# Perturbation of charge density waves in $1T$-TiSe2

Imrankhan Mulani  Umashankar Rajput  Luminita Harnagea  Aparna Deshpande
Indian Institute of Science Education and Research (IISER) Pune,Dr. Homi
Bhabha Road, Pashan, Pune 411008, India

###### Abstract

In this study, using low-temperature scanning tunneling microscopy (STM), we
focus on understanding the native defects in pristine $1T$-TiSe2 at the atomic
scale. We probe how they perturb the charge density waves (CDWs) and lead to
local CDW modulated region formation. These defects influence the correlation
length of CDWs. We establish a connection between suppression of CDWs, Ti
intercalation, and show how this supports the exciton condensation model of
CDW formation in $1T$-TiSe2.

††preprint: APS/123-QED

##  I Introduction

Transition metal dichalcogenides (TMDCs) are versatile materials that exhibit
phenomena like strongly correlated phases to multifaceted tunable properties
for applications in flexible electronics, optoelectronics, spintronics, and
energy harvesting [1, 2, 3, 4]. Many exotic physical phenomena, like spin-
valley interaction, exciton-polariton states, charge density wave (CDW)
states, are observed in the TMDCs. TMDCs have a characteristic layered
structure, tunable bandgap, and strong spin-orbit coupling favoring such
applications. The chemical composition of TMDCs is of the form $MX_{2}$ where
$M$ denotes transition metal (like $Ti$, $Mo$, $W$) and $X$ represents
chalcogen (like $S$, $Se$, and $Te$). Despite having a similar structure,
these materials can be categorized as insulating, semiconducting, semi-
metallic, metallic, or superconducting, depending on their chemical
composition. The non-bonding $d$ band and the extent of its electron filling
endow these compounds with diverse properties. Bulk crystals of TMDCs consist
of two-dimensional (2D) layers bonded by weak van der Waals interactions,
rendering them easy to exfoliate. Depending on the arrangements of atoms,
structural polytypes of 2D TMDCs can be categorized as $1T$ (tetragonal
symmetry with octahedral coordination), $2H$ (hexagonal symmetry with trigonal
prismatic coordination), and $3R$ (rhombohedral symmetry with trigonal
prismatic coordination). The layer degree of freedom provides an additional
parameter to tune the properties of TMDCs [5].

One of the group IV TMDCs hosting interesting properties is $1T$-TiSe2. It is
a semimetal with a small indirect bandgap [6]. At temperature $T_{CDW}\sim
200K$ $1T$-TiSe2 undergoes a second-order phase transition to the charge
density wave (CDW) phase [7]. Charge density wave is a many-body, correlated
electrons phenomenon, where a periodic distortion modulates electron density.
CDWs, depending on the ratio of their wavelength and lattice parameters, are
classified as commensurate (CCDW), nearly commensurate (NCCDW), or
incommensurate (ICDW) [8, 9]. Pristine $1T$-TiSe2 shows CCDW order below
$200K$ [7]. Charge carrier doping via electrochemical ionic gating in
$1T$-TiSe2 leads to suppression of CCDW, emergence of ICDW and superconducting
phase at lower temperature [10]. The CDW transition temperature drops
significantly for $1T$-TiSe2 doped with copper. Superconductivity emerges in
CuxTiSe2 for $x\sim 0.04$, maxima of superconducting transition temperature
reaches at $x\sim 0.08$ [11]. ICDW phase coexists with superconducting phase
under applied pressure [12] and copper doping [13]. The mechanism of formation
of CDWs in $1T$-TiSe2 is still a matter of debate. The two favored mechanisms
are excitonic insulator phase [14] and band type Jahn-Teller effect [15].
Excitons are bound states of electrons and holes. Excitons are bosons and can
form a Bose-Einstein condensate [16, 17, 18, 19]. The excitonic insulator
phase can arise in a small gap semiconductor or semimetal, where excitons form
spontaneously because of low carrier density to screen the attractive Coulomb
interaction between electrons and holes. In the Jahn-Teller effect, the
lattice spontaneously distorts to lift the degeneracy and reach the lower
symmetry state because of the interaction between phonons and degenerate
electron states. Jahn-Teller like mechanism is independent of free carrier
concentration.

$1T$-TiSe2 is a nonstoichiometric compound. The concentration of $Ti$ in a
crystal depends on the temperature at which the crystal is grown. Commonly
used method for TMDC crystal growth is chemical vapour transport (CVT). Higher
growth temperature generates higher chalcogen pressure resulting in a crystal
with intercalation of metal [20]. An indirect inference of presence of excess
$Ti$ can be carried out through temperature-dependent resistivity
measurements. $1T$-TiSe2 shows anomalous resistivity peak near CDW phase
transition. As the growth temperature increases, the crystal becomes more
metallic due to excess $Ti$ and hence suppresses the anomalous resistivity
peak [7, 21]. $1T$-TiSe2 shows a small indirect band gap above the transition
temperature $T_{CDW}\sim 200K$ [22, 23]. Angle resolved photo emission
spectroscopy (ARPES) experiments show a small band gap below CDW transition
[24]. Recent experimental evidence suggests excitonic insulator mechanism [25,
26, 27, 28, 29, 30]. Theoretical investigations suggest that excitonic
condensation can be either superfluid [31] or an insulator [32]. Chiral nature
of CDWs in pristine and Cu-doped $1T$-TiSe2 has been reported using polarized
optical reflectometry and STM [33, 34, 35]. Suppression of CDW results in
superconducting phase transition with application of pressure[36] via
electrostatic gating [10] or with Cu intercalation [11] or Pd intercalation
[37]. $1T$-TiSe2 becomes insulating after Pt doping[38]. This implies that
doping affects the nature of CDWs and electronic properties of $1T$-TiSe2.

A correlated phenomenon, like CDWs, is sensitive to the presence of defects.
Here, using low-temperature STM, we focus on the behavior of CDWs at the
atomic scale in the presence of intrinsic defects. We investigated the
relationship between growth conditions and intrinsic defects in crystals grown
by different synthesis routes. We referred to them as sample A (crystal was
grown by chemical vapor transport (CVT) using iodine (I2)) and sample B
(crystal was grown using $Se$ as self-flux) in the subsequent discussion.

##  II Methods

The experiments were performed by using an Omicron ultra-high vacuum (UHV) low
temperature(LT) scanning tunneling microscope (STM) at $77K$ with a base
pressure of 5$\cdot 10^{–11}$ mbar. Two $1T$-TiSe2 crystals were studied.
Sample A was grown using CVT with $I_{2}$ as the transport agent. Sample B
synthesized using the flux zone method by 2D semiconductors, USA [39]. For STM
studies of both the samples, $1T$-TiSe2 single crystal surface was prepared by
cleaving it in the UHV sample preparation chamber. The surface quality was
checked with STM at $77K$. A clean surface at the atomic level was inferred
from several STM images of different scan sizes. Electrochemically etched
tungsten($W$) tips used for imaging. Several trials of imaging were performed
with different tips. Topography measurements were taken in constant current
mode. In our set up bias voltage is applied to the sample. The images were
processed using the image analysis software SPIP 6.0.9 (Image Metrology).
MATLAB (by The MathWorks Inc.) was used for numerical calculations.

##  III Results and discussion

$1T$-TiSe2 crystal is a stack of monolayers bonded by van der Waals
interaction. Each monolayer consists of titanium ($Ti$) atoms sandwiched
between two sheets of selenium ($Se$) atoms (Fig.1(a)). Each $Ti$ atom is
surrounded by six $Se$ atoms in octahedral coordination, as shown in Fig.1(a).
The space group of $1T$-TiSe2 is $P\overline{3}m1(no.164)$. The crystal
cleaves along $(001)$ planes were exposing $Se$ atoms in a hexagonal symmetric
pattern. In constant current topography images, STM tip probes the topmost
$Se$ atoms [40, 41]. The tunneling current is mainly due to the contribution
from the $4p$ orbital of $Se$ atoms. The preferred site for $Ti$ atom
intercalation is schematically shown in Fig.1(b), whereas $O$ substitution
site shown in Fig.1(c). Iodine is used as a transport agent in CVT method of
crystal growth. Substitution of $Se$ atom by $I$ is another defect commonly
found in CVT grown samples. $I$ substitution occupies same sites as $O$
substitution. Defects in the STM topography image of sample A in Fig.1(d) and
sample B Fig.1(e) show up as localized bright regions. Note that the defects
are less in sample B than in sample A.

Figure 1:  (a) Schematics of $1T$-TiSe2 unit cell viewed from an angle, $Ti$
atoms in octahedral coordination, $Se$ atoms form a triangular lattice. (b)
Schematics of $Ti$ intercalation site (c) Schematic of $O$ substitution site
(d) Empty-states constant current STM image of $30nm\times 30nm$ area of
sample A, Enlarged view of $Ti$ intercalation sites are in yellow rounded
rectangle inset and that of $O$ substitution sites are in blue rounded
rectangle inset. $V_{bias}=300$mV and $I_{set}=300$pA. (e) Empty-states
constant current STM image of $20nm\times 20nm$ area of sample B
,$V_{bias}=300$mV and $I_{set}=500$pA.

In Fig.1(d) the inset in yellow rounded rectangle shows the enlarged atomic-
scale image of the defect, which can be identified as $Ti$ intercalation site.
Intercalated atoms occupy octahedral sites or tetrahedral sites in the van der
Waals gap. For $Ti$, favorable sites are octahedral sites because of their
more significant coordination number, and spacing between two nearby
octahedral sites is more considerable than tetrahedral sites [42]. The
characteristic appearance of intercalated $Ti$ sites can be attributed to
change in the local density of states [43]. These defects appear bright in
filled-states STM images (at negative bias voltages).

There were defects with a strongly localized density of states at the topmost
$Se$ atoms surrounded by depleted region in empty-states STM images (at
positive bias voltages). Those defects can be identified with oxygen ($O$)
substitution of the $Se$ atom in the lower layer. $I$ substitution sites are
hard to recognize as they appear very faint compare to $O$ sites [41].
Substitution of $Se$ by $O$ or $I$ on topmost layer is not favored
energetically [41]. Oxygen, present as an impurity in iodine and selenium
precursor during growth, acts as a native dopant. Sample A has been observed
to possess more $Ti$ intercalation sites than sample B (Fig.1(d) and 1(e)).
Using STM images taken in different regions of both samples, the average
number of intercalation sites were counted. Employing this method, we found
that the average count of $Ti$ intercalation sites per unit cell for sample A
is $0.6\%$ as compared to $0.2\%$ sites per unit cell for sample B. This
observation supports the dependence of $Ti$ intercalation sites’ concentration
on sample growth conditions [7].

Figure 2:  (a) Filled-states constant current STM image of 30 nm X 30 nm area
of sample A grown by CVT method, $V_{bias}=-400$ mV $I_{set}=300$ pA.Areas
marked by red are CDW modulated regions, separated by non-modulated regions
induced by Ti intercalation sites. (b) Fourier transform of topography image
in (a). (c) An enlarged image showing 2 X 2 superstructure with CDW peaks
marked by yellow circles (d) Selective inverse Fourier transform of CDW peaks
in (b), (e) Empty-states constant current STM image of 30 nm X 30 nm area of
sample B grown by flux zone method, $V_{bias}=-300$mV $I_{set}=400$pA, (f)
Fourier transform of topography image in (e). (g) Selective inverse Fourier
transform of CDW peaks in (f)

Intercalation and substitution change the material’s carrier concentration,
leading to a change in the material’s electronic properties. Correlated states
like CDWs are sensitive to external factors like mechanical stress, external
pressure, and carrier concentration. Point defects can affect CDWs by changing
carrier density locally and introducing localized distortions in lattice [20].
STM topography images were obtained at different bias voltages, spatially
resolving the CDWs. Fig.2(a) and 2(e) shows STM images of sample A and B with
different $Ti$ intercalation and $O$ substitution densities. Real space
topography shows the clear $2\times 2$ superlattice of CDW. Fig.2(c) is an
enlarged image with atomic resolution showing CDWs forming superstructure
(marked by yellow circles). Fourier transform (FT) of topography images
(Fig.2(b) and 2(f)) yields contributions of lattice and CDW superlattice as
six-fold symmetry peaks. Peaks in FT images affirm that the wavelength of CDW
is twice the lattice spacing (first-order Bragg peak). To reduce background
noise and understand the spatial variation of CDW amplitude, we performed an
inverse transform of CDW FT peaks. Inverse Fourier transform (IFT) image
clearly shows diminishing CDW amplitudes in the presence of $Ti$ defects in
Fig.2(d) and 2(g). In the presence of a large number of defects, CDW modulated
regions can be easily visually identified from IFT images (Fig.2(d)). In
$1T$-TiSe2 with more $Ti$ intercalation density, domains of CDWs are formed by
the presence of defects. 2 X 2 superstructure of CDWs changes to 1 X 1
modulation near intercalation sites, which forms the boundaries of CDW
modulated region (Fig.2(a) and 2(c)). Also, the IFT image is used as a guide
to locate these boundaries. Different CDW modulated regions are marked in red
in Fig.2(a). CDW modulated regions separated by boundaries was not observed in
Fig.2(e) containing fewer intercalation sites. IFT in Fig.(g) shows the
reduced amplitude of CDWs in the presence of intercalation sites but no
formation of separate regions. $1T$-TiSe2 thin flakes are reported to oxidise
quickly in the atmospheric conditions and disrupt the CDW phase [44]. The
oxidation of $1T$-TiSe2 was minimised by cleaving the crystals in our UHV
sample preparation chamber(refer to section II. METHODS). $O$ and $I$
substitution sites increases the density of states in the unoccupied states
away from the Fermi energy [41]. The CDW gap near Fermi level remains
unaffected by $O$ or $I$ substitution.

Figure 3: (a) Inverse Fourier transform image of Bragg peaks and CDW peaks of
STM image shown in Fig.2(a) of sample A. Line sections 1 and 2 passing through
domain boundary (marked by red region) plotted in (b) Red part of line section
indicates no CDW modulation. No phase shift observed across the domain wall.

Metal intercalation of $1T$-TiSe2 at certain doping concentration shows phase
shift of CDW modulation across the domain walls [26, 45].$Cu$ intercalation in
$1T$-TiSe2 results in incommensurate CDW phase with localized commensurate
CDWs separated by domain walls at critical doping concentration [46]. $Ti$,
like $Cu$, is electropositive and suppresses the CDW phase. $Cu$ intercalation
into van der Waals gap of $1T$-TiSe2 was stabilized by transfer of charge from
$Cu$ to $Se$ atoms in the adjacent layers [47]. Similarly, the $Ti$
intercalant gets stabilized by charge transfer. A significant overlap between
the atomic orbitals of intercalated $Ti$ and $Se$ atoms in the layer results
in the intercalant sites’ peculiar appearance in the STM images. To better
understand phase shift of CDWs across domain boundaries, IFT of both lattice
Bragg peaks and CDW peaks of Fig.2(a) were obtained, as shown in Fig.3(a). A
line section of electron density indicates the suppression of CDW modulation
at the boundary (Fig.3(b)).

Interestingly we observed no phase shift in the CDW modulation across the
boundaries. We can infer that the increased concentration of $Ti$
intercalation leads to the disruption of CDW and reduces long-range order to
small regions [48].

The electropositive nature of $Ti$ intercalation defect and formation of CDW
modulated regions and non-modulated boundaries in excess $Ti$ intercalation
may provide some insight into the mechanism of formation of CDWs. Considering
the results as mentioned above, we offer a hypothesis for suppression of CDWs
by $Ti$ intercalation. Semimetals or small gap semiconductors at lower
temperatures can undergo CDW phase transition by excitonic condensation
mechanism. Exciton is a bound state of electrons and holes, mediated by
Coulomb interaction, a composite boson. For $1T$-TiSe2 a small carrier density
and a weakly screened Coulomb interaction can result in exciton formation.
Excitons are formed from electrons in the pocket at the $L$ point in the
Brillouin zone (BZ) and holes from the pocket at $\Gamma$ point in the energy
band stucture of $1T$-TiSe2. Being a composite boson, it undergoes
condensation at $T_{CDW}\sim 200K$ as the number of excitons increases.
Excitonic condensation leads to a periodic modulation of charge density, a CDW
state. The extent to which CDWs are modulated in $1T$-TiSe2 is intercalation
dependent. If we assume excitonic condensation as the mechanism of CDW state
in $1T$-TiSe2 then the suppression of CDWs and formation of domain boundaries
can be explained by the electropositive nature of $Ti$ intercalation.

The first-principles energy band calculations show that the electron pocket at
$L$ point in BZ is derived mainly from $Ti$ $3d$ states. Hole pockets at
$\Gamma$ are derived from $Se$ $2p$ states hybridized with $Ti$ $3d$ states
[49]. At Fermi energy, more than half the density of states (DOS) results from
$Ti$ states, and the rest comes from $Se$ [47]. As $Ti$ intercalation’s
character is electropositive, the addition of $Ti$ results in the acquisition
of electrons. Because of $Ti$’s larger DOS, electrons added to the system go
to $Ti$ derived states around the Fermi level.$Ti$ intercalation now causes
the enhancement of electronic states at Fermi energy, resulting in the strong
screening of Coulomb interaction between electrons and holes. This screening
results in the reduction of excitonic pair and suppression of charge density
waves.

Figure 4:  (a) Autocorrelation of 30 nm x 30 nm area of sample A , (b)
Autocorrelation of 30 nm x 30 nm area of sample B, (c) and (d) Line section of
autocorrelation of sample A shown in (a) and autocorrelation of sample B shown
in (b) along the CDW direction respectively, red curve indicates the rate of
decay of CDW amplitude.

To further characterize the CDW order, we have calculated the translational
correlation. Calculating the correlation length of CDWs using topographic
images can shed light on the interplay between CDWs and intercalant density.
Non modulated boundaries of CDW modulated regions can be easily identified
visually from STM topography images of areas with large defect concentration
as in Fig.3(a). Correlation length can be used to gauge the disruption of CDW
order because of intercalation. Correlation length measurement can pick up the
change in CDW order where it is not easily discernible from STM images
visually. The presence of intercalants disprupts the CDW order hence results
in a reduced translational correlation length.

CDW peaks in the Fourier transform of topography images were selected and IFT
was performed. IFT image displays CDW amplitudes as function of position (as
in Fig.2(d) and 2(g)). Autocorrelation of this image provides information
about the translational correlation length. The correlation length was
obtained by calculating the change in amplitude of CDW from line sections in
the auto correlation of IFT images, using following equation [50],

$A(x)=A_{0}e^{-x/\xi}\cos(k_{\text{cdw}}x)+B_{0}$ (1)

Where $A(x)$ is CDW amplitude, $\xi$ is correlation length, $k_{\text{cdw}}$
wave vector of CDW in $x$ direction, $B_{0}$ is average random noise in
autocorrelation obtained from IFT and $A_{0}=1-B_{0}$. The finite correlation
length is a consequence of the in-homogeneity and finite scanning area [50].
Autocorrelation of STM topography image of area with more defects in Fig.2(a)
is shown in Fig.4(a) and area with less number of defects in Fig.2(e)is shown
in Fig.4(b). Line section along the axis in autocorrelation image that is the
amplitude of CDW as a function of $x$ was plotted. Correlation length was
calculated by a fitting curve to the decay rate of amplitude in the line
section. In Fig.4(b) and (d) blue curve is line section along CDW direction
indicates its amplitude and red curve fitted using (1) shows rate of change of
CDW amplitude. The correlation length obtained from fitting indicates that the
correlation length is smaller for images with more intercalant sites than
images with fewer intercalant sites. This implies the long-range coherence of
CDWs has broken into small domains because of intercalation. A sharp decline
of CDW amplitudes means a smaller correlation length, which is a sign of the
reduced long-range order.

Figure 5: Correlation length obtained by fitting the CDW amplitudes for
topography images of different Ti intercalation concentration. Sample A
contains more intercalation sites than sample B. (The average density of
intercalation sites for sample A is $0.6\%$ and for sample B is $0.2\%$).
Correlation length of CDWs in sample B seems to be higher than that of CDWs in
sample A

The correlation length of CDW is a function of the concentration of Ti
intercalation. A better picture of the exact dependence of correlation length
on Ti intercalation density can be obtained by plotting the correlation
length, as shown in Fig.5. Atomically resolved images were taken, and the
number of Ti intercalation sites were counted. The correlation length of CDWs
was calculated from the autocorrelation of IFT images, as mentioned above. Ti
intercalation concentration in Sample B is lesser than Sample A and shows a
large correlation length. The overall trend indicates that the correlation
length of CDWs decreases as the Ti intercalation increases. The dependence of
correlation length on electron doping nature of Ti intercalant strongly
supports our hypothesis discussed in the earlier part of the excitonic origin
of CDWs. However, there are certain features in the graph that require close
examination. The correlation length tends to increase gradually at defect
concentration of $0.3\%$, reaching maxima at $0.5\%$ then decrease again. This
feature could be connected to the in-equivalent $Ti$ defect sites for each
defect that are a consequence of CDW modulation [48]. We believe our
experimental result will motivate further detailed Density Functional theory
(DFT) based calculations to explain this pattern in our correlation length.

##  IV Conclusions

We have explored defects at the atomic scale in two crystals of $1T$-TiSe2
grown using different methods. Ti atoms get intercalated in the van der Waals
gap. We find that the density of Ti intercalation affects the long-range order
of CDWs. An increased concentration of Ti intercalation forces CDWs to form
domains. There is no phase shift across the domain walls. The correlation
length of CDWs is inversely proportional to the density of intercalated Ti
atoms. Intercalation of Ti in $1T$-TiSe2 suppresses CDWs and forms domain
boundaries that can be explained by electron doping nature of Ti. Intercalated
Ti atoms reduce the number of holes in the system. In the excitonic
condensation model of CDW formation, the reduction of hole concentration
depletes excitonic bound states. CDW suppression is a direct consequence of
excitonic bound state depletion. Thus we demonstrate that our study of the
dependence of correlation length on Ti intercalation supports the excitonic
condensation model of formation of CDWs in $1T$-TiSe2.

###### Acknowledgements.

LH acknowledges the financial support from the Department of Science and
Technology(DST), India [Grant No. SR/WOS-A/PM-33/2018 (G)] and IISER Pune.
This work was financially supported by the Indian Institute of Science
Education and Research (IISER), Pune. IM is grateful to CSIR, Government of
India, for SRF and IISER Pune, for financial support in the form of a research
fellowship, and UR is thankful to UGC, Government of India, for SRF.

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