Phenomenological approach to study the degree of the itinerancy of the $5f$ electrons in actinide ferromagnets with spin fluctuation theory
Naoyuki Tateiwa Jiri Pospisil Yoshinori Haga Hironori Sakai Tatsuma D. Matsuda Etsuji Yamamoto

APS/123-QED






tateiwa.naoyuki@jaea.go.jp 








^1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Naka, Ibaraki 319-1195, Japan

^2Charles University, Faculty of Mathematics and Physics, Department of Condensed Matter Physics, Ke Karlovu 5, 121 16 Prague 2, Czechia

^3Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan



             



Actinide compounds with 5f electrons have been attracting much attention because of their interesting magnetic and electronic properties such as heavy fermion state, unconventional superconductivity, co-existence of the superconductivity and magnetism. Recently, we have reported a phenomenological analysis on 80 actinide ferromagnets with the spin fluctuation theory originally developed to explain the ferromagnetic properties of itinerant ferromagnets in the 3d transition metals and their intermetallics (N. Tateiwa et al., Phys. Rev. B 96, 035125 (2017)). Our study suggests the itinerancy of the 5f electrons in most of the actinide ferromagnets and the applicability of the spin fluctuation theory to actinide 5f system. In this paper, we present a new analysis for the spin fluctuation parameter obtained with a different theoretical formula not used in the reference. We also discuss the results of the analysis from different points of views.

 


Phenomenological approach to study the degree of the itinerancy of the 5f electrons in actinide ferromagnets with spin fluctuation theory[This paper was presented at international conference ACTINIDE 2017 held in Sendai Japan on July 2017. It will be published in Nuclear Science and Technology.]
    Etsuji Yamamoto^1
    December 30, 2023
========================================================================================================================================================================================================================================================================================================







§ INTRODUCTION

The nature of the 5f electrons in actinide compounds has been extensively studied for many years from both theoretical and experimental points of views<cit.>. Despite numerous theoretical studies, the behavior of the 5f electrons has not been fully understood yet. One of remaining issues is whether the 5f electrons should be treated as being itinerant or localized in various actinide interemetallic compounds. 

Recently, we have reported the result of analyses of magnetic data in 69 uranium, 7 neptunium and 4 plutonium ferromagnets with the spin fluctuation theory developed for the magnetic properties in the 3d metals and their intermetallics<cit.>. The basic and spin fluctuation parameters of the actinide ferromagnets have been determined using our experimental data as well as those from literature. The analysis suggests the itinerancy of the 5f electrons in most of the actinide ferromagnets and the applicability of the spin fluctuation theory to the actinide 5f system. 

In this paper, we discuss the result of the analysis from different points of views. In addition, we show a new analysis for the spin fluctuation parameters obtained with a different theoretical expression in the spin fluctuation theory not used in the reference.




§ RESULTS AND DISCUSSIONS



 §.§ Spin fluctuation theory

We briefly summarize the spin fluctuation theory developed by Takahashi<cit.>. The spin fluctuation spectrum for itinerant ferromagnets is described by double Lorentzian distribution functions in small energy ω and wave vector  spaces.
    
   
    Imχ(,ω)    =   χ(0) 1+q^2/κ^2ωΓ_qω^2+Γ_q^2

Here, q ≡ ||, κ represents the inverse of the magnetic correlation length, and Γ_q (= Γ_0q(κ^2+q^2)) is the damping constant. The spectrum is represented in a parameterized form by introducing two energy scales T_0 (=Γ_0q_ B^3/2π) and T_ A [=N_0q_ B^2/(2χ(0)κ^2)], where q_ B is the zone-boundary wave vector for the crystal with N_0 magnetic atoms with the volume V (=6π^2N_0/q_ B^3). These parameters represent the distribution widths of the spin fluctuation spectrum in the energy and wave-vector spaces, respectively. 

 In the Takahashi's spin fluctuation theory, T_0 and T_ A are expressed in following relations. 
   
    (T_ CT_0)^5/6 = p_ s^25g^2C_4/3(15cF_12T_ C)^1/2
          (T_ CT_ A)^5/3 = p_ s^25g^2C_4/3(2T_ C15cF_1)^1/2

  Here, g represents Lande's g factor and C_4/3 is a constant (C_4/3 = 1.006089···). p_ s is the spontaneous magnetic moment and T_ C is the ferromagnetic transition temperature. F_1 is the mode-mode coupling term: the coefficient of the M^4 term in the free energy that can be evaluated experimentally from the inverse slope of the Arrott plots (M^2 versus H/M plot) at low temperatures.  
 
  Generally, actinide ferromagnets have anisotropic magnetic properties. We analyzed the magnetic data taken on single crystal samples under magnetic field applied along the magnetic easy axis for most of the ferromagnets. But experimental data on polycrystalline samples have been used for several compounds. This could cause uncertainty in obtained parameter values that is reflected in error bars of the data points. Readers are referred to Ref. 4 for the details of our analysis.
  
  

 §.§ Application of the spin fluctuation theory to actinide ferromagnets

    

 In this subsection, we summarize our analysis on actinide ferromagnets with the spin fluctuation theory reported in Ref. 4. An important result in the Takahashi's spin fluctuation theory is generalized Rhodes-Wohlfarth relation between p_ eff/p_ s and T_ C/T_0<cit.>.
   
    p_ eff/p_ s ≅ (T_ C/T_0)^-3/2

Here, p_ eff is the effective magnetic moments in the magnetic susceptibility. This relation has been confirmed in itinerant ferromagnets of the 3d electrons systems<cit.>. As shown in Fig. 1, we find that the same relation holds in most of actinide ferromagents for T_ C/T_0< 1.0<cit.>. This suggests the itinerant nature of the 5f electrons. Here, a ratio T_ C/T_0 characterizes the degree of itinerancy of magnetic electrons in the spin fluctuation theory. At T_ C/T_0 ≪ 1, the magnetic electrons have a strong itinerant character. Both quantities approach unity when the degree of itinerancy becomes small. The local magnetic moment is responsible for the ferromagnetism when T_ C/T_0 = 1. Note that several data points deviate from the relation for T_ C/T_0 ∼ 1.0. This deviation may be due to several other effects not included in the spin fluctuation theory such as the crystalline electric field effect on the 5f electrons from ligand atoms. We note that the spin fluctuation theory neglects the orbital moment oriented antiparallel to the spin moment of the 5f electrons. Further elaborate theoretical consideration is necessary for the present result.



 §.§ Relation between T_ C and T_0, and T_ A

We show new result of the analysis not reported in Ref. 4. Figure 2 shows relations between T_ C and (a) T_0 and (b) T_ A  for uranium, neptunium and plutonium compounds, and transition 3d metals plotted as closed circles, squares, triangles and anti-triangles, respectively. The parameters for the 3d systems are cited from Ref. 5. A general tendency is that the widths of spin fluctuation spectra in the actinide ferromagents are about one order magnitude smaller than those in the 3d systems. This suggests smaller energy scales of magnetic excitations for the actinide systems. This could be attributed to the larger spatial extent of the 3d wave functions than that of the 5f ones. An interesting feature is that the values of T_ A for the plutonium ferromagnets are generally larger than those of the uranium and neptunium compounds and the values are comparable to those of the 3d systems. The spin fluctuation spectra for the plutonium ferromagnets spread to the higher momentum  space, similar to the 3d systems.
  





 §.§ Uranium ferromagnetic superconductors UGe_2, URhGe, and UCoGe

Many studies have been done for the superconducting properties in uranium ferromagnetic superconductors UGe_2, URhGe, and UCoGe since the findings of the superconductivity<cit.>. Meanwhile, there have been only a few studies for the dynamical magnetic property of the ferromagnetic superconductors. Figure 3 shows relations between the spontaneous magnetic moment p_ s and T_ C/T_0 for UGe_2, URhGe, and UCoGe. The data points for several itinerant ferromagnets in the 3d system are also plotted. In this figure, right and lower region corresponds to the local moment system, and the left and upper region does the weak ferromagnetic state. The dynamical magnetic property differs depending on each ferromagnetic superconductor. The value of UGe_2 is 0.571, indicating that this compound is located close to the local moment system (T_ C/T_0 = 1). On the other hand, the value for UCoGe is 0.0065, suggesting the weak ferromagnetic state similar to those in Y(Co_1-xAl_x)_2. The values of the parameters (T_0 = 362 K and T_ A = 5.92 103  K) in UCoGe are significantly larger than those (T_0 = 92.2 K and T_ A = 442 K) in UGe2. The spin fluctuation spectrum Imχ(,ω) in UCoGe spreads to the higher energy and momentum spaces. URhGe is located in an intermediate region between the two limiting cases. The value of T_ C/T_0 for URhGe is 0.121, similar to those in MnSi (T_ C/T_0 = 0.131) and Ni (T_ C/T_0 = 0.237). 


   




 §.§ Determination of  T_ A^* from the M(T)-T curve  

 In Ref. 4, the value of the spin fluctuation parameter T_ A was determined from Eqs. (2) and (3) with the mode-mode coupling term F_1 that is obtained from the slope of the magnetization at low temperatures<cit.>. Here, we estimate T_ A of uranium ferromagnets classified as “group I” in Ref. 4 by a different theoretical formula in the Takahashi's spin fluctuation theory<cit.>. The spontaneous magnetization M(T) of itinerant ferromagnets is expressed by a following relation in the theory.
     
    (M(T)M_0)^2 = 1-a_0p_s^2(TT_ A^*)^2
 
, where M_0 is the spontaneous magnetization at 0 K, a_0 is a constant 50.4, and p_s = M_0/μ_ B. The value of T_ A^* can be determined by the fit of the data M(T) with Eq. (5). Figure 4 shows temperature dependencies of the magnetization for UGa_2, UCu_2Ge_2, URh_6Ge_4, URhAl, URhGe_2, URhSi, UIr, and URhGe in magnetic fields of 0.01 ∼ 0.5 T applied along the magnetic easy axes. The Curie temperatures T_ C are denoted by arrows. Solid lines are result of fits to the data with Eq. (5). Table I shows the values of T_ A^* for the uranium ferromagnets in the group I determined in this method. The values of T_0^0 and T_ A^0 determined from the mode-mode coupling term F_1 using Eqs. (2) and (3) are also shown in the table. Generally, the values of T_ A^0 and T_ A^* obtained by the two different methods are consistent with each other. This suggests the effectiveness of the theoretical expression (Eq. (5)). However, there is difference between T_ A^0 and T_ A^* for UCu_2Ge_2, URh_0.2Co_0.8Ge, and URh_0.1Co_0.9Ge where the parameters are larger than those of the other ferromagnets. The coefficient of the T^2 term in Eq. (5) becomes smaller for the larger value of T_ A^*. The discrepancy may arise from ambiguity in the determination of T_ A^* with Eq. (5). Note that the lowest temperature for the magnetic measurement is 2.0 K. The magnetization data M(T) below 2.0 K may be necessary for the accurate determination of T_ A^* when the parameter is larger.



§ CONCLUSION

   

We have analyzed 80 actinide ferromagnets using the spin fluctuation theory and found the itinerancy of the 5f electrons in most of the actinide ferromagnets. The analysis also suggests the smaller energy scales of the magnetic excitation spectra of the actinide ferromagnets. We discuss relations between the spontaneous magnetic moment p_s and the spin fluctuation parameter T_ C/T_0 in uranium ferromagnetic superconductors UGe_2, URhGe, and UCoGe. We determine the parameter T_ A^* using a different theoretical formula not used in our previous study<cit.>. The obtained values of T_ A^* are basically consistent with T_ A^0 determined using the mode-mode coupling constant F_1. The magnetization data at very low temperatures may be necessary for the accurate determination of T_ A^* when it is larger.


 

§ ACKNOWLEDGMENTS

 This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 16K05463, 16K05454, 16KK0106, JP15H05884(J-Physics) and 26400341. 







santini1P. Santini, R. Lémanski, and P. Erudős, Adv. Phys. 48, 537 (1999).
santini2P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, and G. H. Lander, Rev. Mod. Phys. 81, 807 (2009).
mooreK. T. Moore and G. van der Laan, Rev. Mod. Phys. 81, 235 (2009).
tateiwa1N. Tateiwa, J. Pospíšil, Y. Haga, H. Sakai, T. D. Matsuda, and E. Yamamoto, Phys. Rev. B 96, 035125 (2017).
takahashi1Y. Takahashi, J. Phys. Soc. Jpn. 55, 3553 (1986).
takahashi2Y. Takahashi, J. Phys.: Condens. Matter. 13, 6323 (2001).
takahashi3Y. Takahashi, Spin Fluctuations Theory of Itinerant Electron Magnetism (Springer-Verlag, New York, 2013).
saxenaS. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley, I. Sheikin, D. Braithwaite, and J. Flouquet, Nature 406, 587 (2000).
huxley0 A. Huxley, I. Sheikin, E. Ressouche, N. Kernavanois, D. Braithwaite, R. Calemczuk, and J. Flouquet, Phys. Rev. B 63, 144519 (2001).
aoki0D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J. P. Brison, E. Lhotel, and C. Paulsen, Nature 413, 613 (2001).
huy N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P. Klaasse, T. Gortenmulder, A. de Visser, A. Hamann, T. Görlach, and H. v. Löhneysen, Phys. Rev. Lett. 99, 067006 (2007). 
 




