**8.3. UGe**<sup>2</sup>

*m*,

20 Will-be-set-by-IN-TECH

been observed. This can be considered as an indication that *Ts* is very small and does not produce a measurable effect. So the generic superconducting temperature will be estimated on the basis of the following arguments. For *Tf*(*P*) > *Ts* we must have *Ts*(*P*) = 0 at *P* ≥ *Pc*, where *Tf*(*P*) ≤ 0, and for 0 ≤ *P* ≤ *P*0, *Ts* < *TC*. Therefore for materials where *TC* is too small

As far as the shape of FM-FS transition line is well described by Eq. (17), we will make use of additional data from available experimental phase diagrams for ferroelectric superconductors. For example, in ZrZn2 these are the observed values of *TFS*(0) and the slope *ρ*<sup>0</sup> ≡ [*∂TFS*(*P*)/*∂P*]<sup>0</sup> = (*Tf* 0/*P*0)*ρ*˜0 at *P* = 0; see Eq. (17). For UGe2, where a maximum (*T*˜*m*) is observed on the phase-transition line, we can use the experimental values of *Tm*, *Pm*, and *P*0*c*. The interaction parameters *γ*˜ and *γ*˜1 are derived using Eq. (17), and the expressions for *T*˜

*<sup>m</sup>*, and *ρ*˜0, see Table 1. The parameter *κ* is chosen by fitting the expression for the critical-end

Experiments for ZrZn2 [13] gives the following values: *Tf* <sup>0</sup> = 28.5 K, *TFS*(0) = 0.29 K, *P*<sup>0</sup> ∼ *Pc* = 21 kbar. The curve *TF*(*P*) ∼ *Tf*(*P*) is almost a straight line, which directly indicates that *n* = 1 is adequate in this case for the description of the *P*-dependence. The slope for *TFS*(*P*) at *P* = 0 is estimated from the condition that its magnitude should not exceed *Tf* 0/*Pc* ≈ 0.014 as we have assumed that is straight one, so as a result we have −0.014 < *ρ* ≤ 0. This ignores the presence of a maximum. The available experimental data for ZrZn2 do not give clear indication whether a maximum at (*Tm*, *Pm*) exists. If such a maximum were at *P* = 0 we would have *ρ*<sup>0</sup> = 0, whereas a maximum with *Tm* ∼ *TFS*(0) and *Pm* � *P*<sup>0</sup> provides us with an estimated range 0 ≤ *ρ*<sup>0</sup> < 0.005. The choice *ρ*<sup>0</sup> = 0 gives *γ*˜ ≈ 0.02 and *γ*˜1 ≈ 0.01, but similar values hold for any |*ρ*0| ≤ 0.003. The multicritical points A and C cannot be distinguished experimentally. Since the experimental accuracy [13] is less than ∼ 25 mK in the high-*P* domain (*P* ∼ 20 − 21 kbar), we suppose that *TC* ∼ 10 mK, which corresponds to *κ* ∼ 10. We employed these parameters to calculate the *T* − *P* diagram using *ρ*<sup>0</sup> = 0 and 0.003. The differences obtained in these two cases are negligible, with both phase diagrams being in

Phase diagram of ZrZn2 calculated directly from the free energy (7) for *n* = 1, the above mentioned values of *Ts*, *P*0, *Tf* 0, *κ*, and values of *γ*˜ ≈ 0.2 and *γ*˜1 ≈ 0.1 which ensure *ρ*<sup>0</sup> ≈ 0 is shown in Fig. 2. Note, that the experimental phase diagram [13] of ZrZn2 looks almost exactly as the diagram in Fig. 2, which has been calculated directly from the model (7) without any approximations and simplifying assumptions. The phase diagram in Fig. 2 has the following coordinates of characteristic points: *PA* ∼ *Pc* = 21.42 kbar, *PB* = 20.79 kbar, *PC* = 20.98 kbar,

The low-*T* region is seen in more detail in Fig. 3, where the A, B, C points are shown and the order of the FM-FS phase transition changes from second to first order around the critical end-point C. The *TFS*(*P*) curve, shown by the dotted line in Fig. 3, has a maximum *Tm* = 0.290 K at *P* = 0.18 kbar, which is slightly above *TFS*(0) = 0.285 K. The straight solid line BC in Fig. 3 shows the first order FM-FS phase transition which occurs for *PB* < *P* < *PC*. The solid AC line shows the first order N-FS phase transition and the dashed line stands for the N-FM

*TA* = *TF*(*Pc*) = *TFS*(*Pc*) = 0 K, *TB* = 0.0495 K, *TC* = 0.0259 K, and *TFS*(0) = 0.285 K.

to be observed experimentally, *Ts* can be ignored.

excellent agreement with experiment.

phase transition of second order.

*P*˜

point *TC*.

**8.2. ZrZn**<sup>2</sup>

The experimental data for UGe2 indicate *Tf* <sup>0</sup> = 52 K, *Pc* = 1.6 GPa (≡ 16 kbar), *Tm* = 0.75 K, *Pm* ≈ 1.15 GPa, and *P*0*<sup>c</sup>* ≈ 1.05 GPa [2–5]. Using again the variant *n* = 1 for *Tf*(*P*) and the above values for *Tm* and *P*0*<sup>c</sup>* we obtain *γ*˜ ≈ 0.0984 and *γ*˜1 ≈ 0.1678. The temperature *TC* ∼ 0.1 K corresponds to *κ* ∼ 4.

Using these initial parameters, together with *Ts* = 0, leads to the *T* − *P* diagram of UGE2 shown in Fig. 5. We obtain *TA* = 0 K, *PA* = 1.723 GPa, *TB* = 0.481 K, *PB* = 1.563 GPa, *TC* = 0.301 K, and *PC* = 1.591 GPa. Figs. 6 and 7 show the low-temperature and the high-pressure parts of this phase diagram, respectively. There is agreement with the main experimental findings, although *Pm* corresponding to the maximum (found at ∼ 1.44 GPa in Fig. 5) is about 0.3 GPa higher than suggested experimentally [4, 5]. If the experimental plots are accurate in this respect, this difference may be attributable to the so-called (*Tx*) meta-magnetic phase transition in UGe2, which is related to an abrupt change of the magnetization in the vicinity of *Pm*. Thus, one may suppose that the meta-magnetic effects, which are outside the scope of our current model, significantly affect the shape of the *TFS*(*P*) curve by lowering *Pm* (along with *PB* and *PC*). It is possible to achieve a lower *Pm* value (while leaving *Tm* unchanged), but this has the undesirable effect of modifying *Pc*<sup>0</sup> to a value that disagrees with experiment. In SFT (*n* = 2) the multi-critical points are located at slightly higher *P* (by about 0.01 GPa), as for ZrZn2. Therefore, the results from the SFT theory are slightly worse than the results produced by the usual linear approximation (*n* = 1) for the parameter *t*.

framework of the phenomenological theory (6, this *T* − *P* phase diagram can be explained after a modification on the *Tf*(*P*)-dependence is made, and by introducing a convenient nontrivial pressure dependence of the interaction parameter *γ*. Such modifications of the present theory are possible and follow from important physical requirements related with the behavior of the *f*-band electrons in URhGe. Unlike UGe2, where the pressure increases the hybridization of the 5 *f* electrons with band states lading to a suppression of the spontaneous magnetic moment *M*, in URhGe this effects is followed by a stronger effect of enhancement of the exchange coupling due to the same hybridization, and this effect leads to the slow but stable linear increase in the function *TF*(*P*)[8]. These effects should be taken into account in the modeling the pressure dependence of the parameters of the theory (7) when applied to

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 437

Another ambient pressure FS phase has been observed in experiments with UCoGe [9]. Here the experimentally derived slopes of the functions *TF*(*P*) and *TFS*(*P*) at relatively small pressures are opposite compared to those for URhGe and, hence, the *T* − *P* phase diagram of this compound can be treated within the present theoretical scheme without substantial

Like in UGe2, the FS phase in UIr [12] is embedded in the high-pressure/low-temperature part of the ferromagnetic phase domain near the critical pressure *Pc* which means that UIr is certainly a U-type compound. In UGe2 there is one metamagnetic phase transition between two ferromagnetic phases (FM1 and FM2), in UIr there are three ferromagnetic phases and the FS phase is located in the low-*T*/high-*P* domain of the third of them - the phase FM3. There are two metamagnetic-like phase transitions: FM1-FM2 transition which is followed by a drastic decrease of the spontaneous magnetization when the the lower-pressure phase FM1 transforms to FM2, and a peak of the ac susceptibility but lack of observable jump of the magnetization at the second (higher pressure) "metamagnetic" phase transition from FM2 to FM3. Unlike the picture for UGe2, in UIr both transitions, FM1-FM2 and FM2-FM3 are far from the maximum *Tm*(*Pm*) so in this case one can hardly speculate that the *max* is produced by the nearby jump of magnetization. UIr seems to be a U-type spin-triplet ferromagnetic

Finally, even in its simplified form, this theory has been shown to be capable of accounting for a wide variety of experimental behavior. A natural extension to the theory is to add a *M*<sup>6</sup> term which provides a formalism to investigate possible metamagnetic phase transitions [45] and extend some first order phase transition lines. Another modification of this theory, with regard to applications to other compounds, is to include a *P* dependence for some of the other GL parameters. The fluctuation and quantum correlation effects can be considered by the respective field-theoretical action of the system, where the order parameters *ψ* and *M* are not uniform but rather space and time dependent. The vortex (spatially non-uniform) phase due to the spontaneous magnetization *M* is another phenomenon which can be investigated by a generalization of the theory by considering nonuniform order parameter fields *ψ* and *M* (see, e.g., Ref. [28]). Note that such theoretical treatments are quite complex and require a number of approximations. As already noted in this paper the magnetic fluctuations stimulate first

order phase transitions for both finite and zero phase-transition temperatures.

URhGe.

modifications.

superconductor.

**9. Final remarks**

## **8.4. Two types of ferromagnetic superconductors with spin-triplet electron pairing**

The estimates for UGe2 imply *γ*1*κ* ≈ 1.9, so the condition for *TFS*(*P*) to have a maximum found from Eq. (17) is satisfied. As we discussed for ZrZn2, the location of this maximum can be hard to fix accurately in experiments. However, *Pc*<sup>0</sup> can be more easily distinguished, as in the UGe2 case. Then we have a well-established quantum (zero-temperature) phase transition of second order, i.e., a quantum critical point at some critical pressure *P*0*<sup>c</sup>* ≥ 0. As shown in Sec. 6, under special conditions the quantum critical points could be two: at the lower critical pressure *P*0*<sup>c</sup>* < *Pm* and the upper critical pressure *P*� <sup>0</sup>*<sup>c</sup>* < *Pm*. This type of behavior in systems with *Ts* = 0 (as UGe2) occurs when the criterion (23) is satisfied. Such systems (which we label as U-type) are essentially different from those such as ZrZn2 where *γ*<sup>1</sup> < *γ* and hence *TFS*(0) > 0. In this latter case (Zr-type compounds) a maximum *Tm* > 0 may sometimes occur, as discussed earlier. We note that the ratio *γ*/*γ*<sup>1</sup> reflects a balance effect between the two *ψ*-*M* interactions. When the trigger interaction (typified by *γ*) prevails, the Zr-type behavior is found where superconductivity exists at *P* = 0. The same ratio can be expressed as *γ*0/*δ*0*M*0, which emphasizes that the ground state value of the magnetization at *P* = 0 is also relevant. Alternatively, one may refer to these two basic types of spin-triplet ferromagnetic superconductors as "type I" (for example, for the "Zr-type compounds), and "type II" – for the U-type compounds.

As we see from this classification, the two types of spin-triplet ferromagnetic superconductors have quite different phase diagram topologies although some fragments have common features. The same classification can include systems with *Ts* �= 0 but in this case one should use the more general criterion (29).

### **8.5. Other compounds**

In URhGe, *Tf*(0) ∼ 9.5 K and *TFS*(0) = 0.25 K and, therefore, as in ZrZn2, here the spin-triplet superconductivity appears at ambient pressure deeply in the ferromagnetic phase domain [6–8]. Although some similar structural and magnetic features are found in UGe2 the results in Ref. [8] of measurements under high pressure show that, unlike the behavior of ZrZn2 and UGe2, the ferromagnetic phase transition temperature *TF*(*P*) ∼ *Tf*(*P*) has a slow linear increase up to 140 kbar without any experimental indications that the N-FM transition line may change its behavior at higher pressures and show a negative slope in direction of low temperature up to a quantum critical point *TF* = 0 at some critical pressure *Pc*. Such a behavior of the generic ferromagnetic phase transition temperature cannot be explained by our initial assumption for the function *Tf*(*P*) which was intended to explain phase diagrams where the ferromagnetic order is depressed by the pressure and vanishes at *T* = 0 at some critical pressure *Pc*. The *TFS*(*P*) line of URhGe shows a clear monotonic negative slope to *T* = 0 at pressures above 15 kbar and the extrapolation [8] of the experimental curve *TFS*(*P*) tends a quantum critical point *TFS*(*P*� *oc*) = 0 at *P*0*<sup>c</sup>* ∼ 25 − 30 kbar. Within the framework of the phenomenological theory (6, this *T* − *P* phase diagram can be explained after a modification on the *Tf*(*P*)-dependence is made, and by introducing a convenient nontrivial pressure dependence of the interaction parameter *γ*. Such modifications of the present theory are possible and follow from important physical requirements related with the behavior of the *f*-band electrons in URhGe. Unlike UGe2, where the pressure increases the hybridization of the 5 *f* electrons with band states lading to a suppression of the spontaneous magnetic moment *M*, in URhGe this effects is followed by a stronger effect of enhancement of the exchange coupling due to the same hybridization, and this effect leads to the slow but stable linear increase in the function *TF*(*P*)[8]. These effects should be taken into account in the modeling the pressure dependence of the parameters of the theory (7) when applied to URhGe.

Another ambient pressure FS phase has been observed in experiments with UCoGe [9]. Here the experimentally derived slopes of the functions *TF*(*P*) and *TFS*(*P*) at relatively small pressures are opposite compared to those for URhGe and, hence, the *T* − *P* phase diagram of this compound can be treated within the present theoretical scheme without substantial modifications.

Like in UGe2, the FS phase in UIr [12] is embedded in the high-pressure/low-temperature part of the ferromagnetic phase domain near the critical pressure *Pc* which means that UIr is certainly a U-type compound. In UGe2 there is one metamagnetic phase transition between two ferromagnetic phases (FM1 and FM2), in UIr there are three ferromagnetic phases and the FS phase is located in the low-*T*/high-*P* domain of the third of them - the phase FM3. There are two metamagnetic-like phase transitions: FM1-FM2 transition which is followed by a drastic decrease of the spontaneous magnetization when the the lower-pressure phase FM1 transforms to FM2, and a peak of the ac susceptibility but lack of observable jump of the magnetization at the second (higher pressure) "metamagnetic" phase transition from FM2 to FM3. Unlike the picture for UGe2, in UIr both transitions, FM1-FM2 and FM2-FM3 are far from the maximum *Tm*(*Pm*) so in this case one can hardly speculate that the *max* is produced by the nearby jump of magnetization. UIr seems to be a U-type spin-triplet ferromagnetic superconductor.
