Phase separation in the vicinity of Fermi surface hot spots

Spatially inhomogeneous electronic states are expected to be key ingredients for the emergence of superconducting phases in quantum materials hosting charge-density waves (CDWs). Prototypical materials are transition-metal dichalcogenides (TMDCs) and among them, $1T\text{−}{\mathrm{TiSe}}_2$ exhibiting intertwined CDW and superconducting states under Cu intercalation, pressure, or electrical gating. Although it has been recently proposed that the emergence of superconductivity relates to CDW fluctuations and the development of spatial inhomogeneities in the CDW order, the fundamental mechanism underlying such a phase separation (PS) is still missing. Using angle-resolved photoemission spectroscopy and variable-temperature scanning tunneling microscopy, we report on the phase diagram of the CDW in $1T\text{−}{\mathrm{TiSe}}_2$ as a function of Ti self-doping, an overlooked degree of freedom inducing CDW texturing. We find an intrinsic tendency towards electronic PS in the vicinity of Fermi surface (FS) “hot spots,” i.e., locations with band crossings close to, but not at the Fermi level. We therefore demonstrate an intimate relationship between the FS topology and the emergence of spatially textured electronic phases which is expected to be generalizable to many doped CDW compounds.

Figure 1

(a) Side view of two $1T\text{−}{\mathrm{TiSe}}_2$ layers separated by a van der Waals gap. ${\mathrm{Se}}_{\mathrm{up}}$ and ${\mathrm{Se}}_{\mathrm{down}}$ respectively refer to atoms of the top and bottom Se atomic planes of a $1T\text{−}{\mathrm{TiSe}}_2$ layer. (b) Three-dimensional Brillouin zone (BZ) of $1T\text{−}{\mathrm{TiSe}}_2$. IAlso shown is one CDW $q$ vector, ${\mathbf{q}}_{CDW}$, that connects the $\mathrm{\Gamma }$ and one of the $L$ points at the phase transition. (c) RT Fermi surface (FS) of pristine $1T\text{−}{\mathrm{TiSe}}_2$ as measured using He I radiation ($h\nu =21.22$ eV). The $k_x$ axis depicted in (b) and (c) by the black arrow is the $A\text{−}L$ direction of the BZ. Also indicated is the in-plane component of the CDW $q$ vector. (d) Large–energy scale ARPES spectrum of pristine along the $k_y=0$ ${\AA }^{-1}$ cut of the FS. The blue- and red-dashed boxes respectively indicate the energy and $k_x$ windows shown in (e) and (f) for the hole and electron pockets. (e),(f) RT ARPES intensity maps, as false color plots (blue colors represent strong intensity) as a function of $x$ at the $A$ and $L$ points of the BZ. He I radiation is used and the doping concentrations, $x$, are $<0.20\%$, $0.75\pm 0.10\%$, $1.21\pm 0.14\%$, $1.98\pm 0.15\%$, and $2.57\pm 0.22\%$ from left to right. Blue-, respectively red-, dashed lines and triangles in (e) and (f) correspond to parabolic fits of the maxima of the MDC curves (triangles). (g) Linear dependence of the experimental chemical-potential shift $\mathrm{\Delta }\mu$ with $x$ as extracted from the evolution of the hole and electron band extrema.

Figure 2

(a) $T$-dependent EDC maps measured at $L$ as a function of Ti self-doping (from left to right). For each doping, lower and upper panels respectively show the RT and LT EDCs. The red-dashed lines locate the critical CDW transition temperatures ($T_c$) extracted from the energy shift of the Se $4p$ backfolded band (blue dots) $\mathrm{\Delta }{E}_{\mathrm{backf}}$ as a function of $T$ using a BCS-type gap equation as shown in (b). (c) $T\text{−}x$ phase diagram of Ti-doped $1T\text{−}{\mathrm{TiSe}}_2$. The CDW transition temperature $T_c$ is linearly decreasing with $x$ towards a QCP. The inset shows $T$-dependent resistivity curves measured on the pristine (red curve) and Ti-doped crystals. The curves are normalized to the resistivity measured at RT and the vertical dashed lines show the location of $T_c$ for each doping as determined using ARPES. (d) $5.5\times 5.5{\mathrm{nm}}^2$ STM images taken at 4.5 K [see dark-blue points on (c)] of pristine ($x<0.20\%$) and Ti-doped crystals ($x=1.21\%$, $1.98\%$, and $2.57\%$). $I=0.2$ nA, $V_{\text{bias}}=150$ mV (pristine), 50 mV ($1.21\%$), 100 mV ($1.98\%$), and $-50$ mV ($2.57\%$). Added meshes on the $x=2.57\%$ STM image and white-dashed lines on the $x=1.21\%$ and $1.98\%$ ones highlight CDW domains and phase shifts between them, respectively.

Figure 3

DFT-calculated band structures of $1T\text{−}{\mathrm{TiSe}}_2$ along the $M\text{−}\mathrm{\Gamma }\text{−}A\text{−}L$ path of the 3D BZ for the $1\times 1\times 1$ normal state (red points) and of a slightly-distorted atomic structure ($10\%$ of the full lattice distortion) according to the PLD as proposed by Di Salvo et al. [15] (blue points) and shown in the inset. The color scale from white to blue indicates the spectral weights obtained using an unfolding procedure (see Appendix pp2). Also shown are the $2\times 2\times 2$ folded band dispersions (gray-dotted lines) and superimposed our RT ARPES measurements that mainly probe the $A\text{−}L$ direction of the BZ after matching the top of the calculated and experimental Se $4p_{x,y}$ bands at $A$.

Figure 4

(a) Normal-state near-$E_F$ folded dispersions of $1T\text{−}{\mathrm{TiSe}}_2$. The three-equivalent electron pockets (red lines, normally located at the $L$ points) are shifted here to $\mathrm{\Gamma }$ on the hole pocket (blue line) by the three CDW $q$ vectors. The $T$ dependence of the chemical-potential shift $\mathrm{\Delta }\mu \left(T\right)$ allows for raising $\mu$ at the electron- and hole-bands crossing at $T_c$ (black line). The colored dashed lines show the position of the chemical potential for the different Ti doping extracted from ARPES. (b) $T$-dependent shift of the bottom of the Ti $3d$ band for pristine as extracted from Fig. 2 and which relates to $\mathrm{\Delta }\mu \left(T\right)$. (c) Sketch of the pristine $1T\text{−}{\mathrm{TiSe}}_2$ FS at $T=T_c$ and showing FS hot spots at the intersections of the folded hole (blue circle) and three-equivalent electron pockets (red ellipses). (d)–(g) Sketches of normal-state FSs for $x=0.75\%$ (d), $1.21\%$ (e), $1.98\%$ (f), and $2.57\%$ (g). The colored dashed lines and arrows correspond to the $k_y=0$ ${\AA }^{-1}$ cuts of Fig. 4.

Figure 5

(a) LT ARPES spectra evolution as a function of $x$ at $L$. Are shown, in full lines, near-$E_F$ dispersions of the $1T\text{−}{\mathrm{TiSe}}_2$ CDW state calculated within an effective Hamiltonian model of four interacting bands with the LT order parameter ${\mathrm{\Delta }}_0$ matching the experiments (see Appendix pp3). (b) Mean-field evolution of ${\mathrm{\Delta }}_0$ as a function of the Ti concentration $x$. The blue-dashed line is a fit to the ${\mathrm{\Delta }}_0$ values with ${\mathrm{\Delta }}_0/{\mathrm{\Delta }}_0(x=0)={(1-x/x_c)}^{\beta }$ with ${\mathrm{\Delta }}_0(x=0)=103\pm 3$ meV, $\beta =0.5$, and $x_c\sim 2.99\pm 0.15\%$.

Figure 6

(a), (b) $5.5\times 5.5$ ${\mathrm{nm}}^2$ STM images above ($T=176$ K) and below ($T=111$ K) $T_c$; $V_{\text{bias}}=50$ mV, $I=2$ nA (a) and 1 nA (b). The white arrow in (a) points to a CDW droplet. The white and green rhombi indicate $2\times 2$ and $1\times 1$ surface unit cells. (c) $5.5\times 5.5$ ${\mathrm{nm}}^2$ zoom-in of a $20\times 20$ ${\mathrm{nm}}^2$ STM image (d) close to $T_c$; $V_{\text{bias}}=50$ mV, $I=1$ nA. The white-dashed lines highlight that CDW domains located by the two black squares are phase shifted. (e)–(g) FFT-amplitude plots obtained from (a), (b), and (d). The white circles in (f) show the extra spots originating from the CDW. (h) ARPES spectra evolution at $L$ across the CDW phase transition. (i) BCS-like $T$ dependence of $\mathrm{\Delta }{E}_{\mathrm{backf}}$ and $2\times 2$/$1\times 1$ FFT spot ratios as obtained using ARPES and VT-STM. The blue-dashed line is BCS-type gap fit to the $\mathrm{\Delta }{E}_{\mathrm{backf}}$ values and gives a $T_c$ of $164\pm 6$ K. The red-dashed circles correspond to the ARPES spectra of (h).