arXiv:1112.1864v2 [cond-mat.supr-con] 29 Feb 2012 trs( temper- transition atures superconducting higher even revealed elements, a ruis(11tp)srcuewt pc group space with tetrago- structure primitive P (1111-type) the ZrCuSiAs in nal crystallize compounds These esfidatrtedsoeyo uecnutvt t2 K LaFeAsO 26 compound at the superconductivity in of discovery the after tensified eaosfr qaeltiei the in lattice square a form atoms Fe aeerheeet aessakdaogthe along stacked layers element) earth rare neo ihtmeauesprodciiy si the in high- as long-range appear- cuprate superconductivity, the the layered high-temperature of for suppression of necessary state This Suchance appears oxidation transitions atoms. formal F. ordering Fe nonintegral by the a O for in substituting results partially doping by doping upon density transitions, spin cou- (SDW) (AF) show wave antiferromagnetic compounds and from parent structural equidistant undoped and pled The of side plane. Fe either on an planes As in coordinating lie the atoms where tetrahedra, As by coordinated ecnutn ttmeaue pt bu . K. 4.5 about to up temperatures su- at becomes perconducting LaNiPO analogue Ni-P self-doped 1111-type 4/ h erhfrhg eprtr uecnutr in- superconductors temperature high for search The usqety h aetcompounds parent the Subsequently, nmm tutrl hra,Mgei n lcrncTasotPr Transport Electronic and Magnetic Thermal, Structural, T . 3,4 c msLbrtr n eateto hsc n srnm,Io Astronomy, and Physics of Department and Laboratory Ames 6 pnrpaigL ihsalrrr earth rare smaller with La replacing upon ) hyhv lentn esand FeAs alternating have They iliga yielding eetn ag est fsae tteFrieeg hti that energy Fermi the at the states for of found density ues large a reflecting il ahrlreSmefl lcrncseicha coe heat specific electronic Sommerfeld large rather a yield prxmtl ierywith linearly approximately 0 ,idctn htalcmoiin nti ytmaeme are system this in compositions all that indicating K, 300 eprtr-needn aaantcbhvo costh LaNi across behavior paramagnetic temperature-independent The temperature hs opud rsalz ntebody-centered-tetrago the in crystallize compounds these observed I fteDbetmeauedtrie rmthe from determined temperature Debye the of experimental ru esneeto-hnnrssiiyfntosversus functions resistivity th Gr¨uneisen electron-phonon representing Pad´e correla approximants High-accuracy antiferromagnetic system. short-range from arise may that ρ la vdnefrbl uecnutvt rayohrlong other any LaNi or the superconductivity of bulk any for evidence clear hi rprisivsiae yxrydffato XD me (XRD) capacity diffraction x-ray heat by by investigated properties their ASnmes 47.a 26.d 52.n 65.40.Ba 75.20.En, 02.60.Ed, 74.70.Xa, numbers: PACS 4/ o l ape r eldsrbdoe this over well-described are samples all for oyrsaln ape fLaNi of samples Polycrystalline mmm .INTRODUCTION I. 2 ρ Ge ( T esrmnssoe oiietmeauececetfor coefficient temperature positive a showed measurements ) 2 ihcmoiindpnetltieprmtr htslig that parameters lattice composition-dependent with ) ρ hr ra eki bevdat observed is peak broad a where , T 30K ausaelre hncluae rmti oe.As A model. this from calculated than larger are values K) (300 .J otc,V .Aad bihkPne,adD .Johnston C. D. and Pandey, Abhishek Anand, K. V. Goetsch, J. R. T c c T 7–9 superconductors. C f5 o SmFeAsO for K 55 of p 2 1 rm18t 5 .Revl enmnso odrXDpatterns XRD powder of refinements Rietveld K. 350 to 1.8 from ( (Ge − T hc r ohsuppressed both are which C x A and ) F p 1 Fe antcsusceptibility magnetic , x − . x 2 1,2 As P ρ x 2 usqetstudies Subsequent ( ) T 2 ls fmtraswt h aecytlsrcue The structure. crystal same the with materials of class ab x aa epciey o 1 for respectively, data, ) opstosstudied. compositions A o741 JmlK mJ/mol 7.4(1) to LaNi 5,7,8,10–13 ln n are and plane Fe 2 c 2 (Ge Dtd eebr7 2011) 7, December (Dated: ai.The -axis. As R ( O 1 2 1 2 − − (Ge 14,15 x x ( A R P F The x x ) = = T . 1 2 5 − χ C ag yteBohG¨nie oe,atog the although Bloch-Gr¨uneisen model, the by range ( x ≈ n lcrclresistivity electrical and , x p P ( 2 0 = 0 from K 300 T yecmons o xml,tesmcnutn AF semiconducting the BaMn example, compound For compounds. type et ihspin with ments uecnutr n ocaiytemtrasfeatures materials new the for search high- clarify for to to necessary structure and 122-type superconductors the with pounds neetoi/antcmcaimrte hnfo the interaction. from electron-phonon than conventional rather mechanism electronic/magnetic an ecnutvt nteehigh- these in perconductivity ssuppressed. is uae ob nucett edt h bevdhigh observed the to lead to insufficient cal- T be been to has culated interaction electron-phonon conventional onsabove pounds reigtastosaesuppressed. magnetic and are crystallographic transitions long-range to ordering these up as temperatures K at 38 appears again Superconductivity 2 K. 625 nadto,te hwSWadsrcua transitions structural and temperatures SDW high show compounds. at 1111-type they the addition, in as In layers FeAs of type same a r a n u eeinvestigated. were Eu) and Ba, Sr, Ca, tutr ihsaegroup space with structure ThCr body-centered-tetragonal the in lize nte11-yecmonsadteeaesmlrysup- the similarly on are substituting these upon and pressed compounds 1111-type the in x for aawsosre o ahcmoiin No composition. each for observed was data ) ) c s n togA utain tl cu nteecom- these in occur still fluctuations AF strong and ’s, h bv icvre oiae tde fohrcom- other of studies motivated discoveries above The 2 T , . K 8 x System 0 r rsne n r sdt nlz our analyze to used are and presented are in naqaitodmninlmagnetic quasi-two-dimensional a in tions a ThCr nal . nierneo opstosecp for except compositions of range entire e ey atc etcpct n Bloch- and capacity heat lattice Debye e 25 fficient .The 1. = 30–33 ≤ rnepaetasto a on for found was transition phase -range , srmnsa omtmeaueand temperature room at asurements aSaeUiest,Ae,Iw 50011 Iowa Ames, University, State wa alc h low- The tallic. 0 T oprbewt h ags val- largest the with comparable s . χ 50 ercnl oe hscmon ihKto K with compound this doped recently We ( ≤ γ T 2 , 2 tydvaefo eadsLaw. Vegard’s from deviate htly T Si esrmnsu o10 K 1000 to up measurements ) 12 = 0 .The K. 300 0 h osnu steeoeta h su- the that therefore is consensus The c 2 χ . 75 tp tutr saegroup (space structure -type S 2 vnatrteln-ag Fordering AF long-range the after even As T esrmnsso nearly show measurements l ape rm18Kto K 1.8 from samples all . , ()m/o K mJ/mol 4(2) c 5 = 2 16–25 )wr yteie and synthesized were 1) ignificant uecnutvt nthe in superconductivity ρ otislclM antcmo- magnetic Mn local contains / esrmnsversus measurements C T n N´eel temperature and 2 htaesmlrt hs seen those to similar are that T priso the of operties p dpnecsof -dependences I T 4/ measurements T c ofimthat confirm -dependence mmm γ opud rssfrom arises compounds A 2 eado ssites. As and/or Fe , decreases for 26–29 x n oti the contain and 0 = 2 2 2 Si hycrystal- They oee,the However, 2 (122-type) A Fe T 2 As N 2 = 2 - 2 form a new series of metallic AF Ba1−xKxMn2As2 com- unit. This D(EF) is comparable to the largest values re- pounds containing the Mn local magnetic moments,34 ported for the FeAs-based 122-type superconductors and 2 but no superconductivity has yet been observed in this parent compounds. Three bands were found to cross EF, 34,35 system. This may be because the TN was not suffi- with two Fermi surfaces that were hole-like (0.16 and ciently suppressed by the K-doping levels used. Studies of 1.11 holes/f.u.) and one that was electron-like (0.27 elec- 122-type compounds in which the Fe and As in AFe2As2 trons/f.u.), and therefore with a net uncompensated car- are both completely replaced by other elements have also rier charge density of 1.00 holes/f.u. Thus for the hypo- been carried out. For example, LaRu2P2 becomes super- thetical compound ThNi2Ge2 one can assign formal ox- 36 +4 +2 −4 conducting at Tc =4.1 K. SrPd2Ge2 was recently found idation states Th , Ni and Ge . Then substituting 37 to become superconducting with Tc =3.0 K, with con- trivalent La for tetravalent Th yields a net charge carrier ventional electronic and superconducting properties.38 concentration of one hole per formula unit. The electron The Fe-based phosphides not containing magnetic Fermi surface is a slightly corrugated cylinder along the c- 39 40 rare earth elements such as CaFe2P2, LaFe2P2, axis centered at the X point of the Brillouin zone, indicat- 40 SrFe2P2, and BaFe2P2 (Ref. 41) show Pauli paramag- ing quasi-two-dimensional character, similar to the elec- netic behavior. The 122-type Co-based phosphides ex- tron Fermi surface pockets in the FeAs-based 122-type 2 hibit varying magnetic properties. SrCo2P2 does not compounds. On the other hand, the two hole Fermi sur- order magnetically, although it has a large Pauli sus- faces are three-dimensional and are centered at the X and 2 ceptibility χ ∼ 2 × 10−3 cm3/mol that has variously Z (or M, depending on the definition ) points of the Bril- been reported to exhibit either a weak broad peak at louin zone. These calculated Fermi surfaces were found ∼ 110 K attributed to “weak exchange interactions be- to satisfactorily explain the results of the above dHvA 51 tween itinerant electrons” (Ref. 40), or a weak broad measurements, including the measured band masses. peak at ∼ 20 K attributed to a “nearly ferromagnetic From a comparison of the calculated D(EF) with that Fermi liquid” (Ref. 42). LaCo2P2 orders ferromagnet- obtained from experimental electronic specific heat data, 40,43 ically at a Curie temperature TC ≈ 130 K, and Yamagami inferred that many-body enhancements of the 53 CaCo2P2 is reported to exhibit A-type antiferromag- theoretical band masses are small. netism at TN = 113 K in which the Co spins align Hall effect measurements on single crystals of ferromagnetically within the basal plane and antiferro- LaNi2Ge2 are consistent with the occurrence of multi- magnetically along the c-axis.44 These differing magnetic ple electron and hole Fermi surfaces, with the weakly T - properties of the Co-based phosphides are correlated with dependent Hall coefficients given by a positive (hole-like) −10 3 the formal oxidation state of the Co atoms, taking into value RH ∼ +3 × 10 m /C for the applied magnetic account possible P-P bonding.44 Compounds where the field H parallel to the a-axis and a negative (electron- −10 3 Co atoms have a formal oxidation state of +2, ∼ +1.5, like) value RH ∼−2 × 10 m /C for H parallel to the 54 and ∼< +1 show no magnetic order, ferromagnetic order c-axis. The thermoelectric power obtained on a poly- 44 55 and antiferromagnetic order, respectively. None of the crystalline sample of LaNi2Ge2 is negative. above 122-type Fe or Co phosphides were reported to Herein we report our results on the mixed system become superconducting. LaNi2(Ge1−xPx)2. For x = 0 or 1, alternating La Among Ni-containing 122-type compounds, supercon- and NiGe or NiP layers, respectively, are stacked along ductivity has been reported with Tc = 0.62 K in the c-axis. We wanted to investigate whether any new 45 SrNi2As2, Tc = 0.70 K in the distorted structure of phonomena occur with Ge/P mixtures that do not oc- 46 BaNi2As2, Tc = 1.4 K in the orthorhombically dis- cur at the endpoint compositions, such as happens when torted structure of SrNi2P2 (Ref. 47) and Tc = 3.0 K in the parent FeAs-based compounds are doped/substituted 48 BaNi2P2. The Pauli paramagnet LaNi2P2 is reported to form high-Tc superconductors. In addition, Yam- 49 not to become superconducting above 1.8 K. There are agami’s electronic structure calculations for LaNi2Ge2 conflicting reports about the occurrence of superconduc- discussed above53 indicated some similarities to the elec- 50,51 tivity in LaNi2Ge2 with either Tc = 0.69–0.8 K, or tronic structures of the FeAs-based 122-type compounds. 52 no superconductivity observed above 0.32 K. We report structural studies using powder x-ray Several studies have been reported on the normal state diffraction (XRD) measurements at room temperature, properties of LaNi2Ge2. de Haas van Alphen (dHvA) together with heat capacity Cp, magnetic susceptibility measurements at 0.5 K indicated moderate band ef- χ, and electrical resistivity ρ measurements versus tem- ∗ fective masses m /me = 1.2 to 2.7, where me is the perature T from 1.8 to 350 K for five compositions of 51 free electron mass. Electronic structure calculations LaNi2(Ge1−xPx)2 with 0 ≤ x ≤ 1. Our low-T limit were subsequently carried out by Yamagami using the of 1.8 K precluded checking for superconductivity with 50,51 all-electron relativistic linearized augmented plane wave Tc < 1 K reported for LaNi2Ge2, but we did find ev- method based on the density-functional theory in the idence for the onset of superconductivity below ∼ 2 K in 53 local-density approximation. The density of states at two samples of LaNi2P2 from both ρ(T ) and χ(T ) mea- the Fermi energy EF was found to be large, D(EF) = surements. However, it is not clear from our measure- 5.38 states/(eV f.u.) for both spin directions, arising ments whether this onset arises from the onset of bulk mainly from the Ni 3d orbitals, where f.u. means formula superconductivity or is due to an impurity phase. 3

Also presented in this paper is the construction of Pad´e button was flipped and remelted five times to ensure 56 approximants for the Debye and Bloch-Gr¨uneisen func- homogeneity. Next, the samples (except for LaNi2Ge2 tions that describe the acoustic lattice vibration contri- which was prepared following the general procedures out- bution to the heat capacity at constant volume CV(T ) lined in Ref. 64) were throughly ground and mixed with of materials and the contribution to the ρ(T ) of met- the necessary amount of P powder in a glove box un- als from scattering of conduction electrons from acous- der an atmosphere of ultra high purity He. The pow- tic lattice vibrations, respectively. These Pad´eapproxi- ders were cold-pressed into pellets and placed in 2 mL mants were created in order to easily fit our respective alumina crucibles. The arc-melted button of LaNi2Ge2 experimental data using the method of least-squares, but was wrapped in Ta foil. The samples were then sealed they are of course more generally applicable to fitting the in evacuated quartz tubes and fired at 990 ◦C for ≈ 6 corresponding data for other materials. The Debye and d. Samples containing phosphorus were first heated to Bloch-Gr¨uneisen functions themselves cannot be easily 400 ◦C to prereact the phosphorus. used for nonlinear least-squares fits to experimental data After the first firing, the phase purities of the sam- because they contain integrals that must be evaluated ples were checked using room temperature powder x- numerically at the temperature of each data point for ray diffraction (XRD) with a Rigaku Geigerflex pow- each iteration. Several numerical expressions represent- der diffractometer and CuKα radiation. The x-ray pat- 57–61 62 ing the Bloch-Gr¨uneisen or Debye functions have terns were analyzed for impurities using MDI Jade 7. If appeared. However, they replace the integrals in these necessary, samples were thoroughly reground and repel- functions with infinite series, use very large numbers of letized (except for LaNi2Ge2 which was just rewrapped terms, and/or use special functions. These approxima- in Ta foil) and either placed back in alumina crucibles tions are therefore not widely used for fitting experimen- or wrapped in Ta foil and resealed in evacuated quartz tal data. One paper presented a method for approxi- tubes. Samples were again fired at 990 ◦C for 5–6 d. The 63 mating the Debye function using the Einstein model. LaNi2P2 sample was arc-melted with additional La and P This method is also of little use for fitting because it uses in order to achieve a single phase sample. This may have a different equation for each temperature range and it been necessary because the compound may not form with becomes inaccurate at low temperatures. However, as the exact 1:2:2 stoichiometry. After arc-melting, part of we demonstrate, the Debye and Bloch-Gr¨uneisen func- the sample was annealed for 60 h at 1000 ◦C. Throughout tions can each be accurately approximated by a sim- this paper, the annealed LaNi2P2 sample will be referred ple Pad´eapproximant over the entire T range. To our to as x = 1.00a and the as-cast sample as x = 1.00b knowledge, there are no previously reported Pad´eap- where x is the composition of LaNi2(Ge1−xPx)2. As seen proximants for either of these two important functions. in the XRD patterns and fits in Sec. IV A, all final sam- The T -dependences of ρ for all samples discussed here ples were single-phase except for two samples showing are well-described by the Bloch-Gr¨uneisen prediction, al- very small concentrations of impurities. though the observed ρ(300 K) values are larger than cal- Rietveld refinements of the XRD patterns were carried culated. A significant T -dependence of the Debye tem- out using the FullProf package.65 Magnetization mea- perature determined from the Cp(T ) data was observed surements versus applied magnetic field H and temper- for each composition. ature T were carried out using a superconducting quan- The remainder of this paper is organized as follows. tum interference device (SQUID) magnetometer (Quan- An overview of the experimental procedures and appara- tum Design, Inc.). Gel caps were used as sample holders tus used in this work is given in Sec. II. The construction and their diamagnetic contribution was measured sepa- of the Pad´eapproximants for the Bloch-Gr¨uneisen and rately and corrected for in the data presented here. Debye functions is described in Sec. III and Appendix A. The C (T ) and ρ(T ) measurements were carried out The structural, thermal, magnetic, and electrical resis- p using a Quantum Design Physical Property Measure- tivity measurements of the LaNi (Ge − P ) system and 2 1 x x 2 ment System (PPMS). Samples for heat capacity mea- their analyses are presented in Sec. IV and Appendices B surements had masses of 15–40 mg and were attached to and C. A summary and our conclusions are given in the heat capacity puck with Apiezon N grease for ther- Sec. V. mal coupling to the platform. The ρ(T ) measurements utilized a four-probe ac technique using the ac transport option on the PPMS. Rectangular samples were cut from II. EXPERIMENTAL DETAILS the sintered pellets or arc-melted buttons using a jew- eler’s saw. Platinum leads were attached to the samples Polycrystalline samples of LaNi2(Ge1−xPx)2 (x = 0, using EPO-TEK P1011 silver epoxy. The sample was 0.25, 0.50, 0.75, 1) were prepared using the high purity attached to the resistivity puck with GE 7031 varnish. elements Ni: 99.9+%, P: 99.999+%, and Ge: 99.9999+% Temperature-dependent ρ measurements were recorded from Alfa Aesar and La: 99.99% from Ames Laboratory on both cooling and heating to check for thermal hys- Materials Preparation Center. Stoichiometric amounts of teresis. No significant hysteresis was observed for any of La, Ni, and Ge were first melted together using an arc fur- the samples. In addition, the vibrating sample magne- nace under high-purity Ar atmosphere. The arc-melted tometer (VSM) option on the PPMS was used to measure 4

the high-T magnetization of the LaNi2Ge2 sample up to 2.5

1000 K. Bloch-Grüneisen Formula

Fit

2.0

III. PADE´ APPROXIMANT FITS TO THE ) ¨ 1.5

BLOCH-GRUNEISEN AND DEBYE FUNCTIONS R (

A Pad´eapproximant g(T ) is a ratio of two polynomi- 1.0 (a) als. Here we write these polynomials as series in 1/T (T)/ according to

0.5

N1 N2 N0 + + 2 + ··· g(T )= T T . (1) D1 D2 0.0

··· D0 + T + T 2 + ) R

The first one, two or three and last one, two or three 10

( in each of the sets of coefficients N and D in g(T ) can ) i i 5

5 both (b) be chosen to exactly reproduce the low- and high-T (T)/ (10 limiting values and power law dependences in T and/or 0 1/T of the function it is approximating. This is a very

3 important and powerful feature of the Pad´eapproximant. Then the remaining terms in powers of 1/T in the numer- (T) 2

(c)

1

ator and denominator have freely adjustable coefficients (%) (T)/ that are chosen to fit the intermediate temperature range 0 of the function. A physically valid approximant requires -1 that there are no poles of the approximant on the positive 0.0 0.5 1.0 1.5 2.0 real T axis. T/ R

FIG. 1: (Color online) (a) Normalized Bloch-Gr¨uneisen elec- A. Bloch-Gr¨uneisen Model trical resistivity in Eq. (5) (one in every five points used for fitting are plotted, open circles) and the Pad´eapproximant The temperature-dependent electrical resistivity due fit (solid red curve), (b) Residuals (Pad´eapproximant value to scattering of conduction electrons by acoustic lat- minus Bloch-Gr¨uneisen formula value), and (c) Percent error tice vibrations in monatomic metals is described by the (residual divided by value of the Bloch-Gr¨uneisen formula), Bloch-Gr¨uneisen (BG) model according to66 all versus temperature T divided by the Debye temperature ΘR. T 5 ΘR/T x5 ρ(T )=4R(ΘR) dx Θ (ex − 1)(1 − e−x)  R  Z0 In practice, one fits the T dependence of an experi- (2) mental ρ(T ) data set by the BG model using an inde- where pendently adjustable prefactor ρ(ΘR) instead of 4R(ΘR) 2 in Eq. (2), because accurately fitting both the magnitude ¯h π3(3π2)1/3¯h R(ΘR)= , (3) and T dependence of a data set cannot usually be done e2 2/3 " 4ncellaMkBΘR # using a single adjustable parameter ΘR. One therefore normalizes Eq. (2) by ρ(T = Θ ). When T = Θ , the Θ is the Debye temperature determined from resistiv- R R R integral in Eq. (2) is ity measurements, ¯h is Planck’s constant divided by 2π, ncell is the number of conduction electrons per atom, 1 x5 M = (atomic weight)/NA is the atomic mass, NA is x −x dx =0.2366159, 1/3 0 (e − 1)(1 − e ) Avogadro’s number, a = (volume/atom) , kB is Boltz- Z mann’s constant, and e is the elementary charge. These yielding variables map a monatomic metal with arbitrary crystal structure onto a simple-cubic lattice with one atom per ρ(ΘR)=0.9464635 R(ΘR). (4) unit cell of lattice parameter a. To calculate R(Θ ) in R Equations (2)–(4) then yield the normalized T - units of Ω cm, one sets the prefactor (¯h/e2) in Eq. (3) to dependence of the BG function (2) as 4108.24 Ω in SI units and calculates the quantities inside the square brackets in cgs units so that the quantity in ρ(T ) T 5 square brackets has net units of cm. If one instead has = 4.226 259 ρ(Θ ) Θ a polyatomic solid, one can map the parameters of that R  R  solid onto those of the monatomic solid described by the ΘR/T x5 × dx. (5) Bloch-Gr¨uneisen model as explained in Sec. IV D below. (ex − 1)(1 − e−x) Z0 5

Fig. 1(b), the difference between the BG function and the TABLE I: Values of the coefficients in the Pad´eapproximant Pad´eapproximant is less than 1 × 10−4 at any T in the in Eq. (8) that accurately fits the normalized Bloch-Gr¨uneisen function in Eq. (7). range 0

A set of ρn(Tn) data points calculated from Eq. (7) is plotted in Fig. 1(a). These values were then used as a set of “data” to fit by a Pad´eapproximant as described B. Debye Model next. In order to construct a Pad´eapproximant function that The Debye model67 is widely used for fitting experi- accurately represents ρn(Tn) in Eq. (7), the power law mental heat capacity Cp(T ) data taken at constant pres- Tn dependences of the latter function must be computed sure p arising from acoustic lattice vibrations. It is some- at high and low temperatures and the coefficients of the times useful to fit a large T range of experimental Cp data Pad´eapproximant adjusted so that both of these limiting using the Debye function. Here we describe the construc- T dependences are exactly reproduced (to numerical pre- tion of an accurate Pad´eapproximant function of T that cision). Then the remaining coefficients are determined can easily be used in place of the Debye function (10) by fitting the approximant to a set of data obtained by for least-squares fitting experimental Cp(T ) data over an evaluating the function over the entire relevant tempera- extended T range. It can also be conveniently used to cal- ture range. This procedure is described in Appendix A 1, culate the T dependence of the Debye temperature from which constrains the values of D1, D2, D3, and D8. The experimental lattice heat capacity data over an extended resulting approximant is T range. The lattice heat capacity at constant volume V per N1 N2 N3 2 3 67 N0 + Tn + n + n mole of atoms in the Debye model is given by ρ (T )= T T . (8) n n D1 D2 D7 D8 + 2 + ··· + 7 + 8 Tn Tn Tn Tn T 3 ΘD/T x4ex CV(T )=9R x 2 dx, (10) where the fitted coefficients in the Pad´eapproximant (8) ΘD 0 (e − 1) are listed in Table I. The denominator in Eq. (8) was   Z checked for zeros and all were found not to lie on the where R is the molar gas constant and ΘD is the De- positive Tn axis. This insures that the approximant does bye temperature determined from heat capacity measure- not diverge at any (real positive) temperature. As seen in ments. The CV and T can be normalized to become di- 6

3.5 TABLE II: Values of the coefficients in the Pad´eapproximant Debye Function

3.0 in Eq. (13) that accurately fits the normalized Debye function Fit in Eq. (12).

2.5 Coefficient Value N0 226.684 46

2.0

/(R) N1 64.752 051 1 V N 17.285 710 5

2 C 1.5 N3 1.052 246 63 N4 −0.035 843 776 1 1.0 N5 0.027 925 482 7 D0 75.561 486 7 0.5 D1 21.584 017 0

D2 9.539 977 83 0.0 (a) ) 4 D3 1.427 243 1

2 D4 0.337 538 084

1 D5 0.034 609 046 3

0 /R) (10 D 0.007 440 025 83

6 V

-1 D7 −0.000 210 411 972

(C

(b) -2 D8 0.000 119 451 046

0.2

0.1 (%) 0.0 V

-0.1

/C approximant does not deviate from the normalized De-

V − -0.2 4 (c) ×

C bye function in Eq. (12) by more than 2 10 at any -0.3 T as seen in Fig. 2(b). By construction, the deviation 0.0 0.5 1.0 1.5 2.0 goes to zero at both low and high T . The percent error

T/ D in Fig. 2(c) has its maximum magnitude of 0.3% at low T , and occurs because Cn(Tn → 0) → 0 and numerical FIG. 2: (Color online) (a) Plot of normalized Debye function precision becomes an issue there. For another example in Eq. (12) (One in every two points used for fitting are plot- of the high accuracy and use of this Pad´eapproximant, ted) (open circles) and Pad´eapproximant (red line). (b) Plot see Fig. 9 below, where the ΘD versus T is calculated of residuals (Pad´eapproximant minus Debye function). (c) directly for each of our samples of LaNi (Ge − P ) us- Percent error (residual divided by value of the Debye func- 2 1 x x 2 tion). ing the Debye function in Eq. (10) and compared with that found using the Pad´eapproximant in Eq. (13); only small differences are found. mensionless according to To fit experimental Cp(T ) data by the Pad´eapprox- imant Cn(Tn) in Eq. (13), one fits both the magnitude CV(Tn) Cn(Tn) = , (11) and T dependence of Cp simultaneously using R T Tn = . Cp(T )= nRCn(T/ΘD), (14) ΘD Equation (10) then becomes where n is the number of atoms per formula unit and ΘD is the only fitting parameter. Here one substitutes T/ΘD 1/Tn 4 x 3 x e for Tn, according to Eqs. (11), in the Pad´eapproximant Cn(Tn)=9Tn x 2 dx. (12) 0 (e − 1) function Cn(Tn) in Eq. (13). For a metal, one can add to Z Eq. (14) a linear specific heat term γT giving The Cn was calculated for a representative set of Tn val- ues using Eq. (12). The resulting data are plotted as C (T )= γT + nRC (T/Θ ). (15) open red circles in Fig. 2(a). p n D As discussed in Appendix A 2, the Pad´eapproximant that fits both the high- and low-T power law asymptotics The γ is the Sommerfeld electronic specific heat coeffi- cient that can be experimentally determined from a prior of Cn(Tn) in Eq. (12) and has additional terms in pow- 67 ers of 1/T in the numerator and denominator to fit the separate fit to Cp(T ) data at low T according to intermediate T range is Cp(T ) 2 N1 N2 N5 = γ + βT (16) 2 ··· 5 N0 + Tn + n + + n T C (T )= T T , (13) n n D1 D2 D7 D8 D0 + + 2 + ··· + 7 + 8 Tn Tn Tn Tn as in Fig. 6 below, where βT 3 is the low-T limit in where the coefficients are given in Table II. The result- Eq. (A8) of the Debye heat capacity. In Eq. (15), the ing fit and error analyses are shown in Fig. 2. The Pad´e only fitting parameter is ΘD. 7

10.0 1000

LaNi P c LaNi (Ge P )

2 2 9.9 2 1-x x 2

a As Cast 213

800

9.8

Annealed (Å) c

9.7

600

9.6

9.5 105

Annealed As Cast 204 400 116 211

4.2 220

4.1 200 (Å)

4.0 Intensity (arb. units) a

(a)

0

3.9

0.00 0.25 0.50 0.75 1.00

50 55 60 65 70

Composition x

180

2 (degree)

LaNi (Ge P )

2 1-x x 2

175 FIG. 4: (Color online) Comparison of a section of the room temperature powder XRD pattern of LaNi2P2 before and af-

170 ter annealing. Numbers above the peaks are their (hkl) Miller indices. ) 3

165

(Å

cell deviations were found from the value of unity. However, V 160 the Rietveld fits were not very sensitive to changes in site occupation.

155 Annealed The lattice parameters a and c and the unit cell volume

(b) As Cast Vcell are plotted versus composition x in Fig. 3. As the 150 concentration of P increases, the lattice parameters and 0.00 0.25 0.50 0.75 1.00 unit cell volume all decrease monotonically while, from Composition x Table III, the c/a ratio increases. The composition de- pendences of the quantities in Fig. 3 deviate slightly from FIG. 3: (Color online) (a) Unit cell lattice parameters a the linearities expected from Vegard’s Law. From Ta- and c and (b) unit cell volume Vcell versus composition x ble III, the zP/Ge c-axis position parameter of the P/Ge for LaNi2(Ge1−xPx)2. The error bars are smaller than the site has a small overall increase with increasing P con- symbol size and the solid curves are guides to the eye. centration. An interesting effect was observed in the x-ray data for LaNi2P2. The as-cast sample after arc-melting showed IV. EXPERIMENTAL RESULTS AND broadening of the diffraction peaks with large c-axis con- ANALYSES tributions. After annealing the arc-melted sample for 60 h at 1000 ◦C, those peaks became sharp and shifted A. Structure and Chemical Composition to lower 2θ angles reflecting an increased c-axis lattice pa- Determinations rameter. These effects are shown on an expanded scale in Fig. 4 for the (105) and (116) reflections. The c-axis peak broadening may arise from disorder in the interlayer The starting parameters for the Rietveld refinements of 73 the powder XRD patterns were those previously reported stacking distances along the c-axis. for LaNi2P2 (Refs. 68, 69, 70) and LaNi2Ge2 (Refs. 53, 64, 71, 72) that are presented in Table III. All samples in B. Heat Capacity Measurements the LaNi2(Ge1−xPx)2 system were found to crystallize in the body-centered-tetragonal ThCr2Si2 structure (space group I 4/mmm) as previously reported for the compo- Plots of Cp(T ) for our samples of LaNi2(Ge1−xPx)2 sitions x = 0 and 1, and our refined values for the lat- from 1.8 to 300 K are shown in Fig. 5. The heat capac- tice parameters are in agreement with reported values, ity at 300 K ranges from 118.3–123.0 J/molK for these as shown in Table III. The refinements of the powder samples. These values are approaching with increasing T XRD patterns are shown in Figs. 15–18 in Appendix B the expected classical Dulong-Petit value CV = 3nR = and the crystal data are listed in Table III. All samples 124.7 mJ/mol K for the heat capacity due to acoustic were also refined for site occupancy and no significant lattice vibrations, where n is the number of atoms per 8

TABLE III: Crystallographic parameters of the body-centered-tetragonal LaNi2(Ge1−xPx)2 system at room temperature (space group I 4/mmm). Atomic coordinates are: La: (0, 0, 0), Ni: (0, 1/2, 1/4), P/Ge: (0, 0, zP/Ge). Listed are the lattice parameters a and c, the c/a ratio, the unit cell volume Vcell, and the z-coordinate of the P/Ge site zP/Ge. The quality-of-fit parameters 2 Rp, Rwp and χ are also listed.

3 2 Compound a(A)˚ c(A)˚ c/a Vcell (A˚ ) zP/Ge Rp (%) Rwp (%) χ Ref. LaNi1.70(1)P2 4.018(2) 9.485(6) 2.361 153.1 0.3716(2) 68 LaNi2P2 4.010(1) 9.604(2) 2.395 154.5 0.3700(2) 69 4.007 9.632 2.404 154.6 70 (as cast) 4.0276(3) 9.5073(9) 2.3605(4) 154.22(4) 0.3692(7) 15.7 20.9 4.24 This work (annealed) 4.0145(2) 9.6471(6) 2.4031(3) 155.47(3) 0.3681(6) 13.2 17.7 3.01 This work LaNi2(P0.75Ge0.25)2 4.0550(1) 9.7289(3) 2.3992(1) 159.97(1) 0.3685(3) 11.5 16.4 9.05 This work LaNi2(P0.50Ge0.50)2 4.08132(7) 9.7424(2) 2.38707(9) 162.281(9) 0.3678(2) 10.8 14.4 3.80 This work LaNi2(P0.25Ge0.75)2 4.1353(4) 9.8012(9) 2.3701(4) 167.61(5) 0.3668(2) 9.40 13.4 6.35 This work LaNi2Ge2 4.18586(4) 9.9042(1) 2.36610(6) 173.535(6) 0.3678(1) 10.2 13.2 8.66 This work 4.1860(6) 9.902(1) 2.366 173.51 0.3667(2) 71 4.187 9.918 53 4.187(6) 9.918(10) 2.369 173.8 64 4.1848(2) 9.900(1) 72

225 70

x

x = LaNi (Ge P ) LaNi (Ge P )

2 1-x x 2 2 1-x x 2 200 0.00 0.00

60

0.25 0.25

175

0.50

0.50 50

0.75 )

150 2

0.75

1.00(b)

40 1.00(b) 1.00(a)

125

1.00(a)

100 30 (J/mol K)

p 75

20 /T (mJ/mol K C p

50 C

10

25

0

0

0 10 20 30 40 50 60 70 80 90 100

0 50 100 150 200 250 300 350

2 2

T (K) T (K )

FIG. 5: (Color online) Heat capacity Cp versus temperature T FIG. 6: (Color online) Heat capacity Cp divided by tempera- 2 for samples in the LaNi2(Ge1−xPx)2 system (open symbols). ture T versus T (open symbols) and linear fits (solid curves of 2 Fits of the data by Eq. (15), which is the sum of electronic corresponding color) to the lowest-T data by Cp/T = γ+βT . and lattice contributions, are shown as solid curves with the Table IV lists the temperature ranges of the fits and the values respective color. For clarity, each plot is offset vertically by of γ and β obtained. 15 J/mol K from the one below it.

The values obtained for γ, β, and ΘD for each sam- formula unit (n = 5 for our compounds). ple are listed in Table IV. A plot of γ versus x in To determine the Sommerfeld electronic specific heat LaNi2(Ge1−xPx)2 is shown in Fig. 7, where a nearly lin- coefficient γ and a low temperature value of the Debye ear decrease in γ with increasing x is seen. Using this γ temperature ΘD for each sample, the lowest temperature values in Table IV, we then fitted the data from 1.8 to linear Cp(T )/T data for each sample, plotted in Fig. 6, 300 K in Fig. 5 by Eq. (15) and obtained the ΘD fitting were fitted by Eq. (16) and the values of γ and β ob- parameters listed in Table IV. The fits are shown as the tained. A value of ΘD can be calculated from each value solid curves in Fig. 5, which are seen to agree rather well of β using67 with the respective data. However, small T -dependent deviations between the data and fit for each sample are 1 3 12π4Rn / seen, which we address next. ΘD = . (17) 5β Deviations of a Debye model fit from experimental lat-   9

TABLE IV: Values of γ and β obtained from the low-T fits of the data in Fig. 6 by Eq. (16) are listed together with the density of states at the Fermi energy D(EF) in units of states/(eV f.u.) for both spin directions calculated from γ using Eq. (24). Also shown are the ΘD values calculated from the low-T β values using Eq. (17) and from a global fit to all the lattice Cp(T ) data from 1.8 to 300 K for each sample. Available values from the literature are also listed.

Sample Low-T γ D(EF) β ΘD ΘD Ref. − Fit Range (mJ/mol K2) (eV f.u.) 1 (mJ/mol K4) Low-T fit All-T fit (K) (K) (K) LaNi2P2 (as-cast) 1.81–5.34 7.7(2) 0.086(7) 483(14) 369(2) This work (annealed) 1.81–7.31 5.87(2) 2.49 0.126(2) 426(3) 365(3) This work LaNi2(P0.75Ge0.25)2 2.93–8.60 7.4(1) 3.13 0.159(3) 394(3) 348(2) This work LaNi2(P0.50Ge0.50)2 1.82–7.13 9.3(2) 3.94 0.194(7) 369(5) 326(2) This work LaNi2(P0.25Ge0.75)2 2.59–6.52 11.27(6) 4.78 0.261(2) 333.9(9) 301(2) This work LaNi2Ge2 2.93–6.41 12.4(2) 5.26 0.371(9) 297(2) 287(2) This work 14.5 0.273 328 52 12.7 (calc) 5.38 53 a 13.5 (obs) 53

aNo experimental evidence or reference citation was given for this quoted observed value.

on the right-hand side of Eq. (16)].

LaNi (Ge P ) • Assumptions are made that the speed of acous- 12 − 2 1 x x 2 tic sound waves in a material is temperature- independent, is isotropic and is the same for lon- gitudinal and transverse acoustic sound waves. In 8 general, these assumptions are too simplistic and the parameters are temperature dependent.

• The Debye model only accounts for acoustic lattice 4 vibrations and does not take into account optic lat- tice vibrations arising from opposing vibrations of atoms with different masses in the unit cell. The contribution of these to Cp(T ) can be modeled by 0 adding Einstein terms to the fit function.67 0.0 0.2 0.4 0.6 0.8 1.0 x Because of these approximations and assumptions of the Debye model, the lattice heat capacity of a material FIG. 7: (Color online) Sommerfeld electronic specific heat is never precisely described by the Debye model over an coefficient γ versus composition x in LaNi2(Ge1−xPx)2 from extended temperature range such as from 2 K to 300 K. Table IV. For x = 1, the datum in Table IV for the annealed The most serious approximation in our T range is the sample is plotted. The error bars are smaller than the data second approximation. Within the Debye model ΘD is symbols. The line is a guide to the eye. independent of T . One can therefore parameterize the deviations of a fit from the data by allowing ΘD to vary with T .74 tice heat capacity data are due to the following assump- The value of ΘD was calculated for each data point tions and approximations of the model. in Fig. 5, after subtracting the contribution from γT ac- • The system is assumed to be at constant volume as cording to Eq. (15), using the Debye function in Eq. (10). T changes, rather than at constant pressure which The resulting T dependences of ΘD are shown in Fig. 8, is the experimental condition. This deficiency can where ΘD is seen to vary nonmonotonically and by up to 30% with increasing T . The plots have a similar shape be corrected for if the T -dependent compressibility 75 and thermal expansion coefficient are known for the to that for sodium iodide. The ΘD is expected to be constant below ΘD0/50 and above ΘD0/2, where ΘD0 compound of interest. 74 is the zero-temperature value of ΘD. Our ΘD(T ) data • A quadratic density of phonon states versus energy qualitatively agree with these expectations. is assumed, which terminates at the Debye energy Considerable scatter in the ΘD(T ) data in Fig. 8 oc- kBΘD. For actual materials, this assumption can curs at temperatures above 250 K and also below 7 K only be accurately applied at temperatures T ≪ (although not as clearly visible in the figure). In these 3 ΘD, which gives the Debye T law [the second term T ranges, Cp is becoming nearly independent of T , so in 10

1.25

LaNi (Ge P ) 500

2 1-x x 2 LaNi (Ge P )

2 1-x x 2 x = 0.00

1.00

450 0.25 x= 1.00(b) 1.00(a) 0.75

0.75

0.50

400

0.50

0.75

350 (K) (K) D 1.00(b) 0.25 D

1.00(a) 300

0.00

250

-0.25

x= 0.00 0.25 0.50

200 -0.50

0 50 100 150 200 250 300 0 50 100 150 200 250 300

T (K) T (K)

FIG. 8: (Color online) Debye temperature ΘD versus temper- FIG. 9: (Color online) Difference ∆ΘD between the ΘD cal- ature T (open symbols) obtained by solving Eq. (10) for each culated from the Debye function [Eq. (10)] plotted in Fig. 8 Cp(T ) data point after subtracting the electronic contribu- and calculated from the Pad´eapproximant [Eqs. (13) and tion γT . The error bars plotted are calculated using Eq. (18). (14)]. Solid curves are guides to the eye. For clarity, each Solid curves are guides to the eye. plot is offset vertically upwards by 0.2 K from the one below it. Horizontal dotted lines are at ∆ΘD = 0 for the data set with the corresponding color. these T ranges any error in the value of CV is greatly amplified when ΘD is calculated. The error bars on the values of ΘD(T ) therefore increase significantly in these T regions. We now consider such errors and for this dis- Using the low-T approximation in Eq. (A8) and the cussion ignore the difference between CV and Cp. same procedure as described above, the error in ΘD is The scatter dΘD in the derived ΘD versus T depends on the statistical error dCV in CV, 4 dC 5ΘD V dΘD ≈− dCV. (T ≪ ΘD) dΘD = . (18) 36Rπ4T 3 dCV/dΘD   This expression was used to obtain the error bars plotted in Fig. 8. The denominator dCV/dΘD was calculated Therefore, similar to the situation at high-T , a small er- using the Debye function in Eq. (10). ror in Cp at low T is greatly amplified when calculating In order to more clearly see why the error in ΘD sub- ΘD(T ). stantially increases above ≈ 250 K, the high-T approxi- mation in Eq. (A7) can be used, yielding To verify the applicability of the Pad´eapproximant for the Debye function developed in Sec. III B, the ΘD(T ) 2 3RΘD was calculated for each data point in Fig. 5 using the CV ≈ 3R − . (T ≫ ΘD) 20T 2 Pad´eapproximant in Eq. (13) instead of by directly us- ing the Debye function in Eq. (10). The electronic γT Taking the derivative with respect to ΘD gives contribution was again subtracted from the Cp(T ) data first. The difference between these ΘD(T ) values and dC −3RΘ V ≈ D ≫ those calculated using the Debye function in Eq. (13) is 2 . (T ΘD) dΘD 10T plotted versus T in Fig. 9. The values calculated from the Pad´eapproximant do not deviate by more than 0.35 K Inserting this result into Eq. (18) gives the approximation from those calculated using the Debye formula. This er- ror is of order 0.1% of ΘD, which is usually small com- 2 10T pared to the error in ΘD itself and is negligible compared dΘD ≈− dCV. (T ≫ ΘD) 3RΘ to its T dependence. Therefore the Pad´eapproximant  D  provides a viable alternative for calculating ΘD(T ) that This result shows that the error in ΘD is proportional to does not require evaluation of the integral in the Debye T 2 at high T , which results in a dramatic increase in the function (10) or the use of a look-up table for each data scatter and error in ΘD(T ) at high T . point. 11

30 2.5

1.92

LaNi (Ge P ) (a) LaNi Ge

2 1-x x 2 2 2 /mol) 3

2.4 x (as measured)

1.88 25 cm 4

0.00

2.3 (10

1.84

0 10 20 30 20

T (K)

2.2 /mol) /mol) 3 3

0.25

15

cm 2.1 4 cm 5

10 Measured in a 3 T field 2.0 (10 0.50 (10

0.75

SQUID 1.9

5

1.00(a)

VSM

1.00(b)

1.8

0 0 200 400 600 800 1000

0 50 100 150 200 250 300 350 400

T (K)

T (K)

30 FIG. 11: (Color online) Expanded plot along the vertical FM and PM corrections (b) axis of the magnetic susceptibility χ versus temperature T

x for LaNi2Ge2 up to 1000 K after correction for paramagnetic 25 and ferromagnetic impurity contributions. The data below 0.00 350 K were measured with a SQUID magnetometer and the

20 data above 300 K were measured with a VSM, both at ap- plied fields of 3 T. Inset: Expanded plot of the SQUID data

/mol) at low temperatures. The small downturn below about 10 K 3

0.25 15 is most likely spurious, arising from an imperfect correction

cm for the paramagnetic impurities. 5

10 Measured in a 3 T field

0.50 (10

0.75 presence of saturating paramagnetic and/or ferromag-

5

1.00(a) netic impurities in the samples. In order to determine

1.00(b) the intrinsic behaviors, it is necessary to correct for these

0 impurities. To determine the individual contributions to 0 50 100 150 200 250 300 350 400 the susceptibility data, the M(H) curves were fitted by

T (K) gµBH M(T,H)= M0 + χH + fMsatB , (19) S k (T − θ) FIG. 10: (Color online) Magnetic susceptibility χ versus tem-  B  perature T for the LaNi2(Ge1−xPx)2 system. (a) Measured χ ≡ M/H data (uncorrected). (b) Intrinsic χ obtained af- where M0 is the saturation magnetization of the ferro- ter correcting for both ferromagnetic (FM) and paramagnetic magnetic impurities, χ is the intrinsic susceptibility of (PM) impurities. Solid symbols of corresponding shape and the sample, f is the molar fraction of paramagnetic im- color are values of χ found from fitting M(H) isotherms. purities, g is the spectroscopic splitting factor of the im- purities which was fixed at g = 2 to reduce the number of fitting parameters, µB is the Bohr magneton, kB is C. Magnetization and Magnetic Susceptibility Boltzmann’s constant, θ is the Weiss temperature of the Measurements paramagnetic impurities (included for consistency with a possible Curie-Weiss law behavior at low H/T ), Msat is the saturation magnetization of the paramagnetic im- Magnetization versus applied magnetic field M(H) purities, and BS is the Brillouin function. The Brillouin isotherms were measured for the LaNi2(Ge1−xPx)2 sys- function is tem for H = 0–5.5 T and the results are plotted in Figs. 19–21 of Appendix C. The M versus T for the 1 x x B (x)= (2S + 1)coth (2S + 1) − coth , samples were also measured from 1.8 to 300 K at a fixed S 2S 2 2 field H = 3 T and the resulting susceptibilities χ ≡ M/H n h i  o(20) are plotted in Fig. 10(a). As evident from the nonlinear where behavior at low fields in the M(H) plots and the upturn that follows the Curie-Weiss-like behavor [χ = C/(T −θ)] gµBH x = in the observed χ(T ) at low temperatures, we infer the kB(T − θ) 12

10 in Fig. 11 below where our χ(T ) data for LaNi2Ge2 are plotted on an expanded vertical scale for temperatures 0 up to 1000 K.

2.6 K The χ(T ) data in Fig. 10 show that the samples in the -10 − 2.2 K LaNi2(Ge1 xPx)2 system exhibit nearly temperature-

-20 independent paramagnetism over the composition region H=20 G x = 0.25–1. This trend does not extend to LaNi2Ge2 ZFC /mol) -30 3 (x = 0), for which a broad maximum appears to occur at ≈ 300 K, which is close to the upper temperature

cm -40 4 limit of the SQUID magnetometer. In order to deter-

-50 mine whether a maximum in χ near 300 K does occur, (10 M(H,T ) data for the sample of LaNi2Ge2 were measured -60 Annealed using a VSM from T = 300 to 1000 K. The M(H)

As-Cast isotherm data are plotted in Fig. 21(b) of Appendix C -70 and the χ(T ) ≡ M(T )/H data at H = 3 T are plotted in

0 5 10 15 20 Fig. 11. The magnetic contributions due to the sample holder and paramagnetic impurities are corrected for in T (K) the plots. The data in Fig. 11 clearly show that a broad peak FIG. 12: (Color online) Zero-field-cooled (ZFC) low- occurs in χ(T ) of LaNi2Ge2 at about 300 K. The plot in temperature measurement of the magnetic susceptibility χ Fig. 10(b) shows the peak plotted on a less expanded ver- versus temperature T for LaNi2P2 with applied field H = tical scale. The cause of this peak is not clear, but may be 20 G. due to low-dimensional antiferromagnetic correlations.77 Further investigation is needed. The inset in Fig. 11 and the molar saturation magnetization Msat of the para- shows an expanded view of the low temperature behav- magnetic impurities is ior. Due to its smooth nature, the small downturn in the data below about 10 K is most likely spurious due to a Msat = NAgSµB, slight error in correcting for the susceptibility contribu- tion of the paramagnetic impurities in the sample. where S is the spin of the impurities and N is Avo- A The onset of superconductivity was observed in both gadro’s number. the annealed and as-cast samples of LaNi P at 2.6 and To determine the values of f, S, and θ for the param- 2 2 2.2 K, respectively, from zero-field-cooled (ZFC) magne- agnetic impurities, a global two-dimensional surface fit of tization measurements in a field of 20 Oe. Figure 12 the M(H) data from 1–5.5 T taken at both 1.8 and 5 K ≡ MATLAB shows these low-field χ(T ) M(T )/H measurements. It was done using Eq. (19) with ’s Surface Fitting is not clear whether the data represent the onset of bulk Tool. For this H range, the ferromagnetic impurities are superconductivity in LaNi2P2 or if the diamagnetism expected to be nearly saturated, as assumed by Eq. (19). arises from a superconducting impurity phase. The ρ(T ) The parameter values obtained are given in Table V. Fix- data in Fig. 13 below do not clarify this issue. ing the variables f, S, and θ at the respective values for We now analyze the normal-state χ of the samples. each sample, each M(H) curve at higher temperatures The magnetic susceptibility of a metal consists of the was fitted by Eq. (19) in the range 1–5.5 T to obtain val- sum of the spin and orbital contributions ues for M0(T ) and χ(T ). The M(H) fits are shown in Figs. 19–21 in Appendix C. A plot of M0 versus T is pre- χ = χspin + χorb. (21) sented in Fig. 22 in Appendix C and the fitted values of the intrinsic susceptibility χ are plotted as solid symbols In the absence of local magnetic moments, the spin con- in Fig. 10(b). tribution χspin is the Pauli susceptibility χPauli of the In order to correct the M(T ) data at H = 3 T in conduction electrons. One can estimate χPauli using2 Fig. 10(a) for the paramagnetic impurities, the above val- 2 ues of f, S, and θ obtained by fitting the M(H) isotherms Pauli g 2 χ = µ D(EF), (T =0) (22) at 1.8 and 5 K were inserted into the last term of Eq. (19) 4 B to calculate their contributions versus T at H = 3 T. where g is the spectroscopic splitting factor and D(E ) These contributions were then subtracted from the re- F is the density of states at the Fermi energy E . Setting spective M(T ) data that were already corrected for ferro- F g = 2 gives magnetic impurities to obtain the intrinsic susceptibility Pauli −5 of the samples as plotted in Fig. 10(b). Previously re- χ = (3.233 × 10 )D(EF), (23) −5 3 ported values of χ for LaNi2Ge2 are 24 × 10 cm /mol −5 3 Pauli 3 at 300 K (Ref. 50) and 28 × 10 cm /mol at 296 K where χ is in units of cm /mol, D(EF) is in units (Ref. 76). The former value is essentially the same as our of states/eV f.u. for both spin directions and f.u. means value 23.2×10−5 cm3/mol at 300 K, as seen more clearly formula unit. 13

TABLE V: Values obtained by simultaneous fitting of the 1.8 and5K M(H) isotherms. The parameters listed are f [molar fraction of paramagnetic (PM) impurities], S (spin quantum number of the PM impurities), θ (Weiss temperature of the PM impurities), M0 (saturation magnetization of the ferromagnetic impurities), and χ (intrinsic magnetic susceptibility of the compound). The negative signs of θ indicate antiferromagnetic interactions between the magnetic impurities.

f S θ M0 χ − − Sample (10 5) (K) (Gcm3/mol) (10 5 cm3/mol) LaNi2P2 (annealed) 3.89 3.48 −5.88 0.0478 2.92 LaNi2(P0.75Ge0.25)2 2.33 1.92 −1.02 0.1175 5.53 LaNi2(P0.50Ge0.50)2 4.12 2.42 −1.89 0.4128 7.49 LaNi2(P0.25Ge0.75)2 5.73 2.57 −1.54 0.1834 16.0 LaNi2Ge2 2.35 2.43 −1.08 0.0700 19.5

TABLE VI: Contributions to the magnetic susceptibility χ. The parameters listed are the observed value χobs of χ averaged over the temperature range measured and the contributions from the Pauli spin susceptibility χPauli and the orbital core − susceptibility χcore, Landau susceptibility χLandau, and Van Vleck susceptibility χVV. All values are in units of 10 5 cm3/mol.

Sample χobs χPauli χcore χLandau χVV LaNi2P2 (as-cast) 1.49 10.5(3) −18.8 −3.50(1) 13.3(4) LaNi2P2 (annealed) 2.82 8.04(3) −18.8 −2.68(1) 16.3(4) LaNi2(P0.75Ge0.25)2 5.28 10.1(2) −19.3 −3.37(7) 17.9(3) LaNi2(P0.50Ge0.50)2 7.32 12.7(3) −19.7 −4.2(1) 18.5(4) LaNi2(P0.25Ge0.75)2 14.36 15.44(8) −20.1 −5.15(3) 24.7(1) LaNi2Ge2 21.01 17.0(3) −20.5 −5.7(1) 30.4(4)

Pauli ∗ To calculate χ , one can obtain an estimate of For our samples, it is assumed that m = me, therefore Landau Pauli D(EF) from the Sommerfeld electronic linear specific χ can be calculated from the χ obtained above. heat coefficient γ according to An estimate of χcore was obtained from the sum of the Hartree-Fock diamagnetic atomic susceptibilities.79 Us- γ = γ (1 + λ ), 0 ep ing the observed average value of the susceptibility (χobs) where67 over the temperature range 2–300 K and the calculated Pauli core Landau 2 2 sum of the values χ , χ and χ , the Van Vleck π kB γ0 = D(EF)=2.359 D(EF) (24) susceptibility is calculated from Eqs. (21) and (26) via 3 VV obs Pauli core Landau is the bare Sommerfeld coefficient in the absence of χ = χ − (χ + χ + χ ). electron-phonon coupling, λ is the electron-phonon ep The values obtained in the manner described for χPauli, coupling constant, and in the right-hand equality of χcore, χVV and χLandau are listed in Table VI. Eq. (24) D(EF) is in units of states/eV f.u. for both 2 spin directions and γ0 is in units of mJ/mol K . Tak- ing λep = 0 yields D. Electrical Resistivity Measurements − χPauli = (1.370 × 10 5)γ, (25) − Pauli 3 The ρ versus T data for our LaNi2(Ge1 xPx)2 sam- where χ is in units of cm /mol and γ is in units of ples are plotted in Fig. 13(a). The magnitudes of the 2 D Pauli mJ/molK . The values of (EF) and χ obtained data may not be reliable due to the polycrystalline na- using Eqs. (24) and (25), respectively, and using the γ ture of the samples and the resulting grain boundary values given in Table IV, are listed in Table VI. scattering. Such scattering is expected and found to be The orbital susceptibility consists of three contribu- smallest for the samples of LaNi2Ge2 and LaNi2P2 that tions were cut from arc-melted buttons and therefore had a χorb = χcore + χVV + χLandau, (26) higher density than the other samples cut from sintered pellets. The values of ρ at 1.8 and 300 K are listed in core where χ is the diamagnetic contribution from the Table VII along with previously reported values from the VV atomic core electrons, χ is the paramagnetic Van Vleck literature.51,55,80,81 The positive slopes of the ρ(T ) data Landau susceptibility, and χ is the diamagnetic Landau indicate that all samples are metallic. The maximum re- susceptibility of the conduction electrons. The Landau sistivities at 1.8 K and at 300 K both occur for x =0.50 78 susceptibility is approximated by where the disorder on the Ge/P sublattice is largest, as 1 m 2 might have been anticipated, so this may be an intrinsic χLandau = − e χPauli. 3 m∗ effect.   14

TABLE VII: Values of the Debye temperature determined from resistivity ρ measurements (ΘR), the resistivity at the Debye temperature [ρ(ΘR)] and the residual resistivity (ρ0) obtained from least-squares fits of the ρ(T ) data in Fig. 13(a) by Eq. (9). Also listed are ρ values at ∼ 2 K and at 300 K, and literature values parallel to the c-axis (ρc) and parallel to the a-axis (ρa) of a single crystal. The systematic errors in our ρ values due to uncertainties in the geometric factors are of order 10%.

Sample ΘR (K) ρ(ΘR) ρ0 ρ(∼ 2 K) ρ(300 K) Ref. (µΩcm) (µΩcm) (µΩcm) (µΩcm) LaNi2P2 (as-cast) 211(2) 44.9(3) 83.15(3) 83 148 This work (annealed) 265(3) 109.(1) 26.0(1) 25 152 This work LaNi2(P0.75Ge0.25)2 242(1) 85.8(4) 191.51(5) 191 300 This work LaNi2(P0.50Ge0.50)2 208(1) 102.5(7) 249.79(9) 249 401 This work LaNi2(P0.25Ge0.75)2 119(7) 37.(2) 95.9(3) 95 189 This work LaNi2Ge2 148(5) 38.(1) 6.8(2) 6.1 85 This work 0.4 51 ∼ 1 ∼ 26 (ρa), ∼ 38 (ρc) 80 ∼ 5 ∼ 80 81 ∼ 1–2 ∼ 43 55

As seen in the expanded plot at low temperatures down • The calculation is carried out at constant volume to our low-T limit of 1.8 K in Fig. 13(b), there appears instead of at constant pressure as in experiments. to be an onset of superconductivity occurring at ≈ 2.1K for both the annealed and as-cast samples of LaNi2P2, • The conduction electron energy is assumed not to consistent with the above measurements of the low-field change due to scattering off phonons. magnetic shielding susceptibilities in Fig. 12. It is not • known if these results indicate the onset of bulk super- The phonons are assumed to be in thermal equilib- conductivity or whether they are due to a superconduct- rium. ing impurity phase. As a result of these assumptions and approximations, the The T dependence of ρ was fitted by Eq. (9), which 82 T dependence of ΘR is much stronger than that of ΘD. includes our Pad´eapproximant for the Bloch-Gr¨uneisen Zimon concluded, “The actual observed value of ΘR does function. From these fits shown in Fig. 13(a), estimates of 82 not have any great significance.” Thus ΘR is less mean- the Debye temperature (ΘR), ρ at the Debye temperature ingful than ΘD. Furthermore, differences between ΘR [ρ(ΘR)], and residual resistivity (ρ0) were obtained as and ΘD are not unexpected, given the different assump- listed in Table VII. The ΘR for the LaNi2(P0.25Ge0.75)2 tions of the BG and Debye models enumerated here and sample is lower than the values for the adjacent compo- in Sec. IV B, respectively. sitions. We speculate that this arises from inaccuracies Now we compare our experimental result for the mag- introduced by the polycrystalline nature of the samples. nitude of ρ(300 K) for the annealed sample of LaNi2Ge2 The ΘR obtained from ρ(T ) measurements is usually with the value predicted by the Bloch-Gr¨uneisen theory. different from that obtained by heat capacity measure- 74 We chose to use the ρ data for this compound for com- ments (ΘD) although in some cases agreement found. parison with the theory because it has the lowest resid- For our samples in the LaNi2(Ge1−xPx)2 system, while ual resistivity of our five samples. The BG theory de- the magnitudes of ΘR differ considerably from sample scribes monatomic materials. In order to use this theory to sample, both sets of ΘR and ΘD data in Tables VII to predict the resistivity of polyatomic compounds such and IV, respectively, indicate an overall decrease in the as LaNi2Ge2, it is necessary to slightly modify Eq. (3) as Debye temperature with increasing Ge content. Like the Debye model, the BG model makes several ¯h π3(3π2)1/3¯h2 1 assumptions and approximations in order to obtain an R(ΘR)= , (27) e2 2/3 M analytic formula for ρ(T ): " 4ncellakBΘR  ave#

• A strong approximation is that the lattice vibra- where the variables have the same meaning as before, tions have a Debye spectrum. except n 1 N 1 • Another strong approximation is that Umklapp = A (28) conduction electron scattering processes are ig- M n Mi  ave i=1 nored. Umklapp scattering is expected to be T - X dependent in our T range.82 is the average inverse mass of the atoms in a f.u., n is the number of atoms per f.u. and Mi is the atomic weight of 1/3 • Only longitudinal phonons are assumed to con- element i. In this case, a = (Vcell/nZ) because there tribute to electron-phonon scattering. are Z = 2 f.u. of LaNi2Ge2 with n = 5 per body-centered 15

450 10 smaller than the experimental value of 79 µΩ cm [after

(a) LaNi (Ge P ) subtracting ρ(1.8 K)] from Table VII, which is a large dis- 2 1-x x 2 400 agreement in magnitude even though the T dependence

350 for ρ for LaNi2Ge2 was well-fitted by the BG theory. This large disagreement is typical of the BG theory as applied 300 x = 0.50 to transition metals and alkaline-earth metals66 and illus-

250 trates why it is necessary to use an adjustable prefactor 0.75 cm)

ρ(ΘR) in Eq. (9) in addition to the fitting parameter ΘR

200

0.25 in order to fit both the magnitude and T dependence of (

150 experimental data using the Bloch-Gr¨uneisen model. 1.00(b) The reason for the enhanced electron-lattice resistivity 100 in transition metals with both s- and d-bands crossing

1.00(a) the Fermi energy is that the s-electrons carry most of 50

0.00 the current due to their much lower effective mass than

0 the d-electrons, and electron-phonon scattering from an 0 50 100 150 200 250 300 s-band into a much higher density of states d-band acts

as a conduction electron sink, thus strongly enhancing T (K) the resistivity due to electron-phonon scattering.82 90

(b) LaNi P

2 2

85 V. SUMMARY AND CONCLUSIONS As-Cast

Single- or nearly single-phase polycrystalline samples

80 of LaNi2(Ge1−xPx)2 with the compositions x = 0, 0.25, 0.50, 0.75, and 1 were synthesized and their properties cm) measured. Rietveld refinements of the powder XRD

30

( patterns showed that all samples have the tetragonal I mmm Annealed ThCr2Si2 structure with space group 4/ . The refined crystal data are presented in Table III. A possi- 25 ble stacking disorder was observed along the c-axis in the as-cast sample of LaNi2P2 as revealed by the c-axis line

20 broadening in Fig. 4.

0 5 10 15 20 25 30 35 40 Electrical resistivity ρ measurements showed a posi- tive temperature coefficient for all samples from 2 K T (K) to 300 K, indicating that all compositions in this sys- tem are metallic. Consistent with this result, the low-T FIG. 13: (Color online) (a) Electrical resistivity ρ versus tem- Cp measurements yield a rather large Sommerfeld elec- perature T for the LaNi2(Ge1−xPx)2 system (open symbols). 2 Solid curves of corresponding color are least-squares fits by tronic specific heat coefficient γ = 12.4(2) mJ/molK Eq. (9), which includes the Pad´eapproximant in Eq. (8) that for x = 0 reflecting a large density of states at the is used in place of the normalized Bloch-Gr¨uneisen function Fermi energy comparable to the largest values found for in Eq. (7). Values of the fitting parameters obtained are listed the AFe2As2 class of materials with the same crystal in Table VII. (b) Expanded plots at low temperatures of ρ(T ) structure.2 The γ decreases approximately linearly with 2 for as-cast and annealed LaNi2P2. x to 7.4(1) mJ/molK for x = 1. New Pad´e approximants for the Debye and Bloch- Gr¨uneisen functions were presented. They have the dis- 2 tetragonal unit cell with volume Vcell = a c where a and tinct advantage of being able to precisely reproduce the c are given in Table III. power law dependences at both the low- and high-T lim- The number of carriers per f.u. in LaNi2Ge2 predicted its of the function they are approximating. The Pad´e 53 by band structure calculations is 1.54. Therefore, approximants presented here for each of CV(T ) and ρ(T ) ncell = 1.54/5=0.308 carriers/atom because the car- cover the entire T range and have a good balance of high riers are modeled as being evenly distributed among the accuracy and low number of terms. The T dependences of 5 atoms in the f.u. As described previously, to calculate ρ for all samples were well-fitted by the Bloch-Gr¨uneisen 2 R(ΘR) in units of Ωcm in Eq. (27), the prefactor ¯h/e model and values of the Debye temperature ΘR were ob- is set to 4108.24 Ω and the part in the square brackets tained (Table VII), although the measured magnitudes is calculated in cgs units. Using ΘR = 148 K from Ta- were larger than calculated on the basis of this model. ble VII, R(ΘR) is calculated to be 3.763 µΩ cm. Inserting Fitting the T dependences of Cp revealed significant T this result into Eq. (2) for T = 300 K, ρ(300 K) is pre- dependences of the Debye temperatures ΘD. dicted to be 7.53 µΩ cm. This value is about a factor of Magnetization and magnetic susceptibility measure- 16 ments of our LaNi2(Ge1−xPx)2 samples revealed nearly At low temperatures, ΘR/T → ∞ and the upper limit to temperature-independent paramagnetic behavior (except the integral in Eq. (7) can be set to ∞. The integral in for LaNi2Ge2), with increasing susceptibility as the Ge Eq. (7) is then concentration is increased. For LaNi Ge , a broad peak ∞ 2 2 x5 was observed in χ(T ) at ≈ 300 K. A possible explanation dx = 5!ζ(5) ≈ 124.431 331, (ex − 1)(1 − e−x) is that the peak arises from the onset of strong antifer- Z0 romagnetic correlations in a quasi-two-dimensional mag- where ζ(z) is the Riemann zeta function. Inserting this 77 netic system. There may be a correlation here between result into Eq. (7) yields the χ(T ) behavior and the corrugated electron cylinder 5 Fermi surface found by Yamagami53 that was mentioned ρn = CTn . (T ≪ ΘR) (A2) in the Introduction. However, the heat capacity data C = 525.8790 were well-fitted from 1.8 to 300 K for all samples by an electronic γT term plus a lattice contribution described There are no additional terms in powers of Tn in the Tay- by the Debye model as shown in Fig. 5 and Table IV, with lor series expansion of ρn(Tn) about Tn = 0, because it no clear evidence for an additional magnetic contribution is not possible to express the exponentials in the integral in Eq. (7) as Taylor series in 1/x about 1/x = 0. in LaNi2Ge2. An interesting finding is the onset of a superconduct- Based on the above low- and high-T expansions of the ing transition at T ≈ 2.2 K in both the annealed and normalized BG function, the coefficients of the Pad´eap- proximant are now chosen so that the limiting T depen- as-cast samples of LaNi2P2. This onset was observed in both the resistivity and magnetic susceptibility measure- dences of the approximant exactly match the required ments. However, it is not clear if these onsets are due to power law T dependences in Eqs. (A1) and (A2). In bulk superconductivity or to an impurity phase. Apart addition, intermediate power terms were added in pairs (one in the numerator and one in the denominator) until from the broad peak in χ(T ) for LaNi2Ge2 and the po- there were enough for the approximant to accurately fit tential bulk superconductivity in LaNi2P2, there were no other signs of structural, magnetic, or superconducting the intermediate temperature range of the BG function. transitions down to 1.8 K in the samples. The resulting approximant has the form N1 N2 N3 N0 + + 2 + 3 ρ (T )= Tn Tn Tn . (8) n n D1 D2 D7 D8 + 2 + ··· + 7 + 8 Acknowledgments Tn Tn Tn Tn In order that Taylor series expansions of the Pad´eap- Work at the Ames Laboratory was supported by the proximant (8) at high- and low-T correctly reproduce Department of Energy-Basic Energy Sciences under Con- the coefficients of the limiting T dependences of the nor- tract No. DE-AC02-07CH11358. malized BG function in Eqs. (A1) and (A2), some of the coefficients Ni and Di in Eq. (8) are not independent. Ex- panding Eq. (8) as a Taylor series in 1/Tn about 1/Tn =0 Appendix A: Pad´eApproximants gives

N0 1 −D2N0 + D1N1 0 ρn(Tn) = Tn + 2 Tn 1. Bloch-Gr¨uneisen Model D1 D1 2 2 D2 N0 − D1D3N0 − D1D2N1 + D1 N2 −1 In order to construct a Pad´eapproximant function that + 3 Tn D1 accurately represents another function, the power law T −2 dependences of the latter function must be computed at + O(Tn ). (A3) high and low temperatures and the coefficients of the Equating the coefficients in Eq. (A3) with the respective Pad´eapproximant adjusted so that both of these limit- coefficients in the high-T Taylor series expansion of the ing T dependences are exactly reproduced (to numerical normalized BG function in Eq. (A1) yields precision). For the BG function at high T ,ΘR/T ≪ 1 in Eq. (7). Therefore, the integrand in Eq. (7) can be N0 N1 AN2 − BN0 D1 = ,D2 = ,D3 = . (A4) expanded in a Taylor series about x =0 as A A A2 The Taylor series expansion of the approximant (8) about x5 x5 ≈ x3 − + O(x7). (x ≪ 1) Tn = 0 is (ex − 1)(1 − e−x) 12 N3 5 6 ρn = Tn + O(Tn ). (A5) Equation (7) then becomes D8 Equating the coefficient of this T 5 term with that of the 1 0 −1 −3 ρn(Tn) ≈ ATn +0 Tn + BTn + O(Tn ) (T ≫ ΘR) Bloch-Gr¨uneisen function in Eq. (A2) yields A = 1.056565 (A1) N3 − D8 = . (A6) B = 0.05869804 C 17

Using the four constraints in Eqs. (A4) and (A6), one can and the corresponding Taylor series expansions of the 1 0 −1 exactly reproduce the Tn , Tn , and Tn dependences of Pad´eapproximant (13) yield the constraints 5 the high-T expansion in Eq. (A1) and the Tn dependence of the low-T expansion in Eq. (A2) for the normalized BG N0 D0 = (A9) function. E A table of values generated from Eq. (7) and plotted N1 D1 = above in Fig. 1(a) was least-squares fitted by Eq. (8) E using the constraints in Eqs. (A4) and (A6) for D1, D2, −FN0 + EN2 D2 = D3, and D8. The final values of the coefficients in the E2 Pad´eapproximant (8) are listed in Table I. N5 D8 = . G

2. Debye Model These constraints on D0, D1, D2 and D8 guarantee that 0 −1 −2 the Tn , Tn , and Tn terms in the high temperature 3 series expansion and the Tn term in the low-T expansion At high temperatures 1/Tn ≪ 1, the integrand in Eq. (12) can be expanded in a Taylor series about x = 0, of the Pad´eapproximant (13) exactly match to within numerical accuracy the corresponding terms in the Tay- yielding lor series expansion of the Debye function in Eqs. (A7) x4ex x4 and (A8), respectively. = x2 − + O(x6). The Debye function data in Fig. 2(a) were least-squares (ex − 1)2 12 fitted by the Pad´eapproximant in Eq. (13) using the con- After evaluating the integral in Eq. (12) using this ap- straints in Eqs. (A9). After fitting, the denominator of the approximant was checked for zeros and all were found proximation, the Cn(Tn) becomes to be at non-real Tn. This insures that the approximant 0 −1 −2 −4 does not diverge at any (real positive) temperature. The Cn(Tn)= ET +0 T + FTn + O(Tn ) (T ≫ ΘD) (A7) final coefficients in the Pad´eapproximant are listed in where Table II.

E = 3, Appendix B: Rietveld Refinement Figures 3 F = − . 20 In this Appendix the Rietveld refinement Figs. 14–18 In the limit of low temperatures 1/Tn → ∞, the integral referred to in Sec. IVA are shown. in Eq. (12) can be evaluated with an upper limit of ∞ to obtain

3 Cn(Tn) = GTn (T ≪ ΘD) 12π4 G = . (A8) 5 This T 3 dependence of the lattice heat capacity at low temperatures is universal and is known as the Debye T 3 law. Similar to the Bloch-Gr¨uneisen function, it is not possible to obtain additional terms in the Taylor series expansion of Cn(Tn) in Eq. (12) about Tn = 0. Since the Pad´eapproximant must follow Eqs. (A7) and (A8) at high and low temperatures, respectively, the ap- proximant was set up in the form

N1 N2 N5 N0 + + 2 + ··· + 5 C (T )= Tn Tn Tn . (13) n n D1 D2 D7 D8 D + + 2 + ··· + 7 + 8 0 Tn Tn Tn Tn

The number of intermediate power terms was increased in pairs (one in the numerator and one in the denominator) until the final fitted approximant accurately matched the Debye function in the intermediate T range. Using the same procedure as described in Sec. IIIA, the high- and low-T limits of the Debye function in Eqs. (A7) and (A8) 18

3000 16000

(a) LaNi P (As Cast) LaNi (P Ge )

I 2 2 14000 2 0.75 0.25 2

obs.

2500 I

obs.

I 12000

calc.

I

2000 calc.

I I 10000

obs. calc.

I I

obs. calc.

1500 8000 Bragg Positions

Bragg Positions

6000

1000

4000

500

2000

0

0 Intensity (arb. units) Intensity (arb. units)

-2000 -500

-4000

-1000

10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 100

2 (degree) 2 (degree)

3000

(b) LaNi P (Annealed) FIG. 15: (Color online) Room temperature powder XRD pat- 2 2

2500

I tern (red circles) of LaNi2(P0.75Ge0.25)2, Rietfeld refinement obs. fit (solid black line), difference profile (lower solid blue line),

2000 I calc. and positions of Bragg peaks (vertical bars).

I I

1500 obs. calc.

Bragg Positions

1000

500

0 Intensity (arb. units)

-500

7000

LaNi (P Ge ) I

-1000 2 0.50 0.50 2 6000 obs.

10 20 30 40 50 60 70 80 90

I

calc. 5000

2 (degree) I I

obs. calc.

4000

Bragg Positions

FIG. 14: (Color online) Room temperature powder XRD pat- 3000

Ni Ge tern (red open circles) of LaNi2P2, Rietfeld refinement fit 5 3 (solid black line), difference profile (lower solid blue line), and 2000 positions of Bragg peaks (vertical bars). The two panels show 1000 the XRD pattern obtained (a) before and (b) after annealing the sample. 0 Intensity (arb. units)

-1000

-2000

10 20 30 40 50 60 70 80 90 100

2 (degree)

FIG. 16: (Color online) Room temperature powder XRD pat- tern (red circles) of LaNi2(P0.50Ge0.50)2, multi-phase Rietfeld refinement fit (solid black line), difference profile (lower solid blue line), and positions of Bragg peaks (vertical bars; upper: LaNi2(P0.50Ge0.50)2, lower: Ni5Ge3). The refinement reveals that the phase composition is 99.6 mol% LaNi2(P0.50Ge0.50)2 and 0.4 mol% Ni5Ge3. 19

12000

LaNi (P Ge ) I

2 0.25 0.75 2

obs.

10000

I

calc.

8000

I I

obs. calc.

Bragg Positions 6000

4000

2000

0 Intensity (arb. units)

-2000

-4000

10 20 30 40 50 60 70 80 90 100

2 (degree)

FIG. 17: (Color online) Room temperature powder XRD pat- tern (red circles) of LaNi2(P0.25Ge0.75)2, Rietfeld refinement fit (solid black line), difference profile (lower solid blue line), and positions of Bragg peaks (vertical bars).

25000

I LaNi Ge

obs. 2 2

20000

I

calc.

I I

15000

obs. calc.

Bragg Positions

10000

La O

2 3

Ni Ge

19 12 5000

0 Intensity (arb. units)

-5000

-10000

10 20 30 40 50 60 70 80 90 100

2 (degree)

FIG. 18: (Color online) Room temperature powder XRD pattern (red circles) of LaNi2Ge2, multi-phase Rietfeld re- finement fit (solid black line), difference profile (lower solid blue line), and positions of Bragg peaks (vertical bars; up- per: LaNi2Ge2, middle: La2O3, lower: Ni19Ge12). The re- finement indicates that the phase composition of the sam- ple is 93.9 mol% LaNi2Ge2, 5.7 mol% La2O3, and 0.4 mol% Ni19Ge12. 20

1.00 4.0

300 K 300 K LaNi P as-cast LaNi (P Ge )

2 2 2 0.75 0.25 2

3.5

200 K 200 K

100 K 100 K

0.75 3.0

50 K 50 K

20 K 2.5

10 K /mol) /mol) 3 0.50 3 2.0

5 K

1.8 K

1.5

0.25 M (G cm

1.0 M (G cm

0.5

(a)

(a)

0.00

0.0

0 1 2 3 4 5

0 1 2 3 4 5

H (T)

H (T)

3.0

6

300 K LaNi P annealed

2 2

300 K LaNi (P Ge )

2 0.50 0.50 2 200 K

2.5

200 K

5 100 K

100 K

50 K

2.0

50 K

20 K 4

20 K

10 K /mol) 3 1.5 10 K

5 K /mol) 3 3

5 K

1.8 K

1.0 1.8 K

2 M (G cm M (G cm 0.5

1

(b)

0.0 (b)

0 1 2 3 4 5

0

0 1 2 3 4 5

H (T)

H (T) FIG. 19: (Color online) Magnetization M versus applied field 11 H isotherms (symbols) at the listed temperatures for LaNi2P2 300 K LaNi (P Ge )

10 (a) before and (b) after annealing. The solid curves of the 2 0.25 0.75 2

200 K

corresponding colors are fits by Eq. (19). 9

100 K

8

50 K

7 Appendix C: Magnetization versus Field Isotherms 20 K

10 K 6 /mol) 3

In this Appendix the M(H) isotherms in Figs. 19–21 5 K 5

and the dependence of the ferromagnetic impurity satu- 1.8 K ration magnetization on temperature in Fig. 22 that are 4 discussed in Sec. IV C are shown. 3 M (G cm

2

1

(c)

0

0 1 2 3 4 5

H (T)

FIG. 20: (Color online) Magnetization M versus applied mag- netic field H isotherms (symbols) at the listed temperatures for LaNi2(Ge1−xPx)2 samples with (a) x = 0.25, (b) x = 0.50, and (c) x = 0.75. In each figure, the solid curves of the cor- responding colors are fits by Eq. (19). 21

14 0.45

LaNi P as-cast 300 K LaNi Ge

2 2 2 2 0.40

12 200 K LaNi P annealed

2 2

0.35

100 K

LaNi (P Ge )

2 0.75 0.25 2

10 50 K

0.30

LaNi (P Ge )

2 0.50 0.50 2

20 K

0.25

8 LaNi (P Ge ) /mol)

10 K 2 0.25 0.75 2 3 /mol)

3

0.20 LaNi Ge 5 K

2 2

6

1.8 K

0.15 (G cm 0

4 M 0.10 M (G cm

0.05

2

(a) 0.00

0 50 100 150 200 250 300 350 0

0 1 2 3 4 5

T (K)

H (T)

25 FIG. 22: (Color online) Saturation magnetization of ferro- magnetic impurities M0 in the LaNi2(Ge1−xPx)2 samples ver- 300 K (SQUID) LaNi Ge 2 2 sus temperature T . The data at 350 K are extrapolated. The 300 K (VSM) data for x = 0.5 suggest the presence of a ferromagnetic im- 20 500 K (VSM) purity phase with a Curie temperature of ≈ 100 K. The solid

700 K (VSM) lines are guides to the eye.

15 /mol) 3

10 M (G cm

5

(b)

0

0 2 4 6 8 10

H (T)

FIG. 21: (Color online) Magnetization M versus applied field H isotherms (symbols) at the listed temperatures for LaNi2Ge2 (a) using a SQUID magnetometer and (b) using a VSM magnetometer. The solid curves of the corresponding colors are fits by Eq. (19). 22

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