High-pressure transport properties of TiS₂ and TiSe₂

R H Friend†, D Jérome‡ and A D Yoffe†

† Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, UK

‡ Laboratoire de Physique des Solides, Université Paris-Sud, 91405 Orsay, France

Received 13 August 1981

**Abstract.** Measurements of the pressure dependence of the Hall coefficient and conductivity of TiS₂ indicate that TiS₂ is an extrinsic semiconductor. The strong temperature dependence of the conductivity is matched by a strong pressure dependence that is consistent with electron-phonon scattering. In contrast, TiSe₂ is a semimetal, with a strongly pressure dependent p-d band overlap. The transition temperature of the structural phase transition in TiSe₂ falls at -1.2 K kbar⁻¹.

**1. Introduction**

The transport properties of the group IV layer structure dichalcogenides TiS₂ and TiSe₂ have been the subject of recent interest. Both materials exhibit metallic properties with carrier concentrations in the range 10²⁰–10²¹ cm⁻³, whereas band-structure calculations have until recently predicted semiconducting behaviour for both materials, with an energy gap greater than 1 eV between the top of the chalcogen p valence band and the bottom of the titanium d conduction band (Murray and Yoffe 1972, Myron and Freeman 1974, Krusius et al 1975, Isomaki et al 1979). Two models have been advanced to explain the metallic nature of TiS₂ and TiSe₂; the extrinsic semiconductor model in which there is partial filling of the d conduction band in the slightly metal-rich material that is usually grown, with the excess Ti residing between the layers and transferring electrons to the conduction band, and the semimetal model in which slight overlap between the p and d bands results in p-band hole conduction, and partial occupation of the Ti d band. There is now firm experimental evidence to favour the extrinsic semiconductor model for TiS₂ and the semimetal model for TiSe₂. We have previously shown that TiS₂ has a pressure-independent Hall coefficient in contrast to TiSe₂ in which the carrier concentration rises rapidly with pressure as the p-d overlap is increased (Friend et al 1977), and angle-resolved photoemission experiments show directly the overlap of the p band at the zone centre and the d band at the zone edges by a few tenths of an electron volt in TiSe₂, and a small p-d gap for TiS₂ (Traum et al 1978, Chen et al 1980). These photoemission results are in good agreement with the self-consistent LCAO band-structure calculations of Zunger and Freeman (1977, 1978).

The Hall-effect-determined carrier concentration in TiS₂ ranges from 10²⁰ to about 3 × 10²¹ cm⁻³ for crystals grown by vapour transport at temperatures from 600 to 800 °C, with lower carrier concentrations obtained for crystals grown at low temperatures. Logothetis et al (1979, 1980) and Klipstein et al (1981) have recently made detailed
studies of the variation in carrier mobility with temperature and carrier concentration. As emphasised by Thompson (1975), the temperature dependence of the mobility follows a law of the type $\mu \propto T^{-p}$ with $p \sim 2$ for a carrier concentration $n \sim 3 \times 10^{20} \text{ cm}^{-3}$. This temperature dependence is stronger for $n \sim 10^{20} \text{ cm}^{-3}$, with $p \sim 2.3$. Several models have been advanced to explain this strong temperature dependence; Thompson (1975) suggested electron-electron scattering in view of the $T^2$ dependence of the scattering rate that is expected from the two $(kT/E_F)$ factors that determine the allowed final states of the scattered electrons, and several electron-phonon scattering mechanisms have been proposed. These include scattering with the homopolar optic phonon (Wilson and Mahajan 1977, Wilson, 1978), second-order scattering involving low-frequency acoustic modes (Kaveh et al 1981), and acoustic phonon scattering, in which the combination of intrapocket and interpocket scattering has been shown to give a strong temperature dependence below room temperature (Klipstein et al 1981).

TiSe$_2$ undergoes a structural phase transition at a temperature of 200 K (Di Salvo et al 1976). The low temperature superstructure involves transverse atomic displacements with wavevector $q = (\frac{1}{2}, 0, \frac{1}{2})$, and the TA phonon branch with L$_2$ symmetry is seen to soften at the L point of the Brillouin zone boundary (Moncton et al 1978). The doubling of the lattice periodicity in the plane has the effect of coupling the electron pockets at the L points with the hole pocket at $\Gamma$. Furthermore, large changes in transport properties are seen at the transition, with indications from the Hall effect and magnetic susceptibility (Di Salvo et al 1976) for a substantial loss of carriers, to a density in the distorted state a factor of ten lower than the room temperature value of about $3 \times 10^{20} \text{ cm}^{-3}$. This electron-hole coupling has been considered as the driving force for the distortion (Di Salvo et al 1976, Wilson and Mahajan 1977, Wilson 1977, 1978, Woo et al 1976), with a reduction in electronic energy for the reconstructed bands. This mechanism is similar to the models for the Fermi-surface-driven charge density wave-periodic lattice distortions observed in the group V dichalcogenides (Williams et al 1974, 1975, Wilson et al 1974, 1975). However, the number of carriers involved in the distortion in TiSe$_2$ is low so that it is arguable whether electron-hole coupling of this type is sufficient to drive the transition, and in view of the large infrared effective charge in TiSe$_2$, White and Lucovsky (1977) consider that there is an intrinsic softening of the TA phonon branch at the L point, and that the transition can be described as a phonon-driven antiferroelectric distortion.

In this paper we present measurements of conductivity and Hall coefficient at high pressure in the temperature range 4.2–300 K. Some of our results have been previously reported (Friend et al 1978). The high pressure transport measurements on TiS$_2$ favour electron-phonon scattering, and the behaviour of the structural phase transition in TiSe$_2$ under pressure is argued to favour the phonon-driven model for the distortion of White and Lucovsky (1977).

## 2. Experimental

Experiments to follow the in-plane resistivity ($j$ and $E$ perpendicular to $c$) and the Hall coefficient ($j$ and $E$ perpendicular to $c$, $B$ parallel to $c$) as a function of pressure were performed using a variety of hydrostatic pressure techniques (Malfait and Jérôme 1969, Delaplace et al 1976), with either helium gas or isopentane as the high pressure medium. The Hall effect was measured in fields of up to 20 kG provided by a superconducting solenoid around the pressure system.
The crystals of TiSe$_{2}$ used in these experiments were grown by J C Mikkelsen. TiSe$_{2}$ crystals were grown using I$_{2}$ vapour transport at 750°C with excess Se present. These crystals show a resistivity anomaly at low temperatures similar to that reported by Di Salvo *et al* (1976), with $ρ(165 K)/ρ(300 K) \sim 4$. The room temperature resistivity was measured using the Van der Pauw (1958) geometry to be $1.32 \pm 0.13 \times 10^{-3} \Omega$ cm, and the Hall coefficient was measured to be $3.25 \pm 0.3 \times 10^{-2} \text{ cm}^3 \text{ C}^{-1}$. The conditions for the growth of TiS$_{2}$ as close as possible to stoichiometry are discussed by Mikkelson (1977), and the crystals used here showed a resistance ratio $ρ(300 K)/ρ(4.2 K) \sim 15$. The room temperature resistivity was $3.25 \pm 0.1 \times 10^{-3} \Omega$ cm, and the Hall coefficient $-5.5 \pm 0.3 \times 10^{-2} \text{ cm}^3 \text{ C}^{-1}$.

### 3. Results

#### 3.1. TiS$_{2}$

Resistivity against temperature curves for TiS$_{2}$ at ambient and at elevated pressures are shown in figure 1. The data are replotted in the form $log(ρ - ρ_0)$ against T in figure 2,

<figure><img src="image_1.png" /><figcaption>**Figure 1.** Temperature dependence of the resistivity of TiS$_{2}$ at ambient and at elevated pressures. A, 1 bar; B, 10 kbar; C, 20 kbar; D, 30 kbar.</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>**Figure 2.** $log(ρ - ρ_0)$ against $log T$ for TiS$_{2}$ at ambient and at elevated pressures: ▼, 1 bar; ■, 10 kbar; ▲, 20 kbar; ●, 30 kbar.</figcaption></figure>

when $ρ_0$ is the extrapolated resistivity at $T = 0$ for each pressure. The temperature dependence is seen clearly in figure 2, and we note that although a simple power law with a temperature exponent of 2.29 provides an excellent fit to the ambient pressure data, such an agreement is not obtained for the high-pressure resistivity curves, for which there is a stronger temperature dependence at low temperatures. A line of slope 2.46 is shown in figure 2 against the resistivity curve obtained at 30 kbar, to indicate that the average slope for these high-pressure curves is larger than for the ambient pressure data. Fitting the resistivity data to a power law over a more limited temperature range, we find slopes of 2.6 at about 80 K and 2.2 at about 150 K for the data at 30 kbar. We have used the data shown in figures 1 and 2 to determine the temperature dependence
of the pressure coefficient of resistivity d ln ρ/dp, which we define as

$$ \frac{d \ln \rho}{dp} = \sum_i \frac{1}{p_i N} \{ \ln \rho(p_i, T) - \ln \rho(1 \text{ bar}, T) \} \quad (1) $$

where $p_i = 10, 20$ and $30$ kbar and $N = 3$. This is shown in figure 3, and the room temperature value of $-2.5\% \text{ kbar}^{-1}$ is in good agreement with the room temperature results shown previously (Friend et al 1977). Note that at low temperatures d ln ρ/dp

<figure><img src="image_3.png" /><figcaption>Figure 3. -d ln ρ/dp and -d ln (ρ - ρ₀)/dp for TiS₂ as a function of temperature. These quantities as defined in § 3.</figcaption></figure>

does not tend to zero, but reaches a value of $-0.56\% \text{ kbar}^{-1}$, indicating that the residual resistivity, $\rho_0$, is pressure dependent. We may express $\rho_0$ as

$$ \rho_o = \frac{m^*}{ne^2 \tau_i} \quad (2) $$

where $\tau_i$ is the relaxation time for impurity scattering. Since we know from the pressure independence of the Hall coefficient that $n$ is independent of pressure, and do not expect the impurity scattering time to be pressure dependent, we must attribute this pressure dependence to a decrease in effective mass, $m^*$ with pressure.

$$ \frac{d \ln \rho_0}{dp} = \frac{d \ln m^*}{dp}. \quad (3) $$

This could arise from an increase in d bandwidth under pressure, or an increase in p-d hybridisation as the p-d bandgap is reduced from its ambient pressure value of ~0.5 eV, in a similar way to the increase in p-d overlap in TiSe₂ under pressure. We may express the total resistivity as

$$ \rho_{\text{total}} = \rho_T + \rho_0 = \frac{m^*}{ne^2} \left( \frac{1}{\tau_i} + \frac{1}{\tau_T} \right). \quad (4) $$

If we now assume the pressure dependence of $m^*$ as determined from (3), and we define
the pressure dependence of $\rho - \rho_0$ as

$$ \frac{d \ln (\rho - \rho_0)}{dp} = \sum_i \frac{1}{N p_i} \{ \ln[\rho(p_i, T) - \rho(p_i, 0K)] - \ln[\rho(1\text{bar}, T) - \rho(1\text{bar}, 0K)] \} \quad (5) $$

then

$$ \frac{d \ln (\rho - \rho_0)}{dp} = \frac{d \ln m^*}{dp} - \frac{d \ln \tau_T}{dp}. \quad (6) $$

This quantity is also plotted in figure 3. Note that the value of $-d \ln (\rho - \rho_0)/dp$, which defined in this way is an average pressure dependence over the pressure range 0-30 kbar, increases at low temperature. We shall argue in § 4 that this correlates with the stronger temperature dependence of the resistivity seen below about 100 K at high pressure. To obtain the intrinsic pressure dependence of the temperature dependent relaxation time, $\tau_T$ it is necessary to subtract the value of $d \ln m^*/dp$ from $d \ln (\rho - \rho_0)/dp$. At room temperature we obtain a value for $d \ln (\tau_T)/dp$ of 2.1% kbar⁻¹.

## 3.2. TiSe₂

Resistivity against temperature curves for TiSe₂ at ambient and at elevated pressures are shown in figure 4 in which the resistivity scale is logarithmic in order to accommodate the wide range of resistivities found at high pressure. It is clear that the phase transition is suppressed by the application of pressure, and that the associated resistivity anomaly becomes much less prominent. Results for the temperature dependence of the Hall coefficient are shown in figure 5. The ambient pressure results are similar to those reported by Lévy and Froidevaux (1979) and Lévy (1979). Under pressure the sign reversal at the transition does remain, but the carrier concentration remains higher below the transition. This is also evident in the greatly increased low-temperature

<figure><img src="image_4.png" /><figcaption>Figure 4. Resistivity against temperature for TiSe₂ under pressure: A, 1 bar; B, 5 kbar; C, 10 kbar; D, 15 kbar; E, 20 kbar; F, 30 kbar.</figcaption></figure>
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 5.** (a) Hall coefficient for TiSe₂ as a function of temperature at 1 bar and 8 kbar, (b) at 15 kbar and 20 kbar.</figcaption></figure>

conductivity seen in figure 4; between 1 bar and 30 kbar the residual resistivity falls by a factor of 50.

We do not consider that the sign reversal of $R_H$ is a useful means of determining the transition temperature of this second-order transition. The sign reversal at 1 bar occurs at a lower temperature (~180 K) than for samples measured by Di Salvo *et al* (1976) of about 200 K, although this may be explained by the greater degree of compensation between electrons and holes in the more heavily doped samples of Di Salvo *et al*, which have a lower room temperature value of $R_H$. Di Salvo *et al* report that the transition temperature seen in neutron diffraction experiments coincides with the peak value in $d\rho/dT$ at 200 K at ambient pressure, and we have used this criterion to estimate the

<figure><img src="image_7.png" /><figcaption>**Figure 6.** The pressure dependence of the distortion temperature of TiSe₂ determined from the resistivity data in figure 4.</figcaption></figure>
pressure dependence of $T_d$ in figure 6 for data up to 20 kbar; at 30 kbar the transition is evident as a break in slope in $\rho(T)$, much as the PLD-CDW transition in 2H-TaSe₂ (Di Salvo 1977). The initial rate of depression of $T_d$ is -1.2 K kbar⁻¹. This value is in agreement with that reported by Ogasawara et al (1979) who have made pressure measurements up to 5 kbar. There are discrepancies in their transport measurements, however, particularly in the low-temperature behaviour of $R_H$, which they report to be little changed at 5 kbar and 77 K.

# 4. Discussion

## 4.1. Pressure dependence of the carrier mobility in TiS₂

The evidence in favour of electron-phonon scattering as the origin of the strongly temperature-dependent mobility in TiS₂ has recently been detailed by one of the authors (Klipstein et al 1981), based on the comparison of carrier mobility in samples of TiS₂ with differing carrier concentrations. The pressure results presented in this paper are consistent with this view, and it is not possible to explain the relatively high value for the pressure coefficient of resistivity by the electron-electron scattering model of Thompson (1975) since, as seen in the pressure-independent Hall coefficient, there are no significant changes in the nature of the partially occupied conduction band under pressure.

For first-order electron-phonon collisions, we may express the phonon contribution to the resistivity as

$$ \rho = \frac{a}{\theta} \left(\frac{T}{\theta}\right)^n \qquad (7) $$

where $k_B\theta$ is a characteristic phonon energy, and the temperature dependence follows a $T^n$ law. The term $a/\theta$ comes from the electron-phonon matrix element, and the $(T/\theta)^n$ term from the phonon Bose occupation factor (see, e.g. Ziman 1972). We expect the pressure dependence of $\rho$ to arise through an increase in $\theta$ with pressure. Defining the Grüneisen constant for the appropriate phonons, $\gamma = -\partial \ln \theta / \partial \ln V$, we have for the pressure dependence of the resistivity

$$ \frac{d \ln \rho}{d p} = \frac{1}{B} \gamma(n + 1) \qquad (8) $$

when B is the bulk modulus. We expect, therefore, a simple correlation between $d \ln \rho/dp$ and the temperature dependence of the mobility. This is shown clearly in figures 2 and 3 where it is seen that the stronger temperature dependence of the resistivity at low temperatures under pressure is matched by an increased value of $-d \ln (\rho - \rho_0)/dp$ in this temperature range. Such a correlation has been known for some time in the alkali metals, for which the pressure coefficient of resistivity increases in the low temperature regime in which the $T^5$ resistivity law holds (Dugdale and Phillips 1965, Dugdale 1969). The room-temperature value of $d \ln \rho/dp$ of -2.5% kbar⁻¹ for the samples of TiS₂ we have measured which show $p = 2.3$ at ambient pressure is consistent with typical values of $d \ln \rho/dp$ for metallic layer materials which show a linear temperature dependence, such as 2H-NbSe₂ for which Chu et al (1976) report a value of $d \ln \rho/dp = -1\% kbar^{-1}$.

We note that whilst a simple power law for the temperature dependence of the resistivity with temperature exponent 2.3 provides a good fit over the range 20-300 K at
ambient pressure, this behaviour is not found under pressure. Thus this simple power law behaviour is, we consider, accidental in that it is particular to ambient pressure and also to a narrow range of carrier concentrations (10²⁰⁻³ × 10²⁰ cm⁻³) (Klipstein *et al* 1981).

In the model proposed by Klipstein *et al* (1981) for the electron-phonon scattering, the strong temperature dependence in TiS₂ with low carrier concentrations such as the material used in these experiments, arises below about 250 K from the combined effects of electron-longitudinal acoustic phonon scattering both within the electron pockets and between pockets, the latter process requiring the excitation of a zone boundary phonon. Under pressure, the minimum phonon energy for this intervalley scattering to take place will be increased, and this will lead to an increase in temperature dependence for temperatures less than this phonon energy, as is seen by comparison of curves A and B in figure 8 of Klipstein *et al* (1981). The pressure results presented here cannot, however, be used to distinguish between different phonon scattering processes since these will all follow a pressure dependence of the form given in equation (8).

## 4.2. Pressure dependence of the structural phase transition in TiSe₂

The form of the resistivity anomaly associated with the structural phase transition in TiSe₂ changes substantially with pressure. It is clear from both the behaviour of the resistivity and Hall coefficient that an increasingly smaller fraction of the carriers present a high temperatures are removed by the distortion with increasing pressure; indeed the resistivity curve at 30 kbar is very similar to that observed for group V materials 2H-TaS₂ and 2H-TaSe₂ at their PLD-CDW distortion temperatures of 75 K and 122 K respectively (Di Salvo 1977). From the fall in Pauli susceptibility below these distortions (Di Salvo 1977) it is possible to estimate that less than 30% of the conduction electrons are removed by the distortions in these materials. The gradual removal of the peak in resistivity below the transition in TiSe₂ (165 K at ambient pressure) as the p-d band overlap increases with pressure is consistent with the model for the resistivity anomaly suggested by Di Salvo *et al* (1976) in which the resistivity peak at ambient pressure is the result of the competition between the fall in the carrier concentration as the distortion amplitude builds up, and the increase in carrier mobility with falling temperature as the various scattering processes are frozen out.

The rapid increase in p-d band overlap with pressure will bring about a large change in the shape of the Fermi surface, and conduction band minima such as M and Γ, besides that at L, may well fall below the Fermi level. The relatively slight change in transition temperature with pressure does therefore indicate that the transition is not critically dependent on the detailed shape of the Fermi surface, as is also the case for the PLD-CDW transitions in the group V materials (Doran 1980). It is of importance to confirm that under pressure the superlattice in TiSe₂ retains the commensurate 2x superstructure seen at ambient pressure; if there is a transition to a different superstructure this would argue for a Fermi-surface-driven instability in TiSe₂. This can only be resolved by high-pressure diffraction experiments, but from the continued second-order character of the transition and the absence of any discontinuity in the pressure dependence of the transition we suspect that the superstructure will remain unchanged under pressure.

It is not easy to determine unambiguously the rate at which the p-d band overlap is increased under pressure from our transport measurements. We need to know both the ambient pressure carrier concentration, *n* in TiSe₂ and its fractional rate of increase under pressure, (1/*n*)(dn/dp). Values of *n* between 10²⁰ and 10²¹ cm⁻³ have been sug-
gested by Di Salvo *et al* (1976) and Di Salvo and Waszczak (1978), and a value in this range has been estimated from photoemission experiments (Traum *et al* 1978, Chen *et al* 1980). Samples of TiSe₂ (and V doped TiSe₂) that we have measured (Sarma *et al* 1981) show a higher p-type mobility at room temperature of about 22 cm² V⁻¹ s⁻¹ than the values reported by Di Salvo *et al*. Since this mobility is considerably higher than the expected d-band mobilities, seen for example in the n-type mobility of ~5 cm² V⁻¹ s⁻¹ for hydrazine intercalated TiSe₂ (Sarma *et al* 1981), we consider that hole transport dominates the room-temperature ambient pressure conduction in TiSe₂ and that the concentration of holes present can be reliably determined from the Hall coefficient, $n_h \sim 1/R_{He}$. This gives a value of $2 \times 10^{20} \text{ cm}^{-3}$ for our samples. We consider that the electron and hole concentrations are similar. The concentration of charged impurities in these samples of TiSe₂ is low since we obtain a high value for the resistance ratio at 30 kbar, of $\rho(295 \text{ K})/\rho(4.2 \text{ K}) = 30$. This may be compared with a ratio for TiS₂ of 15 for a sample with an impurity derived carrier concentration of $1 \times 10^{20} \text{ cm}^{-3}$ (Klipstein *et al* 1981).

In estimating $(1/n)(dn/dp)$ from the transport data it is necessary to know the pressure dependence of the carrier relaxation time and carrier effective mass. The relaxation time will increase with pressure (see discussion for TiS₂), and the indications from the pressure dependence of the conductivity and Hall coefficient (Friend *et al* 1977) is that the carrier effective mass increases under pressure. These two effects tend to cancel each other, and we therefore ignore changes in carrier mobility under pressure, and take the pressure dependence of the conductivity to estimate a value for $(1/n)(dn/dp)$ of about 7% kbar⁻¹. With $n = 2 \times 10^{20} \text{ cm}^{-3}$ this gives $dn/dp = 1.4 \times 10^{19} \text{ cm}^{-3} \text{ kbar}^{-1}$, and a rate of increase in the p-d band overlap of approximately 10 meV kbar⁻¹.

We can compare the effect of pressure on the phase transition in terms of the rate of increase in carrier concentration

$$ \frac{dT_d}{dp} / \frac{dn}{dp} = -8.6 \times 10^{-20} \text{ cm}^3 \text{ K} $$

with effects on the transition of an increase in band filling as a result of interlayer excess Ti or substitutional V and Ta doping. Di Salvo and Waszczak (1978) were not able to find a universal value of $dT_d/dn$ for these different systems, and noticed a weaker effect on the transition when carriers were introduced through the presence of interlayer Ti. They estimate a value of $dT_d/dx$ in Ti₁+xSe₂ of $-35 \pm 5 \text{ K (at. \%)}^{-1}$ which, with the assumption that each excess Ti donates four electrons to the d band, gives $dT_d/dn = -5.8 \times 10^{-20} \text{ cm}^3 \text{ K}$. Thus in spite of the effect of disorder in non-stoichiometric TiSe₂ which must tend to depress the transition, for a similar increase in carrier concentration by the application of pressure, the transition temperature is depressed more rapidly. This indicates the influence of pressure on the transition is not primarily through the change in size and shape of the Fermi surface, but must be through the stiffening of lattice force constants.

We have shown that the pressure dependence of the transition is stronger than that expected from Fermi-surface changes alone, and we therefore favour the phonon-driven model for the structural distortion in TiSe₂ proposed by White and Lucovsky (1977). A similar conclusion has been reached from the independence of the distortion temperature in TiSe₂ to an increase in band filling as the result of hydrazine intercalation (Sarma *et al* 1981). Further progress in the interpretation of the pressure results will require a better understanding of the lattice properties of TiSe₂, including the high infrared effective charge and possible resonance bonding.
Acknowledgments.

We thank Dr J C Mikkelsen for the crystals of TiS₂ and TiSe₂ used in this work, and we
are grateful to Dr W Y Liang and Mr P C Klipstein for useful discussions.

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