Results

To show the Fermi-liquid properties of a metal, the resistivity is usually fitted or shown against T² [186]. Another way of observing this regime is to use the resistivity derivative in Figure 4.10.

The Sr₂RuO₄ resistivity was never reported in such a form, which unveils a very interesting trend. The low temperature derivative is linear with zero intercept which means that indeed below T_FL ρ₁(T) = a₁T²+c₁. At the same time, the resistivity at high temperature has a quadratic form, but with an additional linear componentρ₂(T) =a₂T²+b₂T+c₂; the fitted

(a)

Figure 4.10: ab-plane andc-axis resistivity of Sr₂RuO₄, sample Tc134. (a)The resistivity is fitted using a polynomial equationρ(T) =aT²+bT+cboth at low (1.6 K to 8 K) and high temperature (60 K to 220 K). (b)c-axis transport and derivative. (c)In-plane resistivity derivative is showing the two fitted models. T_FL=25.8 K is defined as the crossing temperature of the two models and shown with a star, reported as a thick point in the upper graph. The double triangle model is shown as a black thick line(d)with corresponding Fermi-liquid temperature.

parameters are shown in Table 4.3 and the models as red dashed lines in Figure 4.10 (a-d).

The crossover is defined by the temperature at which the two models cross and correspond to T_FL at 25.8 K. The transition is rather broad and occurs in a range of about 15 K around T_FL, which explain why theµexponent of Figure 4.8 (a) gets constant at about 10 K.

In section 1.4.4.2 it was shown that Equation 1.138 produces a T²resistivity up toω_c/2πk_B; above, when the susceptibility (bosonic spectral function) remains zero, the regime is linear.

Here, in order to reproduce the exact behavior of theab-plane resistivity, two superimposed linear susceptibilities were used. The model has five parameters: the slope of the two triangles, their respective cutoff, and the constant impurity scattering rate. The data of Tyleret al. were used from 300 K to 1300 K without any scale adjustment[242]. The resulting fit is shown as a black line in Figure 4.10 (c) and Figure 4.12 reproduces exactly the correct temperature behavior from 1.8 K to 1300 K. The resulting fitted susceptibility is shown in Figure 4.11.

The first triangle is extremely narrow, its cutoff energy is 13.8 meV which corresponds to T_FL ω_c/2πk_B=25 K. The second cutoff has is at 500(50)meV, or about 900(100)K. The fit of the high-temperature data is shown in Figure 4.12. The slope of these susceptibilities is proportional

0 0.1 0.2 0.3 0.4 0.5 Photon energy (eV) 0

0.1 0.2 0.3 0.4

α2 χ2(ω)

Figure 4.11: Inplane susceptibility of Sr₂RuO₄

to the density of states at the Fermi energy, which indicates that the second triangle has a smaller impact on the total density of states and the low-energy electronic properties.

One possible scenario of the very low energy cutoff and steep slope of the first triangle is the presence of a Van Hove singularity close to the Fermi energy. Predicted both in cuprates and in the early ruthenates papers[243, 244]this scenario was claimed to change the scattering rate properties fromT² to a T transport at about one-quarter of the energy of the singularity. At that time, the Van Hove singularity of thed_xy orbital was predicted to be at about 20(2)meV above the Fermi energy which was somewhat corresponding to the apparition of linear regime at 58 K in the early samples[206]. Nowadays, recent calculations Figure 4.2 showed that the Van Hove singularity has a strong impact of the resistivity regime, allowing to move from a T² behavior at low temperature to a linear dependence at a temperature above the singularity energy[245]. Additionally, further refinement of the Van Hove singularity energy[246, 247] leads to reduction of this value down to about 10 meV due multi-bands effects (narrowing of the d_xz and d_yz orbitals and transfer of spectral weight to the d_xy orbital). This energy is very close to the cutoff of the first triangle, which is a good indication that below about 25 K, the quasi-2Dγband dominates the electronic properties and displays Fermi-liquid behavior.

This picture is fully consistent with[238]. From another point of view, the susceptibility is a convolution of the occupied and unoccupied states in the density of states. In Figure 4.13, the DOS is almost constant below the Fermi energy which in principle should create a much broader transition and thus, if taken without additional assumptions, contradicts the former scenario.

Another scenario would directly explain the data by coupling to 2D bosonic modes such as spin fluctuations. They have an energy scales either smaller or larger with T_SF=5 meV, or the Debye energy T_D=35 meV [236]. However, based on their dispersion relation in 2D, a linear susceptibility can be reproduced. Assuming that the 2D-modes happen at long wavelength

(a)

Figure 4.12: High-temperature fit of the Sr₂RuO₄ in-plane resistivity using the double triangle model. (a)Resistivity and its derivative(b). The data are reproduced from[242]

(acoustic dispersion)vħhk=ω_k, the density of states can be obtained by summing all thei sates N(ω) = 1

V X

i

δ(ω−ω_k) (4.1)

withV, the volume. Taking the continuous limit ink-space N(ω) = 1

If the electron-mode coupling is independent ofk[248], the susceptibility is

α²χ₂(ω) =α²Cω, (4.3)

where C = _2π¹ _(ħhv)¹². In such a case, the 2D antiferromagnetic fluctuations (paramagnons) are the reminiscent of the physics of³He, which would be an interesting scenario[249].

It is striking to see that the transport could so easily be described at 1000 K. However, this model is a simplification of the physics of Sr₂RuO₄ which is a multi-band system. It is totally believable that like for the cuprates (section 3.5.5.3), the susceptibility may be temperature dependent which would imply smaller cutoff energies. Nevertheless, it was shown that the main structures of the susceptibility persist, so that this analysis would still indicate that the first Fermi-liquid regime is hiding a second regime above.

−2 −1.5 −1 −0.5 0 0.5 1

" (eV) 0

0.5 1

1.5 Φ(")/Φ(0) N(")/N(0)

Figure 4.13: ab-plane transport function Φ(") and DOS N(") of Sr₂RuO₄. These functions are normalized to their Fermi level values. Φ(0) =ε₀ω²_p=0.38×10²¹Ω⁻¹ms⁻¹ (ħhω_p=4.3 eV), in reasonable agreement with previous results[250, 251].