Heavy Fermionic SystemsLeland HarrigerUTK Physics Dept. HW Project Solid State II, Instructor: Elbio Dagotto(Dated: April 27, 2009)I. HEAVY FERMIONS:AN OVERVIEWHeavy fermion systems are a collection of materials whose properties are governed by a lattice that car-ries f-electron magnetic ions at crystalographic sites [Hewson]. The electrons within these f-orbitals interactmagnetically with conduction electrons within the system. This state of affairs is reminiscent of the singleion Kondo problem that arises in other systems. In single ion Kondo systems, magnetic impurities are in-jected into the system which then interact with passing conduction electrons. This results in a direct ex-change coupling J forming between the localized spin impurities and those of the itenerant conduction electrons.FIG. 1: Possible ground states resulting from the competitionbetween the Kondo and RKKY interactions. Tmis the orderingtemperature and JN is the f-d exchange coupling times the fdensity of states at the Fermi energy. [Sanchez]A direct consequence of this interaction is that resistiv-ity within these systems breaks from standard Fermiliquid theory at very low temperatures . Moreover,with decreasing temperature the coupling can screenout the spin impurities by binding conduction elec-trons to them to form a singlet state. The tempera-ture at which this screening occurs is refered to as theKondo Temperature, TK. Jun Kondo was the first tosolve this problem and provide a logarithmic correctionterm to the resistivity [Kondo]. This term accuratelyaccounted for the peculiar upturn in resistivity at lowtemperatues but also asymptotically diverged as thetemperature was suppressed to absolute zero. Furtherwork by others succeded in fixing the divergence and today the solution to the Kondo problem stands as an impressiveachievement in solid state physics.2A fundamental and very important difference between single ion Kondo systems and heavy fermion systems is thatin the former, the ions exist as impurities scattered within the system and, as a result, interactions are isolated shortrange events. However, in heavy electron systems, the f-orbitals are part of the crystalographic structure and forma Kondo lattice of magnetic ions. Although this may at fist appear to be a direct extension of the original Kondoproblem, in practice a solution is much less tractable. Moreover, due to the periodicity of the magnetic lattice, anindirect exchange coupling mediated by the conduction electrons is established between the sites; the so-called RKKYinteraction [Kittel]. Indeed, many models of heavy fermion systems are treated as a competition between a RKKYinteraction that acts to set up long range magnetic order at a temperature TRKK Yand a Kondo effect that beginsscreening the sites as the temperature drops below TK. In the Kondo effect the onset of magnetic screening is givenbyTK= ρ−1e−1ρJ(1)where ρ is the density of states at the Fermi surface and J is the exchange coupling between the conduction electronsand the localized magnetic f-orbitals. However, the onset temperature for magnetic ordering due to the RKKYinteraction goes asTRKK Y∝ J2ρ (2)Consequently, as the temperature is supressed, the moments associated with the long range magnetic order will beginto be screened away (Fig 1) as the Kondo interaction begins to dominate the system.Aside from TRKK Yand TK, there exists a third temperature that plays a vital role in determining the onset ofproperty changes within heavy fermion systems. This temperature, which is referred to as T∗, corresponds to apoint where the bound f-electrons become (at least partially) itenerate. Interestingly, it is for temperatures belowT∗that the f-electrons begin to unbind. T∗is typically very low, around 1-10K depending on the system [Fisk]. Acalculation of the change in entropy over this temperature range reveals a sharp climb which is attributed to thisunbinding process. The magnitude of this change is fairly consistent from system to system, around Rln(2) where Ris Rydbergs constant, and is accompanied by significant changes in properties such as reduced resistivity, modifiedspin sucseptibility, an observed Knight shift, etc [Yang, PRL]. Because of this, it is convinient to define T∗as∆S =ZT∗0γdT = Rln(2) (3)Recent work [Yang, Nature] has demonstrated that T∗can be modeled very well asT∗= cJ2ρ (4)3FIG. 2: a) Confirmation of T∗given by the intersite RKKY interaction for a variety of Kondo lattice materials; c = 0.45. b)Updated Doniach diagram for Kondo lattice materials. [Yang]where c is a parameter to be determined. Combining this with (1) gives the relation:[ln(TKρ)]−1=pc−1T∗ρ (5)A value of c = 0.45 was determined by fitting (5) to experimenal values of T∗, TK, and γ for a variety of Kondolattices (Fig 2a). From this, a modified version (Fig 2b) of the Doniach diagram [Doniach] was generated that relatesthe general phase diagram behavior of the system to the fundamental quantities that drive this behavior.II. PROPERTIES AND GROUNDSTATESSurprisingly, there exist a vast number of ground states that support the formation of heavy fermions from sys-tem to system. For instance, systems such as UBe13form a Non-Fermi Liquid which crosses over into an ex-otic unconventional superconductivity at very low temperatures. However, other systems like U P t3begin as anantiferromagnet that transition to a heavy Fermi Liquid phase below Tnbefore ultimately crossing over to su-perconductivity at even lower temperatures. Others like CeAl2and U2Zn17remain as antiferromagnets at verylow temperatures while some like CeNiSn become narrow-gap semiconductors [Misra]. Work on a doped variantof the parent compound URu2−xRexSi2has uncovered new and equally interesting phenomena. For example,URu2−xMxSi2(M = Re, Tc) revealed the first instance of a Fermi surface (FS) instability in a heavy fermionsystem [Bauer]. Re doping, while quickly suppressing the SC phase, also results in a region of non-Fermi-liquid(NFL) behavior. Of particular interest, the NFL behavior is exhibited across a FM quantum critical point (QCP),4which is in direct contrast to the majority of studies that exhibit NFL behavior across SG or AFM QCPs [Aron-son, Wilson]. Moreover, this NFL behavior extends deeply into the ferromagnetic phase, acting as a rare instanceof deep NFL penetration into an ordered magnetic state. Specifically, the electrical conductivity ρ(T), magneticsusceptibility