated
and measured spectra (Fig. 5(c,e)) taking $\Delta=42\pm 10$ meV. This is a
significantly weaker interaction strength than for the sister compound ML-
TiSe2, where $\Delta\approx 100$ meV can be directly estimated from the
experimental data [16], entirely consistent with the weaker nature of the
instability predicted by our calculations of the lattice dynamics discussed
above (Fig. 1(i)) and with the lower $T_{c}$. Crucially, our simulated spectra
well reproduce the suppression of spectral weight in the ordered phase
identified in fits to MDCs of our measured spectra (see green lines in Fig.
3(d,f)), including the pronounced asymmetry in spectral weight suppression at
the crossing of $e_{1}$ and $h_{in}$ and $h_{out}$. This, therefore, provides
direct experimental evidence for the symmetry and orbital-selectivity of the
band hybridisation at the CDW phase transition in ML-TiTe2.

Figure 6: (a) ARPES measurements ($T=14$ K) of bulk TiTe2 at $k_{z}$ = $\pi/c$
($h\nu=144$ eV, top) and at $k_{z}$ = 0 ($h\nu=121$ eV, bottom). The orange
lines show the backfolded bands that would be expected from a hypothetical
$(2\times 2\times 2)$ periodic lattice distortion as is observed for the
sister compound TiSe2. No such CDW backfolding is observed in the experiment.
(b) Calculated density of states (DOS) from a model considering such a
hypothetical $(2\times 2\times 2)$ backfolding, and assuming a hybridisation
strength equal to the one estimated for the monolayer. The significant size
mismatch of the backfolded electron and hole bands, evident in (a), leads to a
DOS suppression from the band hybridisation which disperses from below to
above the Fermi level as a function of $k_{z}$, suppressing the associated
electronic energy gain.

### CDW energetics: from monolayer to bulk

Having validated the essential properties of our model, we can use this to
provide new insight on the key question of why the CDW phase becomes stable in
ML-TiTe2, while it is absent for the bulk. First, we note that there is in
fact a significant electronic energy gain that results from the above band
hybridisations at the CDW transition in the monolayer system. While the
hybridisation gaps along $\Gamma$-M occur largely below the Fermi level, at
other angles around the Fermi surface, the gaps open at the Fermi level itself
(Fig. 5(a)) partially gapping the Fermi surface. Indeed, calculating the
density of states for the normal and hybridised states (Fig. 5(f)) it is
evident how the hybridisation in ML-TiTe2 significantly lowers the total
electronic energy over an extended bandwidth comparable to the band overlap in
the normal state. We thus conclude that this electronic energy gain is
ultimately the driver of the CDW transition in ML-TiTe2.

Given this, one might assume that a similar band hybridisation in the bulk
would stabilise a CDW there, where none is known to exist experimentally, nor
is one found as an instability in our DFT calculations (Fig. 1(d)). However,
there are two key distinctions in bulk TiTe2 as compared to the ML case. First
is a sizeable out-of-plane dispersion of the electronic states, for both the
valence and conduction bands, as shown in Fig. 6(a). The conduction band
states yield large Fermi pockets for $k_{z}=\pi/c$, while they are barely
occupied for $k_{z}=0$. Though the valence state dispersion is weaker, the
Fermi wavevectors nonetheless exhibit a notable decrease with increasing
$k_{z}$. For a $(2\times 2)$ ordering as in the monolayer case, the backfolded
electron- and hole-like Fermi pockets are completely mismatched in size, and
thus unlikely to lead to any significant electronic energy gain as for the
monolayer. Even for a $(2\times 2\times 2)$ instability as in the sister
compound TiSe2 [7], the pockets are still significantly mismatched (Fig.
6(a)), causing the energies at which the hypothetical backfolded bands overlap
to disperse strongly as a function of the out-of-plane momentum. Including a
hybridisation between these bands in the same manner as we did for the
monolayer case again leads to characteristic dips in the DOS, but these DOS
suppressions now move from the occupied to the unoccupied states as a function
of the out-of-plane momentum, $k_{z}$ (Fig. 6(b)). As a result, this
significantly reduces the effectiveness of the band hybridisation here to
lower the total electronic energy of the system as compared to the normal
state.

Figure 7: (a,b) DFT band structure projected onto the atomic $p$ orbitals
(red-green colouring) showing (a) an inversion of the orbital character of the
valence bands for the $\Gamma$-plane as compared to the A-plane. (b) Isotropic
$p_{z}$ character at the Fermi level of the outer valence band in the
$\Gamma$-plane. (c) Fermi surface of bulk TiTe2 from our minimal model, sliced
in the $k_{x}-k_{y}$ plane and including a hypothetical $(2\times 2\times 2)$
backfolding (i.e. from $\Gamma$ to L and A to M). The colors indicate the
orbital character of the Ti 3$d$-derived and Te 5$p$-derived electron and hole
pockets, respectively. (d) DOS of bulk TiTe2 assuming such a backfolding and
symmetry-allowed band hybridisations of strength $\Delta$ = 0 meV and $\Delta$
= 42 meV, indicating negligible electronic energy gain for the CDW state in
the bulk.

Moreover, our DFT calculations indicate that the significant out-of-plane
dispersion in fact leads to a $k_{z}$-dependent band inversion of the $d$ and
$p$-derived states along the $\Gamma$-A direction, of the form known from
other families of TMDs where it generates bulk Dirac points and topological
surface states [3]. As shown in Fig. 7(a), this causes the orbital character
of the inner and outer valence bands to become inverted between the $\Gamma$\-
and the A-plane, switching approximately mid-way through the bulk Brillouin
zone along $k_{z}$. Additionally, $p_{z}$ character is also mixed with the
$p_{x}$ states. The $p_{z}$ orbitals exhibit odd parity with respect to the
$C_{2}$ axis, forbidding their hybridisation with the electron pockets as for
the $p_{x}$-derived states. The in-plane rotational symmetry of the $p_{z}$
orbitals is also reflected via an isotropic $p_{z}$-dominated character of the
outer Fermi surface sheet for $k_{z}$ = 0, shown in Fig. 7(b,c). This
suppresses the hybridisation with the electron pockets even away from the
$\Gamma$-M direction. We have incorporated all of these effects
($k_{z}$-induced size mismatch, switch of orbital selectivity mediated by the
band inversion, and the additional incorporation of the $p_{z}$ state) into
our minimal model (see Methods Methods). We find that only minimal changes in
the occupied DOS then occur between the high-temperature state and when
considering a $(2\times 2\times 2)$ structure with the same hybridisation
strength that we found for ML-TiTe2 (Fig. 7(d)). Our calculations thus
demonstrate that the electronic driving force for the CDW instability present
in the monolayer is effectively completely removed in the bulk case.

## Discussion

Our work thus points to a key role of orbital-selective band hybridisation in
dictating whether CDW instabilities can occur in TiTe2. Understanding
microscopic details of electron-phonon coupling will undoubtedly be important
to gain a full understanding of the CDW ordering. Nonetheless, our
experimental measurements and model calculations indicate how significant
electronic energy gains are derived from band hybridisation arising from a
$(2\times 2)$ lattice distortion in the monolayer case, which are almost
completely suppressed due to the three-dimensionality, and associated band
inversions, in the bulk case. This provides a natural picture to explain the
emergence of CDW order in monolayer TiTe2 as well as its intriguing thickness-
dependence, and may bring new insight to understand a recently-observed
enhanced CDW order in Moiré superlattices of TiTe2-based heterostructures
[23]. More generally, the model developed here will provide a natural
framework through which to interpret and understand the low-energy electronic
structure evolution in the sister compounds TiSe2 and ZrTe2, which host the
same crystal structure and similar electronic structures, and where the nature
of their CDW-like order is under intense current debate [8, 24, 25].

Figure 8: RHEED images at different steps of the MBE growth of ML-TiTe2 and
capping.

## Methods

Sample preparation: ML-TiTe2 was grown by molecular beam epitaxy (MBE) on
bilayer graphene-terminated SiC wafers using the method described in [26]. The
bilayer graphene termination of the SiC wafer was achieved by direct current
heating of SiC wafers at $1500^{\circ}$C in a dedicated high-vacuum chamber
equipped with a pyrometer to check the temperature. These substrates were
degassed at $600^{\circ}$C before the growth was commenced. The epitaxy of ML-
TiTe2 was performed in a highly Te-rich environment (Ti:Te flux ratio
$\sim\\!10^{3}$) at a substrate temperature of $400^{\circ}$C as measured by a
thermocouple placed behind the substrate. The growth was stopped after 70
minutes while the graphene RHEED streaks are still visible (see Fig. 8(a,b))
and before the onset of bilayer formation, as confirmed from the lack of any
bilayer splitting in our subsequent ARPES measurements of the electronic
structure. The sample was then capped with Te for 20 min resulting in a spotty
RHEED pattern (Fig. 8(c)), indicating a crystalline Te layer completely
covering the underlying TiTe2 mono-layer. An additional amorphous Se capping
was deposited to protect the Te layer against oxidation in air, and was
deposited until the Te signal disappeared completely (Fig. 8(d)). After
growth, the film was capped with an initial Te layer and a subsequent
amorphous Se layer which were removed by in situ annealing in the CASSIOPEE
endstation at SOLEIL synchrotron immediately prior to performing ARPES
measurements. Additional measurements on bulk TiTe2 single crystals were
performed on freshly-cleaved single-crystal samples.

DFT: DFT calculations were performed using the Vienna Ab Initio Simulation
Package (VASP) [27, 28, 29] in order to investigate the lattice dynamical
properties of TiSe2 and TiTe2 (bulk and monolayer). The interactions between
the core and valence electrons were described using the projector augmented
wave (PAW) method [30], with Ti 3$p^{6}$4$s^{2}$3$d^{2}$, Se 4$s^{2}$4$p^{4}$
and Te 5$s^{2}$5$p^{4}$ treated as valence electrons. The HSE06 (Heyd-
Scuzeria-Ernzerhof) [31, 32] screened hybrid functional was employed to
describe the exchange and correlation functional. Within HSE06, the exchange
interaction is split into short-range (SR) and long-range (LR) parts, with
25$\%$ of the SR exchange modelled using the exact non-local Fock exchange and
the remainder of the contributions coming from the PBE (Perdew-Burke-
Ernzerhof) [33] functional. The hybrid functional calculations are
computationally very demanding, but are expected to provide a better
description of structure and phonon frequencies [34]. The empirical correction
scheme of Grimme’s DFT-D3 method [35] was also employed along with the hybrid
functional to model the van der Waals interactions between the layers, as it
has been successful in accurately describing the geometries of several layered
materials in the past [36, 37, 38]. For the monolayers, a vacuum layer of 12 Å
was included to avoid interaction between the periodic images. Geometry
optimisations were performed by setting the plane-wave energy cutoff at 350 eV
for each structure, alongside $\Gamma$ centred $k$-point meshes of $16\times
16\times 10$ (TiSe2-bulk), $16\times 16\times 8$ (TiTe2-bulk), and $16\times
16\times 1$ for the monolayers. A full relaxation of atomic positions, cell
volume and shape was performed for the bulk structures until the residual
forces acting on all the atoms were found to be less than 10-4 eV/Å. The same
convergence criterion for forces was used for relaxing the internal atomic
coordinates of the monolayers. The self-consistent solution to the Kohn-Sham
equations was obtained when the energy changed by less than 10-8 eV. The
phonon calculations were performed using the finite-displacement method (FDM)
[39, 40] as implemented in the Phonopy package [41, 42]. The $k$-point
sampling meshes were scaled accordingly as appropriate for the supercell
calculations. We also mapped the potential energy curves spanned by the
imaginary frequency modes in the undistorted structures to estimate the
barrier associated with CDW transition. For this purpose, we used the ModeMap
code to create a series of structures with atomic displacements along the mode
eigenvectors over a range of amplitudes “frozen in”. This was followed by
performing single-point energy calculations on the modulated structures to
obtain the double-well potential spanned by each mode.

Tight binding: The band structure of ML-TiTe2 was calculated adapting the
Slater-Koster tight-binding model used for TiSe2 in Ref. [15]. Accordingly, we
defined the orbital basis in the octahedral coordinates
{$x^{\prime}y^{\prime}z^{\prime}$} in Fig. 4(d), considering the three Te 5$p$
orbitals $p$ {$p_{x}^{\prime},p_{y}^{\prime},p_{z}^{\prime}$} for each of the
two chalcogen sites (Te(1) and Te(2)), and the five Ti 3$d$ orbitals $d\gamma$
{($d_{x^{\prime 2}-y^{\prime 2}},d_{3z^{\prime 2}-r^{\prime 2}})$} and
$d\epsilon$
{($d_{x^{\prime}y^{\prime}},d_{y^{\prime}z^{\prime}},d_{x^{\prime}z^{\prime}})$}.
The hopping parameters and on-site energies reported in Table 1 were
determined by initially fitting the tight binding model along M-$\Gamma$-K to
the DFT band structure, and subsequently refining the fits using our
experimental ARPES data of the undistorted phase (see Supplemental Fig. S3).
An on-site spin–orbit coupling term on the Te sites is included ($\lambda$ =
0.38 eV), optimised from fitting the dispersion on the high temperature ARPES
spectrum.

Table 1: Tight binding parameters for ML-TiTe2 On-site energies (eV)
---
$e(p)$ = -1.958 |  $e(d\epsilon)$ = 0.256 |  $e(d\gamma)$ = 0.721
Hopping parameters (eV)
$t(pp\sigma)_{1}$ = 0.904 |  $t(pp\sigma)_{2}$ = 0.581 |  $t(pd\sigma)$ = -1.618
$t(pp\pi)_{1}$ = -0.139 |  $t(pp\pi)_{2}$ = 0.075 |  $t(pd\pi)$ = 0.992
$t(dd\sigma)$ = -0.427 |  $t(dd\pi)$ = 0.216 |  $t(dd\delta)$ = -0.070

Overlap evaluation: The orbital overlaps in Fig. 4(c) were calculated
considering the tesseral harmonics of the six Te $p$ orbitals at the vertex of
the TiTe6 octahedron defined in the global coordinate system, $\\{x,y,z\\}$
and the Ti $d_{x^{\prime}y^{\prime}}$ orbital at the centre of the TiTe6
octahedron defined in the octahedral coordinate system
$\\{x^{\prime}y^{\prime}z^{\prime}\\}$. The overlap integral was evaluated
numerically on a $(121\times 121\times 121)$ cubic mesh. We simulate the
effect of the phonon mode softening by displacing the atoms along the
direction indicated by the black arrows in Fig. 4(c), following the distortion
pattern reported for TiSe2 in Ref. [7]. Since Ti is lighter that Te, we assume
the Ti displacement is 9 times larger that the Te ones.

Minimal model: As basis for the Hamiltonian in Equation 1 we choose the five
bands involved in the hybridisation near the Fermi level. The three electron
pockets ($e_{i}$, $i={1,2,3}$) are parametrised as 2D elliptic paraboloids,
while the two hole bands as circular non-parabolic bands:

$e_{1}=e_{c}+\mu k_{x}^{2}+\nu k_{y}^{2}$ (2)
$e_{3}=e_{c}+\mu(-\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y})^{2}+\nu(-\frac{\sqrt{3}}{2}k_{x}-\frac{1}{2}k_{y})^{2}$
(3)
$e_{3}=e_{c}+\mu(-\frac{1}{2}k_{x}-\frac{\sqrt{3}}{2}k_{y})^{2}+\nu(\frac{\sqrt{3}}{2}k_{x}-\frac{1}{2}k_{y})^{2}$
(4) $h_{in}=e_{h}+(\sqrt{2}-\sqrt{2+4\alpha_{in}\mu{}k^{2}})/(2\eta_{1})$ (5)
$h_{out}=e_{h}+(\sqrt{2}-\sqrt{2+4\alpha_{out}\mu{}k^{2}})/(2\eta_{2})$ (6)

where $e_{h}$ and $e_{c}$ are the band minimum energy for the valence and
conduction bands respectively; $\mu$ and $\nu$ are the effective masses along
the major and minor axis of the elliptical parabola, and $\alpha_{in}$ and
$\alpha_{out}$ are the non-parabolic terms for the inner and outer valence
band following the Kane model [43]. All these coefficients were determined by
fitting the simulated spectrum with $\Delta=0$ on the experimental data at
$T=160$ K shown in Fig. 5(c).

In order to extend the model to the bulk case we introduced and additional
cosine dispersion in kz for the onsite energies:

$e_{c}=[e_{c}(L)-e_{c}(M)]cos(k_{z})+[e_{c}(L)+e_{c}(M)]/2$ (7)
$e_{h}=[e_{h}(\Gamma)-e_{h}(A)]cos(k_{z})+[e_{h}(\Gamma)+e_{h}(A)]/2$ (8)

where the values of $e_{c}$ at M and L and $e_{h}$ at $\Gamma$ and A were
determined by fitting the ARPES data in Fig 6(a). To ensure charge neutrality,
a shift of the chemical potential across the CDW transition was taken into
account, simulating how the occupied DOS depends on the hybridisation strength
($\Delta$) at constant $T=16$ K (see Supplemental Fig. S4 for details).

ARPES simulation: The ARPES simulations in Fig. 5(c,e) were performed taking
into account the intensities of the 5 bands
($e_{1},e_{2},e_{3},h_{in},h_{out}$) considered in the minimal model plus a
background with third-order polynomial along the momentum $k$ and a linear
dispersion along the energy axis $\omega$. Each band intensity
$I_{B}(k,\omega)$ takes the form of:

$I_{B}(k,\omega)=[M(k)A(k,\omega)\textit{f}(T,\omega)]\ast R(\Delta
k,\Delta\omega)$ (9)

where $M$ is the matrix element, $A$ is the spectral function,
$\textit{f}(T,\omega)$ is the Fermi Dirac distribution at temperature $T$. The
entire expression is then convolved with a 2D Gaussian ($R$) to simulate the
experimental energy and momentum resolution. $M$ depends on the band character
$C_{b}$ and can be approximated to have a linear dependence on $k$:

$M=\sum_{b=1}^{5}m_{b}C_{b}^{2}k+n_{b}C_{b}^{2}.$ (10)

$A$ takes into account the Lorentzian broadening of the band intensity due to
impurity scattering:

$A(k,\omega)=\frac{\Sigma_{b}}{(\omega-e_{b}(k)^{2}+\Sigma_{b}^{2}}$ (11)

where $e_{b}$ is the bare band dispersion calculated by the minimal model and
$\Sigma_{b}$ is the Lorentzian FWHM. For measurements performed at the
Brillouin zone centre, the matrix element can be expected to approximately
follow that of original valence band character [44, 45] (dark blue in Fig.
5(b,d)). We note that for both the high and low $T$ spectra the band
intensities decrease monotonically form -0.16 eV up to the Fermi level. Since
it is not a temperature-dependent feature we do not consider it related to the
CDW transition, while it might be due to the presence of defects in the ML-
TiTe2 or azimuthal disorder in the grown film. In order to have a better
agreement with the experimental data we include a temperature-independent
intensity decay, $I_{dec}$, of the from:

$I_{dec}=\begin{cases}cos\left[\frac{\pi(\omega-
E_{i})}{2(E_{f}-E_{i})}\right],&\text{if $\omega\leq E_{i}$}\\\ 1,&\text{if
$\omega>E_{i}$}\end{cases}$ (12)

where the coefficients $E_{f}$ and $E_{i}$ were determined fitting the high
temperature data and kept constant for all the simulations at different
temperatures. Thus, this contribution is cancelled out when we calculate the
intensity difference between the high and low temperature data as in Fig. 3.

## Acknowledgments

We thank Sebastian Buchberger, Brendan Edwards, Lewis Hart, Chris Hooley,
Federico Mazzola, Martin McClaren, Philip Murgatroyd and Gesa Siemann for
useful discussions and technical assistance. We gratefully acknowledge support
from the Leverhulme Trust and the Royal Society. Via membership of the UK’s
HEC Materials Chemistry Consortium, which is funded by the EPSRC (EP/L000202,
EP/R029431, EP/T022213), this work used the ARCHER2 UK National Supercomputing
Service (www.archer2.ac.uk) and the UK Materials and Molecular Modelling (MMM)
Hub (Thomas – EP/P020194 & Young – EP/T022213). W.R. is grateful to University
College London for awarding a Graduate Research Scholarship and an Overseas
Research Scholarship. O.J.C. and K.U. acknowledge PhD studentship support from
the UK Engineering and Physical Sciences Research Council (EPSRC, Grant Nos.
EP/K503162/1 and EP/L015110/1). I.M. and E.A.-M. acknowledge studentship
support from the International Max-Planck Research School for Chemistry and
Physics of Quantum Materials. S.R.K. acknowledges the EPSRC Centre for
Doctoral Training in the Advanced Characterisation of Materials (CDT-ACM,
EP/S023259/1) for funding a PhD studentship. The MBE growth facility was
funded through an EPSRC strategic equipment grant: EP/M023958/1. We thank
SOLEIL synchrotron for access to the CASSIOPEE beamline (proposal Nos.
20181599 and 20171202). The research leading to this result has been supported
by the project CALIPSOplus under Grant Agreement 730872 from the EU Framework
Programme for Research and Innovation HORIZON 2020.

## Contributions

M.D.W., A.R., O.J.C., K.U., I.M., E.A.M. and P.D.C.K. measured the ARPES data,
which was analysed by T.A.
W. R. and S. R. K. performed the DFT calculations, which were analysed by
W.R., T.A., D.O.S and P.D.C.K.
T.A. performed the tight-binding calculations, ARPES simulations and matrix
element calculations. M.W., T.A. and P.D.C.K. developed the minimal model.
A.R. and A.D. grew the monolayer samples. K.R. grew the single crystal
samples. P.L.F. and F.B. maintained the CASSIOPEE beam line and provided
experimental support. P.D.C.K. conceived and led the project. T.A. and
P.D.C.K. wrote the manuscript with contributions from all authors.

## Competing interests

The authors declare no competing interests.

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1720March 8th, 2023The University of Tokyo, Bunkyo, Japan Institute of Space
and Astronautical Science, JAXA, Sagamihara, Japan Planetary Exploration
Research Center, Chiba Institute of Technology, Chiba, Japan The University of
Aizu, Aizu-Wakamatsu, Japan Osaka University, Toyonaka, Japan The University
of Pisa, Pisa, Italy

# Thermophysical Model Development for Hera Mission to Simulate Non-
Gravitational Acceleration on Binary Asteroid

MasanoriKanamaru<EMAIL_ADDRESS>TatsuakiOkada  2)
HirokiSenshu  3) HirohideDemura  4) NaruHirata  4) YutoHorikawa 5) and
GiacomoTommei 6)

(2020)

###### Abstract

The surface temperature of an asteroid is fundamental information for the
design of an exploration mission and the interpretation of scientific
observations. In addition, the thermal radiation of the asteroid causes a non-
gravitational acceleration that induces secular changes in its orbit and spin.
We have been developing a numerical calculation library for simulating the
dynamics and thermophysics of asteroids. The asteroid dynamical simulator,
Astroshaper, can calculate the temperature distribution based on a
3-dimensional shape model of an asteroid and predict the non-gravitational
acceleration. In recent years, asteroid exploration missions such as Hayabusa2
and Hera have been equipped with thermal infrared imagers. The asteroid
thermography can provide the thermal properties of the surface material of the
target bodies. The functionality of thermophysical modeling in Astroshaper
contributes to simulating the thermal environment on the asteroids, estimating
the thermal properties, and predicting the dynamical evolution controlled by
the non-gravitational effects.

###### keywords:

Asteroid 65803 Didymos, Binary asteroid, Thermophysical model, Yarkovsky
effect, YORP effect

## Nomenclature

$A_{\rm B}$ | : |  Albedo at visible wavelength
---|---|---
$A_{\rm TH}$ | : |  Albedo at thermal radiation wavelength
$a$ | : |  Area of a facet, $\mathrm{m}^{2}$
$C_{p}$ | : |  Heat capacity at constant pressure, $\mathrm{J}\mathrm{/}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{K}$
$c_{0}$ | : |  Speed of light in vacuum, $\mathrm{m}\mathrm{/}\mathrm{s}$
$d\bm{f}$ | : |  Thermal force on a facet, $\mathrm{N}$
$E$ | : |  Sum of emittance of scattered light and thermal radiation from a facet, $\mathrm{W}\mathrm{/}\mathrm{m}^{2}$
$E_{\rm cons}$ | : |  $E_{\rm out}\,/\,E_{\rm in}$
$E_{\rm in}$ | : |  Energy incident on an asteroid, $\mathrm{W}$
$E_{\rm out}$ | : |  Energy emitted from an asteroid, $\mathrm{W}$
$F_{\rm rad}$ | : |  Energy flux by thermal radiation from surrounding facets, $\mathrm{W}\mathrm{/}\mathrm{m}^{2}$
$F_{\rm scat}$ | : |  Energy flux by scattered light from surrounding facets, $\mathrm{W}\mathrm{/}\mathrm{m}^{2}$
$F_{\rm sun}$ | : |  Energy flux by direct sunlight, $\mathrm{W}\mathrm{/}\mathrm{m}^{2}$
$F_{\rm total}$ | : |  Total energy flux into a facet, $\mathrm{W}\mathrm{/}\mathrm{m}^{2}$
$f$ | : |  View factor between two facets
$k$ | : |  Thermal conductivity, $\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{/}\mathrm{K}$
$\bm{\hat{n}}$ | : |  Normal vector of a facet
$\bm{r}$ | : |  Position vector, $\mathrm{m}$
$T$ | : |  Temperature, $\mathrm{K}$
$t$ | : |  Time, $\mathrm{s}$
$z$ | : |  Depth, m

$\alpha$ | : |  Thermal force on an asteroid, $\mathrm{N}$
---|---|---
$\Gamma$ | : |  Thermal inertia, $\mathrm{J}\cdot\mathrm{m}^{-2}\cdot\mathrm{K}^{-1}\cdot\mathrm{s}^{-0.5}$ (tiu)
$\varepsilon$ | : |  Emissivity
$\theta$ | : |  Tilt angle of a facet
$\rho$ | : |  Density, $\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$
$\sigma$ | : |  Stefan–Boltzmann constant, $\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{/}\mathrm{K}^{4}$
$\tau$ | : |  YORP torque on an asteroid, $\mathrm{N}\cdot\mathrm{m}$

Subscripts

$\rm Didy$ | : |  Didymos
---|---|---
$\rm Dimo$ | : |  Dimorphos, the satellite of Didymos
$i$ | : |  Index of a facet of a shape model
$j$ | : |  Index of a facet visible from facet $i$

##  1 Introduction

###  1.1 Thermophysical modeling of an asteroid

Thermophysical modeling (TPM) is a numerical simulation to obtain temperature
distribution on the surface of an asteroid. TPM plays a vital role in a small-
body mission’s science and engineering aspects as follows.

  * •

TPM simulates the thermal environment around the asteroid that is critical for
a proximity operation and a touch-down operation to the surface.

  * •

It is possible to map the asteroid’s thermal inertia and surface roughness by
comparing TPM and thermal infrared spectroscopy or imaging [1, 2, 3].

  * •

TPM can predict the non-gravitational acceleration on the asteroid induced by
anisotropic thermal radiation. The changes in orbit and rotation of asteroids
due to thermal radiation are known as the Yarkovsky and YORP effects,
respectively [4, 5].

  * •

The orbit evolution by the Yarkovsky effect is also important for assessing
the risk of asteroid impact on Earth in planetary defense [6, 7].

  * •

Changes in surface temperatures may cause material ejection from the asteroid
and comet nuclei [8].

  * •

Thermal radiation pressure from the asteroid’s surface affects the trajectory
of the spacecraft or the ejecta particle in the vicinity of the asteroid [9,
10].

  * •

Thermal radiation causes a bias in the infrared spectra of the asteroid. To
interpret the spectra at $\sim 3$ µm or longer wavelength, removing this
“thermal tail” is necessary[11].

###  1.2 Hera mission to explore a binary asteroid

DART and Hera are planetary defense missions to a binary asteroid with a
satellite [12, 13]. The DART spacecraft successfully impacted Dimorphos, a
satellite of the asteroid Didymos, in September 2022 [14]. The momentum
transfer efficiency by the DART impact was estimated from the change in the
mutual orbit period of the binary asteroid [15, 16]. The Hera spacecraft is
scheduled to rendezvous with Didymos and Dimorphos in December 2026 to observe
in detail the crater formed by the DART impact [13]. Japan’s team led by the
Institute of Space and Astronautical Science (ISAS) is developing a thermal
infrared imager (TIRI) onboard the Hera spacecraft. TIRI is the successor to
the thermal infrared imager (TIR) on Hayabusa2, with higher sensitivity and
resolution and six band filters for mid-infrared spectroscopy. Asteroid
thermography by TIRI will provide us with the thermal inertia or density of
the boulders and gravel that make up the target asteroids, which is essential
for assessing the efficiency of the asteroid deflection experiment by DART.

###  1.3 Development of thermophysical models for single/binary asteroids

Several thermophysical models have been developed for single asteroids. One of
the most elaborate models is the Advanced Thermophysical Model (ATPM),
including the effect of small-scaled surface roughness [17]. We have been
developing a numerical simulator for the dynamics and thermophysics of
asteroids, Astroshaper. This simulator was originally developed for YORP
prediction of asteroid Ryugu, a target asteroid of the Hayabusa2 mission [18].
Astroshaper is being developed as an open-source project in the Julia
programming language at GitHubiiiAstroshaper – https://github.com/Astroshaper.
We hereby report on the functionality of thermophysical modeling implemented
in the AsteroidThermoPhysicalModels.jl
packageiiiiiiAsteroidThermoPhysicalModels.jl –
https://github.com/Astroshaper/AsteroidThermoPhysicalModels.jl, one of the
sub-modules of Astroshaper. Some sample codes for TPM simulation are also
available in the repository of Astroshaper-exampleiiiiiiiiiAstroshaper-example
– https://github.com/Astroshaper/Astroshaper-examples. We have extended the
capabilities of TPM for a single asteroid to apply to a binary asteroid for
interpreting the TIRI imagery of Didymos and Dimorphos.

##  2 TPM Functionality of Astroshaper

The thermophysical model implemented in AsteroidThermoPhysicalModels.jl is
based on a 3-dimensional shape model of an asteroid covered with a triangular
mesh. As with other TPMs[17], it can calculate the temperature distribution of
the asteroid considering some fundamental thermophysical processes (See Table
1): the 3D shape of the asteroid, 1-dimensional heat conduction in the depth
direction, shadowing by the local horizon (i.e., self-shadowing), and
reabsorption of scattered light and thermal radiation by interfacing facets
(i.e., self-heating).

Table 1: Thermophysics implemented in Astroshaper.
---
Asteroid 3D shape |  Yes. Triangular mesh models can be imported.
Heat conduction |  Yes. 1D heat conduction in the depth direction is considered.
Self-shadowing |  Yes.
Self-heating |  Yes. Only single scattering is considered.
Mutual-shadowing |  Yes.
Mutual-heating |  Yes.
Surface roughness |  Not yet implemented.

###  2.1 Heat conduction

Our TPM code independently solves a 1-dimensional heat conduction equation on
each shape model facet. Assuming that the thermal conductivity $k$ is constant
regardless of depth $z$, the heat conduction equation becomes as follows.

$\frac{\partial T}{\partial t}=\frac{k}{\rho
C_{p}}\frac{\partial^{2}T}{\partial z^{2}}$ (1)

The boundary condition at the surface of the asteroid ($z=0$) is given by the
balance of incident light to the facet, heat flux to the ground, and thermal
radiation to space (See Fig. 1).

$F_{\rm total}+k\left(\frac{\partial T}{\partial
z}\right)_{z=0}=\varepsilon\sigma T^{4}_{z=0}$ (2)

where $F_{\rm total}$ is the total energy the facet absorbs at each time step.

$F_{\rm total}=(1-A_{\rm B})F_{\rm sun}+(1-A_{\rm B})F_{\rm scat}+(1-A_{\rm
TH})F_{\rm rad}$ (3)

The solar incident $F_{\rm sun}$ is an energy flux that considers the
inclination of the facet concerning the sun’s direction and the shadow of the
surrounding facets. To consider the self-shadowing effect, $F_{\rm sun}$ is
set to zero when the other facet blocks the solar ray. The facet exchanges the
energy flux with other interfacing facets by reabsorbing the scattered light
and thermal radiation. $F_{\rm scat}$ and $F_{\rm rad}$ are the energy fluxes
from the interfacing facets to the facet in question in visible and thermal
infrared wavelengths, respectively. In our model, single scattering is only
considered. The additional flux due to multiple scattering is negligible for a
low albedo body. The boundary condition of insulation is given so that the
temperature gradient is zero at the bottom cell.

$\left(\frac{\partial T}{\partial z}\right)_{z\rightarrow\infty}=0$ (4)

Our TPM code solves the above equations by an explicit Euler scheme. The
radiative boundary condition involving a nonlinear term at Eq. (2) is solved
using the Newton-Raphson method. It is in the process of being implemented to
allow users to select implicit and higher-order solvers.

Figure 1: Basic thermophysical processes on an asteroid.

###  2.2 Non-gravitational force

Non-gravitational perturbations on the asteroid can be calculated from the
temperature distribution[19]. We assume that a facet of the shape model
scatters and radiates isotropically (i.e., Lambertian scatterer and emitter).
The total emittance of scattered light and thermal radiation emitted from
facet $i$ is

$E_{i}=A_{\rm B}F_{{\rm sun},\,i}+A_{\rm B}F_{{\rm scat},\,i}+A_{\rm
TH}F_{{\rm rad},\,i}+\varepsilon\sigma T_{i}^{4}$ (5)

The force exerted by the photon pressure on the element can be expressed as
follows.

$d\bm{f}_{i}=-\frac{2E_{i}a_{i}}{3c_{0}}\bm{\hat{n}}_{i}+\sum_{j\,\in\,{\rm
visible\,from\,facet}\,i}\frac{E_{i}a_{i}}{c_{0}}f_{i,j}\,\frac{\bm{r}_{j}-\bm{r}_{i}}{|\bm{r}_{j}-\bm{r}_{i}|}$
(6)

The first term is a force component normal to the surface element. The
coefficient $-2/3$ is derived from the isotropic emittance. The second term
represents the additional component due to the interaction with visible
facets. The reabsorbed photons exert a force along the direction from facet
$i$ to facet $j$ in proportion to the view factor $f_{i,j}$. The view factor
from facet $i$ to facet $j$ refers to the fraction of absorption by facet $j$
to the emittance from facet $i$[19, 20].

$f_{i,j}=\frac{\cos{\theta_{i}}\cos{\theta_{j}}}{\pi\,|\bm{r}_{j}-\bm{r}_{i}|^{2}}a_{j}$
(7)

where $\theta_{i}$ and $\theta_{j}$ are the angles between each normal vector
and the line connecting the two facets, and $d_{i,j}$ denotes the distance
between the two facets. The summation of Eq. (6) should only be taken for
facets seen from facet $i$. In our code, the visible facets from each facet
are searched and stored before the TPM is performed.

The total force $\alpha$ and torque $\tau$ on the asteroid are obtained by
integrating the thermal force over the entire surface.

$\alpha=\sum_{i}\left(\frac{\bm{r}_{i}}{|\bm{r}_{i}|}\cdot
d\bm{f}_{i}\right)\,\frac{\bm{r}_{i}}{|\bm{r}_{i}|}$ (8)

$\tau=\sum_{i}\bm{r}_{i}\times d\bm{f}_{i}$ (9)

The perturbation to the motion of the asteroid’s center-of-mass causes the
Yarkovsky drift in orbit, and the torque causes the YORP spin evolution.

###  2.3 Binary and additional thermophysics

Some additional thermophysics must be considered for a binary asteroid, as in
Fig. (2). We utilized the functions of ray tracing for detecting local shadows
on a single asteroid to simulate an eclipse by a pair of asteroids (i.e.,
mutual shadowing). Two types of eclipse events can occur: when the satellite’s
shadow falls on the primary asteroid and when the satellite enters the shadow
of the primary. The primary and secondary asteroids exchange energy by thermal
radiation and warm each other. This mutual heating effect is also implemented.
The impact of the thermal infrared beaming by small-scaled surface roughness
will be implemented in the future.

Figure 2: Thermophysics for a binary asteroid. The shape models of Didymos and
Dimorphos based on ground-based observations are shown here.

##  3 TPM for Binary Asteroid Didymos and Dimorphos

###  3.1 Parameter setting

We used the SPICE kernels and 3D shape models provided by the Hera mission for
a thermophysical simulation of the binary asteroid Didymos and
Dimorphos.ivivivShape models used in this study (Version: v140_20230731_001):
• g_50677mm_rad_obj_dida_0000n00000_v001.obj for Didymos •
g_06650mm_rad_obj_didb_0000n00000_v001.obj for Dimorphos Available at
https://s2e2.cosmos.esa.int/bitbucket/projects/SPICE_KERNELS/repos/hera/browse.
The shape models used in this study are based on ground-based observations
before the DART impact experiment. It should be noted that the shape of
Dimorphos is approximated by an ellipsoid.

A thermal inertia of $\Gamma=403$ tiu was given, corresponding to a typical
value for an S-type asteroid[21]. Running TPM over tens of thermal cycles in
advance is necessary to obtain a temperature distribution independent of
initial conditions. In this study, TPM was performed for two months (from
January 1st to March 1st, 2027) after temperatures of 0K were given at all
facets of the shape models and all depth cells, corresponding to $\sim 627$
rotations for Didymos and $\sim 119$ mutual orbit cycles for Dimorphos. We
confirmed that the calculation sufficiently converged in terms of the balance
between the energy input and output on the surface of each asteroid, where
$E_{\rm cons}$ was greater than 0.98 at the final time step. We used the
simulated temperature data for 24 hours on March 1st, 2027, for the later
analysis.

###  3.2 Temperature map of the binary asteroid

The upper and middle panels of Fig. 3 show the temperature maps of Didymos and
Dimorphos at the epochs of the mutual events, respectively. In the upper
panel, Dimorphos cast the shadow around $(20^{\circ}\rm{S},90^{\circ}\rm{W})$
of Didymos at 5:37 a.m. After 5.96 hours or half of the orbit period of
Dimorphos, one can observe Dimorphos hiding in the shadow of Didymos (middle
panel). The lower panel shows the temperature changes over time at the points
indicated by the blue dots on the above maps. It can be seen that rapid
temperature drops of several tens of Ks occurred during the eclipse events. By
observing the eclipse events in addition to diurnal thermal cycles,
thermophysical properties corresponding to different depths can be
investigated by TIRI. Because of the considerable uncertainty in the
inclination of the mutual orbit, it will be turned out after Hera’s rendezvous
how frequently the eclipse events will occur.

Figure 3: Temperature maps of Didymos (upper) and Dimorphos (middle), and
temperature changes at the points indicated by the blue dots (lower).

###  3.3 Non-Gravitational Effects on the binary asteroid

Based on the above temperature distribution, we also calculated the thermal
recoil force on each facet of the shape model. We integrated it over the
surface to obtain non-gravitational force and torque on the binary asteroid.
By averaging over several rotations, the torque components for rotational
acceleration were estimated as $\tau_{\rm Didy}=0.19\,\,{\rm N\cdot m}$ for
Didymos and $\tau_{\rm Dimo}=-1.1\times 10^{-4}\,\,{\rm N\cdot m}$ for
Dimorphos. It suggests that the rotation of Didymos is accelerating at the so-
called YORP time scale of $4.1\times 10^{6}$ years, that is, a time to double
the rotation speed. On the other hand, the negative acceleration of Dimorphos
decelerates its rotation at a time scale of $8.6\times 10^{4}$ years, reducing
the rotation speed by half.

##  4 Discussion

Generally, the resolution of a pre-arrival shape model is insufficient for
YORP prediction sensitive to small-scale topography[22]. We must wait for
Hera’s rendezvous for a more precise prediction of YORP on Didymos and
Dimorphos. The shape model of Dimorphos used in this study is an ellipsoid
based on ground-based observations. The symmetrical shape should cancel out
the thermal torque, but the asymmetry of the temperature distribution results
in the non-zero torque. Cooling due to the eclipse is likely the cause of the
negative acceleration on the satellite. The drastic temperature change may
have the effects of expanding the mutual orbit of the binary asteroid and
shortening its dynamical lifetime.

##  5 Conclusion

We hereby reported on the asteroid dynamical simulator, Astroshaper. We have
developed a thermophysical simulation for the Hera mission applicable to a
binary asteroid. This tool is expected to contribute to the operation planning
of TIRI and investigate the dynamics of the binary asteroid controlled by the
non-gravitational effects.

## Acknowledgments

This study was supported by the JSPS KAKENHI No. JP17H06459 (the Aqua
Planetology project) and No. JP22J00435/JP22KJ0728. This work was also
supported by MEXT Promotion of Distinctive Joint Research Center Program Grant
Number JPMXP0622717003. G. Tommei acknowledges the support from the Italian
Space Agency (grant 2022-8-HH.0).

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