Strong enhancement of the Edelstein effect in f-electron systems
SStrong enhancement of the Edelstein effect in f -electron systems Robert Peters ∗ and Youichi Yanase Department of Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: September 25, 2018)The Edelstein effect occurring in systems with broken inversion symmetry generates a spin po-larization when an electric field is applied, which is most advantageous in spintronics applications.Unfortunately, it became apparent that this kind of magnetoelectric effect is very small in semi-conductors. We here demonstrate that correlation effects can strongly enhance the magnetoelectriceffect. Particularly, we observe a strong enhancement of the Edelstein effect in f -electron systemsclose to the coherence temperature, where the f -electrons change their character from localized toitinerant. We furthermore show that this enhancement can be explained by a coupling between theconduction electrons and the still localized f -electrons. PACS numbers: 71.27.+a, 72.25.-b, 75.20.Hr, 75.85.+t
INTRODUCTION
Spin-orbit interaction, which leads to a coupling be-tween the spin of an electron and its momentum, providesthe possibility to manipulate the spin polarization of amaterial by applying electric fields as desired for spin-tronics. Particular interesting are lattices without inver-sion symmetry, where the antisymmetric spin-orbit cou-pling leads to fascinating transport properties[1] such asthe anomalous Hall effect[2–4], the spin Hall effect[5, 6],and magnetoelectric (ME) effects[7–14]. The latter leadsto a spin polarization without an applied magnetic fieldwhen an electric current flows, which has been also con-firmed in experiments[15–18]. Thus, the spin polarizationcould be controlled by electric fields, which would be atremendous advantage for memory storage devices[19].However, the ME effect in semiconductors with antisym-metric spin-orbit interaction is usually small, so that itcannot be effectively used in spintronic devices.An analysis using Fermi liquid theory has shown thatin interacting systems without inversion symmetry theME effect can be enhanced[20–23]. This is particularlyimportant for f -electron system, where on the one handthe spin-orbit interaction caused by heavy atoms can belarge, and on the other hand electron correlations in par-tially filled f -electron bands can be very strong. Thus, f -electron systems might give rise to a large ME ef-fect. The existence of the ME and the inverse ME effectin f -electron systems has recently been experimentallydemonstrated for the Kondo insulator SmB [24].The previous Fermi liquid analysis how correlationsaffect the ME effect was however based on the Hubbardmodel, which is not applicable for f -electron systems.In f -electron systems, the hybridization between non- orweakly-interacting conduction electrons ( s − , p − , d − or-bitals) and strongly interacting f -orbitals leads to fasci-nating phenomena, which are not described by the Hub-bard model. While at high temperatures the f -electronsare localized and do not participate in the Fermi surface,at low temperatures the Kondo effect leads to the forma- tion of heavy quasi-particles, which are formed by con-duction ( c -)electrons and f -electrons. Thus, at low tem-peratures the f -electrons become itinerant and do par-ticipate in the Fermi surface[25]. This crossover betweenlocalized f -electrons and itinerant f -electrons when thetemperature is decreased, the Kondo effect, and the re-sulting heavy quasi-particles are not included in the pre-vious theoretical works.The aim of this paper is to analyze the ME effect, par-ticular the Edelstein effect, in strongly correlated non-centrosymmetric f -electron systems such as CeRhSi ,CeIrSi , or CePt Si. By using dynamical mean fieldtheory (DMFT), we fully include the Kondo effect andthus the formation of heavy quasi particles and thecrossover between localized and itinerant f -electrons.Furthermore, the combination of DMFT with the nu-merical renormalization group (NRG) enables us to cal-culate transport properties with high accuracy using real-frequency Green’s functions without the need of an ana-lytic continuation.The main results can be summarized as follows: (i) TheME effect can be strongly enhanced in f -electron systemsand exhibits a maximum at the crossover temperature be-tween localized and itinerant f -electrons. This enhance-ment is beyond Fermi liquid theory. (ii) The enhance-ment of the ME effect originates from a coupling betweenthe c -electrons and the localized f -electrons which gen-erates a momentum dependent spin polarization of the c -electrons even at high temperatures, above the forma-tion of heavy quasi-particles. The spin polarization ofthe c -electrons is thereby generated by a virtual hoppingbetween a c -electron orbital and an f -electron orbital.Thus, the main contribution to the enhancement of theME effect comes from the c -electrons. (iii) Besides theintra-orbital Rashba spin-orbit interaction within the f -electron band, the inter-orbital Rashba spin-orbit inter-action between c -electrons and f -electrons is significantfor a large ME effect. a r X i v : . [ c ond - m a t . s t r- e l ] M a r Figure 1: Noninteracting momentum resolved spectral func-tions for (a) α ff /t f = 0 . α cf /t f = 0, V /t f = 0 . α ff /t f = 0 . α cf /t f = 0 . V /t f = 0 . MODEL AND METHOD
To analyze the ME effect in f -electron systems withantisymmetric spin-orbit interaction, we use a periodic Anderson model, which consists of one c -electron bandand one f -electron band and include a local density-density interaction into the f electron band. Besidesa local c - f hybridization, we include the intra-orbitalRashba spin-orbit interaction within the f -electron band,and the inter-orbital Rashba spin-orbit interaction be-tween c -electrons and f -electrons. Due to the hy-bridization between strongly correlated f -electrons and c -electrons, this model includes all essential ingredientsnecessary to describe heavy fermion behavior. Further-more, the inclusion of the intra-orbital and inter-orbitalRashba spin-orbit interaction, which have been derivedfor CePt Si, reflects the situation of a system withoutinversion symmetry[26] which will lead to the emergenceof ME effect.The Hamiltonian can be split in a single-electron part, H (cid:126)k , and the interaction part, H U , so that H = H (cid:126)k + H U . H (cid:126)k reads H (cid:126)k = (cid:88) (cid:126)k c † k, ↑ c † k, ↓ f † k, ↑ f † k, ↓ T t c (cos k x + cos k y ) 0 V α cf (sin k y + i sin k x )0 t c (cos k x + cos k y ) α cf (sin k y − i sin k x ) VV α cf (sin k y + i sin k x ) t f (cos k x + cos k y ) α ff (sin k y + i sin k x ) α cf (sin k y − i sin k x ) V α ff (sin k y − i sin k x ) t f (cos k x + cos k y ) c k, ↑ c k, ↓ f k, ↑ f k, ↓ , (1)where c † k,σ and f † k,σ create a c -electron and an f -electronwith momentum k and spin projection {↑ , ↓} , respec-tively. Our model includes a spin-independent hoppingfor the c - and f -electron with amplitude t c and t f . Wefix the hopping to t f = − . t c . For simplicity we as-sume a band structure corresponding to a square lattice,but note that our results do not depend on the exactband structure. V is the local hybridization between c -electrons and f -electrons, α cf the inter-orbital Rashbainteraction between c -electrons and f -electrons, α ff theintra-orbital Rashba interaction within the f -electronband. The local density-density interaction within the f -electron band reads H U = U (cid:88) i n fi, ↑ n fi, ↓ . (2)The non-interacting spectrum is shown in Fig. 1 for twoparameter sets. Clearly visible are the c -electron and f -electron bands, which hybridize close to the Fermi energy. The main difference between these two parameter sets isthe band splitting due to the Rashba interaction close tothe Fermi energy. Furthermore, it is important to notethat the particle-hole symmetry is generally broken when V , α cf , and α ff are all nonzero.To analyze transport properties of this system, we solvethe Hamiltonian by using the DMFT[27–29]. DMFTmaps the lattice model onto a quantum impurity model,which is solved self-consistently. DMFT thereby fullyincludes local fluctuations, but neglects nonlocal fluctu-ations. The neglect of nonlocal fluctuations is the maindrawback of DMFT. It thus must be noted that all ob-tained results are only valid as long as nonlocal fluctu-ations are small. However, DMFT has proven to accu-rately describe heavy-fermion physics as necessary to an-alyze f -electron materials[30]. For solving the quantumimpurity model, we use the NRG[31–33], which providesreal-frequency spectral functions and self-energies withhigh accuracy around the Fermi energy for a wide rangeof interaction parameters and temperatures. CONDUCTIVITY AND MAGNETOELECTRICEFFECT
The electric current, J x , and the polarization, M y , arerelated to the electric field, E x , via the conductivity, σ xx ,and the ME coefficient, Υ yx , by J x = σ xx E x (3) M y = Υ yx E x , (4)where σ xx and Υ yx can be calculated using the Kuboformula[29, 34].These two equations can be combined togive the spin polarization depending on the electric cur-rent, M y = Υ yx σ xx J x , (5)which we will below use to quantize the strength of theME effect.After having obtained self-consistent self-energies us-ing the DMFT, we use the Kubo formula to calculate theconductivity, σ xx ( ω ), and the ME effect, Υ yx ( ω ), whichare defined as σ xx ( ω ) = iω (cid:88) k,k (cid:48) Tr (cid:104)(cid:104) v x n k , v x n k (cid:48) (cid:105)(cid:105) ( ω ) (6)Υ yx ( ω ) = iω (cid:88) k,k (cid:48) Tr (cid:104)(cid:104) σ y n k , v x n k (cid:48) (cid:105)(cid:105) ( ω ) . (7)We have set (cid:126) = e = µ = 1. The main problemconsists of calculating the two-particle Green’s functions (cid:104)(cid:104) v x n k , v x n k (cid:48) (cid:105)(cid:105) ( ω ) and (cid:104)(cid:104) σ y n k , v x n k (cid:48) (cid:105)(cid:105) ( ω ), where the dif-ference between the conductivity and the ME effect is thechange from the velocity operator, v x , to the Pauli-spinmatrix, σ y . We take the same Pauli matrix in the c - and f − electron bands setting g = 2 and remind the readerthat all operators represent 4 × σ y = − i i − i i (8) Because vertex corrections are neglected within theDMFT approximation, the two-particle Green’s func-tion reduces to the product of two single-particle Green’sfunctions written in Matsubara frequencies as σ xx ( iω ) = 1 ω Π xx ( iω ) (9)Υ yx ( iω ) = 1 ω K yx ( iω ) (10)Π xx ( iω ) = T (cid:88) k (cid:88) iν Tr (cid:104) v x G k ( iν ) v x G k ( iν + iω ) (cid:105) (11) K yx ( iω ) = T (cid:88) k (cid:88) iν Tr (cid:104) σ y G k ( iν ) v x G k ( iν + iω ) (cid:105) (12)(13)where T is the temperature of the system.Having a self-consistent solution for the self-energy,these single-particle Green’s functions are known andΠ xx ( iω ) and K yx ( iω ) could be calculated using Matsub-ara frequencies, which must be followed by an analyticcontinuation at the end of the calculation.However, a significant advantage of combining DMFTwith NRG is the availability of real-frequency spectralfunctions and self-energies. Thus, we can perform thefull calculation using real frequencies, which results in aconsiderable gain of accuracy. For each component of theGreen’s function, we can write G k ( z ) = (cid:90) dω z − ω A k ( ω ) (14)where A k ( ω ) = πi ( G retk ( ω ) − G advk ( ω )) is the densityof states, which is calculated from the retarded and ad-vanced Green’s functions, G retk ( ω ) and G advk ( ω ).Writing the density of states for all components ofthe Green’s function again as a matrix, we can calculateΠ xx ( ω ) and K yx ( ω ) directly on the real-frequency axis.The conductivity σ xx and ME effect Υ yx thus become σ xx ( ω ) = (cid:88) k (cid:90) dω (cid:48) Tr (cid:104) v x A ( ω (cid:48) ) v x A ( ω + ω (cid:48) ) (cid:105) f T ( ω (cid:48) ) − f T ( ω + ω (cid:48) ) ω (15)Υ yx ( ω ) = (cid:88) k (cid:90) dω (cid:48) Tr (cid:104) σ y A ( ω (cid:48) ) v x A ( ω + ω (cid:48) ) (cid:105) f T ( ω (cid:48) ) − f T ( ω + ω (cid:48) ) ω , (16)where f T ( ω ) is the Fermi-function for temperature T . Taking the static limit ω →
0, we obtain the final result σ xx ( ω = 0) = (cid:88) k (cid:90) dω (cid:48) Tr (cid:104) v x A ( ω (cid:48) ) v x A ( ω (cid:48) ) (cid:105) df T ( ω (cid:48) ) dω (cid:48) (17)Υ yx ( ω = 0) = (cid:88) k (cid:90) dω (cid:48) Tr (cid:104) σ y A ( ω (cid:48) ) v x A ( ω (cid:48) ) (cid:105) df T ( ω (cid:48) ) dω (cid:48) . (18) Figure 2: (a) and (b) Noninteracting momentum resolvedspectral functions. Model parameters are written above eachpanel. The Fermi energy corresponds to ω/t f = 0. (c) MEeffect for the parameter shown in (a)-(b).Figure 3: Fermi surfaces of the non-interacting systems shownin Figs. 1 and 2. The arrows in the plot indicate the directionof the spin polarization in these bands, which is induced dueto the Rashba spin-orbit interaction. We note that in these results the temperature dependenceenters via the Fermi function and A ( ω ) which dependson the self-energy calculated self-consistently for a giventemperature. NONINTERACTING SYSTEM
To gain some understanding about this model, wefirstly show results for the noninteracting system in Fig.2. This will help to clarify the effect of the Coulombinteraction below. Close to the Fermi energy, shown in Fig. 2, the visible bands are composed of hybridized f -electrons and c -electrons. Both systems are metallicwith a spin-split Fermi surface. The chemical potentialis adjusted in both systems, so that the system is half-filled, n f = n c = 1. We note that while Fig. 2(a) corre-sponds to a particle-hole symmetric system, the particle-hole symmetry is broken in Fig. 2(b). Figure 2(c) showsthe results for Υ yx /σ xx for the parameter sets in (a) and(b).For the system with α cf = 0, shown in Fig. 2(a), theME effect disappears at half-filling. Due to the Rashbainteraction, two bands with opposite spin polarizationcut the Fermi energy. Therefore, the particle-hole sym-metry, which is conserved in this system, results in aperfect cancellation of the contributions of these bandsto the ME effect. To verify this statement, we show theFermi surface of this parameter set in Fig. 3(a) and in-clude the spin polarization of each band. Clearly visibleis the appearance of bands with identical shape but op-posite spin polarization.On the other hand, the system including all three pa-rameters ( V (cid:54) = 0 , α ff (cid:54) = 0 , α cf (cid:54) = 0), shown in Fig. 1(b)and Fig. 2(b), has a finite ME effect. The inclusion of α cf breaks the particle-hole symmetry and favours bandswith equal spin-polarization close to the Fermi energy.Therefore, the contributions to the ME effect from dif-ferent bands at the Fermi energy do not completely can-cel. The Fermi surface including the spin polarization isshown in Fig.3(b). Due to the breaking of particle-holesymmetry, bands with opposite spin polarization havevanished from the Fermi surface.These results demonstrate the importance of the inter-orbital Rashba interaction for the ME effect which nat-urally arises in noncentrosymmetric f -electron systemsand leads to bands with spin polarization into the samedirection. We note that the exact cancellation for thesystem with α cf = 0 only holds for the half-filled situ-ation. The ME effect becomes finite, when doping thesystem away from half-filling. INTERACTING SYSTEM
We next turn our attention to the interacting sys-tem. Because the system with α cf = 0 which preservesparticle-hole symmetry can be regarded as a special sit-uation, we will focus from now on the metallic systemwith V = α ff = α cf = 0 . t f . The chemical potential isadjusted in all calculations so that the system remainshalf-filled. We note that the qualitative behavior shownhere does not depend on the filling of the conductionelectrons.Let us start the analysis by showing separately theconductivity, σ xx , and ME effect, Υ xy , for different inter-action strengths and temperatures, see Fig. 4. For thenoninteracting system, the conductivity decreases with Figure 4: Conductivity σ xx (panel a) and magnetoelectriceffect Υ xy (panel b) for α ff /t f = α cf /t f = V /t f = 0 . xy /σ xx for α ff /t f = α cf /t f = V /t f = 0 . decreasing temperature in the shown temperature rangedue to the hybridization between c - and f -electrons whichgaps out parts of the Fermi surface. With increasinginteraction strength, the conductivity develops a peakat finite temperature. Overall, the conductivity exhibitsonly a moderate interaction dependence for the showntemperatures. On the other hand, the ME effect shownin Fig. 4(b) is small at weak-coupling, but strongly in-creases with increasing interaction strength. It developsa peak at a finite temperature. The peak position is ata slightly smaller temperature than that in the conduc-tivity. At very low temperature the ME effect stronglydecreases again.Figure 5 shows the temperature dependent ratio of MEeffect and conductivity for different interaction strengths,which can be measured in experiment. We see that evena weak interaction in the f -orbital, U/t f = 0 .
5, enhances
Figure 6: Momentum resolved spectral functions for
U/t f =5, α ff /t f = α cf /t f = V /t f = 0 . the ME effect, particularly at high temperatures. Com-paring with Fig. 4, it becomes clear that this enhance-ment is due to the enhancement of the ME effect, and notdue to a strong change in the conductivity. Increasing theinteraction further, we find a significant enhancement ofthe ME effect and a clear peak in the temperature depen-dence. While the height of this peak increases with theinteraction strength for U/t f <
5, it becomes constantwhen further increasing the interaction. The tempera-ture of this peak decreases monotonically with increasinginteraction and can be identified as the crossover temper-ature between localized and itinerant f -electrons.Comparing to the ME effect of the noninteracting sys-tem, we observe that the maximum value is more thanten times enhanced by the correlations. However, the en-hancement is even more dramatic at high temperatures,where we find a ME effect nearly 40 times the nonin-teracting value. Thus, our results suggest to look at theME effect in heavy fermion systems above their coherencetemperature. This enhancement would be most useful forspintronics application at room temperature.To understand the mechanism behind this enhance-ment, we show momentum resolved spectral function for U/t f = 5 in Fig. 6. The depicted temperatures corre-spond to a low temperature, where the ME effect is small( T /t f = 0 . T /t f = 0 . T /t f = 0 . c -electrons;due to a strong peak in the imaginary part of the f -electron self-energy, the f -electrons are completely local-ized and thus absent from the spectral function. Heavyquasi-particles are not formed at these temperatures. Be-cause there is no Rashba spin-orbit interaction actingwithin the conduction band, a spin splitting of the con- Figure 7: Momentum resolved σ kxx and Υ kyx along the diagonalin the Brillouin zone for U/t f = 5, α ff /t f = α cf /t f = V /t f =0 . duction band is not observed. At temperatures, shortlybelow the peak of the ME effect, we observe the appear-ance of the f -electron band within the spectrum. The f -electrons become itinerant at this temperature and be-gin to form heavy-quasi particles, which are observableas flat band at the Fermi energy. This proves that thepeak of the ME effect is related to the coherence temper-ature of the system. Finally, at very low temperature,we find coherent heavy quasi-particles around the Fermienergy. The spectrum looks similar to the noninteractingspectrum with renormalized energies.To elucidate the reason for the enhancement athigh temperatures, we show the summand of themomentum integration for the conductivity ( σ kxx = (cid:82) dω (cid:48) Tr (cid:104) v x A k ( ω (cid:48) ) v x A k ( ω (cid:48) ) (cid:105) df T ( ω (cid:48) ) dω (cid:48) ) and the ME effect(Υ kyx = (cid:82) dω (cid:48) Tr (cid:104) σ y A k ( ω (cid:48) ) v x A k ( ω (cid:48) ) (cid:105) df T ( ω (cid:48) ) dω (cid:48) ) along the di-agonal of the Brillouin zone in Fig. 7. The conductivityand the ME effect as shown in the previous figures corre-spond to the momentum integral of these functions overthe whole Brillouin zone. The summand for the conduc-tivity is always positive. Its amplitude around the Fermimomentum ( π/ , π/
2) is increasing with decreasing tem-perature due to an increased lifetime, while the width ofthe peak decreases at the same time. The summand ofthe ME effect, on the other hand, shows a more interest-ing behavior. At low temperature,
T /t f = 0 .
05, it showspositive as well as negative contributions.The existence of positive and negative contributionscan also be immediately understood from the Fermisurface of the system ( α ff /t f = 0 . α cf /t f = 0 . V /t f = 0 . U/t f = 5) including the spin polarizationshown in Fig. 8 for three different temperatures. Figure8(a) shows a low temperature, where the f -electrons areitinerant. We observe a complicated Fermi surface made up of several bands. Furthermore, we observe that thesebands have opposite spin polarization, which results in acancellation of the ME effect at this temperature. Thiscancellation is indeed responsible for the suppression ofthe ME effect in most metallic systems.However, the situation is very different at high temper-atures. In Fig. 7, we observe that the negative contribu-tion to the ME effect vanishes with increasing tempera-ture. At temperatures above the coherence temperature,we only find positive contributions. In Fig. 8(b), the f -electron bands become incoherent and are blurred in thedensity of states at the coherence temperature. Never-theless, the f -electrons still contribute to the spin polar-ization. Thus, the spin polarization includes momentumregions with opposite direction. Finally, Fig. 8(c) showsa temperature where the f -electrons are localized andthus are absent from the spectrum. The calculated spinpolarization only includes the clockwise direction. Thereare only positive contributions to the ME effect at thistemperature, see T /t f = 0 . f -electrons.Because the f -electrons are localized, the ME ef-fect above the coherence temperature is solely gener-ated by the c -electrons. It is rather remarkable thatthe c -electrons contribute to the ME effect when the f -electrons are absent from the spectrum, although there isno direct Rashba interaction within the c -orbitals. Thisfact can be understood in the following way: In a vir-tual process, a c -electron can hop onto an f -orbital andreturn to a c -electron orbital. This hopping process in-volves the inter-orbital Rashba spin-orbit interaction andthus will lead to a term describing a spin-dependent cou-pling, which can generate a spin polarization within the c -electron band. Thus, the dependence of the ME ef-fect on the interaction strength and the temperature canbe understood as an interplay between the localizationof f -electrons and a virtual hopping of c -electrons on f -electron orbitals. At high temperature, f -electrons arelocalized due to the Coulomb interaction. Thus, the MEeffect arises due to polarized c -electrons and it is largebecause of an absence of cancellation. With lowering thetemperature, the ME effect firstly increases until the co-herence temperature of the material is reached. At thistemperature the f -electrons become itinerant. At lowertemperatures, the material is described by a renormal-ized band structure of the noninteracting one. Thus, theME effect is small due to cancellation effects. The co-herence temperature, where the f -electrons change fromlocalized to itinerant, decreases thereby strongly with in-creasing Coulomb interaction.Finally, before concluding this paper, we want toshortly address the situation for a hole-doped system.Up to now, we have focused on a half-filled system,which might be regarded as a special situation, although Figure 8: Cuts of the momentum dependent density of states through the Brillouin zone at the Fermi energy. The color plotdenotes the density of states at the fixed energy with a maximum intensity for yellow. The white arrows denote the calculatedspin polarization. The parameters are α ff /t f = 0 . α cf /t f = 0 . V /t f = 0 . U/t f = 5. (a): T /t f = 0 .
01 (b):
T /t f = 0 . T /t f = 0 . α ff /t f = 0 . α cf /t f = 0 . V /t f =0 . U/t f = 5 at T /t f = 1 (left panels) and T /t f = 0 . c -electron band is n c = 0 . f -electron band is half-filled. (a) Momentum-resolvedspectral function for T /t f = 1. (b) Local density of states(DOS) for T /t f = 1. The black (red) lines correspond to the c -( f -) electrons. (c) Momentum-resolved spectral function for T /t f = 0 . T /t f =0 . the particle-hole symmetry is broken for α ff /t f = 0 . α cf /t f = 0 . V /t f = 0 .
5. We here demonstrate that theenhancement of the ME effect does not depend on thefilling of the c -electron band.Figure 9 shows the momentum resolved and local DOSof the system with α ff /t f = 0 . α cf /t f = 0 . V /t f =0 . U/t f = 5 at T /t f = 1 and T /t f = 0 . f -electron band is half-filled,the c -electron band has a filling n c = 0 .
6. We observe forthe doped system qualitatively the same physics as at halffilling. For high temperature,
T /t f = 1, the f -electronsare localized and thus absent from the Fermi energy inthe momentum resolved spectral function and local DOS. Figure 10: ME effect Υ yx /σ xx for α ff /t f = 0 . α cf /t f = 0 . V /t f = 0 . n c = 0 . The conduction electrons show a spectrum correspondingto noninteracting electrons on a square lattice.At low temperatues,
T /t f = 0 . f -electrons be-come coherent and form heavy quasi-particles togetherwith the c -electrons. The f -electrons form a peak in thedensity of states at the Fermi energy. This is exactly thesame physics as described above for the half-filled system.It is thus not surprising to find qualitatively similarbehavior for the ME effect shown in Fig. 10. The MEeffect is enhanced for the interacting system and shows aclear peak, which can be identified again as the transitionbetween localized and itinerant f -electrons. CONCLUSIONS
We have demonstrated that the ME effect can bestrongly enhanced in f -electron systems showing a peakat the coherence temperature, where the f -electronschange from itinerant to localized behavior. Above thecoherence temperature, where a strong peak in the imag-inary part of the f -electron self-energy is formed, theFermi liquid theory breaks down, and a momentum-dependent spin polarization of the c -electrons is created,which causes the large ME effect. Remarkably, a cancel-lation of the ME effect due to spin-split bands with dif-ferent polarization is absent at this temperature, whichis the main reason for the enhancement. Thus, our re-sults suggest to look at the ME effect in noncentrosym-metric f -electron systems such as CeRhSi , CeIrSi , orCePt Si above their coherence temperature. The coher-ence temperature, as defined in our calculation, can bedetermined from experiment by the peak position of themagnetic contribution to the resistivity. For CeRhSi and CeIrSi this peak can be observed at approximately T c = 100K[35] and for for CePt Si at T c = 80K[36].The spin-orbit interaction in CePt Si has been estimatedfrom first principle calculation to 50meV-200meV [37]. Ifwe assume the strength of the spin-orbit coupling to be100meV in our calculations, t f will also be 100meV. Ourcalculations with U/t f = 6 and U/t f = 7 would thencorrespond to coherence temperatures of T c = 120K and T c = 60K, respectively. The enhancement of the ME ef-fect at room temperature due to interaction effects wouldbe approximately 40 for these calculations. Thus, our re-sults suggest that these noncentrosymmetric f -electronmaterials might have a large ME effect even at room tem-perature, which would be most significant for spintronicsapplications.We thank R. Takashima for helpful discussion. 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