Abstract
A single molecular layer of titanium diselenide (TiSe_{2}) is a promising material for advanced electronics beyond graphene—a strong focus of current research. Such molecular layers are at the quantum limit of device miniaturization and can show enhanced electronic effects not realizable in thick films. We show that singlelayer TiSe_{2} exhibits a charge density wave (CDW) transition at critical temperature T_{C}=232±5 K, which is higher than the bulk T_{C}=200±5 K. Angleresolved photoemission spectroscopy measurements reveal a small absolute bandgap at room temperature, which grows wider with decreasing temperature T below T_{C} in conjunction with the emergence of (2 × 2) ordering. The results are rationalized in terms of firstprinciples calculations, symmetry breaking and phonon entropy effects. The observed BardeenCooperSchrieffer (BCS) behaviour of the gap implies a meanfield CDW order in the single layer and an anisotropic CDW order in the bulk.
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Introduction
Titanium diselenide (TiSe_{2}) is a member of a vast family of transitional metal dichalcogenides, many of which show charge density wave (CDW) transitions at low temperatures leading to periodic modulations of the electronic charge density. The resulting superlattices can be either commensurate or incommensurate. The CDW order can compete with other transitions such as superconductivity and antiferromagnetism, and it is a phenomenon of great interest in solid state physics^{1,2,3}. Specifically, TiSe_{2}, with a simple (2 × 2 × 2) CDW transition at 200 K in the bulk^{4}, remains an intensely debated case^{5,6,7,8,9,10}. The transition has been attributed variably to excitonic interaction, bandtype Jahn–Teller effects, and so on^{5,11,12}. A detailed investigation of the electronic structure is complicated by the threedimensional nature of the CDW order. The perpendicular electronic momentum is not necessarily conserved in angleresolved photoemission spectroscopy (ARPES) measurements, making it difficult to pinpoint the gap locations in the Brillouin zone. A single layer of TiSe_{2}, by contrast, has a twodimensional (2D) electronic band structure, and the gap of interest is limited to the one bridging the and points in the Brillouin zone. A recent study by scanning tunneling microscopy of a single layer of TiSe_{2} revealed a (1 × 1) structure at room temperature and a (2 × 2) superstructure at low temperatures, but it offered no information otherwise on the nature and details of the CDW transition^{13}. A detailed mapping of the electronic structure of the singlelayer case will not only help resolve the issues related to the bulk transition, but also reveal the relevant CDW physics at the 2D limit. A broader impetus for our work is the search and discovery of suitable molecular layers for advanced electronics with minimal physical dimensions suitable for integration and easily amenable to quantum engineering.
In this work, we employ molecular beam epitaxy to prepare highquality singlelayer TiSe_{2} on a suitably chosen substrate with a rather inert surface to minimize the substrate effects on the overlayer. Highresolution ARPES measurements as a function of temperature reveal intricate details including gap evolution and band folding in connection with the (2 × 2) CDW ordering. Surprisingly, the measured transition temperature in the single layer is substantially higher than the bulk transition temperature. We explain these results with the aid of firstprinciples calculations. The observed temperaturetunable gap in the singlelayer case indicates that the system is indeed a strong candidate for applications.
Results
Film structure and electron diffraction patterns
Our single layers of TiSe_{2} were grown in situ on a bilayergrapheneterminated 6HSiC (0001) (refs 14, 15). The interfacial interaction is expected to be of the van der Waals type, resulting in a nearly decoupled TiSe_{2} overlayer. The crystal structure (Fig. 1a) consists of a hexagonal planar net of Ti atoms sandwiched between two Se atomic layers. The (1 × 1) and (2 × 2) Brillouin zones are also hexagonal (Fig. 1b). Reflection highenergy electron diffraction (Fig. 1c) reveals a wellordered layer with the same inplane lattice constant as that of bulk TiSe_{2} within our experimental accuracy. Scans of the core levels (Fig. 1d) show a much stronger Se 3d signal than the Ti 3d signal partly because the top atomic layer is made of Se atoms.
ARPES spectra and calculated band structure
ARPES maps taken from the singlelayer sample along the direction (Fig. 1e) for both the (1 × 1) normal phase at room temperature and the (2 × 2) CDW phase at 10 K are compared with the (1 × 1) and (2 × 2) band structure (Fig. 1f) deduced from firstprinciples calculations based on density functional theory using the Heyd–Scuseria–Ernzerhof (HSE) hybrid functional (Methods section). In the normal phase, the band structure near the Fermi level consists of a pair of concave bands centred at the point that are primarily derived from the Se 4p states. At the point, the bottom of a convex band reaches just below the Fermi level, and this band is primarily derived from the Ti 3d states. The top of the Se 4p band and the bottom of the Ti 3d band are separated by a gap of 98 meV at room temperature. The calculation with the HSE functional including the spin–orbit interaction yields instead a negative gap of 0.20 eV. Further calculations with the GW approximation within a manybody perturbation theory reduce the negative gap to 0.18 eV at the G_{0}W_{0} level and to 0.08 eV with a selfconsistent G. The calculation assumes T=0, while the (1 × 1) structure is observed only at high temperatures. For the (2 × 2) CDW phase, the ARPES features become sharper because of the lower sample temperature. The Se 4p bands are now repeated at the point, which is also the zone centre after (2 × 2) zone folding. The folded (or umklapp) Se bands at have a lower intensity as expected because the superlattice distortion is weak. Experimentally, the gap is 153 meV at 10 K, which is substantially larger than the roomtemperature gap value of 98 meV. For comparison, the calculated HSE bandgap for an optimized structure at T=0 is 330 meV. The calculated total energy per chemical unit is lower by 5 meV (4 meV) for the CDW phase relative to the normal phase in the single layer (bulk). The calculated atomic displacements in the CDW phase are 0.09 Å for Ti and 0.03 Å for Se, respectively, as shown in Fig. 2e.
A detailed comparison of the bands near the Fermi level between experiment and theory is presented in Fig. 2a for singlelayer TiSe_{2}, where the calculated bands are shown as blue dashed curves and are aligned in energy by matching the Ti 3d band bottom or Se 4p band top where appropriate. The corresponding results for bulk TiSe_{2} (Fig. 2b) in both the normal and CDW phases reveal similar behaviour: a small positive gap for the normal phase becomes larger for the CDW phase. One notable difference is that folding of the Se 4p bands is evident for the normal phase of the bulk crystal, which has been reported before and attributed to fluctuation effects^{16}. No such folded bands are observed for the singlelayer sample in the normal phase. Constantenergy ARPES contours (Fig. 2c) at energy of −1.0 eV for the bulk and singlelayer samples in the CDW phase are similar because the bulk material is quasi2D, but there are also clear differences. The pronounced warping for the bulk case can be attributed to interlayer coupling effects. We have performed band mapping along k_{z} over a wide range by scanning the incident photon energy over 30–80 eV for the ARPES measurements of the singlelayer sample. The results for the Se 4p states at the point in the CDW phase (Fig. 2d) show no detectable energy dispersion, in agreement with the 2D nature of the system.
Temperature dependence of the bandgap
A temperature scan of the bands near and (Fig. 3) reveals variations of the band positions and the intensities of the folded bands. The energy gap is extracted from the data, and the square of the energy gap, plotted as a function of T (Fig. 4), shows that the gap becomes smaller as T increases and saturates to a constant value above a transition temperature T_{C}=232 K. It is interesting to note that the square of the energy gap follows closely a linear behaviour for T near but below T_{C}, as indicated by the reddashed line:
which suggests a meanfield behaviour. The blue solid curve is a fit to the data using a semiphenomenological BCS gap equation based on the meanfield theory:
where A=1.16 is a proportional constant and T_{C}=232±5 K. Equation (2) reduces to equation (1) for T near T_{C}. The observed meanfield behaviour is not surprising because quantum fluctuation effects are negligible compared with thermal fluctuation effects considering the high T_{C} value for this system.
Discussion
CDW transitions are often attributed to Fermi surface nesting, but with the existence of a bandgap in both the normal and CDW phases there is no nesting in the present case. Our firstprinciples calculations show that the gap widens with an increasing (2 × 2) distortion because of the lifting of the conduction band degeneracy that couples indirectly to the valance bands^{17}. The larger gap pushes the occupied Se 4p states to lower energies, resulting in total energylowering initially, but this process is soon counteracted by other effects such as an increase in elastic energy. The calculated Ti displacement in the CDW phase of monolayer TiSe_{2} is about 2.5% of the calculated lattice constant (a=3.538 Å), which is very close to the value of 2.4% in the bulk as measured by neutron scattering^{10}. The fact that our HSE calculations show the (2 × 2) phase to be the ground state at T=0 indicates that the CDW phase is entirely a band structure effect. There is no need to invoke additional or exotic manybody interactions beyond the HSE exchange and correlation effects.
The question is then what drives the system into the (1 × 1) structure at higher temperatures. Like the Jahn–Teller and ferromagnetic transitions, CDW transitions can be described as a result of spontaneous symmetry breaking. For TiSe_{2}, the transition involves atomic displacements that can be spatially reversed to yield a configuration with the same total energy but separated from the original configuration by an energy barrier. At low temperatures, the system is frozen in one of the two configurations. At higher temperatures, thermal effects (or phonon entropy effects) allow the system to fluctuate between the two configurations, resulting in average zero atomic displacements or a (1 × 1) structure on average. The physics is very similar for the different types of phase transitions, and indeed, the same meanfield behaviour is observed for the singlelayer TiSe_{2}.
The enhanced T_{C} for the singlelayer TiSe_{2} relative to the bulk indicates that the CDW phase in the single layer is more robust^{18}. This is perhaps not surprising in view of the weak van der Waals coupling between layers in bulk crystals. An important implication is an anisotropic order parameter in the bulk. Presumably, the CDW order along z melts at T just above the bulk T_{C}=200 K, but the individual TiSe_{2} molecular layers remain in the (2 × 2) phase. The layers are, however, no longer phase locked, resulting in an overall (1 × 1 × 1) configuration on average. Nevertheless, the persistence of (2 × 2) of the individual layers can give rise to (2 × 2) band folding above bulk T_{C} as seen in Fig. 2b. Note that the singlelayer T_{C}=232 K determined here is for the layer sitting on a grapheneterminated SiC. It is not necessarily the same as that for a freestanding layer or a layer embedded in bulk TiSe_{2}. The random interface potential caused by the lattice mismatch between graphene and TiSe_{2} could suppress the CDW T_{C} relative to the other cases. In our experiment on the single layer, evidence of bandfolding disappears completely at T greater than the singlelayer T_{C} (Fig. 2a). This is consistent with a single order parameter, as opposed to the bulk case.
Singlelayer TiSe_{2} is an interesting candidate as a substitute of graphene or a complementary platform for building novel 2D electronics. Its natural gap is well suited for traditional semiconductor device architecture. The size of the gap is well matched to lowpower designs and furthermore can be tuned by temperature and likely by other environmental effects. Singlelayer TiSe_{2} is also an excellent prototypical system to unravel the longstanding mysteries and debates about the CDW transition in bulk TiSe_{2} and other related materials. For the singlelayer case, the CDW phase is simply a band structure effect based on energy minimization. It is a more robust phase than the bulk case. The latter depends additionally on the layer stacking order, which melts at a lower temperature. This anisotropic order could be a common feature of layered CDW systems.
Methods
Film growth and characterization
Thin films of TiSe_{2} were grown in situ in the photoemission systems at beamlines 12.0.1 and 10.0.1, Advanced Light Sources, Lawrence Berkeley National Laboratory, where ARPES measurements were made. Substrates of 6HSiC(0001) were degassed at 650 °C for several hours and then flashannealed up to 1,300 °C for multiple cycles to form wellordered bilayer graphene as verified by ARPES (ref. 14). Highpurity Ti and Se were evaporated from an electronbeam evaporator and a Knudsen effusion cell, respectively, onto a substrate maintained at 220 °C. The growth process was monitored by a reflection highenergy electron diffraction system and the growth rate was controlled to be 30 min per single layer of TiSe_{2}. Formation of the second layer of TiSe_{2} is evidenced by evolution of the band structure. The bulk TiSe_{2} samples were prepared by cleavage in the same vacuum chamber to expose a fresh surface. ARPES measurements were performed at a base pressure of ∼3 × 10^{−11} mbar. The system energy resolution was <20 meV, and the angular resolution was 0.2°. For band mapping along k_{z}, a series of measurements was made with various photon energies in the range of 30–80 eV (ref. 19). Each sample orientation was precisely determined by constantenergycontour mapping in k space to identify the highsymmetry points and the crystallographic directions.
Theoretical calculation methods
Firstprinciples calculations were performed using the Vienna ab initio package^{20,21,22} with the projector augmented wave method^{23,24}. The monolayer system was simulated with a 17Å vacuum region to suppress the interaction between adjacent layers. A plane wave energy cutoff of 320 eV and an 18 × 18 × 1 kmesh were used for structure optimization. Using the generalized gradient approximation with the Perdew–Burke–Ernzerhof functional^{25}, the equilibrium (1 × 1) lattice constant was found to be 3.538 Å for monolayer TiSe_{2}, and the (2 × 2) CDW phase has a lower energy at T=0. The structure optimization was performed until the forces were reduced to below 0.005 eV Å^{−1}. Once the atomic displacements were determined by the Perdew–Burke–Ernzerhof functional, a more accurate band structure for the fixed geometry was obtained by using the HSE functional^{26} including spin–orbit coupling. The HSE selfconsistent calculations with 25% exact exchange were performed on a 12 × 12 × 1 (6 × 6 × 1) kmesh for the normal (CDW) phase, and the band energy at an arbitrary k point was deduced by interpolating the Hamiltonian on the basis of maximally localized Wannier functions using the Wannier90 package^{27,28}.
Additional information
How to cite this article: Chen, P. et al. Charge density wave transition in singlelayer titanium diselenide. Nat. Commun. 6:8943 doi: 10.1038/ncomms9943 (2015).
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Acknowledgements
This work is supported by the U.S. Department of Energy (DOE), Office of Science (OS), Office of Basic Energy Sciences, Division of Materials Science and Engineering, under Grant No. DEFG0207ER46383 (T.C.C.) and DEFG0297ER45632 (M.Y.C.). The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DEAC0205CH11231. Y.H.C. is supported by a Thematic Project at Academia Sinica.
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P.C. with the aid of Y.Z., X.Y.F., S.K.M., Z.H., A.V.F. and T.C.C. performed molecular beam epitaxy growth, ARPES measurements and data analysis. Y.H.C. and M.Y.C. performed firstprinciples calculations. P.C. and T.C.C. wrote the paper. T.C.C., P.C. and M.Y.C. interpreted the data. T.C.C. and A.V.F. jointly led the experimental project.
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Chen, P., Chan, YH., Fang, XY. et al. Charge density wave transition in singlelayer titanium diselenide. Nat Commun 6, 8943 (2015). https://doi.org/10.1038/ncomms9943
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DOI: https://doi.org/10.1038/ncomms9943
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