Evolution of ferromagnetic and non-Fermi liquid state with doping:the case of Ru doped UCoGe
Michal Vališka, Jiří Pospíšil, Vladimír Sechovský, Martin Diviš, Mohsen M. Abd-Elmeguid
EEvolution of ferromagnetic and non-Fermi liquid state with doping: the case of Rudoped UCoGe
Michal Vališka, ∗ Jiří Pospíšil, Martin Diviš, Jan Prokleška, Vladimír Sechovský, and Mohsen M. Abd-Elmeguid Faculty of Mathematics and Physics, Charles University,DCMP, Ke Karlovu 5, CZ-12116 Praha 2, Czech Republic Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki, 319-1195, Japan II. Physikalisches Institut, Universität zu Köln, 0937 Köln, Germany
We have investigated the impact of Ru substitution for Co on the behavior of the ferromagneticsuperconductor UCoGe by performing x-ray diffraction, magnetization, specific heat and electricalresistivity measurements on polycrystalline samples of the
UCo − x Ru x Ge series ( ≥ x ≤ . ).The initial Ru substitution up to x ≈ . leads to a simultaneous sharp increase of the Curietemperature and spontaneous magnetization up to maximum values of T C = 8 . and M S = 0 . µ B per formula unit, respectively, whereas superconductivity vanishes already for x ≈ . . Furtherincrease of the Ru content beyond x ≈ . leads to a precipitous decrease of both, T C and M S towards a ferromagnetic quantum critical point (QCP) at x cr = 0 . . Consequently the T − x magnetic phase diagram consists of a well-developed ferromagnetic dome. We discuss the evolution offerromagnetism with x on the basis of band structure changes due to varying 5 f -ligand hybridization.This scenario is supported by the results of electronic structure calculations and consideration of thesimplified periodic Anderson model. The analysis of the temperature dependencies of the electricalresistivity and heat capacity at low temperatures of the samples in the vicinity of the QCP revealsa non-Fermi liquid behavior and assigns the ferromagnetic quantum phase transition to be mostlikely of a continuous Hertz-Millis type. PACS numbers: 71.10.Hf, 74.40.Kb, 71.20.LpKeywords: UCoGe, URuGe, Quantum critical point, non-Fermi liquid behavior, ferromagnetism
I. INTRODUCTION
The phenomena emerging near a quantum criticalpoint (QCP) belong to the most intensively studied top-ics of condensed matter physics. Diligent research in thisfield continuously brings brand new materials carryingcompletely novel properties. Such progress boosts devel-opment of new theoretical approaches describing electroncorrelations in these systems. A specific group of thoseintriguing materials comprises the uranium based ferro-magnetic superconductors (FM SC)
UGe , URhGe and UCoGe . In these compounds superconductivity anditinerant ferromagnetism are carried by the same ura-nium 5 f electrons. It is a novelty distinguishing themfrom previously reported ZrZn . UGe , the first dis-covered case, is a model example of superconductivity(SC) induced by external pressure. Here SC appearsand reaches a maximum T SC on a boundary between twodifferent FM phases under high pressure. URhGe andUCoGe are ambient pressure FM SC where both phe-nomena naturally coexist. A lot of effort both in theoryand experiment has been done to explore the underlyingmechanisms of the coexistence of FM and SC. Ferromag-netic spin fluctuations which appear in the vicinity ofthe QCP have been considered as the main essence forinducing unconventional spin-triplet SC state .Quantum phase transitions (QPTs) were experimen-tally studied for a broad spectrum of materials likehigh- T C superconductors , ordinary metals or heavy-fermion compounds . Most of such investigationshave been carried out on antiferromagnets which by rule exhibit second-order QCP. Prominent examples are CeCu − x Au x with an antiferromagnetic quantum criticalpoint (AF QCP) which is induced by chemical doping or YbRh Si where the AF QCP is achieved by applyingexternal magnetic field . Studies of quantum criticalityin ferromagnets have been less frequent and manifest thathere the situation may be much more complex. The fer-romagnetic phase transition at finite Curie temperature( T C ) is by rule of a second order type. T C of itinerantelectron ferromagnets is often easily suppressed to byexternal pressure p or chemical composition x . However,detailed experimental investigation of archetypal ferro-magnetic metals such as MnSi , ZrZn or UGe hasrevealed that the ferromagnetic phase is suppressed tozero temperature at a first-order transition which wouldmean that no QCP is observed. This can be eluci-dated theoretically either in terms of additional fermionicmodes which may couple to the critical ferromagneticfluctuations driving the phase transition to a first-ordertype or that a first-order magnetic phase transitionmay be induced by strong magneto-elastic coupling .No generic scenario can be drawn for the QPTs of theabove mentioned materials because of rather individuallydifferent phenomena appearing in the quantum criticalregion. In particular MnSi becomes long-period helimag-net (showing ferromagnetism only locally) in which thethermal phase transition is weakly first-order , UGe ex-hibits strong uniaxial anisotropy and ZrZn exhibits amarginal Fermi liquid ground state .UCoGe, the subject of the present study, is uniquein the group of FM SC due to the much lower energy a r X i v : . [ c ond - m a t . s t r- e l ] J un scale on which the magnetism appears . The low Curietemperature of UCoGe is only together with thetiny spontaneous magnetization of . µ B per formulaunit (f.u.) indicate that UCoGe is close to a ferromag-netic instability . It has been observed, however, thatthe Ru and Fe substitution for Co rapidly stabilizes theferromagnetic state , despite the fact that URuGe andUFeGe behave like Pauli paramagnets down to the low-est temperatures . Similar increase of T C was reportedin the case of the initial substitution of Co and Ru forRh in URhGe with the development of a non-Fermiliquid (NFL) state on the higher doping boundary of theFM dome . These observations motivated us to inspectthe development of the magnetic as well as electrical andthermal transport properties in the UCo − x Ru x Ge se-ries over the entire concentration range ( ≥ x ≤ . ).Our study is based on extensive investigation of the crys-tal structure, magnetization, AC magnetic susceptibility,specific heat and electrical resistance of numerous poly-crystalline samples with various Ru content. The resultsare discussed and compared with theoretical calculationsand related models considering the leading role of the5 f -ligand hybridization. II. EXPERIMENTAL DETAILS
In order to study the development of the magneticstate in the
UCo − x Ru x Ge system we have prepared aseries of polycrystalline samples with different Ru concen-trations x between 0 and 0.9. All samples were preparedby arc-melting of the stoichiometric amounts of the ele-ments (purity of Co 4N5, Ge 6N and Ru 3N5). U waspurified by the Solid State Electrotransport technique(SSE) following previous experience with preparationof UCoGe . The arc melting process was realized underprotective Ar (6N purity) atmosphere on a water cooledCu crucible. Each sample was three times turned upsidedown and subsequently re-melted in order to achieve thebest homogeneity. All samples were separately wrappedinto a Ta foil (99.99%), sealed in a quartz tube under thevacuum of · − mbar , subsequently annealed at °Cfor 14 days and then slowly cooled down to room tem-perature to avoid creation of the internal stresses. Eachsample was characterized by X-ray powder diffraction(XRPD) at room temperature on a Bruker D8 Advancediffractometer. The obtained data were evaluated by Ri-etveld technique using FullProf/WinPlotr software with respect to the previously published crystallographicdata of the UCoGe and URuGe compound. Thechemical composition of our samples was verified by ascanning electron microscope (SEM) Tescan Mira I LMHequipped with an energy dispersive X-ray detector (EDX)Bruker AXS. Samples were afterward properly shapedfor individual measurements with a fine wire saw to pre-vent induction of additional stresses and lattice defects.The electrical resistivity ( ρ ) was measured by the 4-probe method on bar-shape samples ( × . × ) and heat-capacity ( C p ) measurements were performed onthin plates ( × × . ) by the relaxation methodon PPMS9T and PPMS14T devices using a He insert.Magnetization ( M ) measurements were done on cubicsamples ( × × ) using a MPMS7T device. Themagnetization was evaluated in µ B / f . u . . For simplicitywe omit “ / f . u . ” everywhere throughout the paper.The electronic structure calculations were performedon the basis of the density-functional theory (DFT)within the local-spin-density approximation (LSDA) and the generalized gradient approximation (GGA) .For this calculations we have used the full-potentialaugmented-plane-wave together with the local-orbitalsmethod (APW+lo) as a part of the latest version(WIEN2k) of the original WIEN code . III. RESULTSA. X-ray diffraction
Both UCoGe and URuGe crystalize in the orthorhom-bic TiNiSi-type structure (space group Pnma) withthe room-temperature cell parameters a = 6 . Å, b =4 . Å, c = 7 . Å and a = 6 . Å, b = 4 . Å, c = 7 . Å, respectively . The unit cell volume ofUCoGe ( V = 208 . Å ) is about 5% smaller than thatof the URuGe compound ( V = 219 . Å ) . The XRPDpatterns confirmed the orthorhombic TiNiSi-type struc-ture of samples over the entire concentration range of the UCo − x Ru x Ge series.The evaluated lattice parameters are listed in Table I.The concentration dependence of all three lattice param-eters and the unit volume reveals a linear behavior, i.e.obeying Vegard’s law (see Fig. 1).While the lattice parameters b and c increase with in-creasing x , the lattice parameter a simultaneously de-creases. The volume expansion seems to reflect the in-crease of the covalent radii from Co (126 pm) to Ru (146pm) . Refinement of the diffraction patterns showed,that the Ru atoms really substitute the Co ones on theirsites.Although the unit cell volume expands with increas-ing Ru concentration the distance between the nearest-neighbor U ions d U − U contracts (see Fig.1). This resultis not surprising because the d U − U lines form a chainmeandering along the a -axis. B. Magnetization and AC-Susceptibility
We have measured the magnetization of each sampleas a function of temperature and applied magnetic field.The values of M S have been estimated from the magne-tization curves measured at .
85 K (the lowest availabletemperature in our MPMS7T) by extrapolating the mag-netization from high magnetic fields to . Table I. The lattice parameters and the unit cell volume ofthe
UCo − x Ru x Ge samples as obtained from the refinementof the X-ray powder diffraction patterns. x a (cid:0) Å (cid:1) b (cid:0) Å (cid:1) c (cid:0) Å (cid:1) V (cid:16) Å (cid:17) The values of T C have been determined by severalmethods. Arrott plot analysis of magnetization data iswidely considered as the most reliable method . For thispurpose the magnetization curves were measured at sev-eral temperatures in the vicinity of the expected T C . TheArrott plots obtained from our magnetization data arestrongly nonlinear. These curves can be approximatedby a third degree polynomial function (see a model ex-ample in Fig. 2). T C is determined as the temperatureof the Arrott plot isotherm that would cross the M axisat 0. An example of the relevant construction is shownin the inset of Fig. 2.The nonlinearity of the Arrott plots (the cubic M vs H/M dependence) suggests presence of a magneti-zation component linearly dependent on the magneticfield. This is related to the fact that UCoGe and theother U
T X compounds crystallizing in the orthorhombicTiNiSi-type structure exhibit strong uniaxial anisotropywith easy magnetization direction along the c -axis. Thehard magnetization directions within the a − b plane arecharacteristic by a weak temperature-independent para-magnetic response with the magnetization proportionalto the magnetic field. We have observed the same typeof magnetocrystalline anisotropy for the ferromagnetic UCo − x Ru x Ge single crystals which we have grown asa part of another study (see Ref. ). Consequently thepolycrystalline samples should show a corresponding lin-ear component also in the ferromagnetic state. By sub-tracting a suitable linear term from measured magneti-zation data we obtain the corrected magnetization values M ∗ = M − a · µ · H . For a = 0 . µ B · T − the Arrottplots M vs H/M ∗ are indeed linear except the low-fieldpart due to low-field magnetization processes and influ- Figure 1. (Color online) - Concentration dependence ofthe lattice parameters and the unit cell volume of the
UCo − x Ru x Ge samples. The lines serve as guides to the eye.Figure 2. (Color online) - Arrott plots for the UCo . Ru . Ge compound. Solid lines are the third orderpolynomial functions. The inset shows that T C is taken asthe value for which the intersection with the M axis wouldbe zero. ence of a demagnetization field as can be seen for examplein the case of the UCo . Ru . Ge sample in Fig. 3.The obtained T C and M S values are listed for all sam-ples in Table II and plotted in the complex phase diagramin Fig. 9(a). T C steeply increases with the initial Ru sub-stitutions for Co which is in agreement with the resultspublished in previous work . This trend terminates at Figure 3. (Color online) - Revised Arrott plots after subtrac-tion of the linear term with the slope a = 0 . µ B · T − fromthe magnetization data measured on the UCo . Ru . Ge sample. x max ≈ . where the ordering temperature reaches amaximum value of T C , max ≈ . . This value is al-most three times higher than T C = 3 K of the parentcompound and is comparable with the value found byHuang et al. in the case of the corresponding substitutionof Fe for Co in UCoGe . Increasing Ru concentrationbeyond x ≈ . is accompanied by a simultaneous de-crease of T C and M S towards zero at the critical concen-tration x cr ≈ . . Thus, the ferromagnetic dome of theconcentration dependence of T C in the T − x magneticphase diagram is intimately connected with a correspond-ing change of M S (see Fig. 9(a)).The M ( T ) curves measured on selected samples withconcentration above x ≥ . displayed in Fig. 4 alsomanifest the collapse of ferromagnetism with increasingRu content. The estimated T C values as derived fromthe temperature of the inflection point in the M ( T ) de-pendence (measured in low external field of
10 mT ) arein good agreement with ordering temperatures obtainedfrom the Arrott plot analysis (see Table II and Fig. 9(a)).We have also measured the AC magnetic susceptibil-ity ( χ ) for different Ru concentration above x ≥ . attemperatures down to .
85 K using a MPMS device. Formeasurements at lower temperatures (down to
400 mK )a custom-made coil system attached to the He insertin PPMS and a lock-in amplifier were utilized (the samesetup as that used in Ref. ). T C is usually identified asthe temperature of the maximum of the real part of χ (seeFig. 5). While the low temperature AC susceptibility ofthe sample with x = 0 . reveal a well-developed peak at .
44 K indicating the onset of ferromagnetism, no clearpeak maximum is observed for the sample with x = 0 . ,which might be at approximately
350 mK as the lowest- T point was measured at
400 mK . For the sample with x = 0 . no trace of χ anomaly has been detected downto
400 mK which seems to be in the immediate vicinity of
Table II. Values of the spontaneous magnetization M S and Curie temperature derived from Arrott plot analy-sis ( T C , Arrott ), temperature dependence of AC susceptibility( T C ,χ ), magnetization ( T C ,M ) and specific heat ( T C ,C p ), andSommerfeld coefficient ( γ ) as determined from the specificheat data at low temperatures for samples with various con-centration of Ru ( x ). x M S T C , Arrott T C ,χ T C ,M T C ,C p γ ( µ B ) ( K ) ( K ) ( K ) ( K ) (cid:0) mJmol · K (cid:1) ≈ . - - 0.15980.40 0.0011 - - - - 0.15230.50 0.0001 - - - - 0.1490Figure 4. (Color online) - Temperature dependence of themagnetization of selected UCo − x Ru x Ge compounds mea-sured in an external magnetic field of
10 mT . The arrowsmark T C for each composition. the critical Ru concentration for existence of ferromag-netism in the UCo − x Ru x Ge compounds. C. Specific heat
To analyze the different contributions to the specificheat we have subtracted from experimental data thephonon contribution using the fit of the phonon specificheat as a C ph ( T ) = βT . We typically obtain values of Figure 5. (Color online) - Temperature dependence of the realpart of the AC susceptibility of selected
UCo − x Ru x Ge com-pounds. The arrows mark T C for each composition. Data areplotted in arbitrary units and normalized because the homemade coil for measurement in He (used for measurement ofsamples with x = 0 . − . ) provides only relative data.Some curves are not shown for clarity of the figure. β ≈ (0 . − . · − J · mol − K − which correspondto Debye-temperature values of −
155 K . The re-maining part of the specific-heat C represents the sumof the electronic and magnetic contributions C e and C m ,respectively.Fig. 6 displays the specific heat C divided by tem-perature T versus T on a log scale for selected samplesbetween x = 0 . and 0.31. The anomaly at T C is gradu-ally smeared out and shifted to lower temperatures withincreasing Ru concentration. Samples with x ≤ . showclear anomalies that are coincident with the onset of fer-romagnetic order and are in reasonable agreement withthe T C values derived from magnetization and AC sus-ceptibility (see Table II and Fig. 9 (a)). For sampleswith x = 0 . and 0.31 C/T versus log T exhibits nearlylinear dependence between 1 and ∼
10 K but graduallylevels off at lower temperatures. This indicates a non-Fermi-liquid (NFL) behavior C ( T ) /T = c ln ( T /T ) that is expected for concentrations in the vicinity of theferromagnetic QCP. We note that our data do not followthis dependence in the whole temperature range similarto that recently reported on UCo − x Fe x Ge system .We further calculate the magnetic entropy S m inte-grated over the temperature range from . up to the T C for each sample and find a steady decrease of S m with increasing x from .
13 R ln 2 for x = 0 . down to .
006 R ln 2 at x = 0 . (see Fig. 9(c)). This is consis-tent with the observation of a gradual disappearance ofthe itinerant magnetic moment by approaching the QCP( x cr ≈ . ). As the system approaches the critical con-centration we observe a large increase of the value of Som-merfeld coefficient γ with a maximum near x cr ≈ . which reflects an enhancement of the effective mass of Figure 6. (Color online) -
C/T versus log T plot for selected UCo − x Ru x Ge compounds. Black arrows indicate T C forsamples with x = 0 . , 0.22 and 0.24, respectively.Figure 7. (Color online) - Estimation of the critical concen-tration for ferromagnetism in the UCo − x Ru x Ge system byapplying the T / vs x plot and the T C values derived fromthe Arrott plots. the quasiparticles in the region where ferromagnetism issuppressed. This finding is consistent with the presenceof a strong spin fluctuation near the ferromagnetic QCP.According to the prediction for the dependence of T C ona control parameter ( x ) for itinerant ferromagnets QCPby Millis and Hertz the ordering temperature shouldobey the relation T C ∼ ( x cr − x ) / i.e. a linear T / vs x plot. As we show in Fig. 7 a linear fit of T C values forthe samples with x from 0.2 to 0.3 reveals that T C van-ishes at the critical concentration x cr ≈ . consistentwith this model. D. Electrical resistivity
The low-temperature resistivity data measured on se-lected polycrystalline samples are plotted in Fig. 8.Anomalies connected with the transition from paramag-netic to ferromagnetic state are not clearly visible. It isevident, that increasing Ru content leads to considerablechanges of the low temperature resistivity behavior. The ρ ( T ) data below T C reasonably follow the ρ = ρ + AT dependence usual for ferromagnets. Data above T C werefitted to the relation: ρ = ρ + AT n (1)The inflection point of the ρ ( T ) dependence was takenas an upper limit for the fitting. The exponent ( n ) grad-ually decreases as the Ru content approaches the criticalconcentration x cr . The minimum value of n ≈ . for x = 0 . is close to the proposed linear temperature de-pendence from the theory of Millis and Hertz forNFL behavior of a clean 3-dimensional itinerant ferro-magnets rather than to the scaling with the exponent n = 5 / which follows from the spin-fluctuation theoryof Moriya . The samples with higher concentration ofRu ( x > x cr ) seem to exhibit gradual recovery towards aFL state which is documented by increasing the value of n exponent with increasing x above x cr .Development of the exponent n is summarized in the T − x phase diagram (Fig. 9 (b)). In order to see the ex-ponent n as a function of temperature we have calculatedthe logarithmic derivative of the electrical resistivity ac-cording to the Eq. (2). n = d ln ( ρ − ρ )d ln T (2)The results of this analysis are displayed in the coloredpart of the phase diagram in Fig. 9 (a). One can see asignificant change of the exponent between the region offerromagnetic ordering ( T < T C ) where n = 2 and inthe nonmagnetic state where n < . The sharp decreaseof the value n near x cr down to the lowest temperaturesis surrounded by regions of higher n (rapidly increasingon the FM side for x < x cr and slower increase on theparamagnetic side). E. Theoretical calculations
In order to better understand the changes in theelectronic structure of the
UCo − x Ru x Ge compoundsacross the ferromagnetic QCP, we have performed first-principles theoretical calculations on the paramagneticcompound URuGe. As a matter of fact, while the den-sity of states (DOS) of the parent compound UCoGe isknown (Ref. ) the information about the DOS of URuGeare missing. The calculated total and partial DOS of theURuGe are plotted in Fig. 10.We used the calculated URuGe band structure by con-sidering the simple model of Silva Neto et al. which is Figure 8. (Color online) - Temperature dependence of theelectrical resistivity for selected polycrystalline samples of
UCo − x Ru x Ge . The vertical arrows denote T C values ob-tained from AC susceptibility data. Dashed lines are fits todata above T C according to Eq. (1). Each curve is arbitraryvertically shifted for better clarity of the figure. based on the periodic Anderson model . This simpli-fied model proposes the key role of the nd − f hybridiza-tion ( V df in Eq. (3)), where n is the number of d electronsin the observed non-monotonous evolution of T C in the URh − x Co x system. They described the evolution of T C with increasing x as a consequence of the broadening ofthe nd and f bands ( W d , W f ), respectively, and the mu-tual shift of their centers ( C T d − C Uf ) that are relatedas : V df = W d W f C T d − C Uf (3)If we apply this model to our UCo − x Ru x Ge systemwe can qualitatively describe the non-monotonous evo-lution of T C with Ru concentration. The concentrationdependence of the broadness of the nd band is assumedto be linear according to 4 Figure 9. (Color online) - Panel a) shows the T − x phase dia-gram based on measurements of polycrystalline samples. Thediagram is supplemented by the results of the electrical resis-tivity measurement revealing occurrence of superconductivityin the parent UCoGe compound and in UCo . Ru . Ge - thetwo data points are taken from Ref. (green triangle). Theblack solid line is only guide to the eye while the red dashedpart is a fit of T C ∼ ( x cr − x ) / . The right axis denotesthe spontaneous magnetization M S (dashed line in the plotis only a guide to the eye). The color plot shows local expo-nents of the resistivity obtained as n = d ln( ρ − ρ )d ln T . The blackfilled circles show the temperature where resistivity starts todeviate from the T dependence. Panel b) shows the evolu-tion of the coefficients n from the fitting of the low temper-ature dependence of the electrical resistivity with equation ρ = ρ + AT n for T > T C . The right vertical axis shows RRR = ρ
300 K /ρ . as a function of x . Panel c) shows de-velopment of C/T (extrapolated to ) and the magneticentropy S m (value for the parent UCoGe is taken from Ref. and is marked by a star). W d ( x ) = W Cod (1 − x ) + W Rud ( x ) (4)where W Cod = 6 . (Ref. ) and W Rud = 8 . (seeFig. 10) and W f = 0 .
43 eV (Ref. ). Such a behav-ior is consistent with other U T X ( T = transition metal, X = p element) compounds where the d band broadenswhile we move from the 3 d to the 4 d transition metals . Figure 10. (Color online) - Total and partial density of states(DOS) for U– f states and Ru– d states in URuGe. Width ofthe d -band ( W Rud ) and its center ∆ C Rudf are marked by dashedarrows. Inset shows that the contribution of the Ge– p statesis far from the Fermi level. Consequently ( C T d − C Uf ) ( x ) = (cid:52) C df ( x ) deviates fromlinearity ∆ C df ( x ) = ∆ C Codf (1 − x ) + ∆ C Rudf ( x ) + (5) δ ´ x (1 − x ) + δ ´´ x (1 − x ) We used the values from calculated DOSes, i.e. ∆ C Rudf = 0 .
65 eV and ∆ C Codf = 1 . (Ref. ) and ad-justable parameters were taken as δ ´ = 2 · − and δ ´´ =2 . Such an approach leads to a non-monotonous depen-dence of the d − f hybridization term V df ; starting with V df ( x = 0) ≈ . for UCoGe (in agreement with Ref. ), V df ( x = 1) ≈ . for URuGe and V df ( x ≈ . ≈ . as estimated for the ferromagnetic QCP . The overall V df ( x ) dependence starts with its decrease and therebycauses an enhancement of the density of f states atthe Fermi level N f ( E F ) . In case of itinerant ferro-magnets we can estimate the ordering temperature asa function of the density of states at the Fermi level T C ∼ ( IN ( E F ) − / where I is the Stoner integraland N ( E F ) is the total density of states at the Fermilevel . In this respect we can attribute the initial in-crease of T C to the enhanced N ( E F ) . At x ≈ . the d − f hybridization reaches its minimum value V df = 1 . and starts to increase with increasing x . This point qual-itatively corresponds to the position of the maximum T C in the experimental data at x ≈ . . As the Ru concen-tration increases the d -band is shifted closer to the po-sition of the f -band and the hybridization increases andthereby results in a reduction of the contribution of the N f ( E F ) to N ( E F ) . For the reason mentioned abovethe ordering temperature decreases and reaches zero near x cr ≈ . . IV. DISCUSSION
The 5 f electron magnetism in uranium compounds iscontrolled by the degree of overlap of the 5 f wave func-tions of neighboring U ions and by the hybridizationof the U-ion 5 f -electron states with states of the lig-and valence electrons (5 f -ligand hybridization). Thesetwo mechanisms cause that the 5 f -electron orbitals loosetheir atomic character which they exhibit in the U freeion. Thus, the 5 f -5 f overlap and/or strong 5 f -ligandhybridization lead to delocalization of the 5 f -electrons,their participation in metallic bonding , and conse-quently a washout of the U magnetic moment . In ad-dition, the spin-orbit interaction in the U ion plays animportant role in electronic structure. Accordingly, anorbital magnetic moment antiparallel to the spin mo-ment is induced by the strong spin orbit coupling inthe spin-polarized energy bands of itinerant 5 f electronmaterials . The magnitude of the U 5 f -electron mag-netic moments is thus further strongly reduced due tothe mutual compensation of the orbital and spin com-ponents. The orbital moment is by rule larger than thespin moment considering results of so far done relevantexperiments (see relevant references in Ref. ).On the other hand, the 5 f -ligand hybridization playsa dual role in U compounds. Besides washing out the5 f -electron magnetic moment it mediates an indirect ex-change interaction which couples the uranium magneticmoments to promote the magnetic ordering and simulta-neously causes very strong magnetocrystalline anisotropyeven in very weak itinerant magnets . Within thisprocess the hybridized ligand valence states become po-larized and as a result the ligand ion (especially transitionelement ion) exhibits a small induced magnetic momentwhich is usually parallel to the dominant 5 f -electronorbital component (see relevant references in Ref. ).This scenario apparently holds for UCoGe as evidencedfrom a recent X-ray magnetic circular dichroism (XMCD)study and polarized neutron diffraction (PND) exper-iments on UCo . Ru . Ge and UCo . Ru . Ge sin-gle crystals . These experiments confirm that the 5 f -electron orbital moment dominates the antiparallel spincomponent. A much smaller Co magnetic moment is in-duced by the 5 f -3 d hybridization.Considering the change of the U-U distance d U − U be-tween the nearest U neighbor ions (overlap of 5 f orbitals)within the UCo − x Ru x Ge series, we find that d U − U de-creases with increasing Ru concentration from ≈ . Åin UCoGe to . Å in URuGe (see Fig. 1 and Fig.11). Both values fall rather on the “nonmagnetic side”of the Hill plot . On the other hand one should bearin mind that each U ion has only 2 nearest U neigh-bors on the d U − U chain meandering along the a -axis.If the 5 f -5 f overlap was the only mechanism control-ling magnetism then a gradual washout of U magneticmoment and monotonously decreasing of T C with in-creasing Ru content would be expected. On the con-trary, however, we observe an initial rapid increase of T C to a maximum followed by a suppression of ferro-magnetism with further increasing x . We note that ourobservation of a ferromagnetic dome in magnetic phasediagram in UCo − x Ru x Ge (see Fig. 9 (a)) is similarto those observed for UCo − x Fe x Ge , URh − x Ru x Ge and URh − x Co x Ge .Apparently an additional mechanism, namely the 5 f -ligand hybridization must be taken into account for con-ceiving the complex evolution of ferromagnetism in thesesystems. The increase of T C and U magnetic momentwith increasing x up to 0.12 is accompanied by increas-ing the 5 f electron orbital moment . The increaseof the orbital moment is usually considered as a signof partial localization of 5 f electrons because the or-bital moment density is distributed closer to the nucleusthan the spin density as it has been demonstrated ona detailed study of the U 5 f electron form factor in UFe . Nevertheless, the µ L /µ S ratio of ≈ . in-dicates still a significant delocalization of the 5 f elec-tron states for x = 0 . . As we mention above ourtheoretical band structure calculation provide the ba-sis for understanding the mechanism responsible for theferromagnetic dome in the magnetic phase diagram of UCo − x Ru x Ge by following the simple model treatingthe changes of 5 f − nd hybridization with variations ofthe widths and mutual positions on the energy scale ofthe transition metal d bands and U 5 f bands . Ac-cordingly, the non-isoelectronic substitution of Co by Rucauses broadening of the d band from 3 d to the 4 d tran-sition metal-like. Together with the mutual movementof the d and f bands on energy scale itself we can qual-itatively conceive the dome-like dependence of the or-dering temperature T C . This is an important confirma-tion of the trend. Variations of the 5 f − nd hybridiza-tion most likely cause analogous non-monotonous varia-tion of the magnetic ground state of UCo − x Fe x Ge and URh − x Ru x Ge exhibiting also a ferromagnetic dome inthe magnetic phase diagram. It is worth to mention thatthe non-monotonous evolution of magnetic ground statecausing a ferromagnetic dome in the magnetic phase di-agram is not only specific to the U T Ge compounds pos-sessing the orthorhombic TiNiSi-type structure. Analo-gous trends reflecting the varying 5 f − nd hybridizationare observed also in U T X compounds with the hexag-onal ZrNiAl-type structure. Here UFeAl , URuAl and UCoAl are paramagnets. The latter compoundis, however, close to a ferromagnetic instability. Amagnetic field of only . induces in UCoAl itinerantelectron metamagnetism . URhAl and URhGe are ferromagnets. Ferromagnetic domes are observedin the magnetic phase diagrams of UCo − x Ru x Al , URh − x Ru x Al , URh − x Ru x Ga and anticipated fromthe results reported on UCo − x Fe x Al .The observed strong delocalization of the 5 f electronsin UCo − x Ru x Ge at higher Ru concentrations is reflectedby a dramatic decrease of the magnetic entropy S m downto the .
006 R ln 2 for x = 0 . which points to the itiner-ant nature of the weak ferromagnetism in the vicinity ofthe critical concentration. Note that a magnetic entropyequal to zero is expected for an ideal itinerant electronferromagnet . Our results of the temperature depen-dence of the electrical resistivity provide evidence for aNFL behavior in the vicinity of x cr most likely caused bythe possible presence of the FM QCP. We have observeda drop of the n exponent in the temperature dependenceof resistivity ρ = ρ + AT n and an almost logarithmicdependence of the heat capacity C ( T ) /T = c ln ( T /T ) in a limited interval at lowest temperatures that wouldbe in agreement with the theoretical predictions of Mil-lis and Hertz . Further evidence for the FM QCPis offered by the rapid increase of the effective mass ofthe quasiparticles near x cr . The proposed scenario isalso corroborated by scaling of the ordering temperaturewith the control parameter itself which obeys the formula T C ∼ ( x cr − x ) / and provides estimation of the criticalconcentration x cr ≈ . .The FM transition of UCo − x Ru x Ge compounds inthe vicinity of x cr is apparently of a second order typein contrast to the first order transition reported for 3-dimensional ferromagnets in the vicinity of a QCP .Microscopic NQR studies of UCoGe suggest a first ordertransition to the FM state . The second order transitionin UCo − x Ru x Ge compounds near x cr can be conceivedas a consequence of the substitution- induced disorderin the system which may blur the first order transitiontowards a continuous second order transition. In thiscontext, we would like to mention the experimental andtheoretical arguments regarding the observed anoma-lies related to the existence of a ferromagnetic QCP in UCo − x Ru x Ge should be considered with a proper cau-tion. Disorder caused by substitution can in some casesemulate NFL behavior and may be one of the rea-sons of the lacking superconductivity in UCo − x Ru x Ge in the proximity of the QCP. Thorough the investiga-tion of single crystals of UCo − x Ru x Ge compounds near x cr at ambient and high pressures is highly desired inorder to clarify the origin of the NFL state and the char-acter of the ferromagnetic quantum phase transition in UCo − x Ru x Ge . V. CONCLUSIONS
We have successfully prepared series of the polycrys-talline samples of UCoGe doped with Ru in a wide rangeof concentration
UCo − x Ru x Ge ( ≥ x ≤ . ). TheRu substitution leads to development of a FM dome be-tween x = 0 − . with the maximum of T C = 8 . and M S = 0 . µ B appearing at the x ≈ . . Further in- crease of the Ru content up to the critical concentration x cr ≈ . leads to the disappearance of the ferromag-netic state at a QCP. Using electronic structure calcula-tions we were able to explain the evolution of ferromag-netism with x for UCo − x Ru x Ge in terms of changes ofthe density of states at the Fermi level due to varying5 f -ligand hybridization. The analysis of the critical ex- Figure 11. (Color online) - Illustrative plot showing the de-pendence of the ordering temperature of the U T Ge com-pounds ( T = transition metal) on the shortest distance be-tween two nearest uranium atoms ( d U − U ). Shaded regionspreads around Hill limit ( . Å) valid for uranium. Posi-tion of UFeGe is exceptional because UFeGe does not keepthe TiNiSi-type structure . ponents of the electrical resistivity and heat capacity atlow temperatures revealed a non-Fermi liquid behaviorfor the samples in the vicinity of the QCP. The NFLstate can be influenced by the substitution-induced dis-order of the system because of the non-isoelectronic mix-ture of the 3 d (Co) and 4 d (Ru) bands. Further study ofthe region around the critical concentration including themeasurements under the external pressure performed onhigh quality single crystals is highly desired for a betterunderstanding the physics underlying the ferromagneticquantum phase transition. ACKNOWLEDGMENTS
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