Excitonic condensate and Jahn-Teller effect

1T-TiSe2 undergoes a second-order phase transition at Tc ~200 K into a commensurate CDW phase with accompanying (2 x 2 x 2) superlattice. [2, 4]. The the electronic structure has been studied extensively both experimentally [6, 25, 63, 64]

and theoretically [23, 65, 66], whichexhibits a maximum of Se 4p valence band and a minimum of Ti 3d conduction band. In view of a semimetal, the overlap between the uppermost 4p band at Г and the lowest 3d band at L has been carefully estimated to be smaller than 120 meV [6]. However, the mechanism of CDW formation in 1T-TiSe2 is still controversial, several possible mechanisms have been proposed.

Nesting effect: In the case of 1T-TiSe2, the Г point hole pockets and the L point electron pockets should be nesting to induce electron-lattice interaction and produce periodic distortion. However, the Fermi surface nesting was ruled out from the calculated band structure, and the experimental results of ARPES indicates low possibility of nesting near the Fermi level, mostly due to the different shapes of electron and hole pockets [25, 16]. The Fermi surface does not show clear evidence of large parallel components in the BZ of 1T-TiSe2 when CDW emerges.

Jahn-Teller effect: Band-type Jahn-Teller effect has been proposed to explain the

periodic lattice distortion under the CDW state in 1T-TiSe2 [10]. The overall density of states of the ‘d_z²’ band is slightly lower for trigonal prismatic than for octahedral coordination in TiSe6 (Fig. 2.15). The periodic lattice distortion (PLD) appears that the upper and lower faces of the trigonal prismatic coordination (Fig. 2.16 (a)). Whangbo and Canadell suggested that the structural modulation originates from the p band energy lowering is associated with the Ti–Se bond shortening in the neighboring octahedron

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(Fig. 2.16 (b)) [12]. Rossnagel et al. observed energy lowering is much stronger for the p band (from Se) than for the d band (from Ti) in ARPES, which favors the proposal of Whangbo and Canadell [11]. Furthermore, Kidd et al. proposed that the 1T-TiSe2 is a very small indirect gap semiconductor and the unconventional CDW phase is driven by an indirect Jahn-Teller effect combined with electron-hole coupling [13].

Excitonic insulator phase: In a semimetal with very small number of electrons and

holes near the Fermi level, the Coulomb interaction between different carriers is only weakly screened, a bound states between electron and hole is possible and called exciton [3, 5]. This semimetal-semiconductor transition is necessarily accompanied with the periodic lattice distortion, a small band gap or band overlap corresponds to the exciton binding energy [67, 68]. The experimental results of angle-resolved photoemission spectroscopy (ARPES) study [15] and theoretical calculations [16]

support the scenario of excitonic condensate. Recent studies preferred the excitonic scenario as the origin of CDW state.

2.4.4 The intrinsic character of 1T-TiSe

without Se vacancy defects

Except for the debate on the origin of the CDW state, whether the normal state of 1T-TiSe2 is a semiconductor or a semimetal remains to be controversial. The temperature dependence of resistivity usually shows a metallic behavior, except below the CDW transition. In the ARPES investigation, Rasch et al. showed that 1T-TiSe2 has a narrow band gap by adsorbing H2O on TiSe2 crystal surface to bend the conduction and valence band below the Fermi level [18]. However, why is the transport behavior of 1T-TiSe2 exhibiting a metallic behavior experimentally always? We believe doping from the defect formations could affect the intrinsic behavior, i.e., 1T-TiSe2 should not be assumed to be stoichiometric. There are two types of defects in 1T-TiSe2, either due

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to Ti excess to form Ti1+xSe2 or Se vacancy to form TiSe2-δ. How much Se is lost in 1T-TiSe2 depends on the temperature and time used on post-annealing. The influence of defect type and level on 1T-TiSe2 will be discussed in the following chapters.

Fig. 2.15: The densities of states for octahedral and trigonal prismatic layered materials that the ‘d_z²’ band is slightly lower for trigonal prismatic than for octahedral coordination in TiSe6 [10].

Fig. 2.16: There are two kinds of Ti environments for TiSe6 octahedron: (a) top and bottom face of Se atoms rotate around Ti, and (b) Ti moves closer to Se-Se edge of the octahedron [10, 12].

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Chapter 3

Sample preparation and experimental methods

3.1 Polycrystalline sample synthesis and single crystal growth

1T-TiSe2 polycrystalline sample was synthesized using row materials of Ti and Se powders. The precursors with the stoichiometric molar ratio were thoroughly mixed and sealed in an evacuated quartz tube. To avoid Se vapor loss, the heating process used low reaction temperature of 350 ℃ for a prolonged period, with repeated grinding and re-sealing after each 24 hours heating cycle. After the single phase of 1T-TiSe2 was confirmed with X-ray diffraction, the polycrystalline sample was used as the precursor for post-annealing treatments at different temperature, ranging from 350 to 950 ℃ for one day each. Light Se vapor with red brown color was found to deposit on the inner wall of the tube. Some Se vapor was found condensed into droplet-like deposit in shinning black color after the higher temperature annealing.

Using single phase TiSe2 polycrystalline sample as the precursor, single crystals of 1T-TiSe2 were grown with the chemical vapor transport (CVT) method, where iodine as the transport agent and sealed together with the precursor in an evacuated. A two-zone tube furnace was set at 650 ℃ and 550 ℃ for a gradient in 30 cm.

Polycrystalline precursor with transport agent iodine was located at the high temperature zone (Fig. 3.1) initially. Single crystals are grown into 3 x 3 mm² size and 0.1 mm thick (as shown in Fig. 3.2) at low temperature zone in two weeks. The iodine also condensed at the low temperature zone and mixed with TiSe2 crystals. In order to remove the iodine from crystal mixtures, we re-heated up the low temperature zone to about 300 ℃ to allow iodine condenses to the room temperature end.

The as-grown single crystals at 550 ℃ were then separated and vacuum-sealed into

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batched to do additional post annealing, from 550 to 950 ℃ for one day each. Light Se vapor deposit was found in the inner wall of the sealed tube after annealing, the higher annealing temperature the thicker Se deposit.

Fig. 3.1: The single crystal growth of 1T-TiSe2 by the CVT method using iodine as the transport agent. The temperature gradient is TH = 650 ℃ and TL = 550 ℃ in 35 cm.

Fig. 3.2: The as-grown 1T-TiSe2 single crystal deposited at the 550 ℃ cold zone by the CVT method.

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3.2 Powder X-ray diffraction

The X-ray powder diffraction (XRD) is an analytical technique primarily used for phase identification of a crystalline material and unit cell dimensions. The purity of a sample can be determined from its diffraction pattern due to symmetry difference, especially used on identifying the presence of impurities phase.

The powder diffraction equipment used in this study is shown in Fig. 3.3. There are three movable parts for the XRD measurement, the X-ray source tube, the detector, and the sample holder. Usually only two of the three parts are designed as movable/rotatable in practice. For example, for the Bruker D8 X-ray diffractometer, all three parts are rotatable, but for the D2 PHASER X-ray diffractometer, only two rotatable parts, i.e., the 2θ/θ is locked.

The D2 PHASER (Fig. 3.4) is a desktop X-ray diffractometer enabling the analysis of polycrystalline material with all-in-one crystalline phase analysis database. The D8 ADVANCE is a fully extensible modular system and addressing the analytical needs of powder, bulk and thin film samples, at ambient and non-ambient conditions. The D8 diffractometer (Fig. 3.5) can collect with high angular resolution (< 0.008°, 2𝜃) powder XRD patterns of polycrystalline samples. This diffractometer can also be used for routine measurement when samples are large quantity and with automatic sample loader for a series of samples measuring.

Our synchrotron X-ray diffraction measurements were performed in the BL01C beamline in National Synchrotron Radiation Research Center (NSRRC), Hsinchu, Taiwan. Comparied with the in-house X-ray equipment, the synchrotron radiation provides much brighter X-ray source and has the advantage of tunable wavelength. The sketch of the synchrotron X-ray setup is shown in Fig. 3.6. The incident photon flux is 1.4 × 10¹⁰ photons/sec and to photon energy is 12 ~ 33 keV (1.03 ~ 0.37 Å ). Our

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measurements used photons energy of 20 keV (~0.61993 Å in wavelength). The sample was packed in a glass capillary tube of 0.1 mm inner diameter. The incident beam passing through the sample is scattered to arrive an image plate. The diffraction patterns form many concentric lines collected to the image plate and any cross section of the diffraction rings is equal to the in-house XRD pattern as shown in Fig. 3.7.

Fig 3.3: A sketch of X-ray diffractometer

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Fig. 3.4: The Bruker D2 PHASER X-ray diffractometer

Fig. 3.5: The Bruker D8 ADVANCE X-ray diffractometer

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Fig. 3.6: A sketch of synchrotron radiation and the beamline distribution in NSRRC, Taiwan. The powder X-ray diffraction is operated at BL01C beamline. The photon energy of 20 keV (~0.61993 Å ) incident beam passing through the sample and scattered to a detector Mar345 to form a diffraction pattern [69, 70].

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Fig. 3.7: The XRD pattern for 1T-TiSe2 polycrystalline obtained using synchrotron X-ray.

3.3 Rietveld refinement method

A fundamental assumption of the X-ray powder diffraction is the random orientation which means the orientation of each microcrystal is random distribution. In such a condition, each microcrystal scatters rays forming spherical surface in 3-dimensional space (a 2D simplified sketch map is shown in Fig. 3.8). However, the geometry of X-ray diffraction (XRD) allows us to collect only one dimensional patterns; resulting in information losses of the reciprocal lattice vectors. For example, in Fig. 3.8 (b), we get the absolute value of reciprocal vector |𝐾⃑⃑⃑| , but cannot distinguish two different reciprocal vectors 𝐾⃑⃑⃑⃑⃑ and 𝐾₁ ⃑⃑⃑⃑⃑. Usually, using Rietveld refinement method can improve ₂ the analyzing results of the powder diffraction patterns.

The basic concept of the Rietveld refinement method is that all reflecting intensities at each 2θ step should be taken into account in the refinement process, so called full profile refinement. It is different from the old method, which only uses the positions of the constructive interference peaks. The advantage of the Rietveld refinement is that

4 6 8 10 12 14 16 18 20 22 24 26 28 30

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the modeling pattern would be fully consistent with experiment pattern. There are some requirements of the Rietveld refinement method: (i) a good guess of crystal structure, including space group, lattice constants, and atomic position etc. It provides the refractive peak positions in the 2θ plot; (ii) a suitable profile of the refractive peaks, which gives the intensity distribution away from the constructive peaks, and (iii) a mathematical algorithm to fit the modeling result with experimental data and lead to a more reliable set of parameters.

The Rietveld refinement method can be applied on not only X-ray powder diffraction but also neutron diffraction by changing the peaks profile and the equipment parameters.

It also has an advantage to analyze thermal vibration of irons or deal with mixed phase powder diffraction. However, it should be noted that the refinement method only models the experimental data and generates a set of well self-consistent parameters (lattice constants and atomic positions, etc.). It cannot be exactly used to solve the crystal structure with only powder diffraction profile.

Many softwares have been developed to systematically analyze the XRD data base on the Rietveld refinement method. One of the popular software is General Structure Analysis System (GSAS) which was developed by A. C. Larson and R. B. Von Dreele [71]. The complete guide, software download, manual, and training course can be found in web site [72]. There is also a Graphical user interface to use GASA named EXPGUI which was developed by B. H. Toby [73]. Most of our powder refraction data are refined by GSAS and EXPGUI combined package.

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Fig. 3.8: (a) A 2D diffraction patterns forming a reciprocal lattice. (b) The random orientation powder scattering rays forming bright cycles.

3.4 Electron probe micro-analyzer (EPMA)

EPMA has been used as an analytical tool on non-destructively determination of the chemical composition of solid materials. EPMA is basically identical to a Scanning electron microscope (SEM) with added function of chemical analysis. Electron beam excites the sample to emit X-ray of characteristic frequency corresponding to atoms with intensities proportional to their relative quantity. The characteristic wavelengths and energies (frequency) of these X-rays are analyzed to provide information about the chemical elements in a qualitative analysis. The measured X-ray intensities can be compared with the reference materials of known stoichiometric ratios to determine the actual material element.

The principle of incident electron beam and scattered characteristic X-ray of each element is shown in Fig. 3.9. There are two major interaction mechanisms under the influence of electrons bombard on the sample: (1) the inelastic interacting with an inner orbital electron to form “electron hole” and the outer orbital electron filling in to generate discrete wavelengths of X-rays, which is characteristic for different element.

(2) When the incident electron beam does not collide with the orbital electrons but being

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decelerated by the Coulombic field associated with sample atoms, the inelastic scattering produces a continuous X-ray spectrum, called Bremsstrahlung. This continuous energy radiation forms the background for the characteristic X-ray lines.

Other interaction processes include the release of secondary electron, backscattered electrons, and the occurrence of cathodoluminescence. Overall, the effects of electrons-elements interaction allow an integrated imaging, qualitative, and quantitative chemical analysis for the samples. Our EPMA measurements were performed in the

Instrumentation Center, National Taiwan University. The sample thickness should be larger than 100 μm to avoid the electron beam penetrate and scattered the holder signal,

especially for layered single crystals. If the material is a semiconductor or an insulator, the sample surface needs to be coated with carbon to assist the scattered electrons to conduct with holder. The total molecular weight percent of quantitative data should be larger than 95% to make sure the signals mostly come from the sample. In 1T-TiSe2

single crystals and polycrystalline, the EPMA quantitative measurements are obtained and calculated more than 10 dots for each sample.

Fig. 3.9: Different types of interaction: (1) ionization and (2) generation of Bremsstrahlung [74].

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3.5 Magnetic property measurement system (MPMS)

The Quantum Design vibrating sample magnetometer (VSM) for the magnetic property measurement system (MPMS) is a fast and sensitive DC magnetometer. The instrument can be used to study the magnetic properties of small experimental samples over a broad range of temperature and magnetic fields. Extremely sensitive magnetic measurements are performed with superconducting pickup coil and superconducting quantum interference device (SQUID). So the MPMS family of instruments are called SQUID magnetometers. The VSM is vibrated at a known frequency and phase-sensitive detection is employed for rapid data collection and spurious signal rejection. The size of the signal produced by a sample is independent on the frequency of vibration, but depends on the magnetic moment of the sample. The MPMS SQUID-VSM utilizes a superconducting magnet with a solenoid of superconducting wire to subject samples to magnetic fields up to 7 Tesla (70000 Oe). The SQUID and superconducting magnet must be cooled with liquid helium. Liquid helium is also used to cool the sample chamber, providing temperature control of samples from 400 to 1.8 K. The instrument of SQUID-VSM is shown in Fig. 3.10.

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Fig. 3.10: QD-SQUID VSM instrument with the helium cryo-cycle compressor in the Center for Condensed Matter Sciences (CCMS), National Taiwan University.

3.6 Physical Property Measurement system (PPMS)

The PPMS provides a flexible, automated workstation that can perform a variety of experiments requiring precise thermal control to execute magnetic, electric-transport, or thermoelectric measurements (the instrument as shown in Fig. 3.11). The principle and technique of transport measurements are introduced below using a Quantum Design (QD) PPMS.

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Resistivity: In a four-wire resistance measurement, current is passed through a

sample via two current leads, and two separate voltage leads measure the potential difference across the sample (Fig. 3.12). In theory, a perfect voltmeter draws no current whatsoever. Therefore, by using the four-wire method, it is possible to know, to a high degree of certainty, both the current and the voltage drop across the sample and thus calculate the resistance with Ohm’s law. The resistivity 𝜌 is calculated by

𝜌 = ^𝑉𝐴_{𝐼 ℓ} (3.6.1)

where V is the measured potential drop across the sample and I is the current through the sample, the voltage lead separation ℓ and the cross-sectional area through which the current passes A.

Hall coefficient: When charge carriers move perpendicular to magnetic field, a force

is exerted on them perpendicular to both the field and the direction of carrier motion F = q𝑣⃑ × 𝐵⃑⃑. (3.6.2)

Therefore, a transverse current is passed through a sample in a longitudinal magnetic field, charge carriers often build up on one edge of the sample and disappear from the other edge, leading to a potential difference across the sample to be named as the Hall potential (see Fig. 3.13). The signal of the Hall potential generally indicates a conductor/semiconductor is n-type or p-type, and the magnitude of the Hall potential is related to the density of charge carriers in the sample. The Hall coefficient Rh describes these two properties is defined by

RH = ^𝐸_𝑗𝐵^𝐻 = ^𝑉_𝐼ℓ𝐵^{𝐻 𝐴} = _𝑛𝑞¹, (3.6.3)

where 𝐸_𝐻 is the Hall field,VH is the Hall potential, j is the current density given by I/A (which is the current divided by the sample cross section), and ℓ is the separation of the transverse voltage leads. It can be shown that RH = (nq)^-1 (in units of Ω-cm/Oe = 10⁸ cm³/coulomb) with n representing the number of charge carriers per unit volume in

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the sample, and q representing the charge of the carriers.

Thermal transport: Thermal conductivity is a measure of the ability of a material

to conduct heat, so measuring this quantity provides information about scattering of heat-carrying phonons and electrons. The Seebeck coefficient describes the thermal diffusion of free charge carriers (electrons or holes), which creates an electric field inside a material when a temperature gradient is sustained. Much like the electrical resistivity, this property is very sensitive to subtle changes in the electronic scattering processes and can be a powerful probe in that regard.

Fig. 3.11: The QD-PPMS instrument with the helium cryo-cycle compressor in the Center for Condensed Matter Sciences (CCMS), National Taiwan University.

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Fig. 3.12: A sketch of the four probe method for resistivity measurement [75].

Fig. 3.13: A sketch of Hall effect measurement [76].

3.7 Scanning Tunneling Microscopy (STM)

STM is a Surface imaging instrument to resolve the atomic scale of surface structure which was invented by G. Binnig and H. Rohrer in 1981 [77]. The resolution can be <

1 Å lateral resolution and < 0.1 Å depth resolution under the ultra-high vacuum Pressure (P ~3 × 10^-11 torr). It provides highly accuracy microscopy technique and convenient to use, but it is limited to observe the surface structure of the materials. In-depth investigate the bulk structure of the materials still need X-ray or neutron diffraction, or transmission electron microscopy techniques. STM still has a potential to figure out the

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characteristics of materials, and the surface/interface structure on two-dimensional materials where become more interesting and important in future.

STM basic components (also shown in Fig 3.14):

1. A sharp tip as a probe (The tip is often made of tungsten or platinum-iridium) 2. A piezoelectric scanning unit, which can control the lateral and vertical motion

of the tip.

3. A coarse positioning unit, which moves the tip to approach the sample and tip-sample separation (~ few Å ) to be the tunneling range.

4. A vibration isolation stage to avoid acoustic and mechanical vibration to interfere the scanning system.

5. A set of electronics, which can detect the small tunneling current (∼10 pA-1 nA), controls the piezo-tube scanner with feedback, and drives the coarse positioning.

In STM principle, a bias voltage is applied between the tip and the sample. When the tip is approached within about 10 Å of the sample, tunneling electrons from the sample through the 10 Å gap to the tip is a filled state. On the other hand, an empty state means electrons tunnel from tip to sample at positive sample bias. The result of tunneling current varies with tip-to-sample spacing, and the sample must be conductors or semiconductors. Thus, STM cannot image insulating materials. The tunneling current is an exponential function of distance. Based on quantum mechanics, the tunneling current (It) is

𝐼_𝑡 = 𝑒^−𝑘𝑑 (3.7.1)

where d is the distance between tip and sample. If the tip-sample separation changes by the order of 1 Å , the tunneling current will change by an order of magnitude.

where d is the distance between tip and sample. If the tip-sample separation changes by the order of 1 Å , the tunneling current will change by an order of magnitude.