Doping-induced nematic and stripe orders within the charge density wave state of TiSe$_2$

TL;DR

This work addresses conflicting reports on the TiSe$_2$ CDW symmetry by developing a symmetry-guided continuum $\boldsymbol{k}\cdot\boldsymbol{p}$ framework and a minimal lattice model to study how conduction-band doping reshapes the CDW. The authors show that electron doping drives a secondary, $C_3$-breaking nematic $3Q$ CDW and, at higher doping, a $1Q$ stripe CDW, with the sequence surviving in 3D bulk TiSe$_2$. Crucially, the nematic state preserves inversion symmetry and offers a single-parameter mechanism tied to conduction-band ellipticity, potentially reconciling STM and bulk experiments without invoking a chiral order parameter. The results yield clear experimental signatures (transport elastoresistance, Raman mode splitting, ARPES $L$-point weights) and emphasize the role of native doping in sample-to-sample variability, with broad implications for tunable CDW nematicity in layered transition-metal dichalcogenides.

Abstract

In this work, we present a theory to address conflicting experimental claims regarding the charge density wave (CDW) state in TiSe$_2$, including whether there is a single or multiple CDW transitions and whether threefold rotation symmetry ($C_3$) is broken. Using a continuum $\boldsymbol{k}\cdot\boldsymbol{p}$ model coupled to the CDW order parameter, we show how commonplace conduction band doping induces a nematic transition from a $C_3$-symmetric $3Q$ CDW to a $C_3$-breaking $3Q$ CDW, which is favored by the large ellipticity of the conduction bands of TiSe$_2$. We also find that a $1Q$ stripe CDW is generically stabilized for sufficiently high electron doping. We then show how both stripe and nematic CDW states emerge self-consistently from a minimal interacting tight-binding model, for both positive and negative initial gaps. Our theory provides a new scenario in which, as temperature is lowered, a second $C_3$-breaking transition may occur or not depending on the doping level, potentially explaining the experimental variability. These predictions can be further verified with a variety of probes including transport, photoemission and tunneling.

Figures (11)

• Figure 1: (a) Side view of the lattice structure of TiSe$_2$. (b) Real space view of the charge density wave pattern. (c) Low energy band structure and symmetry labels for the bands for the bulk (notation in parenthesis for the monolayer). (d) Fermi surface sketch of the normal state in the semimetallic case. The order parameter $\vec{\Delta}$ coupling conduction and valence bands is also shown.
• Figure 2: Atomic displacements for the different phonon configurations discussed in the text. Panels (a-c) represent the different states predicted in our work: $L_1^-$ phonon in $1Q$ stripe $(\Delta,0,0)$, $C_3$-symmetric $(\Delta,\Delta,\Delta)$, and nematic $3Q$$(\Delta_1,\Delta_2,\Delta_2)$ configurations. (d,e) $L_2^-$ displacements in the $(\Phi,0,0)$ and $(\Phi,\Phi,\Phi)$ configurations. (f) Chiral CDW proposal from Ref. Peng22, which involves the mixing of $L_1^-$ and $L_2^+$ configurations. Note that both (c) and (f) have a twofold symmetry axis $C_{2x}$. In an STM experiment which probes predominantly the upper Se lattice (dark gray), states (c) and (e) cannot be distinguished by symmetry.
• Figure 3: (a,b) Folded bands in the presence of the order parameter $\vec{\Delta}$ for directions (a) $(\Delta,\Delta,\Delta)$, (b) $(\Delta,0,0)$, (c) $(\Delta_1,\Delta_2,\Delta_2)$ with $\Delta_1=\tfrac{1}{2}\Delta_2$, and (d) $(0,\Delta,\Delta)$. Grey dashed lines show the bands for $|\vec{\Delta}|=0$. (e) Constant $|\vec{\Delta}|$ phase diagram for $E_g=+50\mathrm{meV}$ and $|\vec{\Delta}| = {82} \mathrm{meV}$ as a function of electron doping $x$ and ellipticity of the conduction bands $b_d/a_d$. The black dotted line indicates the realistic ellipticity $b_d/a_d = 0.87$. (f) The same as (e) but with $E_g = -100\mathrm{meV}$ and on a larger $x$ range.
• Figure 4: (a) Lowest conduction band along the $\Gamma M$ direction and corresponding DOS in the $C_3$-symmetric $3Q$ phase with $E_g=+50 \mathrm{meV}$ and $|\vec{\Delta}| = {82} \mathrm{meV}$ for constant $a_d$ and ellipticities $b_d/a_d = 0, 0.39, 0.60, 0.78, 0.88, 0.92, 0.95$. A kink in the band is signaled by gray dots, and a van Hove singularity by black dots. (b) Corresponding DOS for each band in (a). (c) Conduction bands along the $\Gamma M_1$ and $\Gamma M_2$ directions ($\Gamma M_3$ is equivalent to $\Gamma M_2$) in the $C_3$-symmetric $3Q$ (blue) and nematic $2Q$ (red) phases with $E_g=+50 \mathrm{meV}$, $|\vec{\Delta}| = {82}\mathrm{meV}$ and $b_d/a_d = 0.87$. Horizontal dotted lines indicate the chemical potentials for $x=0.0182$, where the $C_3$-symmetric $3Q$ phase is unstable towards the $2Q$ state. (d) Corresponding DOS for each band in (c). The three equivalent vHs singularities of the lowest conduction band in the symmetric $3Q$ state disappear in the $2Q$ state, which has higher DOS at lower energy and therefore lower total energy.
• Figure 5: Temperature-doping phase diagrams for positive and negative gaps: (a) $E_g = +25 \mathrm{meV}$, $V_{dp} = 845 \mathrm{meV}$; (b) $E_g = -35 \mathrm{meV}$, $V_{dp} = 716.5 \mathrm{meV}$. The intensity of the color of each phase is proportional to $|\vec{\Delta}|$. Critical temperatures at charge neutrality are (a) $T_c = 459 \mathrm{K}$ and (b) $T_c = 239 \mathrm{K}$. The inset shows the band structure of the tight-binding model in the unfolded unit cell for $E_g=0$.