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Chapter 6

Theory of in Heavy

6.1. Review of theoretical models

In 1936, Neel [ 1] had predicted the conditions under which an antiferro- would show an abrupt decoupling between the direction of antifrromagnetism and the easy axis, known as "-flopping" or " turnover". Later, he extended [2] this analysis to behavior termed "metamagnetic transition" [3] in which an antiferromagnetic array abruptly transforms to a fully aligned ferromagnetic array at a critical field. In this review, we will concentrate on theoretical models applied to metam- agnetism in heavy fermions. Evans [4] used band descriptions to explain the metamagnetic transition in CeCuzSi 2 [5-7]. Suvasini et al. [8] have used a fully relativistic spin-polarized linear muffin- orbital (SPRLMTO) method [9, 10] to make first-principle local density approximation (LDA) band structure calculations with the magnetic field H above the metamagnetic critical field H c. They have tried to explain the metamagnetic transition in UPt 3 [11, 12] using this method. Norman [ 13] has solved the Eliashberg equations for magnetic fluctuations for the case of UPt 3. He has used the neutron data on UPt 3 of Aeppli et al. [ 14] which had indicated antiferromagnetic correlations in UPt 3. The spin-fluctuation theory concentrates almost exclusively on spins, and heavy feature only implicitly via a large linear term in the electronic specific heat calculated from the spin-fluctuation spectrum. Preliminary proposals for unified theories have been made by Coleman and Lonzarich [15] as well as by Edwards [16].

6.2. Strong-coupling spin-fluctuation theory in the high-field state

Later, Edwards [17] had proposed a theory, based on the Anderson lattice model, for the onset of metamagnetic transition, by using a strong-coupling spin-fluctuation theory via , in the high-field state. An outline of Edward's theory is given below. In the Anderson lattice model,

H = ZSknka + EfZnia at- gZniTni$ nt- Zgk(ctkafka + fltaCka ), (6.1) ka i i ka

115 ] ] 6 Chapter 6. Theory of Metamagnetism in Heavy Fermions where the symbols have been explained in earlier chapters. In a Ce system, the f-level Ef is well below the with a doublet ground state separated from the first excited ionic state by considerably more than the impurity Kondo temperature. The spin is a pseu- dospin 1/2 labeling the doublet states but Edwards [17] has considered it as a real spin 1/2. Edwards considered the Anderson lattice in the high-field state, due to an applied field H a, and assumed that the number n~ of $ spin f electrons per was negligible. The 1' spin self-energy ET= 0 and thus the T spin quasiparticles have no mass enhancement. In order to calculate the + spin self-energy E+, and the one-particle properties, the Green function can be written as

1-1: (6.2) Gc~f Gc~c -V k E-~ k

Using techniques similar to that for Hubbard model [18], the perturbation in U can be obtained from

~+(k) = Un T -4-U 2 df (2rc)4d4k' i 7(k+;k'T)GTff (k')zf (k - k'), (6.3) where k = (k, E), j~f is the f-transverse susceptibility and 7 is an irreducible vertex func- tion. The vertex function was approximated by Edwards, using a Ward identity [18, 19], Z+(k) ~,(k ~k'T) = ~,(k$;kT) --- (6.4) (Un T)

Here, /-]7 is the renormalized electron- vertex function, zf(q, CO) is approxi- mated by a single pole at the magnon energy 2#BHa + COq, and COq is neglected as a first approximation. This form of E+ is shown in Figure 6.1, the second order of which is the high-field analog of paramagnon (spin-fluctuation) theory.

l l

Figure 6.1. Diagrams(excluding Hartree-Fock one) for E+. The wavy line is a magnon. Reproduced with the permission of Elsevier from Ref. [ 17]. 6.3. Metamagnetic transition in a small cluster t-J model 117

Edward's calculations [17] on the high-field state of the Anderson lattice can be sum- marized as follows: (i) most of the ~, spin spectral weight is at Ef + U, but some are above the + in the form of flattened quasiparticle bands below an electron- magnon scattering continuum for E > 2#alia; (ii) the + spin quasiparticle mass increases with decreasing field; (iii) the onset of a metamagnetic transition from the high-field side is marked by a steep upturn in the rate of increase of the $ spin Fermi radius with decreas- ing field, which occurs when 2#BHa is of the order of kBTK; (iv) in the small V limit, the conduction-band self-energy reduces to the second-order perturbation result obtained with a Schrieffer-Wolff coupling 2V2/IEfl)S.a; (v) the mass enhancement obtained by this localized spin model is also due to f hybridization at the Fermi surface.

6.3. Metamagnetic transition in a small cluster t-J model

Freericks and Falicov [20] have used the t-J model and calculated the many-body eigen- states of a small cluster in a magnetic field. They have shown that antiferromagnetic superexchange favors low-spin arrangements for the ground state while a magnetic field favors high-spin arrangements. The transition from the low-spin ground state to a high- spin ground state, as a function of magnetic field, passes through a range where there is a peak in the many-body which qualitatively describes the metamagnetic transition. The Freericks-Falicov theory [20] can be summarized as follows: the physics of the heavy system is described by the lattice Anderson impurity model [21 ]

t I HA= Zekatkoako+ef Zfi~fio + UZfitfitfi+fi + k,a i,a i + Z [V~kf~;ako + V~ka" ~of~ ] (6.5) i,k,a in the large-U (U ~ ~) limit. All these parameters and operators have been described in Chapter 2. The hybridization matrix elements are assumed of the form exp(iRi'k)Vg(k) V/k = ~ , (6.6) where g(k) is the form factor, which is a dimensionless function of order 1, and N is the number of lattice sites. The Fermi-level E F is defined as the maximum energy of the filled conduction-band states, in the limit V ~ 0. The origin of the energy scale is chosen at E F = 0 and the conduction-band density of states per site at the Fermi level is defined as p. The Anderson Hamiltonian (6.5) can be mapped onto the large-U limit of the Hubbard Hamiltonian [22] which, in turn, may be mapped onto a t-J model [23-25]:

Ot-j = --Ztij( 1 -- f/~-af/-a)f/~fja( ] -- fJ-af j-a) -3t- ZJijSi'Sj (6.7) i,j,a i,j 118 Chapter 6. Theoryof Metamagnetism in Heavy Fermions

Here, the unrenormalized (bare) hopping matrix tij satisfies

tij = ~ Vi*~Vjkk - e - N (8,g2(k'----~)~,) (6.8)

and the antiferromagnetic superexchange is defined to be Jij -- 41tij 12/U. The canonical transformation which maps the Anderson model onto the t-J model is valid only within a narrow region of parameter space [20]. The lattice Anderson impurity model (equation (6.5)) has been studied for various small clusters with at most four sites and has been discussed extensively in Chapter 2. This approach to the many-body problem begins with the periodic crystal approximation with a small number of non-equivalent sites. The results for the tetrahedral cluster [26-32] (with one electron per site), extensively discussed in Chapter 2, indicate the for- mation of the heavy fermion state and its sensitivity to variations in parameters. When the band structure e~ is such that the bottom of the band is at the F point of the fcc , the ground state is a spin singlet (for a small range of e) with nearly degenerate triplet and quintet excitations. The specific heat has a huge low-temperature peak and the is large. A magnetically ordered heavy fermion state is sometimes observed when F is the top of the conduction band. The mapping of the Anderson model onto the t-J model (equation (6.7)) reduces the Hilbert space by a factor of (3/16) N, which allows larger clusters to be studied. A good example of treating the heavy fermion system in the t-J model lies in an eight-site fcc lattice cluster with seven electrons [33, 34]. The many-body states are degenerate at J = 0 (J~ = J when i,j are first nearest neighbors but J = 0 otherwise) but the degeneracy is lifted for finite J, with low-spin configuration favored (energetically) over high-spin con- figuration. A magnetic field in the z direction lifts the degeneracy even more since the many-body eigenstates have the energy

E(B) = E(O)- mzglgBB -- E(O)- mzbJ, (6.9) where m z is the z-component of spin, B the magnetic field, g the Lande g factor, ~B the , and b dimensionless magnetic field. The high-spin eigenstates are ener- getically favored in a strong magnetic field and level crossings occur as a function of b. The heavy fermion system is described by a ground state with nearly degenerate low- lying excitations of many different configurations. The antiferromagnetic superexchange pushes high-spin states up in energy with splittings on the order of J. The magnetic field pulls down these high-spin states and generates level crossings in the ground state. The magnetization and spin-spin correlation functions both change abruptly at the level crossings. This gives rise to the metamagnetic transition in heavy fermions. We show the results of magnetization at temperature (the details of the method of numerical calculations are given in Ref. [20]) fixed at T = J/k B (high-temperature regime) in Figure 6.2. The magnetization smoothly changes from a value of 0 to 5/2 as a function of magnetic field, showing little structure. The magnetization as a function of magnetic field at temperature T = J/5k B is shown in Figure 6.3. Since the temperature is smaller than the energy-level spacing, the magne- tization shows steps at the various level crossings. 6.3. Metamagnetic transition in a small cluster t-J model 119

n

_

R m

I I I I I I I I I I 0 1 2 3 4 5 6 7 8 9 10

Figure 6.2. Magnetization as a function of magnetic field at T = J/k B for the heavy fermion model. Reproduced with the permission of the American Physical Society from Ref. [20].

These theoretical results obtained by calculating the magnetization from the many- body states qualitatively agree with the numerous experimental data (see Ref. [20] for details) on metamagnetism in heavy fermion systems. Tripathi [35] has proposed, starting from a variant of the Hubbard Hamiltonian, a model for the magnetic-field-induced metamagnetic transition in an itinerant electron system. The coupling of the applied field with the spin magnetic moments of the elec- trons is described by two parts--a part which splits the spin degeneracy and the other which is responsible for spin fluctuations. The damping of the spin fluctuations at the high-field state coupled with the strong electron-electron interactions was shown to be responsible for the onset of metamagnetism. However, there was no attempt to compare the results with any experimental observation. Later, Tripathi et al. [36] extended this model to include electron- interaction which was found to suppress metamagnet- ism in an itinerant electron system. They also considered the effect of electron-phonon interaction on the low-field spin susceptibility by considering electron self-energy as a function of frequency and magnetic field. They found that the modification brought about by the field dependence of the self-energy is cancelled by the mass enhancement arising out of the frequency dependence of the self-energy. Ono [37] has investigated the metamagnetism of heavy fermions using the periodic Anderson model (PAM). His theory includes anisotropic hybridization and electron-lattice coupling. The latter coupling in the model appears to be responsible for an enhancement of metamagnetization and a large magnetostriction effect. Ohara et al. [38] have studied 120 Chapter 6. Theory of Metamagnetism in Heavy Fermions

m

._A

m

:Z

_

0 I I I I I 0 1 2 3 4 5 b

Figure 6.3. Calculated magnetization as a function of magnetic field at temperature T = J/5k B (low-temperature regime). Reproduced with the permission of the American Physical Society from Ref. [20].

the metamagnetic behavior of CeRuzSi 2 by using the periodic Coqblin-Schrieffer [39] model with anisotropic c-f exchange interaction in the mean-field theory. The metamag- netic transition is identified with a singularity of the at the field of metamagnetic transition.

6.4. Competition between local quantum spin fluctuations and magnetic-exchange interaction

Satoh and Ohkawa [40, 4 l] have shown, based on the PAM, that the competition between the quenching of the magnetic moments by local spin fluctuations and a magnetic- exchange interaction caused by a virtual exchange of pair excitations of quasiparticles in spin channels is responsible for the metamagnetic crossover in CeRuzSi 2. They have also included the effect of the electron-lattice interaction. The strength of the exchange inter- action is proportional to the bandwidth of quasiparticles and its sign changes with increasing . They have found that it is antiferromagnetic in the absence of magnetizations but ferromagnetic in the metamagnetic crossover region. They have reproduced the experimental results of static quantities. 6.4. Competition between local quantum spin fluctuations and magnetic-exchange interaction 121

Their results can be summarized as follows. The PAM in a magnetic field H can be written as

H = EE,I (k)a~k~a,~ko + E(Ef - o'O* )f~kafka 2ka ka +E[V;~ (k)a]kafka + H.C.] + 1 UZ?.liarli_a ' (6.10) 2ka io-

t with 2 the band index of conduction electrons, rti~r -- fiaf/o., and H* = m0H, where m 0 is the magnetization per f electron. The kinetic energy of conduction electrons E~(k) and the f-electron level Ef are measured from the chemical potential. In order to treat the local quantum spin fluctuations which are responsible for the quenching of the magnetic moments, Satoh and Ohkawa [41] have used the single-site approximation (SSA) [42] which is rigorous for paramagnetic states in infinite dimensions. Within the SSA, Green's function for f electrons and conduction electrons is given by

Gff~(ie~,k) = 1 , (6.11) ie~ - E e - E~ ( ie, ) - Z (( V~ (k) 2)/(ien - E~ (k)) where Z~ is the single-site self-energy function and

G;~,~,,~(ie,,,k) = 6,~,~,&(ie,,,k) + g,~(ie.,,,k)V,~(k)Gffo(ie.,,k)V~,(k)g,v(ien,k ), (6.12) and &(ien, k) = [ign-E2(k)] -1. Here n is an integer and ie, an imaginary . The single-site self-energy function is obtained by solving a single-impurity Anderson model (SIAM) [42] which has the same Coulomb repulsion U and the same localized electron-level Ef as in equation (6.10). This is called a mapped Anderson model (MAM). The parameters of the MAM are determined through the mapping condition,

dff~(ie,,) = N1 y]Gff~ (ien,k), (6.13) k where N is the number of unit cells, and

Gffa (ien) = 1 , (6.14) i~,, - Ef-~o (ie,, ) - L(i~,, ) with L(ien) = (1/rc)~ deA(e)/(ien - e). Here A(e) is the hybridization energy of MAM. ~(iG) is obtained by solving the MAM numerically once a trial function for A(e) is given. Satoh and Ohkawa [41 ] have used the well-known results of the Kondo problem. They have used a Fermi- description to expand the self-energy function

Z~(ie~) - Z~(+iO) + [1 - @m]ign + "", (6.15) 122 Chapter 6. Theoryof Metamagnetism in Heavy Fermions for small len, where (~m is a mass enhancement factor in the SSA. In the symmetrical case, the coherent part of equation (6.11) can be written as

(C)(ie,,,k) = 1 G fro- ~ (6.16) OmiSn--Z2 (( V2 (k) 2 ) / (i% - E~ (k)))

The quasiparticles are defined as the poles of equation (6.16), i.e., the of quasiparticles are obtained by solving the following equation:

]Vx (k) ~ @mZ -- Z S---'~),?~) --- 0. (6.17) 2

The solutions are written as z = ~v(k) with v representing the branch of quasiparticles. It can be shown that in a magnetic field [41 ],

~ Vx(k) 2 @mz - 6Z~(m) - Z z_E~(k ) = 0, (6.18) 2 with fiZz(m) a magnetic part of the self-energy and m is the magnetization m = E~a(f/J/~). Solutions of equation (6.18) are denoted by ~v~(k, m). Thus, for the finite-field case, all arguments can be developed in parallel with the zero-field case. Comparing equations (6.17) and (6.18), they obtained [41]

~v, (k,m) ~ ~v (k) - ~rAE(m), (6.19) where

~rA E(m) = -6s (m) (6.20) Om "

It can be shown [41 ] that with the use of equation (6.20), the Luttinger's theorem gives

m = Zafde pf[e + aAE(m)], (6.21)

where the polarization of conduction electron has been ignored and p* has been replaced g: with pf, since the right hand side of equation (6.21) is a difference between the contribu- tions of spin from up and down directions and hence only the low-energy part is relevant. AE m, the magnetic part of the self-energy, can be determined as a function of m from equation (6.21). 6.4. Competition between local quantum spin fluctuations and magnetic-exchange interaction 123

Satoh and Ohkawa [41 ] have also studied the magnetic exchange interactions working between quasiparticles. They have shown that in the symmetrical case, the magnetic susceptibility can be written as

Zs (i~ q,m) = Zs (i~ , (6.22) 1 - (1/4)Is(iOol,q,m)~s(iOol,m) where Is(ifo1, q, m) is the intersite interaction and ~,s(iO9~, m) the magnetic susceptibility of the MAM. Equation (6.22) is consistent with the physical picture of the Kondo lattice that local quantum spin fluctuations at each site are connected with one another by the intersite interaction. It can also be shown that the main contribution to I s is divided into two parts [43]

Is(i~ = Js(q) + JQ(i~ (6.23)

Here Js is an exchange interaction caused by the virtual exchange of high-energy spin excitations which do not depend on m. It consists of the conventional superexchange inter- action, an extended superexchange interaction, and a Rudderman-Kittel-Kasuya-Yosida (RKKY) interaction. Since Js depends on the whole band structure, Satoh and Ohkawa [41] have treated it as a phenomenological parameter. JQ is due to the virtual exchange of low- energy spin excitations within quasiparticle bands. They have also studied the static and uniform component of Jo abbreviated to JQ(m, x). They have also shown that the volume dependence of JQ(m, x) is the same as that of T~, i.e., JQ(m, x) = e-xJQ(m, 0) and the mag- nitude of JQ(m, x) is of the order of kBTK. Since JQ(m, x) is scaled with TK, the bandwidth of the quasiparticles, it is possible to do a single-parameter scaling [40]. In Figure 6.4, the theoretical results of magnetization and magnetostriction, calculated by Satoh and Ohkawa [41], for CeRu2Si 2 by using a parameterized model, are compared with the exper- imental results [5.44]. Muoto [45] has studied the PAM away from the half-filling in the framework of the self-consistent second-order perturbation theory. He has studied the concentration dependence of the density of states spectrum, spin and charge susceptibilities, and mag- netization processes. He has shown that the coefficient of the electronic specific heat and the spin susceptibility are both enhanced by strong correlations. Although the study does not directly pertain to metamagnetic transition, the magnetization curve and the above quantities have some bearing with the metamagnetic transition. Meyer et al. [46] have investigated the SIAM, which has gained much interest in the context of the dynamical mean-field theory (DMFT) (discussed extensively in Chapter 3), by introducing a modified perturbation theory (MPT). Their approximation scheme yields reasonable results away from the symmetric case. Meyer and Nolting [47-49] have addressed the problem of field-induced metamagnetic transition and tried to compare the results with that observed in CeRu2Si 2. They have used DMFT in com- bination with MPT to calculate the magnetization and the effective mass. However, their work is not free from ambiguities as regards the choice of the chemical potential. The results are qualitative and the calculated field at the metamagnetic transition is several hundreds of Tesla which is much too large compared with the experimental value. 124 Chapter 6. Theory of Metamagnetism in Heavy Fermions

0.6 0.4 ,qq,~

0.4

0.2

0.2

0 5 10 15 H [T]

Figure 6.4. Magnetization and magnetostriction of CeRu2Si 2 calculated in a parameterized model. Experimental data are shown in dots and circles, respectively, from Refs. [5] and [44]. Reproduced with the permission of the American Physical Society from Ref. [41 ].

6.5. Itinerant electrons and local moments in high and low magnetic fields

The foregoing remarks suggest that in spite of an extensive activity in metamagnetism, no consensus has been reached about mechanisms that contribute to this phenomenon. We shall present a theory of metamagnetism [50] starting from a variant of the periodic Anderson Hamiltonian in the presence of an applied magnetic field. In Section 6.5.1, we consider equations of motion for the magnetization and spin-fluctuation amplitudes in the high-field state where damping effects are considered phenomenologically. We solve these equations in Section 6.5.2 and obtain a non-linear equation for the magnetization as a function of the applied magnetic field. In Section 6.5.3, we analyze the itinerant elec- tron in the presence of local moments and, in the process, derive an expression for the EPR (Knight) shift including the effect of electron-phonon interac- tions. In Section 6.5.4, we parameterize the model in order to compare with the experi- mental results for CeRuzSi 2.

6.5.1. The model

We consider a variant of the periodic Anderson Hamiltonian in the momentum-space representation"

H A = H c + Hf -t- Hhyb, (6.24) 6.5. Itinerant electrons and local moments in high and low magnetic fields 125 where

Hc= Z F'ckCkaCkat at-UcZ C~k+qT Ck TCk~'-q,~ Ck''~, (6.25) ka kk'q

Hf =Z e,fkfkafka t + gf Z fk+qTfkTfk'-qSfk'$'t ~ (6.26) ka kk 'q and

Hhyb = Z gk~ (CkafkaI" + fYk~Cko)" (6.27) ka

Here, H c and He describe the conduction- and the f-electron Hamiltonians. Cka(fkat t ) and Cu~(fk~) are the creation and annihilation operators for the conduction (f) electrons, respectively, eck and efk are the conduction- and f-electron energies while Uc and Uf are the respective correlation energies. The other terms have their usual meanings defined earlier and the f-electron spin is chosen same as the conduction-electron spin. When a magnetic field H is applied, the resulting interacting Hamiltonians for both the systems are

Hnc = #B Hz E (CkTCM"t _ Ckt $Ck $ ) nt- [2B Z (CkTCksHt + + Ckj,t Ck~fH_) (6.28) k k and

mHf ~-- ~BmzZ(ftkTfkT- ftkSfk$)-Jr~B 2(ftkTfksO+ nt- ftkjfkTO_ ). (6.29) k k

Here H_+ = H x +_ iHy; which are responsible for spin fluctuations, and H z is the z-component of the applied field which is responsible for the spin splitting of the energy levels. Assuming a mean-field approximation (MFA) for the electron correlation energies, the total Hamiltonian can be derived from equations (6.23)-(6.29) as follows:

HT =Z t3ckTCRTCM"f at- Z eck'[Ck'[Ck'["t "~- E efkTfITfkTt + Z e'fk+fi+fk+t k k k k -}-E[gT(CtkTfk ~ -Jr"f~,Ck, ) nt- gj,(ftkjCk+ nt- Ck+fk+)t ] k "Jl"~/lB Z [(Ck~TCk~, -Jr-ftkTfk $)H+ + (Ck+CkTt nt- ftk,fk T)H_ 1, (6.30) k where

eckT --8ck + Ucnc~k+ +/aBHz, (6.31) 126 Chapter 6. Theory of Metamagnetism in Heavy Fermions

eck~, -- /3ck -~- Ucnc~kt - ltBHz, (6.32)

efk T = elk + Ufn~k ~ -t- #BHz, (6.33)

et'k+ = ~fk + Ufn~k T -- #BHz (6.34)

Here Vt~ are assumed to be k-independent and have an implicit H z dependence. To the first order in H z,

Vr+ = V +_ V'H z. (6.35)

In fact, the hybridization parameter, Vk~, depends on the magnetic field through both the spin (a) and the vector operator (to) which can be expressed [51, 52] as

(6.36)

However, the magnetic field dependence through K is an orbital effect and is generally small compared to the effect of spin. In equations (6.31)-(6.35),

n ckl"+c = (Ckl"+Ckl"$), (6.37)

n~kt+ = (fkl"+fkt+)' (6.38) and

(6.39) 0

We have followed a non-perturbative approach [50] within the MFA and assumed

= 6k',6. O, (6.40) and

t t t t = (CkT+C!k+T)6.,O. (6.41) 6.5. Itinerant electrons and local moments in high and low magnetic fields 127

The validity of above approximations results from the fact that in a spatially homoge- neous system, the conservation of momentum holds good [53]. Since is a non-perturbative phenomenon, a non-perturbative approach within the MFA seems to be reasonable. As has been done in our previous work [35, 36], the down-spin electron anni- hilation operators Ck+ andfk + are assumed to be equivalent to the down-spin hole creation operators dr_k+ and bt_u+, respectively, for the conduction (c) and f electrons. Equation (6.30) can be written in the electron-hole representation as

ST -- Z/3ckT CkTCkT? -- Z/3ck,Ld_tk,Ld_k,L nt- Z~fk,ftkTfk,r - ~~-~e'~lbtkl_ ,,-,,-J,b k k k k k

+Z[VT(CtkTfkT-]-fgtCkT ) + V$(d - k ,~ blk- ], + b_ksd -k~,)]I" k + Z/t B[H+~,'-'k,(C,td -k~,t +ribJkT -kSt ) "1- H-(d-MCkT + b-ksfkT)]' (6.42) k and the necessary changes in equations (6.31)-(6.34) are

eck 1, : eck + Uc[1 - nc~ (-k)] + #BHz, (6.43)

/3ck~, : /3ck + Ucl'/eT (k)- #BHz, (6.44)

efkT = en, +Uf[1 - nfh+(-k)] + #aHz , (6.45) and

efM = em + Ufn~T(k ) - t2BHz . (6.46)

In equations (6.43)-(6.46),

neT (k) = (Ck~TCk ), (6.47)

nc~ (-k)= (d1"k-~,dk-,L)' (6.48)

n~l, (k) = (f~tfk~), (6.49) and

nfh~(-k)= (b* k b k ). (6.50) 128 Chapter 6. Theoryof Metamagnetism in Heavy Fermions

These are distribution functions for c- and f-type electrons and holes. The convenience of equation (6.42) is that the entire Hamiltonian is expressed in the k-space. Here Ctktd*__k and Ju+r, b*-k+ are the spin-fluctuation operators for c bands and f bands. Particle-hole symmetry occurs when particles are mapped onto holes. However, we have mapped only the down-spin electrons on down-spin holes, keeping the up-spin electrons intact. Therefore, negative sign appears before the down-spin energies. The up-spin energies involve quantities like (1 - n~). The mapping only becomes a symmetry when both n~ = 1/2 and the number of holes equals the number of particles.

6.5.2. High-field ferromagnetic case The equations of motion for the number and spin-fluctuation operators can be obtained, using Heisenberg's equation [50]:

d i ---7--F = - 2-[F, H T] (6.51) dt h as

d ? 7 (CkTCkl. + d1k_,d_ ~k ) i = --~[2#B(/C*\ k* d' -kS )H 1 -(d_kj, CkT)H_)+ {VT(ckTfkT ~ -- fk?CkT} + ~ (b-V*kS d -k* -d*-k,b-k* )}] + ~cn (CkTfk* T + dtk-,dk$ )' (6.52)

d(fkt?fkT + btk b k ) dt -*-~ t -- __ h[2#Bgrt\JkT b-k* "r }H+- (b_ksfk T)H_ + {__(__CkTV,fktT __ CkT.,,Tf._/ , + V$ (d!ksb_k$I" -- b -k~d-k~}}]t + ~fn (fktTfkT + btk b k ), (6.53)

d t , i t t -~t(CkTd:k$) -- ---h[{(gk~ -'[" V,l)- (/3kT + VT)}(CkTd~-k,],> (6.54) + #BH_{(CtkTCkT + dtk_tdk_+ )- 1}] + ])c,eh(CkT r -k,), and

d t , = --~[{(gfki ~ + V$)- (gfkT -~- VT)}(fkrT -kS>t + #BH_{(ftkTfkT + btk_j,_$b k )-- 1}] + ~f,eh\dkl'/r b -k,L/" t \ (6.55) 6.5. Itinerant electrons and local moments in high and low magnetic fields 129

Here, equations (6.52)-(6.55) have been obtained from equation (6.51) by making the assumption [50] that the average value of the four types of electron-hole operators is the same since we are interested in the variation of the magnetization as a function of the magnetic field:

c,* d t {r b t /c,t b t {r d t -k,/ = \Jk? -k, / = \'-'kT -k~ / = \JkT -k~ /" (6.56)

In addition, the constants 7cn, 7fn represent dampings of the c and f magnetizations and 7c,eh, '~f,eh represent the dampings of the spin-fluctuation amplitudes. Metamagnetic transition is associated with a sharp rise in the magnetization and/or a hysteresis which is a non-equilibrium phenomenon. However, the system relaxes through the damping of the number and the spin-fluctuation amplitudes. These dampings occur partly due to the collision processes among the fermions (electrons and holes) and among the spin (electron-hole pairs). Since the damping mechanisms are sta- tistical in nature and are due to random processes, these have been included phenom- enologically [50], which can also be justified from first principles by coupling the system to a heat bath [54]. Since equations (6.52)-(6.56) denote equations for physical quantities, for simplicity, we assume [50]

(fktTfkT "4- b_~k,l,b_k,l, } -- "efT -Jr- r/hf,L' (6.57)

(CktTCkT 2r-d~k- $d - k)-rtecT$ + nhc$, (6.58)

JkTr -k~ }=A~ (6.59) and

C kTt d -kT)t = ac*0" (6.60)

Here nec T and nhc $ describe the distribution functions of up-spin electrons and down- spin holes in the conduction band; and t/ef T and nhf $ represent the distribution functions for corresponding electrons and holes in the f band. The explicit time dependence of these quantities is

A [o = a tfb eimt , At = t,._.'f.,iogttcd~ , (6.61) H~ = h_T_e+-i~

Using equation (6.61) in equations (6.54) and (6.55), we obtain

/~B h_ (1 -- mck ) 4= (6.62) hco - (1/N)U c Z(1 - mck,) + V~H z + ihTc,eh k' 130 Chapter 6. Theoryof Metamagnetism in Heavy Fermions and

#B h_ (1 - mfk ) (6.63) h09 -- (1/N)Uf ~-'a(1 - mfk, ) -k- VuH z + ihTf,e h k' where

V~, = - 2(V' + #a ), mck = nec T(k) + nhc $(-k), and (6.64) mfk = nef T(k) + nhf $ (-k).

Here, mck and mfk are the average magnetization functions in dimensionless forms for conduction and f bands for strong ferromagnets in which the up-spin electronic bands are occupied and down-spin bands not occupied, i.e., these are filled with down-spin holes. Using equation (6.61) in equations (6.52) and (6.53), one obtains

]~enmc k -- _ -~i 2~B (aJdh+ _ acdh _) _ "~i (WTAhdy nt- WsAhdh ), (6.65) and

i i 7fnmfk = ---~2#B(atfbh+ -- afbh_ ) + -~(VTAhdy + V+Ahdh). (6.66)

Here,

f Ahd~ = A~y - Ah~ ~ (C kTfkT - fiTCkT)~f (6.67) and

Ahdyh = Ahy -- Ahhyt=(b_tk~d_k+ -- dkTbk~t ). (6.68)

We have [50] also made the following assumptions:

Uc(1- mck ) --~ Uf(1- mfk) ~ Uav(1- mav ), (6.69) where

1 Uav -- -~(U c --[- Uf ), (6.70)

1 mav = -~(mck + mfk), (6.71) 6.5. Itinerant electrons and local moments in high and low magnetic fields 131 and

7c,eh "- ~f,eh -- ~)eh" (6.72)

From equations (6.62), (6.63), (6.65), and (6.66), we obtain

E 4/t27ehh2Y~k[2- mt(k)] ])m kmt(k) = [(VuHz/h])eh)-((U c + Uf)/4h])eh)(1/N)~k{2--mt(k)}] 2 + 1' (6.73) where

mt(k) = 2mav(k ), (6.74)

h 2 = H 2 + By2 =h 2 - H z2 , (6.75) and

~m = ])en + 7fn, (6.76) where ~m is the damping of the magnetization. From equations (6.72)-(6.75), we obtain[ )2] 4#2h 2 : 4#2_____ 2z2z + m 1 + V~Hz {Uc -]- Uf } {ms _ m} (6.77) h2])eh])m h 2 ])eh~m (m s - m) hTe h 4hTe h where

1 m - ~ mt(k) (6.78) N#B k and 1 Z m s - - 2. (6.79) NPB k

Here, N is the number of unit cells, and m and m S are the total and the saturation mag- netic moments, respectively, of the atom in the units of #B- We note from equation (6.77) that h --~ m is non-linear even in the absence of H z, and is driven by the transverse part of the magnetic field. For low H z, m is related to the Pauli spin susceptibility. In the next section, we analyze the low-field paramagnetic susceptibility of itinerant electrons in the presence of c-1 hybridization. 132 Chapter 6. Theory of Metamagnetism in Heavy Fermions

6.5.3. Low-field paramagnetic susceptibility

We shall analyze the low-field paramagnetic susceptibility in the limit of H z ~ 0, in the presence of many-body and conduction-electron moment and local-moment interactions. The one-electron Green's function ~(r, r', H z, p, (1) in the presence of magnetic field H z, localized magnetic moments/t z satisfies the following equation:

(~t - 7-[)~(r,r',Hz,lX,~l) = 6(r - r'), (6.80)

where

7"[ = 7[ 0 + 7"Is_f + ]~(r,r',Hz,#,(l ) + 7-[nz. (6.81)

In equation (6.81), 7% is the unperturbed Hamiltonian, ~l = (2/+ 1)i~/2 + #; 1 = 0, ___1, _ 2 ..... # is the chemical potential; the mean-field hybridization Hamiltonian,

1

'~"['s-f= 7 Z J(r -- R i )(7 z (Siz), (6.82)

7-~ z = #a azHz , (6.83)

and ]~r is the proper self-energy which, when expanded in powers of H z and (Siz), becomes

~_.~, = IF.,0 + Hz~.. 1 -JI- Z(aiz)~_.2i -~- HzE2 3 , (6.84) i

where =(O2r Zl t-D-h-z ).z o (6.85)

0ECt ]~2i m (6.86) O( Si z ) (S/z)-,o

and

( 02Z~ ) Z3 = OH2 nz (6.87) ~0

Assuming

y (S/z) = - 1 ZcwHz , (6.88) i 2#B 6.5. Itinerant electrons and local moments in high and low magnetic fields 133 equation (6.81) can be written with the help of equations (6.82)-(6.84) and (6.88) as

7-/= ~o + 7-/', (6.89) where

p2 7-/0 - + leo, (6.90) 2m and

7"/r = []ABO'z -[- ]~1 "[- 271{2 B Zcw ~'(r- R)~rz + s 2 }1H z + H2z~3 . (6.91)

Here Zcw is the Curie-Weiss susceptibility, ,.Tthe exchange integral, and ~rz the Pauli spin matrix. We now consider the thermodynamic potential [55-59]

f~ = ~[Tr ln(-~r ) - TrE~,~ l + q)(~l )]' (6.92) where (~l) is the abbreviated notation for the one-particle Green's function. Tr is defined as EITr, where trace refers to summation over a complete one-particle set, and the func- tional ~b(~r ) is defined as [55-59]

go(Gr ) = lim,~ 1Tr Z 2n (6.93) 2n n

Here ]~(n)(~r is the nth-order self-energy part, where only the interaction parameter 2 occurring in equation (6.93) is used to determine the order, q~(~r is defined through the decomposition of E(n~(~r into the skeleton diagrams. There are 2nGct lines for the nth- order diagrams in qg(Gr Differentiating qg(~r ) with respect to ~l has the effect of "opening" any of the 2n lines of the nth-order diagram and each will give the same con- tribution when Tr is taken. The spin susceptibility is given by

Zs = aHz2 = Zqp + Zcorr, (6.94) H z --*0 where, from equations (6.92) and (6.94), one obtains

Zqp = 0nz2 rr ln(-~ l ) , (6.95) 134 Chapter 6. Theoryof Metamagnetism in Heavy Fermions and

1 [ 02Zr (6.96) Xco~ =~r~ OHz~ ~,+ OHz OHz

It can be easily shown [57-59] that

1 Tr ln(-~r ) 1 (~ .... tr qg(~)G(~)a~, (6.97) fl 2rti J C where C encircles the imaginary axis in a counterclockwise direction and

rp(~) = - 1 ln[1 + e -~(~-~') ] (6.98)

After some lengthy algebra [50], we obtain

kp

+ Zcw (kp ] #Baz + ~1 I kp)(kplVaz + E2 ] kP)f'(~kp)]

1 (Zcw)2Z(kPlazVe-i(k-k'"l~' + Z2 I k,p)(k,plazVei(k-k').,: 2pB2 kk'p

+Z21kP ) f(gko) . (6.99) /3kp -- ~k'p

Here, V = 1/2,.7, ]kp) = ~kp(r) = pe ik'r and is a plane wave state for spin p. f(ekp) is the Fermi function andf'(ekp ) denotes its derivative with respect to energy. The free-electron basis is valid only when the bare electron mass is replaced by the effective mass of the system [50]. The last term in equation (6.99) is an oscillatory term and arises from an RKKY type of interaction. Similarly, we can derive an expression for gcorr from equation (6.96) after some lengthy algebra [50],

Xco~ = Z[ 2 (kp I ~3 I kp>I(~) + kp • (kp t ~, + ~ 1 Zcw~:= I kp)f%,)+ 2--~(kpZcw I ~, I kp) • (kp3W~x + ~ I kp)f'(~,)]

+ 1 (~cw )2Z(kp i Z 2 i k,p)ik,plazVei(k_k,).R: 2/./B 2 kk'p

+Z2lkP ) f(gkp) . (6.100) gkp -- gk'p 6.5. Itinerant electrons and local moments in high and low magnetic fields 135

From equations (6.94), (6.99), and (6.100), after some more algebra, we obtain [50],

Zs = Z~ + Z~, (6.101) where

1 Z~ - Z~ (6.102) 1-o~

z ~ : -~-~(kP I ~,.~z [kp)(kpl#.az I kP)f'(ek.), (6.103) kp

)~ = XcwPs, (6.104) and

Ps -- 1 ~(kpla z i kp)(kplVa z I kP)f'(ekp) 1--~ kp 1_____~Zc___~w Z[(kp l azg l k,p}(k,P l azV I kp)ei(k-k').(al-a2~+c.c.] (1 -- 7) 4#~ kk'p x f(ekp) (6.105) ~kp -- ~k'p where

=-uj) f'( ek'P ), (6.106) k'p

7 = -- U Z f(ekp) -- f(~k'p) (6.107) k,k',p ekP -- /3k'P and U is defined through

1 - (6.108) Ct

The relation between U, E 1, and E 2 is clearly defined in equations (4.32)-(4.34) of Ref. [50]. Here c~ and ,/are the intraband and interband exchange enhancement parameters, )~ is the spin susceptibility of only interacting itinerant electrons, and ~ the contribution to the spin susceptibility from the c-1 hybridization. Ps is the EPR shift [60, 61] which is 136 Chapter 6. Theory of Metamagnetism in Heavy Fermions very similar to the Knight shift [57, 58]. The second term in Ps is due to an RKKY type of oscillatory interaction.

6.5.4. Results and discussion

Introducing dimensionless parameters and the relation m = Zr,I-Iz where Zp -- Zs[N~B, one obtains from equation (6.77),

______2___m b2 = gc2m2 + (m s - m) [1 + {(a - d)m + dms}2], (6.109)

where

b2= 4# 2h2 2#B h2~eh~ m ,c -- h~ehZP ,d = df ~- dc, (6.110) Uf/c 2V' ~eh df/c - ,a = c + v,v - ~,and g - 4h~eh hTehZP ~m

In order to compare the theoretical results with the experimental results observed in CeRu2Si 2, this model has been parameterized [50]. In Figure 6.5, the m (#B) versus the magnetic field h (T) has been plotted. The damping con- stants ~m and 7eh have been chosen as 5.71 • 107 s -1 and 1.5 • 1015 s -~. The values of c and df have been chosen to be 1.0 • 10 -4 and 2.8 as to give average value of ZP to be 1.17 • 10 -4 in cgs units and Uf to be 11.05 eV. v is chosen to be 0.015 so that V' = 150 #B. In Figure 6.5, the m versus h curve has been plotted for two values of Uc and the the- oretical results come closer to the experiment when Uf and Uc are of the same order which implies that both f electron and conduction-electron correlations are important. In Figure 6.5, we observe that the sudden rise in magnetization takes place roughly at 6.4 T, while the experimental value is about 7.8 T. In addition, in the low-field regime, the theoretical curve should look linear, in contrast to what it looks now. One of the rea- sons is that while a detail analysis of the paramagnetic susceptibility has been done in Section 6.5.3, there has been no numerical estimate of it. Instead, ~, has been used as a parameter even though the itinerant and the local characters of the electrons as well as the electron-electron correlations have been included in the theory. The parameterization for CeRuzSi 2 has been done mainly since different heavy fermion systems show different behavior. In addition, a unified picture for a clear and quantitative understanding of the effective mass as a function of the magnetic field is necessary before Zp can be calculated for a particular heavy fermion system. Despite qualitative and phenomenological treatments of some mechanisms, our model [50] explains some basic features of metamagnetism in CeRu2Si 2. The weakness of the model is the MFA and attempts are being made to improve the model by going beyond the MFA. References 137

1.4 - ,..__

Z A n

1,2 -

? . [ A n I, n A n ql

C An o 1 gm 0.8- i -

m~ 0.6-

0.4

0.2

0 L - I I I I I I I 0 2 4 6 8 10 12 14 h (Tesla)

Figure 6.5. Magnetic moment m (#B) versus magnetic field h (Tesla). The dotted curve represents the meta- magnetic behavior for U c = 8.68 eV and the bold curve for U c = 10.48 eV. The experimental curves are taken from Refs. [5, 12], and represent two temperatures (As represent T = 1.35 K and I--]s represent T = 4.2 K). Reproduced with the permission of Elsevier from Ref. [50].

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