Structural, Thermal, Magnetic and Electronic Transport Properties of the LaNi(GeP) System
Polycrystalline samples of LaNi(GeP) () were synthesized and their properties investigated by x-ray diffraction (XRD) measurements at room temperature and by heat capacity , magnetic susceptibility , and electrical resistivity measurements versus temperature from 1.8 to 350 K. Rietveld refinements of powder XRD patterns confirm that these compounds crystallize in the body-centered-tetragonal ThCrSi-type structure (space group I4/mmm) with composition-dependent lattice parameters that slightly deviate from Vegard’s Law. The measurements showed a positive temperature coefficient for all samples from 1.8 K to 300 K, indicating that all compositions in this system are metallic. The low- measurements yield a rather large Sommerfeld electronic specific heat coefficient mJ/mol K for reflecting a large density of states at the Fermi energy that is comparable with the largest values found for the FeAs class of materials with the same crystal structure. The decreases approximately linearly with to 7.4(1) mJ/mol K for . The measurements show nearly temperature-independent paramagnetic behavior across the entire range of compositions except for LaNiGe, where a broad peak is observed at K from measurements up to 1000 K that may arise from short-range antiferromagnetic correlations in a quasi-two-dimensional magnetic system. High-accuracy Padé approximants representing the Debye lattice heat capacity and Bloch-Grüneisen electron-phonon resistivity functions versus are presented and are used to analyze our experimental and data, respectively, for K. The -dependences of for all samples are well-described over this range by the Bloch-Grüneisen model, although the observed K) values are larger than calculated from this model. A significant -dependence of the Debye temperature determined from the data was observed for each composition. No clear evidence for bulk superconductivity or any other long-range phase transition was found for any of the LaNi(GeP) compositions studied.
pacs:74.70.Xa, 02.60.Ed, 75.20.En, 65.40.Ba
The search for high temperature superconductors intensified after the discovery of superconductivity at 26 K in the compound LaFeAsOF.Kamihara2008 (); Johnston2010 () Subsequent studies revealed even higher superconducting transition temperatures () upon replacing La with smaller rare earth elements,Kito2008 (); Ren2008b () yielding a of 55 K for SmFeAsOF.Ren2008a () These compounds crystallize in the primitive tetragonal ZrCuSiAs (1111-type) structure with space group P4/nmm.Quebe2000 () They have alternating FeAs and RO (R rare earth element) layers stacked along the -axis. The Fe atoms form a square lattice in the plane and are coordinated by As tetrahedra, where the coordinating As atoms lie in planes on either side of and equidistant from an Fe plane. The undoped parent compounds show coupled structural and antiferromagnetic (AF) spin density wave (SDW) transitions,GChen2008a (); Dong2008 (); Klauss2008 () which are both suppressed upon doping by partially substituting O by F. Such doping results in a nonintegral formal oxidation state for the Fe atoms. This suppression of the long-range ordering transitions appears necessary for the appearance of high-temperature superconductivity, as in the layered cuprate high- superconductors.XChen2008 (); GChen2008a (); Ren2008a (); Yang2008 (); Bos2008 (); Dong2008 (); Giovannetti2008 () The 1111-type self-doped Ni-P analogue LaNiPO becomes superconducting at temperatures up to about 4.5 K.Tegel2008a (); McQueen2009 ()
Subsequently, the parent compounds AFeAs (A Ca, Sr, Ba, and Eu) were investigated.Johnston2010 () They crystallize in the body-centered-tetragonal ThCrSi (122-type) structure with space group I4/mmm and contain the same type of FeAs layers as in the 1111-type compounds. In addition, they show SDW and structural transitions at high temperaturesRotter2008a (); Krellner2008 (); Ni2008a (); Yan2008 (); Ni2008b (); Ronning2008 (); Goldman2008 (); Tegel2008 (); Ren2008c (); Jeevan2008a () that are similar to those seen in the 1111-type compounds and these are similarly suppressed upon substituting on the A, Fe and/or As sites.Johnston2010 () Superconductivity again appears at temperatures up to 38 K as these long-range crystallographic and magnetic ordering transitions are suppressed.Rotter2008b (); GChen2008b (); Jeevan2008b (); Sasmal2008 () However, the conventional electron-phonon interaction has been calculated to be insufficient to lead to the observed high ’s, and strong AF fluctuations still occur in these compounds above even after the long-range AF ordering is suppressed.Johnston2010 () The consensus is therefore that the superconductivity in these high- compounds arises from an electronic/magnetic mechanism rather than from the conventional electron-phonon interaction.Johnston2010 ()
The above discoveries motivated studies of other compounds with the 122-type structure to search for new superconductors and to clarify the materials features necessary for high- superconductivity in the AFeAs-type compounds. For example, the semiconducting AF compound contains local Mn magnetic moments with spin and Néel temperature K.Singh2009a (); An2009 (); Singh2009b (); Johnston2011 () We recently doped this compound with K to form a new series of metallic AF BaKMnAs compounds containing the Mn local magnetic moments,Pandey2012 () but no superconductivity has yet been observed in this system.Pandey2012 (); Bao2012 () This may be because the was not sufficiently suppressed by the K-doping levels used. Studies of 122-type compounds in which the Fe and As in AFeAs are both completely replaced by other elements have also been carried out. For example, becomes superconducting at K.Jeitschko1987a () was recently found to become superconducting with K,Fujii2009 () with conventional electronic and superconducting properties.Kim2012 ()
The Fe-based phosphides not containing magnetic rare earth elements such as CaFeP,Jia2010 () LaFeP,Morsen1988 () SrFeP,Morsen1988 () and BaFeP (Ref. Arnold2011, ) show Pauli paramagnetic behavior. The 122-type Co-based phosphides exhibit varying magnetic properties. SrCoP does not order magnetically, although it has a large Pauli susceptibility that has variously been reported to exhibit either a weak broad peak at K attributed to “weak exchange interactions between itinerant electrons” (Ref. Morsen1988, ), or a weak broad peak at K attributed to a “nearly ferromagnetic Fermi liquid” (Ref. Jia2009, ). LaCoP orders ferromagnetically at a Curie temperature K,Morsen1988 (); Kovnir2011 () and CaCoP is reported to exhibit A-type antiferromagnetism at K in which the Co spins align ferromagnetically within the basal plane and antiferromagnetically along the c-axis.Reehuis1998 () These differing magnetic properties of the Co-based phosphides are correlated with the formal oxidation state of the Co atoms, taking into account possible P-P bonding.Reehuis1998 () Compounds where the Co atoms have a formal oxidation state of +2, , and show no magnetic order, ferromagnetic order and antiferromagnetic order, respectively.Reehuis1998 () None of the above 122-type Fe or Co phosphides were reported to become superconducting.
Among Ni-containing 122-type compounds, superconductivity has been reported with K in ,Bauer2008 () K in the distorted structure of ,Ronning2008d () K in the orthorhombically distorted structure of (Ref. Ronning2009, ) and K in .Mine2008 () The Pauli paramagnet is reported not to become superconducting above 1.8 K.Jeitschko1987 () There are conflicting reports about the occurrence of superconductivity in with either –0.8 K,Wernick1982 (); Maezawa1999 () or no superconductivity observed above 0.32 K.Kasahara2008 ()
Several studies have been reported on the normal state properties of . de Haas van Alphen (dHvA) measurements at 0.5 K indicated moderate band effective masses to 2.7, where is the free electron mass.Maezawa1999 () Electronic structure calculations were subsequently carried out by Yamagami using the all-electron relativistic linearized augmented plane wave method based on the density-functional theory in the local-density approximation.Yamagami1999 () The density of states at the Fermi energy was found to be large, states/(eV f.u.) for both spin directions, arising mainly from the Ni orbitals, where f.u. means formula unit. This is comparable to the largest values reported for the FeAs-based 122-type superconductors and parent compounds.Johnston2010 () Three bands were found to cross , with two Fermi surfaces that were hole-like (0.16 and 1.11 holes/f.u.) and one that was electron-like (0.27 electrons/f.u.), and therefore with a net uncompensated carrier charge density of 1.00 holes/f.u. Thus for the hypothetical compound one can assign formal oxidation states Th, Ni and Ge. Then substituting trivalent La for tetravalent Th yields a net charge carrier concentration of one hole per formula unit. The electron Fermi surface is a slightly corrugated cylinder along the -axis centered at the X point of the Brillouin zone, indicating quasi-two-dimensional character, similar to the electron Fermi surface pockets in the FeAs-based 122-type compounds.Johnston2010 () On the other hand, the two hole Fermi surfaces are three-dimensional and are centered at the X and Z (or M, depending on the definitionJohnston2010 ()) points of the Brillouin zone. These calculated Fermi surfaces were found to satisfactorily explain the results of the above dHvA measurements,Maezawa1999 () including the measured band masses. From a comparison of the calculated with that obtained from experimental electronic specific heat data, Yamagami inferred that many-body enhancements of the theoretical band masses are small.Yamagami1999 ()
Hall effect measurements on single crystals of are consistent with the occurrence of multiple electron and hole Fermi surfaces, with the weakly -dependent Hall coefficients given by a positive (hole-like) value m/C for the applied magnetic field parallel to the -axis and a negative (electron-like) value m/C for parallel to the -axis.Sato1998 () The thermoelectric power obtained on a polycrystalline sample of is negative.Schneider1983 ()
Herein we report our results on the mixed system LaNi(GeP). For or 1, alternating La and NiGe or NiP layers, respectively, are stacked along the -axis. We wanted to investigate whether any new phonomena occur with Ge/P mixtures that do not occur at the endpoint compositions, such as happens when the parent FeAs-based compounds are doped/substituted to form high- superconductors. In addition, Yamagami’s electronic structure calculations for discussed aboveYamagami1999 () indicated some similarities to the electronic structures of the FeAs-based 122-type compounds.
We report structural studies using powder x-ray diffraction (XRD) measurements at room temperature, together with heat capacity , magnetic susceptibility , and electrical resistivity measurements versus temperature from 1.8 to 350 K for five compositions of LaNi(GeP) with . Our low- limit of 1.8 K precluded checking for superconductivity with K reported for ,Wernick1982 (); Maezawa1999 () but we did find evidence for the onset of superconductivity below K in two samples of from both and measurements. However, it is not clear from our measurements whether this onset arises from the onset of bulk superconductivity or is due to an impurity phase.
Also presented in this paper is the construction of Padé approximantsPade () for the Debye and Bloch-Grüneisen functions that describe the acoustic lattice vibration contribution to the heat capacity at constant volume of materials and the contribution to the of metals from scattering of conduction electrons from acoustic lattice vibrations, respectively. These Padé approximants were created in order to easily fit our respective experimental data using the method of least-squares, but they are of course more generally applicable to fitting the corresponding data for other materials. The Debye and Bloch-Grüneisen functions themselves cannot be easily used for nonlinear least-squares fits to experimental data because they contain integrals that must be evaluated numerically at the temperature of each data point for each iteration. Several numerical expressions representing the Bloch-GrüneisenDeutsch1987 (); Mamedov2007 (); Ansari2010 (); Cvijovic2011 (); Paszkowski1999 () or DebyeNg1970 () functions have appeared. However, they replace the integrals in these functions with infinite series, use very large numbers of terms, and/or use special functions. These approximations are therefore not widely used for fitting experimental data. One paper presented a method for approximating the Debye function using the Einstein model.Listerman1979 () This method is also of little use for fitting because it uses a different equation for each temperature range and it becomes inaccurate at low temperatures. However, as we demonstrate, the Debye and Bloch-Grüneisen functions can each be accurately approximated by a simple Padé approximant over the entire range. To our knowledge, there are no previously reported Padé approximants for either of these two important functions. The -dependences of for all samples discussed here are well-described by the Bloch-Grüneisen prediction, although the observed K) values are larger than calculated. A significant -dependence of the Debye temperature determined from the data was observed for each composition.
The remainder of this paper is organized as follows. An overview of the experimental procedures and apparatus used in this work is given in Sec. II. The construction of the Padé approximants for the Bloch-Grüneisen and Debye functions is described in Sec. III and Appendix A. The structural, thermal, magnetic, and electrical resistivity measurements of the LaNi(GeP) system and their analyses are presented in Sec. IV and Appendices B and C. A summary and our conclusions are given in Sec. V.
Ii Experimental Details
Polycrystalline samples of LaNi(GeP) ( = 0, 0.25, 0.50, 0.75, 1) were prepared using the high purity elements Ni: 99.9+%, P: 99.999+%, and Ge: 99.9999+% from Alfa Aesar and La: 99.99% from Ames Laboratory Materials Preparation Center. Stoichiometric amounts of La, Ni, and Ge were first melted together using an arc furnace under high-purity Ar atmosphere. The arc-melted button was flipped and remelted five times to ensure homogeneity. Next, the samples (except for LaNiGe which was prepared following the general procedures outlined in Ref. Rieger1969, ) were throughly ground and mixed with the necessary amount of P powder in a glove box under an atmosphere of ultra high purity He. The powders were cold-pressed into pellets and placed in 2 mL alumina crucibles. The arc-melted button of LaNiGe was wrapped in Ta foil. The samples were then sealed in evacuated quartz tubes and fired at 990 C for d. Samples containing phosphorus were first heated to 400 C to prereact the phosphorus.
After the first firing, the phase purities of the samples were checked using room temperature powder x-ray diffraction (XRD) with a Rigaku Geigerflex powder diffractometer and CuK radiation. The x-ray patterns were analyzed for impurities using MDI Jade 7. If necessary, samples were thoroughly reground and repelletized (except for LaNiGe which was just rewrapped in Ta foil) and either placed back in alumina crucibles or wrapped in Ta foil and resealed in evacuated quartz tubes. Samples were again fired at 990 C for 5–6 d. The LaNiP sample was arc-melted with additional La and P in order to achieve a single phase sample. This may have been necessary because the compound may not form with the exact 1:2:2 stoichiometry. After arc-melting, part of the sample was annealed for 60 h at 1000 C. Throughout this paper, the annealed LaNiP sample will be referred to as a and the as-cast sample as b where is the composition of LaNi(GeP). As seen in the XRD patterns and fits in Sec. IV.1, all final samples were single-phase except for two samples showing very small concentrations of impurities.
Rietveld refinements of the XRD patterns were carried out using the FullProf package.Rodriguez1993 () Magnetization measurements versus applied magnetic field and temperature were carried out using a superconducting quantum interference device (SQUID) magnetometer (Quantum Design, Inc.). Gel caps were used as sample holders and their diamagnetic contribution was measured separately and corrected for in the data presented here.
The and measurements were carried out using a Quantum Design Physical Property Measurement System (PPMS). Samples for heat capacity measurements had masses of 15–40 mg and were attached to the heat capacity puck with Apiezon N grease for thermal coupling to the platform. The measurements utilized a four-probe ac technique using the ac transport option on the PPMS. Rectangular samples were cut from the sintered pellets or arc-melted buttons using a jeweler’s saw. Platinum leads were attached to the samples using EPO-TEK P1011 silver epoxy. The sample was attached to the resistivity puck with GE 7031 varnish. Temperature-dependent measurements were recorded on both cooling and heating to check for thermal hysteresis. No significant hysteresis was observed for any of the samples. In addition, the vibrating sample magnetometer (VSM) option on the PPMS was used to measure the high- magnetization of the LaNiGe sample up to 1000 K.
Iii Padé Approximant Fits to the Bloch-Grüneisen and Debye Functions
A Padé approximant is a ratio of two polynomials. Here we write these polynomials as series in according to
The first one, two or three and last one, two or three in each of the sets of coefficients and in can be chosen to exactly reproduce both the low- and high- limiting values and power law dependences in and/or of the function it is approximating. This is a very important and powerful feature of the Padé approximant. Then the remaining terms in powers of 1/ in the numerator and denominator have freely adjustable coefficients that are chosen to fit the intermediate temperature range of the function. A physically valid approximant requires that there are no poles of the approximant on the positive real axis.
iii.1 Bloch-Grüneisen Model
The temperature-dependent electrical resistivity due to scattering of conduction electrons by acoustic lattice vibrations in monatomic metals is described by the Bloch-Grüneisen (BG) model according toBlatt1968 ()
is the Debye temperature determined from resistivity measurements, is Planck’s constant divided by , is the number of conduction electrons per atom, is the atomic mass, is Avogadro’s number, , is Boltzmann’s constant, and is the elementary charge. These variables map a monatomic metal with arbitrary crystal structure onto a simple-cubic lattice with one atom per unit cell of lattice parameter . To calculate in units of cm, one sets the prefactor () in Eq. (3) to 4108.24 in SI units and calculates the quantities inside the square brackets in cgs units so that the quantity in square brackets has net units of cm. If one instead has a polyatomic solid, one can map the parameters of that solid onto those of the monatomic solid described by the Bloch-Grüneisen model as explained in Sec. IV.4 below.
In practice, one fits the dependence of an experimental data set by the BG model using an independently adjustable prefactor instead of in Eq. (2), because accurately fitting both the magnitude and dependence of a data set cannot usually be done using a single adjustable parameter . One therefore normalizes Eq. (2) by . When , the integral in Eq. (2) is
This -dependence is only a function of the dimensionless normalized temperature . Therefore we define normalized and variables as
and Eq. (5) becomes
In order to construct a Padé approximant function that accurately represents in Eq. (7), the power law dependences of the latter function must be computed at high and low temperatures and the coefficients of the Padé approximant adjusted so that both of these limiting dependences are exactly reproduced (to numerical precision). Then the remaining coefficients are determined by fitting the approximant to a set of data obtained by evaluating the function over the entire relevant temperature range. This procedure is described in Appendix A.1, which constrains the values of , , , and . The resulting approximant is
where the fitted coefficients in the Padé approximant (8) are listed in Table 1. The denominator in Eq. (8) was checked for zeros and all were found not to lie on the positive axis. This insures that the approximant does not diverge at any (real positive) temperature. As seen in Fig. 1(b), the difference between the BG function and the Padé approximant is less than at any in the range . By construction, the Padé approximant must asymptote to the exact BG dependences at high- and low-, respectively. Figure 1(c) shows the percent error of the fitted approximant. This error is largest at low with a value of at . This is acceptable considering the very small value of at such low . For the most accurate fit of low- experimental data by a power law in , one would directly fit experimental data by the power law rather than using the Padé approximant function.
When fitting experimental data by the Padé approximant in Eq. (8), one fits only the dependence and not the magnitude of by the BG theory, because as noted above, one cannot in general obtain a good fit of both the magnitude and the dependence of a measured data set using only the single fitting parameter . Thus, one fits an experimental data set by
where is the residual resistivity for . The meaning of is that one substitutes for , according to Eqs. (6), in the Padé approximant function in Eq. (8). The three adjustable parameters , and are varied independently to obtain a fit to the data. Once a good fit is obtained and all three parameters are determined, one can compare the measured value of with the value predicted by the BG theory in Eq. (4). Often the agreement is not very good even for - or -metals.Blatt1968 ()
iii.2 Debye Model
The Debye modelKittel2005 () is widely used for fitting experimental heat capacity data taken at constant pressure p arising from acoustic lattice vibrations. It is sometimes useful to fit a large range of experimental data using the Debye function. Here we describe the construction of an accurate Padé approximant function of that can easily be used in place of the Debye function (10) for least-squares fitting experimental data over an extended range. It can also be conveniently used to calculate the dependence of the Debye temperature from experimental lattice heat capacity data over an extended range.
The lattice heat capacity at constant volume V per mole of atoms in the Debye model is given byKittel2005 ()
where is the molar gas constant and is the Debye temperature determined from heat capacity measurements. The and can be normalized to become dimensionless according to
Equation (10) then becomes
As discussed in Appendix A.2, the Padé approximant that fits both the high- and low- power law asymptotics of in Eq. (12) and has additional terms in powers of in the numerator and denominator to fit the intermediate range is
where the coefficients are given in Table 2. The resulting fit and error analyses are shown in Fig. 2. The Padé approximant does not deviate from the normalized Debye function in Eq. (12) by more than at any as seen in Fig. 2(b). By construction, the deviation goes to zero at both low and high . The percent error in Fig. 2(c) has its maximum magnitude of at low , and occurs because and numerical precision becomes an issue there. For another example of the high accuracy and use of this Padé approximant, see Fig. 9 below, where the versus is calculated directly for each of our samples of LaNi(GeP) using the Debye function in Eq. (10) and compared with that found using the Padé approximant in Eq. (13); only small differences are found.
To fit experimental data by the Padé approximant in Eq. (13), one fits both the magnitude and dependence of simultaneously using
where is the number of atoms per formula unit and is the only fitting parameter. Here one substitutes for , according to Eqs. (11), in the Padé approximant function in Eq. (13). For a metal, one can add to Eq. (14) a linear specific heat term giving
The is the Sommerfeld electronic specific heat coefficient that can be experimentally determined from a prior separate fit to data at low according toKittel2005 ()
Iv Experimental Results and Analyses
iv.1 Structure and Chemical Composition Determinations
|(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|
The starting parameters for the Rietveld refinements of the powder XRD patterns were those previously reported for LaNiP (Refs. Bobev2009, , Hofmann1984, , Jeitschko1980, ) and LaNiGe (Refs. Yamagami1999, , Rieger1969, , Hasegawa2004, , Morozkin1997, ) that are presented in Table 3. All samples in the LaNi(GeP) system were found to crystallize in the body-centered-tetragonal ThCrSi structure (space group I4/mmm) as previously reported for the compositions and 1, and our refined values for the lattice parameters are in agreement with reported values, as shown in Table 3. The refinements of the powder XRD patterns are shown in Figs. 15–18 in Appendix B and the crystal data are listed in Table 3. All samples were also refined for site occupancy and no significant deviations were found from the value of unity. However, the Rietveld fits were not very sensitive to changes in site occupation.
The lattice parameters and and the unit cell volume are plotted versus composition in Fig. 3. As the concentration of P increases, the lattice parameters and unit cell volume all decrease monotonically while, from Table 3, the ratio increases. The composition dependences of the quantities in Fig. 3 deviate slightly from the linearities expected from Vegard’s Law. From Table 3, the -axis position parameter of the P/Ge site has a small overall increase with increasing P concentration.
An interesting effect was observed in the x-ray data for LaNiP. The as-cast sample after arc-melting showed broadening of the diffraction peaks with large -axis contributions. After annealing the arc-melted sample for 60 h at 1000 C, those peaks became sharp and shifted to lower 2 angles reflecting an increased -axis lattice parameter. These effects are shown on an expanded scale in Fig. 4 for the (105) and (116) reflections. The -axis peak broadening may arise from disorder in the interlayer stacking distances along the -axis.Johnston1984 ()
iv.2 Heat Capacity Measurements
Plots of for our samples of LaNi(GeP) from 1.8 to 300 K are shown in Fig. 5. The heat capacity at 300 K ranges from 118.3–123.0 J/mol K for these samples. These values are approaching with increasing the expected classical Dulong-Petit value mJ/mol K for the heat capacity due to acoustic lattice vibrations, where is the number of atoms per formula unit ( for our compounds).
To determine the Sommerfeld electronic specific heat coefficient and a low temperature value of the Debye temperature for each sample, the lowest temperature linear data for each sample, plotted in Fig. 6, were fitted by Eq. (16) and the values of and obtained. A value of can be calculated from each value of usingKittel2005 ()
The values obtained for , , and for each sample are listed in Table 4. A plot of versus in LaNi(GeP) is shown in Fig. 7, where a nearly linear decrease in with increasing is seen. Using this values in Table 4, we then fitted the data from 1.8 to 300 K in Fig. 5 by Eq. (15) and obtained the fitting parameters listed in Table 4. The fits are shown as the solid curves in Fig. 5, which are seen to agree rather well with the respective data. However, small -dependent deviations between the data and fit for each sample are seen, which we address next.
|Fit Range||(mJ/mol K)||(eV f.u.)||(mJ/mol K)||Low- fit||All- fit|
|LaNiP (as-cast)||1.81–5.34||7.7(2)||0.086(7)||483(14)||369(2)||This work|
|13.5 (obs)111No experimental evidence or reference citation was given for this quoted observed value.||Yamagami1999,|
Deviations of a Debye model fit from experimental lattice heat capacity data are due to the following assumptions and approximations of the model.
The system is assumed to be at constant volume as changes, rather than at constant pressure which is the experimental condition. This deficiency can be corrected for if the -dependent compressibility and thermal expansion coefficient are known for the compound of interest.
A quadratic density of phonon states versus energy is assumed, which terminates at the Debye energy . For actual materials, this assumption can only be accurately applied at temperatures , which gives the Debye law [the second term on the right-hand side of Eq. (16)].
Assumptions are made that the speed of acoustic sound waves in a material is temperature-independent, is isotropic and is the same for longitudinal and transverse acoustic sound waves. In general, these assumptions are too simplistic and the parameters are temperature dependent.
The Debye model only accounts for acoustic lattice vibrations and does not take into account optic lattice vibrations arising from opposing vibrations of atoms with different masses in the unit cell. The contribution of these to can be modeled by adding Einstein terms to the fit function.Kittel2005 ()
Because of these approximations and assumptions of the Debye model, the lattice heat capacity of a material is never precisely described by the Debye model over an extended temperature range such as from 2 K to 300 K. The most serious approximation in our range is the second approximation. Within the Debye model is independent of . One can therefore parameterize the deviations of a fit from the data by allowing to vary with .Gopal1966 ()
The value of was calculated for each data point in Fig. 5, after subtracting the contribution from according to Eq. (15), using the Debye function in Eq. (10). The resulting dependences of are shown in Fig. 8, where is seen to vary nonmonotonically and by up to 30% with increasing . The plots have a similar shape to that for sodium iodide.Berg1957 () The is expected to be constant below and above , where is the zero-temperature value of .Gopal1966 () Our data qualitatively agree with these expectations.
Considerable scatter in the data in Fig. 8 occurs at temperatures above 250 K and also below 7 K (although not as clearly visible in the figure). In these ranges, is becoming nearly independent of , so in these ranges any error in the value of is greatly amplified when is calculated. The error bars on the values of therefore increase significantly in these regions. We now consider such errors and for this discussion ignore the difference between and .
The scatter in the derived versus depends on the statistical error in ,
In order to more clearly see why the error in substantially increases above K, the high- approximation in Eq. (34) can be used, yielding
Taking the derivative with respect to gives
Inserting this result into Eq. (18) gives the approximation
This result shows that the error in is proportional to at high , which results in a dramatic increase in the scatter and error in at high .
Using the low- approximation in Eq. (35) and the same procedure as described above, the error in is
Therefore, similar to the situation at high-, a small error in at low is greatly amplified when calculating .
To verify the applicability of the Padé approximant for the Debye function developed in Sec. III.2, the was calculated for each data point in Fig. 5 using the Padé approximant in Eq. (13) instead of by directly using the Debye function in Eq. (10). The electronic contribution was again subtracted from the data first. The difference between these values and those calculated using the Debye function in Eq. (13) is plotted versus in Fig. 9. The values calculated from the Padé approximant do not deviate by more than 0.35 K from those calculated using the Debye formula. This error is of order of , which is usually small compared to the error in itself and is negligible compared to its dependence. Therefore the Padé approximant provides a viable alternative for calculating that does not require evaluation of the integral in the Debye function (10) or the use of a look-up table for each data point.
iv.3 Magnetization and Magnetic Susceptibility Measurements
Magnetization versus applied magnetic field isotherms were measured for the LaNi(GeP) system for –5.5 T and the results are plotted in Figs. 19–21 of Appendix C. The versus for the samples were also measured from 1.8 to 300 K at a fixed field T and the resulting susceptibilities are plotted in Fig. 10(a). As evident from the nonlinear behavior at low fields in the plots and the upturn that follows the Curie-Weiss-like behavor  in the observed at low temperatures, we infer the presence of saturating paramagnetic and/or ferromagnetic impurities in the samples. In order to determine the intrinsic behaviors, it is necessary to correct for these impurities. To determine the individual contributions to the susceptibility data, the curves were fitted by
where is the saturation magnetization of the ferromagnetic impurities, is the intrinsic susceptibility of the sample, is the molar fraction of paramagnetic impurities, is the spectroscopic splitting factor of the impurities which was fixed at to reduce the number of fitting parameters, is the Bohr magneton, is Boltzmann’s constant, is the Weiss temperature of the paramagnetic impurities (included for consistency with a possible Curie-Weiss law behavior at low ), is the saturation magnetization of the paramagnetic impurities, and is the Brillouin function. The Brillouin function is
and the molar saturation magnetization of the paramagnetic impurities is
where is the spin of the impurities and is Avogadro’s number.
|Sample||(K)||(G cm/mol)||(10 cm/mol)|
To determine the values of , , and for the paramagnetic impurities, a global two-dimensional surface fit of the data from 1–5.5 T taken at both 1.8 and 5 K was done using Eq. (19) with MATLAB’s Surface Fitting Tool. For this range, the ferromagnetic impurities are expected to be nearly saturated, as assumed by Eq. (19). The parameter values obtained are given in Table 5. Fixing the variables , , and at the respective values for each sample, each curve at higher temperatures was fitted by Eq. (19) in the range 1–5.5 T to obtain values for and . The fits are shown in Figs. 19–21 in Appendix C. A plot of versus is presented in Fig. 22 in Appendix C and the fitted values of the intrinsic susceptibility are plotted as solid symbols in Fig. 10(b).
In order to correct the data at T in Fig. 10(a) for the paramagnetic impurities, the above values of , , and obtained by fitting the isotherms at 1.8 and 5 K were inserted into the last term of Eq. (19) to calculate their contributions versus at T. These contributions were then subtracted from the respective data that were already corrected for ferromagnetic impurities to obtain the intrinsic susceptibility of the samples as plotted in Fig. 10(b). Previously reported values of for LaNiGe are cm/mol at 300 K (Ref. Wernick1982, ) and cm/mol at 296 K (Ref. Zell1981, ). The former value is essentially the same as our value cm/mol at 300 K, as seen more clearly in Fig. 11 below where our data for LaNiGe are plotted on an expanded vertical scale for temperatures up to 1000 K.
The data in Fig. 10 show that the samples in the LaNi(GeP) system exhibit nearly temperature-independent paramagnetism over the composition region .25–1. This trend does not extend to LaNiGe (), for which a broad maximum appears to occur at K, which is close to the upper temperature limit of the SQUID magnetometer. In order to determine whether a maximum in near 300 K does occur, data for the sample of LaNiGe were measured using a VSM from to 1000 K. The isotherm data are plotted in Fig. 21(b) of Appendix C and the data at T are plotted in Fig. 11. The magnetic contributions due to the sample holder and paramagnetic impurities are corrected for in the plots.
The data in Fig. 11 clearly show that a broad peak occurs in of LaNiGe at about 300 K. The plot in Fig. 10(b) shows the peak plotted on a less expanded vertical scale. The cause of this peak is not clear, but may be due to low-dimensional antiferromagnetic correlations.Johnston1997 () Further investigation is needed. The inset in Fig. 11 shows an expanded view of the low temperature behavior. Due to its smooth nature, the small downturn in the data below about 10 K is most likely spurious due to a slight error in correcting for the susceptibility contribution of the paramagnetic impurities in the sample.
The onset of superconductivity was observed in both the annealed and as-cast samples of LaNiP at 2.6 and 2.2 K, respectively, from zero-field-cooled (ZFC) magnetization measurements in a field of 20 Oe. Figure 12 shows these low-field measurements. It is not clear whether the data represent the onset of bulk superconductivity in LaNiP or if the diamagnetism arises from a superconducting impurity phase. The data in Fig. 13 below do not clarify this issue.
We now analyze the normal-state of the samples. The magnetic susceptibility of a metal consists of the sum of the spin and orbital contributions
In the absence of local magnetic moments, the spin contribution is the Pauli susceptibility of the conduction electrons. One can estimate using Johnston2010 ()
where is the spectroscopic splitting factor and is the density of states at the Fermi energy . Setting gives
where is in units of cm/mol, is in units of states/eV f.u. for both spin directions and f.u. means formula unit.
|Sample||(K)||( K)||(300 K)||Ref.|
|( cm)||( cm)||( cm)||( cm)|
|LaNiP (as-cast)||211(2)||83||148||This work|
To calculate , one can obtain an estimate of from the Sommerfeld electronic linear specific heat coefficient according to