ENHANCED PARAMAGNETISM OF TIBE2 AND FERROMAGNETIC TRANSITIONS IN TIBE2-XCUX

Abstract The magnetic behavior of TiBe2 is found to be similar to that of strongly enhanced paramagnets like Pd and Ni3Ga. The susceptibility curves χ(H, T = 0) and χ(H, T = 0) both go through a smooth maximum, at 55 kOe and 10 K, respectively, which might be due to the electron-electron interaction. For TiBe2-xCux compounds the transition from paramagnetism to ferromagnetism is analyzed, starting from Arrott plots of the magnetization. A critical concentration xcr = 0.155 ± 0.005 is obtained. The Stoner-Edwards-Wohlfarth model of itinerant-electron magnetism is well followed near xcr. The low temperature behavior of the specific heat of TiBe2 is tentatively ascribed to the effect of the electron-phonon interaction on the electronic specific heat coefficient γ.


Introduction
There remains some uncertainty in the nature of the magnetism of TiBe2. First, Enz and Matthias rightly predicted [1 ] that this compound would show itinerant-electron magnetism, since it is isostructural and isoelectronic with the weak ferromagnet ZrZn2. The susceptibility of TiBe2 was indeed found [2] to be very large (× ~ 10 -2 emu/mol, at T = 0) and to show a smooth maximum at 10 K, very like the one observed for Pd near 80 K. Because of the negative intercept of the extrapolated X -1 (T) curve for TiBe2 (Pd features that as well) the compound was labelled antiferromagnetic. This choice still influences the present research and leads to difficulties that will be * Partially supported on a grant from the Swiss National Science Foundation. Present address: Institut de Physique Experimentale, Universit6 de Lausanne, 1015 Dorigny, Switzerland. ** Research in La JoUa supported by the National Science Foundation under Grant No. DMR77-08469. *** Work at Los Alamos was performed under the auspices of the Department of Energy. Part of this work was performed while the authors were Guest Scientists at the Francis Bitter National Magnet Laboratory, which is supported at M.I.T. by the National Science Foundation. mentioned below. By contrast, starting from the idea that TiBe2 is a strongly enhanced paramagnet, the occurrence of ferromagnetism in Til_xCuxBe2 was predicted [3] and found [4] in TiBe2_xCux. However, the antiferromagnetic interpretation for TiBe2 survived [5][6][7][8], calling for more experimental evidence.
Here we present and analyze detailed magnetic data for TiBe2 (in particular high field measurements up to 213 kOe) and for TiBe2_xCux (emphasizing the magnetic transition at Xcr). Earlier measurements of the specific heat [9] of TiBe2, which play an important role in recent interpretations [5,7], are also discussed and compared with the results for Pd [10,11].

Experimental techniques
The preparation of the compounds has been described earlier [4]. Measurements of the magnetization were made in three apparatuses. A Faraday balance was used to measure the susceptibility in a wide range of temperature (0.8-300 K). The high field measurements were carried out with a very low frequency vibrating sample magnetometer [12] at the 250 Francis Bitter National Magnet Laboratory. The magnetic transitions in TiBe2_xCux were studied from data taken with a vibrating sample magnetometer where the specimen is held in thermal equilibrium with a large Cu block. This allows the temperature of the sample to be easily controlled (within AT < 10 -2 K between 1.4 and 20 K). Detailed measurements (to 0.1%) of the low field susceptibility of TiBe2 were also taken with this magnetometer. Using the four-probe ac technique the electrical resistance of a TiBe2 sample was measured between 1.27 K and room temperature with a relative precision of 3X 10 -4.

Magnetic behavior of 7~Be2
The magnetic susceptibility, X, of TiBe2 was measured between 0.8 and 300 K. Fig. 1 shows the low field (H< 10 kOe) results for T< 80 K and a large scale plot of the data below 22 K. At the lowest temperature, the magnetic susceptibility of TiBe2 is about 12 times larger than that of Pd. From 0.8 to 10 K, ×(70 increases by about 3% and then decreases steadily. No magnetic transition is suggested. The curves labelled A, B, C and D are calculated and will be described in section 3.3.
Above 50 K, the Curie-Weiss law is only followed in first approximation by the inverse susceptibility.   In fig. 3 the curve labelled x = 0 shows the variation of the magnetic susceptibility of TiBe2 with the external field up to 213 kOe, at 1.24 K. A maximum in X =M/His found at Hmax ~ 55 kOe. Between H= 0 and Hmax, X increases by 38%. The same data are plotted as M 2 vs HIM in fig. 4. This Arrott plot does not become linear even in the highest fields available and shows a characteristic right turn in the low field region.

Magnetic behavior of l~Be2_xCu x compounds
If Cu is partially substituted for Be in TiBe2, the susceptibility gradually increases and ferromagnetism may be observed. We report here on systematic measurements of the magnetization of four TiBe2_xCux samples with x = 0.05, 0.10, 0.16 and 0.20. Data were taken in order to determine whether a critical concentration for ferromagnetism, xcr, could be defined and to find the behavior of X(x, T) and Tc(x) (Curie temperature).
The use of Arrott plots of the magnetization is known to be the best way to deal with nearly-or weakly-ferromagnetic substances fo r which M(H) is nonlinear. Fig. 5 shows such a plot for TiBel,aCuo.2.   The open symbols are from ref. [4]. (M approximation the data points define a set of parallel straight lines which cut the horizontal axis at Tzc. By extrapolation, a (squared) Curie temperature may be formally defined for the paramagnetic compounds (T2c < 0). See ref. [13] for more details about this analysis which is based on the Stoner-Edwards-Wohlfarth model of itinerant-electron magnetism [14]. The values of the inverse susceptibility extrapolated to T = 0 are deduced from fig. 6 and are denoted (-2Xoo) -l. The zero field, zero temperature squared magnetization Mo2o is obtained from the Arrott plots.
In the present analysis, the magnetization hi(H, 7) is described by the relation  I  I  I  I  I  I  I  I  I  I  I  I  I  The low temperature part of our resistance measurements is shown in fig. 8. The resistance ratio R(300 K)/R(0) for this TiBe2 sample is 46. Between 2 and 3 K the resistance varies like T 2. Above 3 K a T n law with n < 2 is followed. No detailed measurements were taken above 4.2 K. A slight anomaly is observed in R(T) around 1.8 K. Below 1.5 K the T 2 law appears to be obeyed.

Discussion
The data presented above all favor the paramagnetic interpretation for TiBe2. The magnitude of the susceptibility, its smooth variation and the nonlinearity of X -1 (7) are characteristic of the enhanced F. Acker et aL / Magnetic behavior of l~Be: and TiBe 2_xCu x paramagnetism found in substances like Pd, YCo2, LuCo2, etc. More strikingly, these materials share a remarkable property with TiBe2: their magnetic susceptibility increases with both temperature and field, at low temperature [15][16][17][18]. At higher temperature ×(T) goes through a maximum. This can be explained by the Fermi liquid model [16][17][18] which further predicts a maximum in ×(H). We believe that TiBe2 is the first compound where both maxima are clearly seen, at Tmax = 10 K and Hmax = 55 kOe. (For comparison, Tmax = 250 K for YCo2; the low temperature susceptibility of this compound bends over above 200 kOe but the maximum is not yet reached at 400 kOe [18].) In fact, Ni3Ga [19,20] first showed such a behavior. Here, unfortunately, the maxima are obscured by an antagonist effect apparently due to magnetic impurities (the presence of 5 ppm Fe can account for it). It seems very likely that for pure and well ordered NiaGa, ×(H) would show a maximum around 40 kOe and that ×(T) would decrease below 10-20 K. The similarity in the magnetic behavior of TiBe2 and Ni3Ga is made even clearer by comparing the values of the low temperature susceptibility, × = 1.65 × 10-: emu/mol for NiaGa [19] and X = 0.90 × 10-2 emu/mol for TiBe:. The comparison may be extended to the temperature variation of X or to the shape and slope of the Arrott plots (field variation).
The introduction of a few atomic percent of impurities in Pd and the deviation from stoichiometry in Ni3Ga tend to suppress the maxima in ×(7) or ×(/-/) [17]. A corresponding behavior is observed for TiBe2_xCux where both maxima disappear for an extrapolated x ~ 0.07.
The relative smallness of Hmax and Tmax in TiBe2 makes this compound the ideal candidate for a detailed study of the surface ×(H, T) [18], in order to check the adequacy of the description based on the Fermi liquid model. We attempted to fit the data shown in figs. 1 and 3 to this model. Curves A, B and C were calculated, using the expression [17] x(T) -X(0) = ~ C~ T ~+1 ln(T/T~), tx=l with n = 1,2 and 3, respectively. The curve A (X -X(0) = -bT: ln(T/Tt)) gives an excellent fit of the low temperature data and deviates from the experimental curve above T~ 2Tm [17]. A fit of x(H) to the lowest order (X -X(0) = -cH 2 In(H/HI)) is shown by the solid curve in fig. 3. The coefficients b and c, which are related to the enhancement factor S [17,20], are both positive here (b = 5.7 × 10 -6 emu/mol K:, c = 1.9 × 10 -6 emu/mol kOe2).
The study of the magnetic transition in TiBe2_xCux brings further support to the paramagnetic interpretation for TiBe:. Fig. 7 shows that the transition from paramagnetism to ferromagnetism at x = 0.155 can be well described by the Stoner-Edwards-Wohlfarth (SEW) model of itinerant-electron magnetism. The Landau parameter A -X~ -1 appearing in the equation of state H = AM + BM 3 is written as A = aT: + /3(Xcr -x). Around xcr this T 2 law is obeyed, in first approximation, as shown in fig. 6. The value of a is practically the same for all the samples and there is no indication of a transition to antiferromagnetism at x = 0.155. The present analysis makes use of the data taken in a "significant" low temperature range. The use [6] of higher temperature susceptibility data in order to define a Curie-Weiss temperature, 0, is not advisable here since 0 has no meaning for an enhanced paramagnet. Note the parabolic variation of Tc with x abo~,e Xcr and the negative values of T2c below Xcr, in our analysis.
The deviation from the SEW model which occurs below x ~ 0.1 seems to be related to the appearance of the maxima in ×(7') and ×(H). If the linear variation with x of the characteristic parameters (--2Xoo) -1 and T2c is extrapolated to x = 0, this gives X -~ (T = 0) = 55 (emu/mol) -1 and T 2 = -720 K: for TiBe:. The figure for the susceptibility compares surprisingly well with the value found by a tentative extrapolation to H = 0 of the high field Arrott plot for TiBe2 in fig. 4. Using X -1 (0) = 55 and T2c = -720 the curve labelled D in fig. 1 was obtained. The difference between curve D and the experimental curve may be viewed as an effect of the electron-electron interaction.

Discussion of the specific heat results
It appears that the unifying theoretical ideas expressed by Enz [21] about itinerant magnetism and superconductivity could weli accomodate a strongly paramagnetic (nearly ferromagnetic) TiBe2. (In ref. [21 ] the "antiferromagnetism" was even viewed as a complication.) Nevertheless, efforts were made [7] to find evidence for antiferromagnetism from the specific heat data which show a low temperature upturn and an intriguing cusp around 2 K [9]. This was done in spite (because [7]) of the fact that no clue for antiferromagnetism can be found from the magnetic behavior of TiBe2 [5,7]. Assuming that spin density wave (SDW) antiferromagnetism occurs in TiBe2, "phasons" were made responsible for the specific heat anomaly. As already outlined in ref. [7], this interpretation bears several difficulties. We shall just mention them here: i) The sharp kink observed in C/T vs. T 2 is not well fitted by a phason term, which has the form of a shoulder, ii) No critical temperature for the SDW can be found from susceptibility measurements (but it was loosely meant [7] that Tm ~" 10 K = Tmax in our notation), iii) The specific heat discontinuity associated with the disappearance of the SDW at Tm is not detected around 10 K or elsewhere, iv) The specific heat cusp at 2 K is still present in the ferromagnetic TiBe2_xCux compounds, where the SDW should, in principle, disappear.
The analogy of TiBe2 with Pd suggests a simpler explanation which is consistent with the magnetic data. Due to the electron-phonon interaction, the coefficient 7 of the electronic specific heat might vary considerably with T, at low temperature. As shown in fig. 9, the decomposition of the specific heat Cv of Pd into a Debye function and a 3'T term with 0D and 7 both constant is impossible. In analyzing the same data, Veal and Rayne [12] chose to keep 7 constant and were left with an increasing 0D(T). On the other hand, Clusius and Schachinger [11 ] noticed earlier that after subtraction of a contribution 3'(T) T = otX(T) T the specific heat of Pd can be described by a nearly constant value of 0D, up to room temperature.
The curves shown in fig. 9 were obtained by subtracting Debye terms with various fixed values of 0 D to the experimental data. They are strongly reminiscent of the curve computed (before the fact) by Grimvall [22] for the temperature variation of 7 due to the electron-phonon interaction. A low temperature upturn of 3' (7')below T ~ 100 K and a maximum around T = 20 K is thus very likely for Pd. For TiBe2, sticking to the analogy with Pd, one may expect to see a similar upturn of C/Tvs. T 2 and possibly a maximum. The fact that the maxima are located at T ~- Tm/4 in both materials might be fortuitous and would require further analysis. The behavior of 0 D in TiBe2 and TiBe2_xCux should be studied by higher temperature specific heat measurements.
Although the entropy associated with the specific heat anomaly appears to be too large to be accounted for by the presence of an impurity phase this explanation cannot be ruled out a priori. Since TiBe3 forms in the center of the grains there could be a superconducting Ti-rich phase (maybe even/~-Ti) at the grain boundaries. Specific heat measurements in a magnetic field might clarify this point. A more remote possibility could be the presence of Magneli phases TinO2n-1 [23][24][25] with n ~> 6. By analogy with VnO2n-1 [26] these still little studied oxides might show large peaks in the specific heat at low temperature, due to metal-insulator transitions.
The resistivity ratio R (300 K)/R (0) = 46 that we found in TiBe2 indicates a high purity. The low temperature data in fig. 8 show the characteristic T 2 variation already found in ZrZn2 [27] and Ni3Al [28]. The anomaly observed near 1.8 K is very small and its possible connection with a maximum ofT(T) at the same temperature or with the presence of an impurity phase should be investigated.

Conclusion References
A coherent picture of the magnetic properties of TiBe2 has been given. TiBe2 is found to be a strongly exchange-enhanced paramagnet and to behave in every respect like Pd or Ni3Ga. The electron-electron interaction can be made responsible for the maxima found in ×(T) and x(H). The low temperature behavior of the specific heat is tentatively attributed to a temperature dependence of the electronic specific heat coefficient 7 due to the electron-phonon interaction.
The transition between paramagnetism and ferromagnetism in TiBel_xCux takes place at the critical concentration Xcr = 0.155 + 0.005. Around Xcr, the Stoner-Edwards-Wohlfarth model of itinerant-electron magnetism is well followed. No transition to antiferromagnetism could be detected below Xcr by our measurements.
The analogies between TiBe2 and Pd suggest further experiments. The possible inducement of giant moments in TiBe2 by Fe or Ni impurities should be investigated. In spite of the fact that the occurrence of superconductivity is not favored by the high susceptibility of the material, the effect of H impurities or irradiation should also be checked. A study of the specific heat below 1.5 K and above 25 K would be of interest and might clarify the origin of the anomaly at2 K.

Note added in proof
Recent measurements of the specific heat of TiBe2 by L.D. Woolf (private communication) show no peak around 1.8 K. Instead, C/T decreases regularly, by about 8%, when/a increases from 0.5 to 5.0 K 2. In the discussion of the susceptibility results we should have mentioned the theoretical result of Beal-Monod et al. who find no T n In T terms (see Phys. Rev. B21 (1980) 5400). According to Doniach and Engelsberg (Phys. Rev. Lett. 17 (1966) 750) T a In T terms should be present in the specific heat of strongly enhanced paramagnets.