Observation of a uniaxial strain-induced phase transition in the 2D topological semimetal IrTe$_2$

Strain is ubiquitous in solid-state materials, but despite its fundamental importance and technological relevance, leveraging externally applied strain to gain control over material properties is still in its infancy. In particular, strain control over the diverse phase transitions and topological states in two-dimensional (2D) transition metal dichalcogenides (TMDs) remains an open challenge. Here, we exploit uniaxial strain to stabilize the long-debated structural ground state of the 2D topological semimetal IrTe$_2$, which is hidden in unstrained samples. Combined angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM) data reveal the strain-stabilized phase has a 6x1 periodicity and undergoes a Lifshitz transition, granting unprecedented spectroscopic access to previously inaccessible type-II topological Dirac states that dominate the modified inter-layer hopping. Supported by density functional theory (DFT) calculations, we show that strain induces a charge transfer strongly weakening the inter-layer Te bonds and thus reshaping the energetic landscape of the system in favor of the 6x1 phase. Our results highlight the potential to exploit strain-engineered properties in layered materials, particularly in the context of tuning inter-layer behavior.

Using external stimuli to manipulate the diverse phenomena observed in quantum materials may allow for tunable control over technologically relevant material properties. Within this context uniaxial strain has recently emerged as a powerful approach to influence the properties of solids [1][2][3][4][5][6] and offers a path to tailor both physical properties and device functionalities, particularly in the 2D TMDs [7][8][9][10] . While efforts to control phase transition behavior with strain have focused predominantly on oxide materials, there also exist many opportunities within the 2D semimetals, which routinely host multiple nearly degenerate structural, electronic and topological phases 11 , thereby making them sensitive to external perturbation.
In this regard, the family of layered tellurides are particularly promising 12 , a prime example of which is 1T-IrTe2. This high-atomic number material is predicted as a type-II bulk Dirac semimetal with a Dirac point slightly above the Fermi level 13 and presents first-order bulk phase transitions to a 5×1×5 structure at 280 K, and to an 8×1×8 structure at 180 K 14,15 . At the surface a complex staircase of nearly degenerate low-temperature phases with periodicity 3n+2 (i.e. 8×1, 11×1, 17×1…) coexist over scales of a few tens of nanometers 15,16 . All of the broken-symmetry phases display characteristic quasi-1D modulations typically identified as Ir dimers 15,17 , although the changes to the in-plane bonding suggest a multi-center bond as a more complete description 18 (for brevity we will continue to use "dimers" throughout the text).
The proposed ground state, a 6×1 phase 15,[19][20][21] , is typically observed only within nanoscale regions, making it all but inaccessible to most techniques. As a result, the electronic structure of the ground state, as well as any influence of the phase transitions on the bulk Dirac states, remains unclear, hindering efforts to elucidate the transition mechanism or exploit the topological properties. Substantial changes to the material behavior are produced by doping: superconductivity is induced by partial substitution of Ir with Pt 22 or Pd 23 , or by temperature quenching 24 , while partial substitution of Te with Se induces charge order 19,20,25 , further emphasizing the metastable nature of the material. The range of competing phases observed in IrTe2 strongly implies that its macroscopic behavior may be tunable via strain, allowing individual phases to be selectively stabilized without the need for external doping.
In this work, by applying a modest uniaxial tensile strain ( ~ 0.1%) to IrTe2 single-crystals, we demonstrate the selective stabilization of a single structural phase transition with domain sizes four orders of magnitude larger than in unstrained samples. Complementary real and momentum space probes reveal this as a 6×1 charge ordered phase, a configuration that maximizes both the formation of Ir dimers and of Ir to Te charge transfer. We show that strain initiates this charge transfer already at room temperature, thereby removing the near degeneracy of the 3n+2 ladder of phases and favoring the 6×1 phase at low temperatures 18 . This energetic bias allows unprecedented spectroscopic access to the ground state of IrTe2, including the previously unobserved bulk Dirac-like states, which undergo a Lifshitz transition due to the charge transfer. Concurrently, charge transfer results in a significant weakening of the majority of inter-layer Te bonds in the unit cell, resulting in a tenfold reduction of inter-layer hopping in the relevant states, and leaving the bulk Dirac states as the dominant inter-layer transport channel. These results demonstrate the power of strain to influence phase transitions, bonding and topology in the layered tellurides, and more broadly in the 2D semimetals. Fig. 1a shows the hexagonal crystal structure, typical of the layered TMDs, in the high-temperature (HT) 1×1 phase of 1T-IrTe2. In comparison, all low-temperature (LT) phases in IrTe2, including the 6×1 (Fig.   1b), are characterized by the formation of Ir dimers stabilized by electronic energy gain 18,19 . The density of these dimers increases in each of the successive charge ordered phases, reaching a maximum in the 6×1 phase 19,26 which is generally considered as the ground state of the system. The band dispersion along the high-symmetry LAL direction of the bulk Brillouin zone (Fig. 1c), as obtained by angle-resolved photoemission spectroscopy (ARPES), is displayed in Fig. 1e for the HT phase in an unstrained sample.
The corresponding Fermi surface (Fig. 1h) shows three-fold rotational symmetry consistent with the literature [26][27][28][29] . Upon cooling the unstrained sample (Fig. 1f), subtle changes occur due to the phase transitions in IrTe2. The overall form of the electronic structure in the LT phase strongly resembles the HT phase, but with increased broadening of the bands and a reduction of spectral weight. The lack of clear features results from the presence of multiple domains and phases (see Fig. 2). While the disappearance of the three-fold symmetry in the Fermi surface (Fig. 1i) implies a dominant domain orientation within the probed region (50 m), the absence of a single-phase domain hinders analysis of the resulting electronic structure. In dramatic contrast, the strained sample (Fig. 1g) displays a rich spectrum of remarkably sharp bands over a wide energy range, implying a uniform signal over the probed region originating from a single phase. Strain was applied along the a-axis of the HT phase using the home-built device pictured in Fig. 1d (see Methods). Of particular interest are sharp hole-like surface states and an apparent bulk-like hyperbolic dispersion close to EF (red arrows, Fig. 1g), discussed in more detail below, which are undiscernible in unstrained samples. The Fermi surface (Fig. 1j), reveals a clear directionality, breaking the rotational symmetry of the HT phase and resulting in a mirror-plane along the kx = 0 line. Cuts along the ky direction ( Supplementary Fig. S2) showing repeated surface and bulk states reveal this phase has a 6×1 in-plane periodicity, which is difficult to discern in Fig. 1j due to the small size of the repeated features and the variation of spectral weight.
The 6×1 periodicity of the strain-induced state is confirmed in Fig. 2, which demonstrates the effect of strain on the real-space surface structure as revealed by low-temperature scanning tunneling microscopy (STM) measurements. Unstrained samples display a mixture of differently oriented domains (Fig. 2a) which form due to the three-fold degeneracy of the HT phase. Within these rotational domains, there exist multiple phases with different 3n+2 periodicities (Fig. 2d). Again, in contrast, strained samples reveal a clear uni-directional domain (Fig. 2b), with a single 6×1 periodicity (Fig. 2f). The line cut in Fig. 2g shows that two non-identical groups of three atoms comprise the 6×1 periodicity in agreement with previous work 15,20 . Although the individual STM images are limited in size, the same 6×1 phasealways with the same orientationis found in multiple images across the sample surface over hundreds of microns: more than half of the sample area ( Supplementary Fig. S3). This macroscopic domain size is also seen in low energy electron diffraction (LEED), which averages over a region of similar dimensions ( Fig. 2c and e). In the strained case, we indeed record a single domain orientation with 6×1 periodicity, contrasting the clear three-fold directionality of the unstrained sample. Further corroboration is obtained from micro-ARPES mapping (Fig. 2h), obtained by integrating the intensity of the sharp surface states characteristic of the 6×1 phase ( Supplementary Fig. S4) across the sample surface. This reveals that the 6×1 phase is found over a continuous region of dimensions ~0.5x0.4 mm 2 .
As is typical of phase transitions in the metal-chalcogenides 30 evidence of charge transfer is observed during the formation of the 6×1 phase. Due to the presence of polymeric bonds, the structure of IrTe2 lies between that of pure 2D or 3D materials 12 . This results in an Ir 3+ configuration and hence a partial charge of Te 1.5on average in the HT phase. In the LT phase, a charge  is transferred which produces modified Ir  and Te  species 19,26 . The electronic energy gain from Ir dimer formation 17 competes with the lattice deformation energy 31 , making a complete dimerization of the surface energetically unstable 19 . As a result, both Ir 3+ and Ir  charge species are present in the LT phase. This can be readily observed in X-ray photoemission spectroscopy (XPS) where the two distinct peaks appear in the Ir 4f spectra 26,28 as shown in Fig. 3b. We note that the higher binding energy of the second peak implies reduced screening of the core potential, consistent with a reduced electronic density on the Ir atom (Ir  ). Further evidence for a charge transfer is discussed below (Fig. 4). The ratio of the peak areas tracks the relative dimer density and indicates a mixture of phases in unstrained samples 26,29 . In contrast, for strained samples an increase of the Ir  peak produces a ratio that accords perfectly with the expectation for a single 6×1 phase (0.67). Crucially, XPS measurements in the HT phase (Fig. 3a) reveal that a small population of the Ir  species is already evident above the transition temperature in strained samples, which is absent for unstrained samples. This implies strain actually induces a charge transfer from Ir to Te, and that this is central to understanding the phase stabilization. The analysis of the peak ratio in the HT phase gives only 0.14, well below the ratio of 0.4 obtained in the 5x1 phase, which has the lowest dimer density of the ordered phases. An open question is whether the appearance of the second charge peak in the HT phase implies the existence of dimers above the phase transition temperature, and indeed whether an ordered phase can be induced at room temperature by increasing the strain level. We note, however, that dimer formation depends on the competition between electronic and lattice energy, and it is therefore possible in the strained system that a charge transfer is induced without dimer formation. Finally, we remark that no evidence for a continuous phase transition is observed in temperature dependent ARPES at this strain level ( Supplementary Fig. S5).
In order to gain more insight into this redistribution of charge in the strain-stabilized phase, in Fig. 3c and d we compare the calculated charge distributions in the 1×1 and 6×1 phases, respectively.
Particularly notable is that there is a clear increase of charge density in the inter-layer Te-Te region as a result of the phase transition, as charge is moved away from the Ir 3+ sites and onto the Te atoms. This implies the strain-induced charge transfer that produces the Ir  signal in the HT phase also redistributes charge into the inter-layer region.

6
The impact of this charge transfer on the out-of-plane Te-Te bonds 25,32 is significant (the effect on the in-plane bonds has been addressed previously 18

). A particularity of the tellurides in comparison to other
TMDs is the presence of 3D polymeric bonding structures 12 , in place of the usual van der Waals gap.
IrTe2 indeed contains a network of weak interlayer covalent bonds in the HT phase 32 . In Fig. 4a we present a calculation in the 6×1 phase highlighting the different inter-layer bond lengths and their strengths relative to the HT phase. Bond strengths are obtained using the integrated crystal orbital Hamiltonian potential (ICOHP) method 33 . We find four inequivalent inter-layer Te-Te bonds in the 6×1 phase, compared with only one type in the HT phase (see Supplementary Table S1). Various bond strengths are changed in the 6x1 phase including a number which strengthen, in line with a multicenter bond description 18 . Three out of the four interlayer bonds are observed to weaken significantly across the phase transition, represented in blue in the figure (see also Supplementary Table S1). The reason for this is that the charge transferred from Ir 3+ populates anti-bonding Te-Te states in the inter-layer region, thereby reducing the overall bond strength. The resulting inter-layer bond weakening has been termed "depolymerization" 12,25,26 , although quantities relevant for bonding such as the out-of-plane bond strengths and hopping have not previously been addressed in detail for the low-temperature phase. We provide a direct experimental quantification of the effect that the bond weakening has on the electronic structure and electronic hopping in the out-of-plane direction. We do so by comparing the out-of-plane (kz) dispersion ( Fig. 4b) in the HT and 6×1 phases, and reemphasize that it is only via the strain stabilization that we are able to access the electronic structure of the pure 6×1 phase. Between the LT and HT phases, the majority of states maintain their small out-of-plane dispersion. However, a sizeable change in the warping of the Fermi contour is observed for bulk states on either side of the Brillouin zone boundary, highlighted by the blue lines in the two panels of Fig. 4b. In general, such warped Fermi surface contours are characteristic of a strong anisotropy in the electronic hopping parameters and are routinely observed in low-dimensional materials. A small warping corresponds to a low coupling between chains (1D) or planes (2D) along the relevant real-space direction 34,35 . In the case of the present out-of-plane dispersion, the narrowing of the warping in the kz direction corresponds to a reduction of the inter-layer hopping in the LT phase. In contrast, significant dispersion is observed for these same states in the (kx, ky) plane, highlighting the quasi-2D behavior of IrTe2 in the 6x1 phase. By applying a tight-binding model 34 (see Methods) we find the inter-layer hopping parameter to reduce by a factor of ten to only tc = 0.014 eV in the LT phase (in comparison, the in-plane ta = 0.53 eV). This prominent reduction of inter-layer coupling in the 6×1 phase therefore strongly enhances the 2D nature of the system. The observed layer decoupling further supports recent calculations that show monolayer IrTe2 has an increased tendency towards the 6×1 phase 18 suggesting that the dimerized phases could potentially be stabilized at much higher temperatures by reducing the sample thickness to the monolayer limit.
These observations strongly implicate the inter-layer bond weakening in the mechanism of the strainstabilized phase transition 25,26,32 . By inducing charge transfer, and hence inter-layer bond weakening already in the HT phase, strain reduces the amount of electronic energy that can be gained through dimerization of the Ir  ions. This therefore destabilizes the nearly degenerate LT phases, and pushes the system to favor the formation of the phase with the highest dimer density and hence highest possible gain in electronic energy i.e. the 6×1 phase. In this way, the less stable 3n+2 phases are removed from the LT phase diagram. The effect of strain is thus twofold: first, by defining a preferential direction, it breaks the degeneracy of the threefold dimer orientation 36 ; second, it biases the energetic landscape of the system in favor of the 6×1 phase. Although the effects of strain are subtle in the HT phase, as expected for the perturbative strain level applied, they pave the way for the stabilized phase transition, with dramatic results at low temperatures. We note that similar bonding behavior is realized in a number of di-and tri-telluride materials spanning the 2D and 3D regimes 12 , suggesting strain or electrical gating as powerful methods to control structural behavior and dimensionality in this class of materials.
In contrast to the discussion above, a state dispersing in kz appears around kx = 0 in the LT phase (red, arrow, Fig. 4b). Due to its strong out-of-plane dispersion ( Supplementary Fig. S6), this state is only observed around kz = 5 Å -1 , i.e. the bulk A-point. This feature corresponds to the triangular block of states observed in Fig. 1g, and shown again at different ky positions in Fig. 4d. The strong kz dependence and "filled-in" nature of these states resulting from the projection of the bulk manifold reveals them as bulk states. Of note is that these bulk states have the cone-like hyperbolic dispersion of massless Dirac fermions ( Supplementary Fig. S7), which occur ubiquitously in the group-10 TMDs and are predicted 8 in IrTe2 13 . However, the location of the type-II Dirac point at room temperature is above EF, hence inaccessible to ARPES, while at low temperature the mixture of phases typically hides their true nature.
The observed shift of the Dirac point to 350 meV below EF in the 6×1 phase occurs as the Dirac states are derived from the out-of-plane Te 5pz orbitals and hence are strongly doped by the charge transfer into Te states as described above. As we have shown, it is only possible to access these states spectroscopically in strained samples. By moving the Dirac point and the electron-like portion of the Dirac cone into the occupied states, strain produces a Lifshitz transition similar to the temperature-driven transitions observed in WTe2 37 and ZrTe5 38 . Such a dramatic change in Fermi surface topology is likely to have a significant impact on the transport properties in this material. In particular, the topological nature of the states involved in the transition may explain the observed large, non-saturating magnetoresistance 39 similar to the behavior in other layered di-tellurides [40][41][42] . The strain-stabilized order may even enhance such effects, paving the way to tunable magneto-resistive behavior. Given the outof-plane character of the 5pz orbitals involved in the Dirac states, it is plausible that particularly large changes in inter-layer transport may be observed, and investigations of the resistivity anisotropy using e.g. focused ion beam methods 43 are highly desirable in this regard. The potentially topological nature of this out-of-plane transport makes IrTe2 layers especially interesting for tuning inter-layer behavior in heterostructure architectures 44,45 .
While a detailed discussion of the topological properties is beyond the scope of the current article, we nonetheless highlight a surprising observation regarding these Dirac-like states: the dispersion is not compatible with a single Dirac cone. This can be seen, for example, from the cut at ky = 0.05 Å -1 in Fig. 4d, which reveals two partially overlapping cone-like dispersions. Indeed, the in-plane dispersion of these Dirac states (Fig. 4c) reveals rich structures related to these bulk states, comprising the central hyperbola around ky = 0 Å -1 , which has a bow tie-like Fermi contour, and additional hyperbolic cones centered at around ky = ±0.15 Å -1 (see also Supplementary Fig. S7) which form asymmetric arcs. The spacing of this latter behavior is compatible with the periodicity imposed by the 6×1 phase, but their unusual distributions and the origin of the additional central bow tie structure remain unclear. Further detailed investigations including theoretical work will be required to clarify the nature of these states.

9
In summary, we have selectively stabilized the 6×1 charge ordered ground state of the layered topological semimetal IrTe2 by employing uniaxial strain. The induced macroscopic domain sizes allow detailed insights into the electronic structure at the surface in both real and momentum space. Charge transfer in the strain-stabilized phase strongly reduces the out-of-plane Te bond strengths, electronically decoupling the layers and resulting in a Lifshitz transition, granting access to a previously inaccessible bulk Dirac dispersion that acts as the main inter-layer channel. Complementary measurements of the transport properties of the 6x1 phase, as well as of monolayer IrTe2, are therefore highly desirable. We note that in contrast to the tensile strain utilized here, uniaxial compression may stabilize superconducting behavior [22][23][24] which, concomitant with the topological states, opens the possibility of strain-tunable topological superconductivity in IrTe2.  The same Ir 4f7/2 core level at 30 K. Again, the Ir  peak is enhanced in the strained case. The ratio of the Ir  peak area to the total weight, Ir  /(Ir 3+ + Ir  ), is 0.67 in the 6×1 phase, as expected for 4 out of 6 atoms being dimerized, implying a pure 6×1 phase in the presence of strain. c, Calculated electronic density (orange isosurface) in the HT 1×1 phase. d, Electronic density in the 6×1 plotted at the same isosurface level as for the 1×1 phase in (a), revealing an increase of charge density in the inter-layer region (highlighted by arrows) that is maximized in the strain-stabilized 6×1 phase. Both calculations are obtained from the sum of electronic states between EF -7 eV and EF. Additional spectra are presented in Supplementary Fig. 7.

Methods
Sample growth and characterization. Single crystals of IrTe2 were grown using the self-flux method 32,46 . Samples were characterized by magnetic susceptibility and resistivity measurements 29 , which confirmed the bulk phase transition temperatures of Tc1 = 278 K and Tc2 = 180 K in unstrained samples. Samples to be prepared for straining were chosen to have large flat areas with minimal cracks or flakes at the surface under an optical microscope to ensure as homogeneous as possible strain application. Bulk samples were initially cleaved with a scalpel to remove thicker layers, and then mounted onto the unstrained device and further thinned by Scotch tape cleaving.
Strain device and characterization. The strain device, shown in Fig. 1d, is a home-built design consisting of three parts: a molybdenum (Mo) base plate, a copper beryllium (CuBe) bridge, and a rounded aluminum (Al) block which is placed under the bridge. The maximum height of the Al block is machined to be slightly larger than the distance from the underside of the top of the CuBe-bridge to the surface of the base plate. Thus, when the pieces are screwed together, the CuBe-bridge if forced to bend by the Al block. Samples were oriented with Laue diffraction such that the bending axis was perpendicular to one of the three-fold symmetric directions in the HT phase i.e. along the a-axis of the HT phase (or equivalently, the b-axis). This is along the in-plane bond direction of half of the Ir-Ir dimers in the 6x1 unit cell. The sample was mounted on the CuBe-bridge of the strain device using a two-part epoxy (EPO-TEK E4110) which was cured and allowed to cool before strain was applied. Strain was applied manually by tightening the screws on the underside of the device, which connect the Mo baseplate with the Cu-bridge. To ensure maximally directed strain, the screws were tightened in pairs. All four screws were tightened loosely, following which the two screws on one side of the bridge were fully tightened. The remaining two screws were then gradually tightened in an alternating fashion, in order to allow as even an application of strain across the device as possible. The strain magnitude was calibrated using commercial strain gauges (Omega Engineering) with nominal resistance 350  and gauge factor, k = 2.2. The gauge was attached to an unstrained device of the design described above using the same epoxy as for the samples, and was then connected to a home-built Wheatstone bridge balanced circuit in a "quarter bridge" configuration. Together with a second (passive) gauge, this constituted one arm of the bridge. The second arm consisted of two 390  resistors and a variable resistor (10 ) was used to balance the circuit. A source voltage of Vs = 5 V was applied. Once balanced, the gauge was strained using the device and the output voltage was recorded with a Keithley digital multimeter. The typical output voltage induced by strain (Vo = 4 mV) was well above the noise level (50 V). The voltage output was converted to a strain value via 47 : where the total strain, is the sum of bending (tensile) strain and perpendicular (compressive) strain.
In such a strain geometry, the perpendicular strain is considerably smaller than the bending strain 4 , hence "strain" in the main text refers to the tensile bending strain. To separate further these components requires additional gauges to be placed on the underside of the device, which is impractical given the geometry and small size. From the above relation, we obtained the strain characteristics of the device.
A maximum strain of up to 0.2 % was initially recorded during tightening due to plastic deformation of the CuBe-bridge. This relaxed to around 0.1 % once all screws were tight, which is therefore the maximum strain that could be applied to the sample using this particular device and Al block combination.
Photoemission spectroscopy. ARPES and XPS measurements were carried out at the I05 beamline 48 of Diamond light source, UK, with additional data obtained at the PEARL beamline 49