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Heavy Fermion Physics Heavy Fermion Physics Hua Chen Course: Solid State II, Instructor: Elbio Dagotto, Semester: Spring 2008 Department of Physics and Astronomy, the University of Tennessee at Knoxville, 37996¤ This article is intended to be a very brief review on the most basic facts and concepts of heavy fermion physics. A general overview is given as an introduction. In the following part, the mechanism leading to the formation of heavy fermions is explained briefly, and its connection to Kondo e®ect is emphasized. Then we talk about heavy fermion theories in more detail, where the concepts like Kondo lattice, slave boson are introduced. Then there are two separate parts on Kondo insulators and heavy fermion superconductivity respectively. Keywords: heavy fermion, Kondo e®ect, Kondo insulator, superconductivity I. INTRODUCTION higher than usual, which is the origin of the name heavy fermion (Fig 1). Heavy fermion martials are those metallic compounds which have enormously large e®ective electron masses be- The ¯rst material of this kind in history is CeAl3 low certain temperatures. It is well known that, at tem- alloy, which was discovered by Andres, Graebner and peratures much below the Debye temperature and Fermi Ott [2] in 1975. Since then, bunch of other heavy fermion temperature, the heat capacity of metals can be written materials have been discovered and studied, which in- as the sum of electron and phonon contributions: clude CeCu2Si2, CeCu6, UBe13, UPt3, UCd11,U2Zn17, NpBe , etc. Among them, there are superconductors, 3 13 C = γT + AT (1) antiferromagnets and insulators. However, one common property of these compounds is that one of the con- and the electron term is linear in T and dominant at stituents of each is a rare-earth or actinide atom with low temperatures. The material-dependent constant γ is partially ¯lled 4f- or 5f-electron shells. At high temper- ¤ proportional to the e®ective electron mass m . In com- atures the f-electrons are localized on their atomic sites mon metals the ratio of this e®ective mass to the real and do not contribute to conduction. However, contrary electron mass is within the order of 10. However, in to conventional materials, some of the f-electrons be- the late seventies, people discovered some metallic com- come itinerant at low temperatures. Crudely speaking, ¤ pounds whose m s are two or three orders of magnitude it is these itinerant f-electrons that lead to the uncom- mon behaviors of heavy fermion materials at low temper- atures. We will go to detail in Sec. II. One distinctive property of heavy fermion materials is that in these systems quantum fluctuations of the mag- netic and electronic degrees are strongly coupled to each other, which may lead to quite a lot of unconventional behaviors. Moreover, as one of the goals of modern con- densed matter physics is to couple magnetic and elec- tronic properties to develop materials with novel proper- ties, such as high temperature superconductivity, colos- sal magnetoresistance (CMR) materials, spintronics, and the recently active area of multiferroic materials, heavy fermion materials can be important to help us under- stand the interaction between magnetic and electronic quantum fluctuations. FIG. 1: Schematic plot of the speci¯c heat C(T ) for CeAl3. For comparison we show also the results for the related com- The remaining of this paper are organized as follows: pound LaAl3. The speci¯c heat coe±cient γ of CeAl3 is of In Sec. II we will explain in more detail the mecha- order J¢mol¡1K¡2, instead of order mJ¢mol¡1K¡2 as in ordi- nisms and physical processes resulting in the unconven- nary metals. [1] tional properties of heavy fermion materials, in which the so-called Kondo e®ect plays the most important role. In Sec. III and Sec. IV we will respectively talk about two important subcategories of heavy fermion systems: ¤Electronic address: [email protected] Kondo insulators and heavy fermion superconductors. 2 II. HEAVY FERMION AND KONDO EFFECT discovered that the second order perturbation gives a log- arithmic dependence of resistivity on temperature: A. Kondo e®ect ½ = a ¡ b ln T; (4) To understand heavy fermion systems, the ¯rst notion which is just the behavior observed in the temperature we must know is the so-called Kondo e®ect [3][4]. Loosely range close to the resistivity minimum in dilute magnetic speaking, the Kondo e®ect refers to certain unconven- alloys. tional phenomena result from the process by which a The second order perturbation include two virtual pro- free magnetic impurity ion becomes \screened" by the cesses all called \superexchange", in which an electron or spins of the Fermi sea at low temperatures and low mag- hole is replaced by another conduction counterpart with netic ¯elds. As the impurity ion is screened, a portion of a di®erent spin. This process can induce an equivalent conduction electrons are bonded to it and hence the con- interaction between conduction electrons and local im- ductivity decreases. However, generally the electronic purity d¡ (orf¡) spins. In most dilute magnetic alloys resistivity due to scattering of conduction electrons by this induced coupling is antiferromagnetic. impurities drops monotonically with decreasing temper- Experiments further showed that when T ! 0K, the ature. So the two competent mechanisms will lead to speci¯c heat of those metals with magnetic impurities a minimal resistivity at low temperatures. This is the deviates dramatically from the law of linear dependence explanation, which was given by Kondo in 1963, to the on T in common metals. This is explained as that when puzzling \minimal conductivity" discovered 30 years be- T ! 0K, magnetic impurities gradually lose their mag- fore [5]. netic moments because of the Kondo screening, which In the remaining of this subsection, for the convenience will lead to a drop of the impurities' entropy and hence of subsequent formulation of heavy fermion physics, we the uncommon behavior of speci¯c heat. So it is nat- will start from the famous Anderson model, and in a step- ural to expect that the ground state of the system at by-step manner, ¯nally go to Kondo's idea and results. T = 0K is the singlet state between conduction and In 1961, Anderson [6] gave the ¯rst microscopic model impurity spins. However, Kondo's perturbation treat- for the formation of magnetic moments in metals. An- ment are not suitable at low temperatures. This can derson's model Hamiltonian is written as: be seen from the divergence of the logarithmic temper- ature dependence of Kondo's result. Thus people must X X ¯nd other ways to strictly prove the existence of Kondo y y H = ²knk¹ + V (k)[ck¹f¹ + f¹ck¹] singlet. K. G. Wilson [8] employed the renormalization k;¹ k;¹ group method and achieved this goal in 1975. Wilson's treatment is numerical. 5 years later, Andrei [9] and + Ef nf + Unf"nf#: (2) Wiegmann [10], using the Bethe ansatz, gave the strict The ¯rst and the third term are respectively the Hamil- analytical solution of single impurity Kondo problem. tonian of conduction electrons and that of impurity atom. New insights as well as complications arise from the sec- ond and the fourth term. The second term describes the B. Kondo lattice and heavy fermion system interaction between conduction electrons and the local impurity spin. And the fourth term represents Coulomb One characteristic of heavy fermion materials is that interaction between f-electrons at the impurity site. every unit cell of the lattice has an ion with localized mag- Anderson model can explain how a localized impurity netic moment. Thus a heavy fermion system is just the magnetic moment is formed in the host metal. Once the periodical version of the Kondo system. So it is tempting local moment is formed, another model is introduced to to construct a periodic Hamiltonian based on the Ander- describe the interaction between local spin and conduc- son model: tion electrons. This is the s-d exchange model, whose X X X H = ² n + E nf + U nf nf Hamiltonian is: k k¹ 0 l¹ l" l# k;¹ l;¹ l X V y ¡ik¢l y ik¢l X X X +p [c fl¹e + f ck¹e ]: (5) J y k¹ l¹ H = ² n ¡ ( ) S^ ¢ σ^ 0 c c 0 0 ; (3) N k k¹ N ¹¹ k¹ k ¹ k;l;¹ k;¹ k;k0 ¹;¹0 Models like this are called Anderson Lattice Model. In which, alternatively, can also be derived from Anderson our model the simpli¯cations include: (1)Neglect the de- Hamiltonian by applying the Schrie®er-Wol® transforma- pendence of the strength of s-f hybridization on k, i.e., tion [7]. Because the absolute value of the coupling con- Vk = V . (2)The f states at every sitep are all nondegen- stant J is always much smaller than the Fermi energy erate. Besides that, the factor (1= N) exp(§ik ¢ l) we of conduction electrons, perturbation techniques can be introduced in the hybridization term is to reflect the fact well adapted. While the ¯rst order perturbation theory that conduction electrons hybridize with the f electron only gives a temperature independent resistivity, Kondo at every site, and every site has a di®erent phase. 3 When there is no hybridization, at every site there are Thus the constraint (12) becomes four eigenstates of f electrons: no electrons(spins) j0i, y y y two antiparallel spins j "#i, and two single spin states bl bl + fl"fl" + fl#fl# = 1: (14) j "i andj #i. Because these four states are orthogonal to y each other and complete, we can use them as basis to Coleman called the bosons bl and bl slave bosons.
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