CeCu ,(tore)
S. H. LIU
3. Theoretical approaches
The goal of this section is to summarize the present status of our understanding of the heavy-fermion problem. In many ways this is an impossible task, because there is no consenus among the community of theorists on the definition of the problem, let alone the proper approach to its solution. At the same time, the number of theoretical publications has grown so large that a mere list of the titles would reach the length of an average scientific article. To make it possible to write a meaningful
PHENOMENOLOGICAL APPROACH TO HEAVY-FERMION SYSTEMS 105 review without being bogged down by details, the author must exercise some choice in what to discuss and what to omit. The reader is forewarned that the choice reflects the author's personal bias, which may or may not stand the test of time. To those investigators whose contributions are not covered, we plead ignorance either of their publications or the future impact of their work.
We begin by reiterating our provisional definition of the heavy-fermion problem as outlined in the last section, that is, how to describe the behavior of the f electrons in various temperature regimes, and how to understand the continuous evolvement from the apparent itinerant properties at low temperatures to the localized properties at high temperatures. We will discuss in separate subsections four classes of theories.
The many variations of the Fermi liquid theory, which are based on the quasi-particle picture, are useful at low temperatures but not at high temperatures. The K o n d o lattice model, which assumes localized f electrons from the start, works well at high temperatures, but fails to shed light on the low-temperature properties. The spin fluctuation resonance model is the modern version of the K o n d o lattice model. It has provided some hope in understanding the narrow band at low temperatures, but we will demonstrate that this band, although fascinating, is most likely not the band of heavy fermions seen in the de H a a s - v a n Alphen measurements. Finally, the electronic polaron model depicts the f electrons as itinerant from the start, but shows that the Coulomb interaction between the f and the conduction electrons causes the former to localize at high temperatures. The shortcoming of this theory is that it does not reproduce the spin fluctuation phenomenology observed at high tempera- tures. When theories are in conflict, we will appeal to experiments as the arbiter.
Since physics thrives on controversy, it is the author's hope that any controversy this chapter may stir up will result in more careful analyses of the data and more refine- ments of the ideas.
3.1. The Fermi liquid models
There are three broad versions of Fermi liquid models in the literature, each is applicable for specific problems. The one-band model has provided the first under- standing of the low-temperature thermodynamic and transport properties. The interpretation of inelastic neutron scattering data requires a two-band hybridization model. The band model based on the local-density functional approximation is by far the most elaborate noninteracting Fermi fluid model. We will discuss only the basic principle of this approach because the details appear elsewhere in this volume.
3.1.1. The one-band Fermi liquid model
F o r the benefit of beginners, we will start the discussion from the most primitive level. This will also serve the purpose of defining a set of basic quantities which will be useful later. Let us consider a band of noninteracting fermions with mass m*. The kinetic energy ek is related to the wave vector k by
~k = k2/2m*. (10)
It is customary to measure the wave vector in momentum units, so the quantity h
will be set equal to unity. There are N fermions per unit volume in two spin states, so the Fermi level #, which is a function of the temperature T, is determined by the implicit relation
f 2N(a)de (11)
N = ,J e x p [ ~ - ~ - ] + 1'
where fl = 1/k~3T, k R is the Boltzmann constant, and N(e) is the density of states per spin per unit volume defined by
( 2 m * ) 3 / 2 8 1 / 2
Y(e) = (12)
4re 2
At T = 0 the Fermi energy has the expression
#(0) = k2v /2m *, (13)
where k v = ( 3 g 2 N ) 2/3 is the Fermi wavevector introduced in section 2. We define a Fermi temperature by Tv = #(O)/k,, then at temperatures much less then Tv, the Fermi level depends weakly on temperature and the density of states at the Fermi level N(#), defined in eq. (3), determines the specific heat and the magnetic suscepti- bility. For temperatures much greater than Tv, the fermion gas is no longer degenerate so that the specific heat reaches the classical limit determined by equipartition of energy:
C ( T ) = ~ N k , . (14)
Since this is not what is observed in heavy-fermion systems, we conclude that a band of free heavy fermions is not a viable model for high-temperature properties.
One can do a little better by postulating a band of finite width. It can be verified that, for rather general forms of the density-of-states curve N(e), the high-temperature specific heat satisfies the T - 2 law and the high-temperature susceptibility goes like T-1. On the other hand, it seems quite difficult to design an N(e) curve which will yield both the total entropy and the Curie constant in the susceptibility. Besides, the T 2 dependence of the low-temperature resistivity argues strongly that the pair interaction between the fermions cannot be ignored if one intends to understand the total problem.
A simple interacting fermion system may be represented by the following Hamiltonian:
I-i = E kcLe,o + E Z ( V o - v,s.s')cLc o . . . . q, ' o c,+ o q, •
k a k k ' q a a '
(15) In the above Hamiltonian the quantities ck~ and c~, are fermion operators in the momentum state k and spin state or. The electron spin operators are denoted by s and s'. The direct interaction term V 0 is independent of the spin, while the exchange term is explicitly spin dependent. The sign of V~ is chosen so that parallel spins are favored. In principle, the potentials are functions of the momenta k, k' and q, but for the purposes of illustrating the physics, it is sufficient to restrict ourselves to the
PHENOMENOLOGICAL APPROACH TO HEAVY-FERMION SYSTEMS 107 momentum-independent s-wave scattering and omit all higher partial wave components.
As discussed in textbooks (Blatt 1968), the interaction term in eq. (15) does not give rise to any resistivity unless the fermion gas is imbedded in a lattice so that umklapp scattering can occur. Furthermore, since at T<< TF only those electrons within an energy shell of width k B T around the Fermi level can conduct electric current, the pair of electrons affected by the interaction must be within this energy shell both before and after scattering. Thus, the phase space argument shows that the resistivity must be scaled by
(T/TF) 2,
a s observed experimentally. Unfortunately, the theory contains too many parameters to allow a reliable determination of the interaction potential from the resistivity data of real materials.The exchange interaction modifies the magnetic susceptibility formula so that at zero temperature
Z(0)- 2N(kL)/~2
1 - V 1 N ( / ~ ) (16)
The denominator, which is less than unity, causes an enhancement of the susceptibility.
This is the reason that on the 7 versus Z(0) plot in fig. 3 the points for some materials appear on the right of the straight line. In case the denominator is less than zero, the paramagnetic state is unstable against the spontaneous splitting of the spin degeneracy. This results in a ferromagnetic state for which the moment per magnetic atom is nonintegral. So far only a few heavy-fermion ferromagnets have been found.
One example is the series of compound CePtxSi for x < 0.97. In particular, the material with x = 0.7 becomes ferromagnetic below 5.5 K with the saturation moment of 0.18#B per Ce atom (Lee et al. 1988). Metallic ferromagnets with nonintegral moments are said to be itinerant. The reader may consult the review by Herring (1966) for further details of the theory of itinerant ferromagnetism.
Even in nonmagnetic materials the exchange enhancement effect produces spin fluctuations which can influence the thermodynamic properties of the material. One defines the dynamical susceptibility function Z(°)(q, co) for the noninteracting Fermi gas by
)(O)(q, co) = ~ nk -- nk+q
k ek+q -- ek -- (co + i~i)' (17)
where nk = {exp [fl(ek --/Z)] + 1} -1 is the Fermi distribution function, 5 = 0 +, and the frequency co is measured in energy units. It is easy to verify that 2#~Z<°)(0,0) is the static susceptibility.of the noninteracting Fermi gas in eq. (4). For the interacting gas the dynamical susceptibility function is enhanced into
Z(°)(q, co)
z ( q , co) - 1 -- V1z(°)(q, co)" (18)
Again, aside from a factor 2#I the quantity Z(0,0) reduces to the enhanced static susceptibility in eq. (16). The dynamical susceptibility is the linear response function
of the interacting gas to an external magnetic field which has the sinusoidal spatial and temporal dependence given by e x p ( i q ' r - icot).
The unenhanced dynamical susceptibility for zero frequency has the expression:
1
(19)2 \ q 2kF~q--q J"
This function is largest at q = 0, as shown in fig. 16, so the uniform susceptibility receives the highest enhancement. In fig. 17 we show qualitatively the real and imaginary parts of Z~°)(q, co) for nonzero co, and in fig. 18 the imaginary part of the enhanced susceptibility for 1 -
V1N(ff)
= 0.5 for three different values of the momen- tum. One can think of z(q, co) as the propagator or Green's function of a spin fluctuation mode, then the imaginary part is the spectral density of the mode. This0.5 1.0
0.0 0.0
I I I
1.0 2.0 3.0 4.0
"S"
Fig. 16. The wave-vector-dependent susceptibility Zl°)(q,0) for the free-electron gas. The wave vector q is measured in units of the Fermi momentum kv, and the susceptibility is normalized by the density of states at the Fermi level.
o = v
s v
E
Y
3.0
2.0
1.0
0.0
-1.0
-2.0 0.00
.•
q = 0 . 1j. l / / J ' / / i
......
I I I
0.05 0.10 0.15 0.20
Fig. 17. The real (solid curve) and imaginary part (dashed curve) of the dynamical suscepti- bility g~°)(q,o)) for the free-electron gas. The energy ~o is measured in units of Fermi energy
#, and the susceptibility components are normalized by the density of states at the Fermi level.
P H E N O M E N O L O G I C A L A P P R O A C H TO H E A V Y - F E R M I O N SYSTEMS 109
E 2.5
2.0
1.5
1.0
0.5
0.0
q = 0 . 1 q = 0.2
(q = 0.3
0.0 0.1 0.2 0.3
Fig. 18. The imaginary part of the enhanced dynamical susceptibility z(q, co) for an interacting electron gas. The wave vector is measured in units of kF, the energy is measured in units of/~, and the susceptibility is normalized by the density of states at the Fermi level.
mode is over-damped, because the linewidth is substantial compared with the energy at the maximum of the spectral density. The peak position shifts linearly with the momentum, giving the mode a linear dispersion relation. In the ferromagnetic system, i.e. 1 - V1N(#)< 0, a permanent moment develops spontaneously and the quantity Im z(q, co) evolves into a 6-function, which is the spectral function of the spin-wave or magnon mode. As a result, the spin fluctuation modes in the paramagnetic system are also called paramagnons.
Just like other boson modes in solids, paramagnons also contribute to the specific heat of the material (Doniach and Engelsberg 1966). A propagating mode with a linear dispersion relation gives a T 3 contribution to the specific heat, but the natural width of the paramagnon mode modifies this contribution so that the entire electronic contribution to the specific heat is
C(T)
-- 7 T + B T 3 In T. (20)The coefficient B in the above expression contains a number of material-dependent parameters. The spin fluctuation contribution to the specific heat is clearly seen in many heavy-fermion compounds such as UPt3 (Stewart et al. 1984) and mixed-valence compounds such as CeSn 3 (Tsang et al. 1984).
It is also possible to excite these modes by neutron scattering, because the magnetic moment of the neutron interacts with the electron spin to produce the spatial and temporal dependent magnetic field. The wavevector q and the frequency co are, respectively, the momentum and energy transfer of the scattering process. The theory predicts a broad inelastic peak near the center of the Brillouin zone. Actual experi- ments carried out o n C e C u 6 (Aeppli et al. 1986), UPt3 (Aeppli et al. 1987) and U B e 1 3
(Mook 1987) reveal such a broad peak near the Brillouin zone boundary. This is evidence that the one-band model is less than adequate for the real material. We will show next that a two-hybridized-band model is needed to understand these results.
3.1.2. The two-band model
Two different two-band models have been proposed by Liu (1988) and Auerbach et al. (1988) to explain the inelastic neutron scattering data in heavy-fermion systems.
The two models differ in how the bands are formed, and both mechanisms will be discussed in detail in later sections. We will show here that the main result, that the inelastic neutron scattering peak m a y appear at the zone boundary, can be understood on the basis of a generic two-band model. The two bands arise from the hybridization of a broad conduction band and a dispersionless f band. The phenomenological Hamiltonian of such a system is
H = • ekC~Ct,~ + ef Z f ~ f k ~ + V E (c*k~f*~ + h.c.), (21)
ka ka ka
where A~ is the f electron annihilation operator, ef is the energy of the f level, and V is the hybridization matrix element between the conduction and f electrons. The parameters are not necessarily those estimated from atomic or band-structure cal- culations, because renormalizations due to m a n y - b o d y correlations are expected.
It is straightforward to obtain the energies of the hybridized bands:
e~ -+) = ½ {e,,, + gr -+ [(ek - el) = + 4V211/2} • (22)
The two bands are shown schematically in fig. 19, in which the original unhybridized bands are shown in dotted lines. The bands become flat on both sides of er, so electrons in these parts of the bands have heavy masses provided that V is sufficiently weak. Assuming that the broad conduction band spans an energy range from 0 at the center of the Brillouin zone to W at the zone b o u n d a r y and the f level is at the middle of the band, ef = W/2, then for the physically interesting case V<< ef, we find that the top of the lower band occurs at the zone b o u n d a r y with energy Ef - A, where A = V2/% and the b o t t o m of the upper band occurs at the zone center with energy
~r + A. An energy gap of the size 2A opens up around ~r. Let N(e) be the density of
+1 a¢
¢ 0
1.1 1.0 0.9 0.8 0.7 0,6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1
0.0
f
[ I ! I
0.2 0.4 0.6 0.8 1.0
k
Fig. 19. Schematic representation of the two hybridized bands (solid curves) for the simple two-band model. The unperturbed broad band is represented by the dashed curve and the f level by the dashed line. The wave vector k is measured in units of the zone b o u n d a r y vector, and the energy is in units of the total bandwidth.
PHENOMENOLOGICAL APPROACH TO HEAVY-FERMION SYSTEMS 111 states of the unhybridized b r o a d band, then N_+(e) for the hybridized bands are
V 2
These functions are shown schematically in fig. 20 for the case of small V. Both peak sharply at or near the edges of the gap, which indicates that the band mass is vastly enhanced over the mass of the broad band, by an a m o u n t measured by ef/A.
In the two publications mentioned earlier the authors proposed different mecha- nisms for the formation of the two bands, and they put the Fermi level within different bands. In our phenomenological discussion we can use either model and obtain the same result, as guaranteed by the symmetry between particles and holes.
Thus, for the sake of argument, we will put g slightly above the b o t t o m of the upper b a n d so that this band is slightly occupied while the lower band is full. The dynamical susceptibility function now has two components, the intraband c o m p o n e n t Z~+ and the interband c o m p o n e n t Z~+ defined by
v~(a) ~(b)
(o) _ '~k - ' ° k + q ( 2 4 )
Za b (q, (_O) -- 2 (b) ~(ka-) S (~.-}- il5)' k g'k+q
where a, b are band indices and ,,Ca) ,,(b) "k ,"k are the occupation numbers of the bands. In particular, Z~+ (0, 0 ) = N+(/~) as found previously, where N +(/~) can be found from eq. (23). The interband c o m p o n e n t at zero energy transfer is large for Q between the center and the b o u n d a r y of the Brillouin zone because the energy d e n o m i n a t o r is small. We can estimate that
Z(°)(Q, O) ~- 1 - N~
2A ' (25)
where N e << 1 is the total occupation of the partially filled upper band. To obtain some feeling for the relative sizes of the two contributions, we consider a very simple band model for which the density of states N(e) is a constant, which we denote by N(0). The b a n d can a c c o m m o d a t e one conduction electron per spin such that N(O)W = 1. We find, after some simple algebra that
Z(_°~+ (Q, 0) 1 - Ue
Z(+°)+ (0, 0) - (1 - 2 N ~ ) 2" ( 2 6 )
8.0
6.0
v 4.0 z
2.0 0.0
-0.1 0.0 0,1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Fig. 20. The density-of-states curve of the two-band model in fig. 19.
The ratio is greater than unity for any finite but small Ne, which means that the total susceptibility Z(°)(q,0)= Z{+°)+(q,0)+ Z~)+(q,0) has a peak at or near Q. For an interacting gas the susceptibility at the peak will receive the highest amount of enhancement. In case Z(°)(q,0) peaks at Q and V1Z(°)(Q,0)> 1, the system will be magnetically ordered at sufficiently low temperatures with a simple antiferromagnetic structure, as seen in most heavy-fermion antiferromagnets.
If the stable state is paramagnetic, i.e.
V1Z~°)(Q,O)<
1, the neutrons can excite paramagnon type of modes with momenta near the zone boundary point Q. The scattering cross-section is given by the imaginary part of the enhanced susceptibility given by eq. (19). The qualitative nature of the line shape can be seen from the unenhanced susceptibilityIm Z(°)+ (Q, 00) = ~ (n~+)Q - n}, +))6(~ +) - ~k+)o - 00). (27) k
If we measure the Fermi level # from the bottom of the lower flat band, the quantity in eq. (27) can be put in the closed form
Z(°)+ (Q, 00) = coA2 [/ ( - A - # - 00/2) - f ( - A - # + 00/2)3, (28) Im
w i t h f ( x ) = 1/(e px + 1),/3 =
1/kBT.
A typical line shape is shown in fig. 21. Both Liu and Auerbach et al. showed that the observed neutron scattering line shape can be well fitted by this theory.The new paramagnon mode determined by the enhanced susceptibility at the zone boundary point Q has little dispersion, so it is not responsible for the T 3 In T contribution to the specific heat. If the broad band has a more complex density of states curve N(g), it is perhaps not unthinkable that for certain values of No or # the enhanced system may prefer the ferromagnetic state rather than the antiferromagnetic state, as has been found in some heavy-fermion materials. This would suggest that suitable adjustment of the Fermi level by doping can cause a material to have different magnetic states. This possibility deserves further investigation.
;)4
E 0.2
0 . 1 - -
0.0 0.0
T = 0 . 1
1.0 2.0 3.0 4.0 5,0
Fig. 21. The theoretical neutron scattering cross section under constant-q scan as predicted by the two-band model. The m o m e n t u m transfer q is at the b o u n d a r y of the Brillouin zone. The temperature is measured in units of A, one-half of the hybridization gap.
PHENOMENOLOGICAL APPROACH TO HEAVY-FERMION SYSTEMS 113 3.1.3.
Local-density-functional band calculations
The band calculations with potentials generated from local density functional approximations give the most detailed one electron band structures. As was discussed in section 2, one obtains Fermi surface dimensions in good agreement with experiments.
We will mention briefly the basic principle of the calculation in order to illustrate that the calculation is surprisingly unsophisticated in terms of many-body correlation effects to be discussed later.
The band calculation is based on the one-electron approximation to the many- electron problem for metallic solids. The one-electron wave functions 6,k(r) and energies e,k are eigenstates and eigenvalues of the Schr6dinger equation:
2too +
V(r)
O,k(r) = e,kO,k(r), (29)where m o is the bare electron mass, the index n labels the bands, and the vector k represents the wave vector or crystal momentum. The potential
V(r)
is the sum of three terms,V(r) = VN(r ) --]- VH(r ) -~- Vxe(r ). (30)
The nuclear potential V N is the electrostatic potential due to the nuclear charges. It is given by the expression
Zle2
(31)VN(r) I