Magnetism and Superconductivity in Ferromagnetic Heavy Fermion System UCoGe under In-plane Magnetic Fields
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Magnetism and Superconductivity in Ferromagnetic Heavy Fermion System UCoGeunder In-plane Magnetic Fields
Yasuhiro Tada,
1, 2
Shintaro Takayoshi, and Satoshi Fujimoto Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan Max Planck Institute for the Physic of Complex Systems, N¨othnitzer Str. 38, 01187 Dresden, Germany Department of Quantum Matter Physics, University of Geneva, Geneva 1211, Switzerland Department of Materials Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
We study the ferromagnetic superconductor UCoGe at ambient pressure under ab -plane magneticfields H which are perpendicular to the ferromagnetic easy axis. It is shown that, by takinginto account the Dyaloshinskii-Moriya interaction arising from the zigzag chain crystal structure ofUCoGe, we can qualitatively explain the experimentally observed in-plane anisotropy for criticalmagnetic fields of the paramagnetic transition. Because of this strong dependence on the magneticfield direction, upper critical fields of superconductivity, which is mediated by ferromagnetic spinfluctuations, also become strongly anisotropic. The experimental observation of “S-shaped” H c k b -axis is qualitatively explained as a result of enhancement of the spin fluctuations due to decreasedCurie temperature by the b -axis magnetic field. We also show that the S-shaped H c is accompaniedby a rotation of the d -vector, which would be a key to understand the experiments not only atambient pressure but also under pressure. PACS numbers: Valid PACS appear here
I. INTRODUCTION
Since the discovery of a ferromagnetic superconductorUGe , a family of ferromagnetic systems, URhGe andUCoGe, has also been found to exhibit superconductiv-ity and they have been extensively studied with specialfocus on relationship between ferromagnetism and super-conductivity . In these compounds, 5 f -electrons areresponsible both for the magnetism and the superconduc-tivity, in sharp contrast to the previously found ferromag-netic superconductors such as ErRh B and HoMo S where the magnetism and superconductivity have dis-tinct origins .Among these uranium compounds, UCoGe has thelowest Curie temperature T C ∼ .
7K and the super-conducting transition temperature T sc ∼ .
6K at am-bient pressure . The ferromagnetism is suppressedby applying pressure and T C seems to approach zeroat a critical pressure p c ∼ . p < p c , ferromagnetism and superconductiv-ity coexist in a microspic way , while the supercon-ductivity alone survives up to p > p c . The experimen-tal pressure-temperature phase diagram of UCoGe couldbe understood from theoretical model calculations whereIsing spin fluctuations mediate superconductivity .Indeed, as revealed by the NMR experiments, spin fluc-tuations in UCoGe have strong Ising anisotropy and thesuperconductivity is closely correlated with them espe-cially under magnetic fields . The experiments showthat the a -axis upper critical field is huge H k ac >
25T inspite of the low transition temperature T sc ∼ .
6K while H c for c -axis is merely less than 1T, which leads to cusp-like field angle dependence of H c in the ac -plane. From atheoretical point of view, the anomalous behaviors of theobserved ac -plane upper critical fields of the supercon- ductivity can be well understood by taking into accountthe experimental fact that the Ising spin fluctuations aretuned by a c -axis component of the magnetic fields .The successful agreement between the experiments andtheories provides strong evidence for a scenario that thepseudo-spin triplet superconductivity is indeed mediatedby the Ising ferromagnetic spin fluctuations in UCoGe.On the other hand, different characteristic behav-iors have been experimentally observed for b -axis mag-netic fields in UCoGe . In the normal (non-superconducting) states, the Curie temperature T C issuppressed by H k b -axis and it seems to become zeroaround H ∗ ∼ H k a -axis in the same experiments. The reduction of T C by H k b -axis is accompanied by an enhancement of thespin fluctuations. Accordingly, at low temperatures, H c is enhanced by the b -axis magnetic field especially around H = H ∗ , resulting in “S-shaped” H c . Interestingly, sim-ilar behaviors have also been found in the isomorphiccompound URhGe, where superconductivity vanishes ata critical H k b -axis but it reappears at a high fieldaround which ferromagnetism is suppressed with a tri-critical point .From a theoretical point of view, based on a scenarioof the spin fluctuations-mediated superconductivity, itis rather natural to expect S-shaped H c or even reen-trant superconductivity, once one simply takes into ac-count enhancement of the spin fluctuations by the reduc-tion of T C . However, within this theoretical approachwhich strongly relies on the experimental observationsof anisotropic behavior with the application of in-planemagnetic fields, it is unclear why T C is unchanged andtherefore H c is not enhanced for a -axis magnetic fields.In order to understand the dependence of H c on thedirection of magnetic field, we should clarify the originof in-plane magnetic anisotropy. Furthermore, even ifone just admits magnetic anisotropy as an experimentalfact, nature of the resulting superconducting state un-der strong b -axis magnetic field is far from trivial. Forsmall magnetic fields, the superconductivity will coexistwith the ferromagnetism as in the zero-field case, andit is robust against the Pauli depairing effect under in-plane magnetic fields due to exchange splitting of theFermi surface as pointed out by Mineev . On the otherhand, for larger magnetic fields H & H ∗ where the super-conductivity survives experimentally, the exchange split-ting is small or even vanishing, and therefore the Mi-neev’s mechanism protecting the superconductivity fromthe Pauli depairing effect does not work. The limitationof the Mineev’s mechanism on the Pauli depairing effectshould also be recognized for understanding H c underhigh pressure where ferromagnetism is suppressed. Evenin paramagnetic states where the Pauli depairing effectsare expected to be important, experiments obtain largein-plane H c ∼ − T sc ∼ . These values of H c were obtained without fine-tuning of the magneticfield directions and H c would be further increased bycareful tuning of the field directions, since it sensitivelydepends on a c -axis component of the magnetic fields inUCoGe . Theoretically, it is expected that the spinfluctuations are large especially around p = p c leading tostrong coupling superconductivity and the orbital depair-ing effect would be less relevant there, while the Pauli de-pairing effect is not suppressed and eventually will breakthe superconductivity.In this study, we investigate anisotropy for in-planecritical magnetic fields of paramagnetic transition andsuperconducting H c in UCoGe. Firstly, we make ananalysis focusing on zigzag chain crystal structure whichis characteristic in UCoGe. Within a minimal spin modelincluding effects of the zigzag chain structure, we showthat the Dyaloshinskii-Moriya (DM) interaction arisingfrom the zigzag structure leads to the in-plane anisotropyfor critical magnetic fields of paramagnetic transitions,which shows a qualitative agreement with the experi-ments. We then examine resulting superconducting H c phenomenologically considering the suppression of ferro-magnetism by application of external fields along b -axis.It is shown that superconductivity can survive above H ∗ where there is no exchange splitting of the Fermi surface.We find that this robustness of superconductivity stemsfrom the d -vector rotation to reduce magnetic energy costas the magnetic field is increased. We also touch on theexperimental observations based on our calculations. II. ANISOTROPY FOR CRITICAL MAGNETICFIELD OF PARAMAGNETIC TRANSITION ANDITS ORIGIN
In this section, we study the origin of strong depen-dence of critical magnetic field H ∗ on the field direc-tion in the ab -plane. As mentioned in the previous sec- tion, the Curie temperature is decreased by magneticfield along the b -axis, while it is unchanged by magneticfield applied parallel to the a -axis within the experimen-tal range . If the magnetic fields are further in-creased, T C will be suppressed for a -axis magnetic fieldsas well. We schematically show an expected magneticphase diagram of UCoGe in Fig. 1. T C decreases rapidly FIG. 1. Expected magnetic phase diagram of UCoGe intemperature( T )-magnetic field( H ) plane. FM and PM referto ferromagnetic phase and paramagnetic phase, respectively. H ∗ a and H ∗ b are critical magnetic fields at zero temperature. by applying magnetic field along the b -axis and even-tually becomes zero at a critical field H ∗ b at zero tem-perature, while it is robust against a -axis field and thecorresponding critical field H ∗ a is much larger than H ∗ b .The purpose of this section is to understand qualitativelywhat causes this anisotropy in the ab plane. Magneticanisotropy generally arises from spin-orbit interactions,and its details depend on strength of the spin-orbit inter-actions and crystal structures. In f -electron compounds,basic magnetic properties could be well understood oncethe local electronic configuration has been fixed by e.g.neutron scattering experiments. For UCoGe, the experi-mentally observed Ising magnetic properties may be dueto a large weight of J = 5 / f -electron systems with low crystalsymmetry such as UGe and UCo(Rh)Ge is very diffi-cult, the resistivity in UCoGe shows rather conventionalheavy fermion behaviors with weak anisotropy in effec-tive mass. This implies that UCoGe is well described byeffective pseudo-spin 1/2 quasi-particles corresponding tothe observed Ising-like magnetism.Here, instead of using local electronic structures, weinvestigate the magnetic anisotropy in UCoGe by focus-ing on its characteristic crystal structure. As seen inFig. 2, UCoGe can be viewed as a composition of one-dimensional zigzag chains along a -axis. The point groupof UCoGe is P nma and the zigzag chains do not havelocal inversion symmetry, although they keep global in-version symmetry. Such a quasi one-dimensional zigzagstructure allows an asymmetric spin-orbit (ASO) inter-
FIG. 2. The crystal structure of UCoGe. Zigzag chains arealong the a -axis. action H ASO ∼ X kss ′ sin k a σ bss ′ [ a † ks a ks ′ − b † ks b ks ′ ] , (1)where a ks ( b ks ) is an annihilation operator of the quasi-particles at A (B) sublattice of the zigzag chain. In or-der to understand effects of the ASO interaction quali-tatively, we focus only on spin degrees of freedom andintroduce a counterpart of the ASO interaction in thespin sector of electrons. Then, in terms of the spin de-grees of freedom, the ASO interaction is mapped to astaggered DM interaction, H DM = X j ( − j D b [ S cj S aj +2 − S aj S cj +2 ] , (2)where S j with j =odd(even) corresponds to the pseudo-spin 1/2 at the A(B) sublattice of the zigzag chain. Notethat the direction of the DM vector D = (0 , D b ,
0) is con-sistent with a general symmetry argument for
P nma ofUCoGe ; there is a mirror symmetry with respect to the( x, / , z ) plane , which results in D k b -axis. In orderto elucidate the effect of this DM interaction in the quasione-dimensional systems, we investigate a single zigzagchain neglecting inter-chain interactions. Under this as-sumption, spin degrees of freedom in UCoGe is describedby the following one-dimensional spin Hamiltonian, H spin = − J X j S cj S cj +1 − X j [ h a S aj + h b S bj ] + H DM . (3)The first term is the Ising ferromagnetic interaction andthe second term corresponds to the ab -plane magneticfields. We have neglected spin-spin interactions of in-plane spin components, since the magnetism of UCoGehas strong Ising nature as verified by the experiments . In this section, we use a unit where J = 1.This spin model should be considered as a variant of thephenomenological Ginzburg-Landau theory developed byMineev, where the free energy is written in terms of mag-netic degrees of freedom only . In the present study, weuse the above spin model as an effective phenomenolog-ical description to capture essential physics behind thecomplicated experimental results with a special focus onthe DM interaction. Although the spin model is oversim-plified for discussing quantitative properties of UCoGe, it is useful for qualitative discussions as a minimal model.Indeed, as will be discussed in the following, the physi-cal mechanism leading to magnetic anisotropy under in-plane magnetic fields revealed within the spin model anal-ysis is applicable also for more realistic models of UCoGe.We also note that, since the Land´e g -factors have notbeen determined in UCoGe, the parameters h a , h b shouldbe regarded as a renormalized magnetic fields which in-clude g -factors. Anisotropy of diagonal components ofthe g -factors in a, b -directions is expected to be small, g aa ≃ g bb , since M - H curves show weak anisotropy be-tween M a and M b for small magnetic fields . Althoughthere may be off-diagonal components of the g -factor wesimply neglected them. If g ca or g cb is large, the ferro-magnetic phase transitions are smeared out and becomecrossover under in-plane magnetic fields. In the presentmodel without such off-diagonal components, there is Z symmetry for general h = ( h a , h b ,
0) with the opera-tion (translation) × (time-reversal) × exp[ iπ P j S cj ] whichtransforms spins as S aj → S aj +1 ,S bj → S bj +1 ,S cj → − S cj +1 . (4)This symmetry is spontaneously broken in the ferromag-netic phase.In order to understand the anisotropy of the crit-ical fields H ∗ , we investigate the Hamiltonian (3) at T = 0 by use of infinite density matrix renormalizationgroup (iDMRG) . In the present one-dimensionalmodel, we find that the calculated ground state preservesthe translational symmetry and the system undergoesa quantum phase transition from a uniform ferromag-netic state with h S cj i 6 = 0 to a disordered state with h S cj i = 0 as h is increased. We have confirmed absenceof a non-uniform magnetic structure by increasing sizesof assumed sub-lattice structures in the numerical calcu-lations. This is essentially due to strong Ising anisotropywhich favor the colinear ferromagnetic structure, andcoplanar magnetic states might be stabilized if one ap-propriately includes inter-chain coupling and the DM in-teraction is sufficiently large. We note that, indeed, sucha coplanar state with a -axis weak antiferromagnetism hasbeen predicted within the Ginzburg-Landau theory .In Fig. 3, we show magnetization as a function of mag-netic fields for different values of the DM interactionwithin our model. For small magnetic fields, the mag-netization does not change from that without the DMinteraction, since the ground state of the Ising Hamilto-nian at h = 0 is at the same time an eigenstate of theDM interaction, H DM | ↑↑ · · · ↑i = 0. For large magneticfields, all the spins are aligned so that they become paral-lel to the applied fields. Interestingly, the magnetizationis not changed by the DM interaction for a -axis magneticfields even at h ∼ J/
2, while it is rapidly suppressed by b -axis magnetic fields as the DM interaction is increased.This anisotropic behavior can be understood as a resultof a competition between the DM interaction and the | M c | h a D b =0D b =0.1D b =0.2D b =0.3 0 0.1 0.2 0.3 0.4 0.5 0.35 0.4 0.45 0.5 0.55 | M c | h b FIG. 3. The c -axis magnetization | M c | for different valuesof the DM interaction under the a -axis magnetic field (leftpanel) and b -axis magnetic field (right panel). h * D b h||ah||b FIG. 4. The critical magnetic fields for a -axis (red squares)and b -axis (blue circles). D b is strength of the DM interactionin unit of J . applied fields; The DM interaction can be rewritten as H DM = D b X j :odd [ ˜ S + j ˜ S − j +2 + ˜ S − j ˜ S + j +2 ] − ( j : even) , (5)˜ S ± j = e ± iπj/ ( S cj ± iS aj ) . (6)The DM interaction alone describes decoupled two copiesof “XY-chains” in the ˜ S -basis and it increases quantumfluctuations of spins in the ac -plane. Classically, the DMinteraction tends to rotate the spins and it frustrates withthe Ising interaction. However, once strong a -axis mag-netic fields are applied, this ac -plane quantum fluctua-tions are pinned and effects of the DM interactions getsuppressed. Therefore, the calculated magnetization isalmost independent of the DM interaction, and in par-ticular, the transition point almost does not change. Onthe other hand, for b -axis magnetic fields h b , the DM in-teraction is not suppressed and the ferromagnetic stateis destabilized by the quantum fluctuations, resulting insmaller critical fields h ∗ b .We summarize stability of the Ising ferromagnetismagainst the DM interaction in Fig. 4. For h k a -axis,the critical field h ∗ a is almost unchanged from h ∗ a ≃ . J as D b is introduced as explained above. It is noted thatthe system is dominated by the DM interaction for largevalues of D b . Around D b ≃ . J , there is a first orderphase transition between the Ising ferromagnetic phaseand a paramagnetic phase at h = 0. The latter state isadiabatically connected to the paramagnetic state with large h a and small D b . On the other hand, for h k b -axis,the critical field h ∗ b is suppressed by D b . If the DM inter-action is sufficiently strong, the ferromagnetism is morefragile against h b than h a at zero temperature. This sug-gests that, at finite temperature, the Curie temperatureis quickly suppressed by h b compared with h a . Therefore,the Hamiltonian (3) qualitatively explains the expectedphase diagram Fig. 1 of UCoGe. Although these re-sults are based on the simple spin model (3), we believethat the mechanism due to the DM interaction basicallyapplies to more realistic models. In general, it is possi-ble that strong magnetic anisotropy remains intact evenwhen one includes itinerant nature of electrons into aspin model, although it may be weakened to some ex-tent. Indeed, UCoGe is an itinerant ferromagnet withstrong Ising anisotropy as verified in experiments .In order to understand the quantitative features of themagnetic anisotropy in UCoGe, one needs to fully includethe on-site electron level scheme together with the ASOinteraction. This issue is left for a future study. III. SUPERCONDUCTING UPPER CRITICALFIELD
In this section, we consider how the reduced Curie tem-perature affects superconducting transition temperature,based on the scenario that the superconductivity is me-diated by the Ising spin fluctuations in UCoGe. Similarproblems were theoretically studied by several authors . Here, we will focus on qualitative properties anduse a simple model to demonstrate effects of the enhancedspin fluctuations on the superconductivity. As was dis-cussed in the previous sections, low-energy properties inUCoGe can be described by quasi-particles interactingthrough the Ising pseudo-spin fluctuations. Therefore,we can approximate our kinetic term as S kin = X [ iω − ε ′ k + ˜ hσ ss ′ ]( a † ks a ks ′ + b † ks b ks ′ ) − X ε k ( a † ks b ks ′ + (h . c . )) + S ASO , (7)where ε k ( ε ′ k ) corresponds to inter-sublattice (intra-sublattice) hopping energy. The action includes spin-dependent terms described by ˜ h = h + h ex , where h = µ B H is the applied Zeeman field and g -factor is sim-ply taken to be g = 2, and h ex is the exchange splittingenergy of the Fermi surface in the ferromagnetic state.From the experiments and the previous sections, it isreasonable to assume that the exchange splitting at zerotemperature h ex k c -axis depends only on b -axis appliedfields. It is phenomenologically approximated as h c ex ( h b ) = (cid:26) h ex (0)tanh (cid:0) . p h ∗ b /h b − (cid:1) , ( h b ≤ h ∗ b ) , , ( h b > h ∗ b ) . (8)This functional form describes a mean field behavior, M c ∼ ( h ∗ b − h b ) / , near the quantum critical point.Note that we have neglected a -axis applied field depen-dence of h c ex , since it is weak as discussed in the pre-vious sections. One can improve the present model byappropriately modifying h ex , e.g. using the critical ex-ponents of the three dimensional Ising ferromagnets orintroducing temperature dependence. The kinetic termalso includes the ASO interaction Eq. (1) between theintra-sublattices. As in the globally noncentrosymmet-ric superconductors, the ASO interaction term tends tofix directions of d -vectors for spin-triplet superconduc-tivity . In UCoGe, however, Cooper pairing betweenthe nearest neighbor uranium sites along the zigzag chainis expected to be stronger than that between the sec-ond nearest neighbor sites. The former is inter-sublatticepairing, while the latter is intra-sublattice pairing. Thissuggests that the ASO interaction between the intra-sublattices will affect the sub-dominant gap functionsonly, while its effects on the dominant inter-sublatticegap functions would be negligible in UCoGe. Therefore,we neglect the ASO interaction and do not explicitly takethe sublattice structure into account in the following cal-culations, which allows us to replace a ks , b ks with a singleoperator c ks . Then, we use a simple isotropic dispersion ε k = − t P j = a,b,c cos k j , ε ′ k = − µ where µ is the chem-ical potential. The model parameters are taken to bethe same as those in the previous study , and inparticular, the exchange splitting is h ex ( h b = 0) = 0 . t which is large enough to suppress Pauli depairing effectsfor small h b . It should be stressed that the ASO is lessimportant for determining directions of d -vector of thepseudo-spin triplet pairing between the nearest neigh-bor sites, but still relevant to understand the magneticanisotropy. The latter effect has already been incorpo-rated in Eq .(8) within the present model for discussingsuperconductivity, by neglecting h a dependence of theexchange splitting..The fermions interact through the Ising spin fluctua-tions which is described by S int = − g X q Z /T dτ dτ ′ χ c ( q, τ − τ ′ ) S cq ( τ ) S c − q ( τ ′ ) , (9) χ c ( q, i Ω n ) = χ δ + q + | Ω n | /γ q , (10)where γ q = vq with v = 4 t is the conventional Lan-dau damping factor and S q = (1 / P k c † k − q,s σ ss ′ c ks ′ .We have neglected interactions arising from in-plane spincomponents, since the Ising spin fluctuations are thedominant fluctuations in UCoGe . Zero-temperaturemass of the Ising spin fluctuations is described by δ ( h b ),and a mean field functional form of δ is used for simplic-ity , δ ( h b ) = (cid:26) ( h c ex ( h b )) ( h b ≤ h ∗ b ) , ( h c ex (2 h ∗ b − h b )) ( h b > h ∗ b ) . (11)Since h b -dependence of δ ( h b ) has not been clarified ex-perimentally, we have assumed that it is symmetric about H c / T sc T/T sc0 H b *H||aH||b FIG. 5. Temperature dependence of H c for a -axis (red curve)and b -axis (blue curve). The dashed line indicates the crit-ical magnetic field H ∗ b . T sc = 0 . t is the superconductingtransition temperature at h = 0. h b = h ∗ b . Details of calculation results depend on func-tional forms of δ , but their overall behaviors are wellcaptured by this simple function.In order to calculate H c , we solve the Eliashberg equa-tion within the lowest order in the Ising interaction. Thelinearized Eliashberg equation reads ∆ ss ( k ) = − T N X k ′ V ( k, k ′ )[ G ss ′ ( k + Π) G ss ′ ( − k )+ G ss ′ ( k ) G ss ′ ( − k − Π)]∆ s ′ s ′ ( k ′ ) , (12)where Π = − i ∇ R − e A ( R ) and A is the vector potentialgiving a uniform magnetic field. The pairing interaction V and the selfenergy in the Green’s function G are eval-uated as V ( k, k ′ ) = − g χ c ( k − k ′ ) + g χ c ( k + k ′ ) , (13)Σ( k ) = T N X qs g χ c ( q ) G ss ( k + q ) , (14)where G is the non-interacting Green’s function and N is the number of k -point mesh in the Brillouin zone. Wehave neglected selfenergies which are off-diagonal in spinspace, since they are much smaller than the diagonalcomponents when h ≪ (band width). In numerical cal-culations, the lowest Landau level is taken into accountfor the orbital depairing effect. We focus on the super-conducting symmetry for which the d -vector is expressedas d ∼ ( c k a + ic k b , c k b + ic k a ,
0) with real coeffi-cients { c j } near the Γ-point in the Brillouin zone, andthe gap function is calculated self consistently by solvingthe Eliashberg equation. We use the same set of modelparameters as in the previous studies , which gives asuperconducting transition temperature T sc = 0 . t at h = 0.In Fig. 5, we show the upper critical field H k bc togetherwith the previous results for H k ac . The horizontallight blue line indicates the critical applied field h ∗ b be-low (above) which the system is ferromagnetic (param-agnetic). In our numerical calculations, we cannot accu-rately compute H k bc near the critical field h b ≃ h ∗ b , sincestrength of the spin fluctuations diverges as ∼ /δ . Asexpected, calculated H k bc is enhanced around h b = h ∗ b ,which is qualitatively consistent with the experiments onUCoGe . Although the superconducting transition tem-perature T sc ( h b = h ∗ b ) seems to exceed T sc ( h b = 0) forthe present model parameters in contrast to the experi-ments, T sc ( h ∗ b ) /T sc (0) strongly depends on δ ( h b = 0) inour model. If δ (0) is sufficiently small and the system atzero field is already very close to the criticality δ = 0, en-hancement of superconductivity due to reduction of δ ( h b )would be moderate . On the other hand, if δ (0) israther away from criticality, enhancement of T sc would bedrastic when δ ( h b ) is tuned. T sc ( h ∗ b ) /T sc (0) also directlydepends on the value of h ∗ b , because b -axis magnetic fieldsnot only tune the magnetic criticality but also break thesuperconductivity at the same time. However, it is notedthat enhancement of T sc due to the field-induced critical-ity is a common qualitative behavior, which is indepen-dent of the details.It is interesting to see that the superconductivity stillsurvives above the critical magnetic field, h b > h ∗ b , wherethe system is paramagnetic. As was discussed by Mi-neev , a large exchange field h ex ≫ T sc is essentiallyimportant to suppress the Pauli depairing effect for equal-spin pairing states. Since the equal-spin pairing state isrealized and ∆ ↑↓ = ∆ ↓↑ = 0 in the present Ising spinfluctuations model, one might expect that the supercon-ductivity is easily destroyed due to the Pauli depairingeffect if the system reaches the paramagnetic state withincreasing the magnetic field along b -axis. To understandthe origin of the robust superconductivity under b -axismagnetic fields, we compare in-plane components of the d -vector, d a and d b , by calculating h| d a |i = s N X k | ∆ ↑↑ ( k ) − ∆ ↓↓ ( k ) | , (15) h| d b |i = s N X k | ∆ ↑↑ ( k ) + ∆ ↓↓ ( k ) | . (16)Note that absolute values of the d -vector cannot bedetermined within the present calculations of the lin-earized Eliashberg equation, but its direction can be self-consistently computed. We show calculation results inFig. 6 together with h ex ( h b ) defined in Eq. (8). The cal-culation results show that h| d b |i / h| d a |i ≃ h b ≃ h b is introduced. This is be-cause Cooper pairing only for the Fermi surface of themajor spin takes place at h b = 0, and that for the minorspin is induced at finite h b >
0. By further increasing h b ,the ratio h| d b |i / h| d a |i sharply decreases around h b ≃ h ∗ b and it becomes nearly zero for h b > h ∗ b . This means thatthe d -vector at h b = 0 is d ∝ (1 , i,
0) and it rotates to d ∝ (1 , ,
0) for large applied fields which is perpendicu-lar to the applied b -axis magnetic field. The rotation of d -vector from the non-unitary state with h| d a |i ≃ h| d b |i at b /H b *h ex /h ex (0)<|d b |>/<|d a |> FIG. 6. In-plane components of the d -vector as a function of H b . The green curve is the exchange field h ex which charac-terizes Fermi surface splitting in the ferromagnetic state. zero field to the nearly unitary state with h| d a |i ≫ h| d b |i at large H b is due to reduction of the exchange split-ting of the Fermi surface and the Pauli depairing effect.As H b is increased, the exchange splitting Eq. (8) getssmaller, which weakens the non-unitarity of d -vector. Atthe same time, in order to reduce Zeeman energy cost,the d -vector favors a configuration d ⊥ H . The resulting d -vector for h b > h ∗ b allows a large Pauli limiting field H k bP at T = 0 which is given by H k bP = ∆ s ρ (0) χ bN − χ bsc , (17)where ∆ and ρ (0) are the gap amplitude and the den-sity of states at the Fermi energy, respectively. χ bN isthe static susceptibility in normal states and χ bsc is thatin superconducting states given by χ bsc = χ bN [1 − h ˆ d b i FS ]within mean field approximations. When the d -vectoris perpendicular to b -axis, the susceptibility is χ bsc ≃ χ bN and the Pauli limiting field becomes large. Therefore, thesuperconductivity can survive up to large b -axis magneticfields h b ≃ h ∗ b ≫ T sc ( h b = 0) in UCoGe. However, wenote that the high field superconductivity is numericallystable only for H b & H ∗ b and H k bc is relatively smallerthan H k ac in the present model. Similar changes of pair-ing states have been discussed in the previous study forURhGe .We think that this mechanism for suppressing Paulidepairing effects under in-plane magnetic fields is im-portant also for the superconductivity under high pres-sure. The superconductivity extends over a wide range ofpressure and it survives in the paramagnetic phase ex-perimentally. Although the Mineev’s mechanism of sup-pressing the Pauli depairing effect does not work in theparamagnetic phase, observed H c for in-plane magneticfields are large, H c & ∼
1T which is naively expected forthe equal-spin triplet pairing with T sc . . H c nearthe magnetic phase transition point p c ≃ . d -vector rotates to suppress Pauli depairing effect,the large spin fluctuations lead to strong coupling super-conductivity around the critical pressure and it can berobust against in-plane magnetic fields. This mechanismwould be relevant for understanding the observed large H c in UCoGe under pressure. IV. SUMMARY
We have investigated ferromagnetism and supercon-ductivity in the heavy fermion compound UCoGe underin-plane magnetic fields. For the magnetic properties,we focused on roles of the DM interaction arising fromthe zigzag chain crystal structure of UCoGe, and qualita-tively explained the experimentally observed anisotropyfor the critical field of the paramagnetic transition. Thenwe incorporated this magnetic anisotropy into a simplesingle-band model for the discussion of superconductiv-ity, where magnetism is tuned by b -axis magnetic fieldsbut is independent of a -axis magnetic fields. Based onthe scenario of the ferromagnetic spin fluctuations medi-ated superconductivity, we demonstrated that H k bc showsS-shaped behaviors in qualitative agreement with the ex-periments, while H k ac is monotonic in temperature. Itwas also numerically found that the superconductivity survives even for large b -axis magnetic fields for whichthe system is paramagnetic and Pauli depairing effectis expected to be relevant. We showed that the super-conductivity survives robustly due to a rotation of the d -vector which reduces the Zeeman energy cost and sup-presses the Pauli depairing effect. The rotation of the d -vector would also be important for understanding large H c under pressure where the Pauli depairing effect isnot suppressed by the exchange splitting of the Fermisurface. ACKNOWLEDGEMENT
We thank K. Ishida, T. Hattori, D. Aoki, K. Hat-tori, Y. Yanase, M. Oshikawa, Y. Fuji, and P. Fulde forvaluable discussions. Numerical calculations were par-tially performed by using the supercomputers at Institutefor the Solid State Physics. This work was supportedby JSPS/MEXT Grant-in-Aid for Scientific Research(Grant No. 26800177, No. 23540406, No. 25220711 andNo. 15H05852 (KAKENHI on Innovative Areas “Topo-logical Materials Science”)) and by a Grant-in-Aid forProgram for Advancing Strategic International Networksto Accelerate the Circulation of Talented Researchers(Grant No. R2604) “TopoNet.” S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche,R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R.Walker, S. R. Julian, P. Monthoux, G. G. Lonzarich,A. Huxley, I. Sheikin, D. Braithwaite, and J. Flouquet,Nature , 587 (2000). D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flou-quet, J. P. Brison, E. Lhotel, and C. Paulsen, Nature ,613 (2001). N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P.Klaasse, T. Gortenmulder, A. de Visser, A. Hamann,T. G¨orlach, and H. v. L¨ohneysen, Phys. Rev. Lett. ,067006 (2007). M. L. Kuli´c and A. I. Buzdin, in
Superconductivity (Springer, Berlin) Chap. 4. N. T. Huy, , D. E. de Nijs, Y. K. Huang, , and A. de Visser,Phys. Rev. Lett. , 077002 (2008). E. Slooten, T. Naka, A. Gasparini, Y. K. Huang, andA. de Visser, Phys. Rev. Lett. , 097003 (2009). A. Gasparini, Y. K. Huang, N. T. Huy, J. C. P. Klaasse,T. Naka, E. Slooten, and A. de Visser, J. Low. Temp.Phys. , 134 (2010). D. Aoki, T. D. Matsuda, V. Taufour, E. Hassinger,G. Knebel, and J. Flouquet, J. Phys. Soc. Jpn. , 113709(2009). D. Aoki, M. Taupin, C. Paulsen, F. Hardy, V. Taufour,H. Kotegawa, E. Hassinger, L. Malone, T. D. Matsuda,A. Miyake, I. Sheikin, W. Knafo, G. Knebel, L. Howald,J. P. Brison, and J. Flouquet, J. Phys. Soc. Jpn. , SB002(2012). D. Aoki and J. Flouquet, J. Phys. Soc. Jpn. , 061011(2014). A. de Visser, N. T. Huy, A. Gasparini, D. E. de Nijs, D. An-dreica, C. Baines, and A. Amato, Phys. Rev. Lett. ,167003 (2009). T. Ohta, T. Hattori, K. Ishida, Y. Nakai, E. Osaki,K. Deguchi, N. K. Sato, and I. Satoh, J. Phys. Soc. Jpn. , 023707 (2010). D. Fay and J. Appel, Phys. Rev. B , 3173 (1980). P. Monthoux and G. G. Lonzarich, Phys. Rev. B , 14598(1999). Z. Wang, W. Mao, and K. Bedell, Phys. Rev. Lett. ,257001 (2001). R. Roussev and A. J. Millis, Phys. Rev. B , 140504(2001). S. Fujimoto, J. Phys. Soc. Jpn. , 2061 (2004). Y. Ihara, T. Hattori, K. Ishida, Y. Nakai, E. Osaki,K. Deguchi, N. K. Sato, and I. Satoh, Phys. Rev. Lett. , 206403 (2010). T. Hattori, Y. Ihara, Y. Nakai, K. Ishida, Y. Tada, S. Fu-jimoto, N. Kawakami, E. Osaki, K. Deguchi, N. K. Sato,and I. Satoh, Phys. Rev. Lett. , 066403 (2012). T. Hattori, Y. Ihara, K. Karube, D. Sugimoto,K. Ishida, K. Deguchi, N. K. Sato, and T. Yamamura,J. Phys. Soc. Jpn. , 061012 (2014). Y. Tada, S. Fujimoto, N. Kawakami, T. Hattori, Y. Ihara,K. Ishida, K. Deguchi, N. K. Sato, and I. Satoh, J.Phys.:Conf. Ser. , 012029 (2013). T. Hattori, K. Karube, K. Ishida, K. Deguchi, N. K. Sato,and T. Yamamura, J. Phys. Soc. Jpn. , 073708 (2014). F. L´evy, I. Sheikin, B. Grenier, and A. D. Huxley, Science , 1343 (2005). Y. Tokunaga, D. Aoki, H. Mayaffre, S. Kr¨amer, M. H.Julien, C. Berthier, M. Horvati´c, H. Sakai, S. Kambe, andS. Araki, Phys. Rev. Lett. , 216401 (2015). K. Hattori and H. Tsunetsugu, Phys. Rev. B , 064501(2013). V. P. Mineev, Phys. Rev. B , 064506 (2014). V. P. Mineev, Phys. Rev. B , 180504 (2010). D. Aoki and J. Flouquet, J. Phys. Soc. Jpn. , 011003(2012). Y. Yanase, J. Phys. Soc. Jpn. , 014703 (2014). T. Moriya, Phys. Rev. , 91 (1960). M. Samsel-Czekala, S. Elgazzar, P. M. Oppeneer, E. Talik,W. Walerczyk, and R. Tro´c, J. Phys. Condens. Matter ,015503 (2009). V. P. Mineev, C. R. Physique , 35 (2006). U. Schollw¨ock, Rev. Mod. Phys. , 259 (2005). U. Schollw¨ock, Ann. Phys. , 96 (2011). I. P. McCulloch, arXiv:0804.2509. M. Oshikawa and I. Affleck, Phys. Rev. Lett. , 2883(1997). E. Bauer and M. Sigrist, eds.,
Non-Centrosymmetric Su-perconductors (Springer, Berlin). M. H. Fischer, F. Loder, and M. Sigrist, Phys. Rev. B ,184533 (2011). Y. Tada, N. Kawakami, and S. Fujimoto, Phys. Rev. B81