Resistivity temperature exponent

So far we have good evidence we are seeing a clean Lifshitz transition in a multiband system that we can study with resistivity, arguably for the first time, but certainly for the first time at this level of purity. Because we have such a clean system, it is important to

Fig. 3.36: Resistivity temperature exponent. The resistivity exponent,

α, for sample 3 plotted against tem-

perature and strain. ρ0 was first

extracted from fits of the typeρ= ρ0+ATα _{and thenαis calculated} as a function of temperature by

dln(ρ₋ρ0)/dlnT. The figure is cut

off below 4 K, due to percolating su- perconducting paths that can affect the resistivity strongly.

−0.8

−0.6

−0.4

−0.2

0.0

0

10

20

30

ε

xx

(%)

Temp

erature

(K)

1.5

1.6

1.7

1.8

1.9

2.0

3.0

d

ln

(

ρ

−

ρ

0

)

/d

ln

T

82 The Physics of Sr₂RuO₄ Approaching a Van Hove Singularity

try and see how much physical significance we can give to the changes in resistivity. When the resistivity starts to deviate from the expected Fermi liquidT2temperature dependence it is usual to

inspect the new temperature exponent to try and help interpret the results. A fit of the formρ=a+bTc can be made to data where

a, b andc are fitting parameters and a log derivative plot of the

resistivity minus the residual resistivity can be made to inspect the change in temperature exponent. This plot can be seen in figure 3.36. Here the change in temperature exponent is presented more straightforwardly; Fermi liquid like T2 behaviour is observed at

both strains below and above the peak inρxxbut a lower power is

observed in the vicinity of the resistivity peak, reducing to∼1.5 at

the lowest point.

Qualitatively similar behaviour is observed when the Lifshitz transition is induced by either La doping or epitaxial biaxial strain in thin films. However the power observed in both these experi- ments decreases to ∼1.4, lower than that observed at first sight

here. This difference might be intrinsic. The Van Hove singularity is reached simultaneously in both the xand y directions of the

Brillouin zone for the other two techniques whereas uniaxial pres- sure only approaches the Van Hove singularity along one direction. Significantly higher levels of disorder are also present in both the La doped system and the MBE films. To be vigilant however, it is worthwhile examining the quality of the fit used for extracting the exponent presented in figure 3.36 before addressing alternative interpretations. For reasons given in appendix B.1 regarding the possible effects of strain inhomogeneity and comparing with alter- native trial fitting functions, at present we can only put an error of 0.1 on the exponent of 1.5 close to the suspected Lifshitz transition.

It is remarkable that for this multiband system in which only one out of its three Fermi surfaces passes through the Van Hove singularity, which itself is only one critical point on the surface at (0,±π/a), such large changes in the temperature dependent

resistivity occur. This shows that the ‘hot’ regions of the Fermi surface are not just shorted out by the unaffected sections and large regions of the Fermi surface must be affected by the approach to the Van Hove singularity.

Qualitatively a T1.5 power law resistivity is also observed in

polycrystalline spin-glass systems once the disorder has been frozen in [158]. The single crystals of Sr2RuO4 used here are not expected to exhibit this sort of behaviour, we note the particular low residual resistivity, ρ0∼0.1µΩcm, and that at zero pressure in comparably

clean samples large quantum oscillation signals have been seen. A temperature dependence of the resistivityρ(T)₋ρ0∝Tαwithα < 2 is evidence for anomalous quasiparticle scattering not captured by the conventional quasiparticle interactions of Fermi liquid theory.

3.4Results and discussions 83

One such example is when a long range interactions increase the cross section for quasiparticle scattering. When I introduced Fermi liquid theory back in section 1.2 I used Fermi’s golden rule to show that the lifetime of the quasiparticles goes like their energy squared so they are stable and well defined. In deriving this I made the assumption that the scattering matrix element is constant. This, however, is not always the case. For example near to a second- order phase transition fluctuations slow down as well as becoming increasingly long range, enhancing the scattering cross section. At a quantum critical point these fluctuations can grow without limit and the form of the scattering matrix element becomes important for determining the exact quasiparticle decay processes.

Quasiparticle-quasiparticle interactions by themselves are un- able to relax the total momentum of the system which is required for a finite resistivity, and in normal Fermi liquid theory it is the umklapp processes that provide this. In the presence of critical fluctuations the scattering rate of fermions near the Fermi surface can still be calculated but the temperature dependence of the scat- tering rate does not necessarily straightforwardly translate to a temperature dependence of the resistivity [159]. In some studies that specifically concentrated on the possible mechanisms for relax- ing the momentum, they found that the temperature exponents can be quite different for the momentum relaxing processes [160,161]. Theory still remains unsettled on how best to account for the quantum-critical fluctuations on the temperature dependent resis- tivity, so instead this leaves us only to make empirical comparisons here.

Many intermetallic heavy fermion compounds host magnetic states at low temperatures and are also susceptible to pressure tuning. The Curie temperature of itinerant-electron ferromagnets MnSi, ZrZn₂and Ni₃Al can be suppressed to absolute zero using hydrostatic pressure and near to the critical pressure power laws in the resistivity lower than 2 are observed, ranging from 1.5 to 1.7 depending on the purity and the material [162–166]. Some heavy fermion antiferromagnets also show similar behaviour. The Néel temperature of both CePd2Si2 and CeIn3 can be driven to absolute zero using hydrostatic pressure whereupon non-Fermi liquid behaviour is observed in the vicinity of the QCP [167,163, 168]. CePd₂Si₂shows an anomalously low power amongst of these materials with a temperature exponent of 1.2_±0.1 for over nearly

two decades in temperature.

In NbFe₂, a material reported to host a low temperature spin density wave near stoichiometry, a QCP can be reached this time not by using hydrostatic pressure but rather through varying its composition away from stoichiometry, suppressing the SDW order before giving rise to a ferromagnetic phase [169]. At the slightly Nb-

84 The Physics of Sr₂RuO₄ Approaching a Van Hove Singularity

rich FM-AFM QCP, aT1.5_{power law dependence of the resistivity}

on temperature was also observed [170].

In Sr2RuO4 the quantum criticality evidenced by the lowered resistivity exponent is suspected to coincide with the strain induced Lifshitz transition and hence the additional scattering processes resulting in the breakdown of Fermi liquid theory may well have their origin in the proximity of the Van Hove singularity to the Fermi level. Band structure calculations of NbFe2have also highlighted this possibility as an explanation for the critical behaviour seen at the magnetic QCP in NbFe2. Neal et al. [171] identified a critical point in the band structure and suggested that the underlying origin of criticality at the magnetic QCP may also be a result of the critical fluctuations associated with a vanishing quasiparticle velocity.

Fig. 3.37: Magnetoresistance. A Transverse magnetoresistance field sweeps at strains below the peak in resistivity for sample 3.BThe same measurements at strains above the peak in resistivity. 0.00 % εxx: −0.10 % −0.20 % −0.30 % −0.40 % −0.49 %

A

Hkc, 5 K

∆

ρ

xx

(

B

)

/ρ

xx

(0)

−0.49 % εxx: −0.56 %−0.60 % −0.66 % −0.69 % −0.73 % −0.78 % −0.82 %−0.92 % −0.94 % −1.01 %

B

H_kc, 5 K

(µ

0

H)

2

(T

2

)

∆

ρ

xx

(

B

)

/ρ

xx

(0)

0

10

20

30

0.0

0.2

0.4

0.6

0.8

0.0

0.2

0.4

0.6

0.8

3.4Results and discussions 85

In document Uniaxial stress technique and investigations into correlated electron systems (Page 92-96)