3 : Presentation of UGe2 3.3 Itinerant Ferromagnet
some of them arguing on the itinerant character and others on the localized one. In this section, the evidence for a degree of delocalization of the f electrons at low temperature is reviewed.
In standard heavy fermion behavior, the f electrons are localized at high temperature and below some characteristic temperature (sometimes called the coherence temperature) the f electrons behave no more independently which leads to strongly renormalized effec- tive masses.
At high temperature, f electrons indeed appear localized :
• Above 450 K the reciprocal susceptibility shows the Curie-Weiss law for all the main crystallographic directions, with effective moments of 3.0, 3.6 and 3.4µB/U for [100], [010] and [001] directions, respectively4 [Galatanu05]5 (see fig. 3.3). The effective magnetic moments are close to a free ion value6 of 3.6µB.
• Above 160 K, the anomalous Hall coefficientRs is 3 orders of magnitude larger than the ordinary Hall coefficientR0 from which it is assumed that the scattering centers of the conduction electrons are the localized 5f electrons [Tran04].
• The results of measurements of positron annihilation radiation in the paramagnetic phase (around 60 K, i.e. above TC = 53 K) are better explained by numerical calculation considering 5f electrons as fully localized [Biasini03].
At low temperature, however, there is evidence for a degree of delocalization of the f electrons :
• Firstly, the value of the ordered moment of 1.4µB/U is smaller than the Curie-Weiss moment above the Curie temperature.
• The delocalization of the 5f electron is also shown by the observation of carriers with high cyclotron masses of (15−25)m0 by dHvA experiments [Onuki91, Satoh92, Terashima01, Settai02, Haga02]. These heavy masses suggest itinerant but strongly correlated 5f electron states.
• Measurements of the optical conductivity also show that the effective mass starts to increase below TC, suggesting an intrinsic coherence temperature T∗ < TC
[Guritanu08].
• In addition, the magnetization does not reach saturation at least up to 27 T applied along the easy magnetization a-axis (see fig. 3.2) [Sakon07]7.
4The corresponding paramagnetic Curie temperatures are 39, −310 and−210 K for [100], [010] and [001] directions, respectively [Galatanu05].
5See also [Onuki92, Saxena00, Huxley01] for values around 200 K.
6We can calculate the magnetic moment of an isolated free uranium ion for U3+ in the configuration 5f3 : we haveS= 32,L= 6 andJ =|L−S|=92. The Lande factor isgJ = 1 +J(J+1)+S(S+1)−L(L+1)
2J(J+1) =
8
11. Thus, the magnetic moment for degenerate5f shell isgJ
pJ(J+ 1)µB= 3.62µB.
For U4+ in the configuration 5f2 : we have S = 1, L = 5 andJ =|L−S|= 4. The Lande factor is gJ =45. Thus, the magnetic moment for degenerate5f shell is 3.58µB.
Crystal electric field effect will split the 5f level and can reduce the magnetic moment. However, no distinct evidence for crystal electric fields are reported, as common for uranium-based compounds. This is certainly because f-electrons of uranium are sufficiently itinerant to reduce the crystal field effect which occurs from localized charges. See also the Rhodes-Wohlfarth plot.
7A small field dependent ordered moment can also arise in the case of localized moments with a mixing of different crystal field levels (see [Huxley03b] and [Wang69]).
3 : Presentation of UGe2 3.3 Itinerant Ferromagnet
• The ordinary Hall coefficient changes its sign from negative to positive below≈20 K and the carrier concentration starts to increase upon cooling. These results can be explained by a delocalization of some part of the 5f electrons [Tran04].
• Another argument is given in ref. [Huxley01] : “at ambient pressure, a degree of delocalization of the f electrons is suggested [Mohn89] by the moderate value of the low-temperature specific heat, C/T = 32 mJ.mol−1K−2, and the ratio of this to the step in C/T (200 mJ.mol−1K−2) at TC.” The same argument is reproduced in ref.
[Pfleiderer09]. The given ref. [Mohn89] indeed compare the results of the Stoner model, which is the pure itinerant case, with a spin fluctuation model introduced by Murata and Doniach [Murata72] which assume fluctuation of a localized mag- netization. It is shown in ref. [Mohn89] that the large specific heat jump at TC in the Stoner model is reduced by the introduction of spin fluctuation. In the Stoner model, the excitations are single particles excitations, and ∆Cm = χM02
0TC where M0 is the spontaneous magnetization and χ0 the initial ferromagnetic susceptibility. It is reduced to ∆Cm = 4χM02
0TC in the spin fluctuation model8. The T-linear magnetic contribution to the specific heat is γm = −2χM00T2C2 in the Stoner model which is in- creased to γm = 152χM02
0TC2 in the spin fluctuation model8. Thus, it is qualitatively true that a degree of delocalization is suggested. However, none of these models can reproduce the experimental values9.
• Finally, during the time of this study, simultaneous analysis of magnetization and heat capacity measurements in the framework of molecular-field theory allowed a derivation of the magnetic heat capacity [Hardy09] (see also section 4.2.3 page 86).
The magnetic entropy at the Curie temperature isSmagn = 0.8Rln2, which is smaller than Rln2, but only by 20%.
The low temperature dependence of the magnetization has an intermediate behavior between itinerant and localized moments [Huxley03b], although the itinerant behavior can be observed if one restricts to the lowest temperatures [Hardy09].
High resolution photoemission spectroscopy PES indicates the presence of a sharp peak in the spectrum just below the Fermi level EF. It is interpreted as the “coherent U 5f” peak originating from the itinerant 5f electrons. A broad shoulder is also attributed to the “incoherent U 5f” representing the localized 5f electrons [Ito02]10.
Finally, we note that the uranium spacing in UGe2 is dU−U= 3.85 Å which is above the Hill limit11 of 3.6 Å. Therefore, without hybridization with other electrons, the f electrons would be localized.
8The parallel and transverse component of the fluctuating moment are considered equal. This is not true for UGe2 where spin fluctuations shows Ising character.
9A comparison of the results from the Stoner model and the spin fluctuation model can be found in [Mohn02], chapter 18.
10Earlier photoemission experiments shows the presence of narrow peak in the density of states below EF [Soda91, Ishii93]
11Hill has realized a systematic study of uranium compounds as a function of uranium spacingdU−U. He found that for dU−U <3.4 Å, the ground state is paramagnetic which is interpreted by the overlap of the 5f orbitals. For dU−U>3.6 Å, the ground state is magnetic, which is interpreted by the result of local magnetic moments. The region 3.4 Å< dU−U<3.6 Å, is the critical region