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with a superconductivity. It looks like the order parameter has an odd parity, which seems to be consistent with the ferromagnetism. An odd parity state suggests, crudely speaking, that the Cooper pairs are mostly triplet, so that a strong magnetic field does not suppress the spin-triplet pairing.
Note that not all of these have been entirely settled down. In fact, for example a superconductivity of UGe2 is found around 2000 [15, 16], and there are a lot of work even recently.
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Sr2RuO4
(a) (b)
Fig. 5.6. (a) Crystal structure of Sr2RuO4 forac-plane (side view). (b) Crystal structure of Sr2RuO4 for ab-plane (top view).
the optimistic hope that we would get a high-Tc superconductor from its similarity in the crystal structure.
5.4.2 Experimental properties of Sr
2RuO
4We briefly review some important experimental properties of Sr2RuO4. For more de-tails, see Ref. [20].
Normal Phase
In the normal state for Sr2RuO4, below T ∼ 25 K, it appears to behave as a Fermi liquid in all three directions. Thus, the specific heat is a combination of the contributions from the electronic and phonon parts
CV ∼ γT +βT3. (5.5)
For the susceptibility, we observe the Pauli susceptibility:
χ∼const. (5.6)
For both the ab-plane and c-axis, the resistivity is
ρ∼A+BT2, (5.7)
which is a characteristic form for a coherent Bloch wave transport limited by the impu-rity and electron-electron Umklapp scattering: the first term comes from the impuimpu-rity scattering, while the second term from the Umklapp scattering. The value ρab itself is considerably smaller compared with that of the cuprates, ρab ∼1µΩ cm. This means that the samples are very pure systems, and indeed it turns out that one can make a very pure sample of Sr2RuO4.
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Table 5.2. Some of the Fermi liquid parameters for Sr2RuO4. Note that the me is the bare electron mass, while mband is the band mass.
α-band β-band γ-band kF (˚A−1) 0.3 0.6 0.75
m∗/me 3.3 7.0 16.0
m∗/mband 3.0 3.5 5.5
Another important feature of Sr2RuO4 is that it is a strongly anisotropic Fermi liquid.
In fact, ρc/ρab ∼ 103, and this is comparable to those of typical cuprates. This strong anisotropy can be also seen in its band structure. We can reconstruct the Fermi surface from the de Haas-van Alphen kinds of experiments7, or we can calculate the band structure by the local density approximation (LDA). Both of them suggest that the Fermi surface consists of three strongly two-dimensional sheets: one hole-like sheet (calledα-sheet), and two electron-like sheets (β- and γ-sheets). This is what we may expect from the strong anisotropy in the resistivity, which indicates that the hopping matrix element along the c-axis is far smaller than that for the inner-plane hopping. In fact, the deviation from the ideal two-dimensional Fermi surface is fairly small, of the order of ∼1%.
The behavior of electrons for each of the sheets is shown in Table 5.2. The Fermi momentum is of the order of a typical value, and it is not so much an exciting value. The effective mass is several times larger than the band mass, which is estimated by neglecting the electron-electron interaction and just solving an one electron problem. This indicates that electrons are strongly correlated, although it can be well described by the Fermi liquid.
Superconducting phase
People have expected that the transition temperature for Sr2RuO4 would be as high as those of the cuprates, but it turns out that Tc is not so high, although it does undergo a superconducting phase transition. The transition temperature for Sr2RuO4 is only about 1.5 K, and in that sense it was disappointing.
It turns out, however, that the susceptibility appears to be χ ∼ const. in the super-conducting state for all directions, which seems to indicate a triplet ESP state. I think this is fairly firm, and most people believe that the superconducting state of Sr2RuO4 is indeed a triplet ESP state.
7Although there are various ways to measure the band structure, if we are faced with some new materials and want to figure out what their band structures are, the most reliable way is the de Haas-van Alphen kinds of experiments, measuring a quantum oscillation with respect to the external magnetic
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Deviation from perfect cylinder
(a) (b)
Fig. 5.7. (a) Crystal structure of Sr2RuO4. (b) Schematic illustration of the Fermi surface of Sr2RuO4.
With that in mind, let us suppose that the order parameter of Sr2RuO4 is indeed of the ESP form:
F(k;σ1, σ2) = F(k;σ1, σ2) =f(k)(| ↑↑i+| ↓↓i). (5.8) A crucial question arises: what is the orbital wave function of the pairs f(k)? Since it is a spin-triplet state, it has to be an odd parity state. This does appear to be consistent with some Josephson experiments, where it is shown that the order parameter seems to change its sign if we reflect it.
The most interesting question is whether the order parameter is real or complex. In the real case, such as f(k)∼kx, the time-reversal symmetry is not broken. On the other hand, if it is complex, for example f(k) = kx+iky8, it can be broken. In the BCS theory, we know a priori, that the amplitude of order parameter |OP|2 want to be as uniform as possible over the Fermi surface, although this is not always the case in a more general theory9.
It turns out that there are various experiments which are in favor of the time-reversal symmetry broken order parameter. Some of the experiments are as follows:
• Muon spin rotation
In the muon spin rotation experiment, we essentially measure an effective magnetic field where the muon is sitting after it is trapped to somewhere. Generally speaking, in the normal state under a zero external magnetic field, the muon sees indeed no local magnetic field. If, on the other hand, we have a system where the time-reversal symmetry is spontaneously broken, then you would expect to see that the muon spin rotates, and indeed we will see it. What is peculiar for Sr2RuO4 is that the signal of
8This state has an angular momentum along z-axis, and you can easily see that the time-reversal symmetry is broken.
93He is one counterexample for this, where we could not exclude the possibility for the polar phase, which does not breaks the time-reversal symmetry.
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the time-reversal symmetry breaking starts to appear at the superconducting phase transition [21]. This experimental fact suggests that the time-reversal symmetry is spontaneously broken in the superconducting state.
There is one worrying factor there: the direction of the local magnetic field. An obvious option for this is in the direction along the symmetry axis, that is along the c-axis. However, this is not the case, and in fact we have a finite component in the ab-plane. This is what remains to be understood.
• Magnetic field dependence and telegraph noise in Ic of Josephson junc-tions interpreted in terms of switching of domains.
If the time-reversal symmetry is broken as kx +iky, there is also an equal pos-sibility for having kx −iky state. There is an argument based on a macroscopic electromagnetic effect that it is likely to have various domains of them. Experimen-talists interpreted the telegraph noise in Ic of Josephson junctions as the switching between these two states [22].
• Kerr effect in zero magnetic field.
Kerr effect is a measurement of the rotation of the polarization of the light when it passes through the material. If this rotation occurs, that is again a direct evidence of the time-reversal symmetry breaking. An experiment has observed the rotation and has confirmed this time-reversal symmetry breaking [23].
So far, so good. For an “ideal”kx+iky state, the energy gap has no node |∆| ∼ |F| ∼ const., so that the number of quasi-particles for T ≪ Tc should be exponentially small.
This should be seen in the measurement of the specific heat, thermal conductivity and so on. However, unfortunately what we find is a power-law behavior for most of these quantities. Whether this power-law is due to the existence of nodes in the gap is somewhat controversial. In principle, there is a critical test for this: Josephson junction test (see Fig. 5.8), which I shall talk in more detail in Sec. 7.4.
Sr
2RuO
4Al
Fig. 5.8. Schematic illustration of the Josephson experiment.
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