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Dielectric phase transitions in the superconducting C15 compounds ZrV2 and HfV2
157
Journal of the Less~ommo~ Metals, 62 (1978) 157 - 166 0 Ekevier Sequoia S.A., Lausanne -- Printed in the Nethedands
DIELECTRIC PHASE TRANSITIONS COMPOUNDS ZrVs AND HfVz*
IN THE SUPERCONDUCTING
V. M. PAN, I. E. BULAKH,
and A. D. SHEVCHENKO
A. L. KASATKIN
Institute of Metul Physics, Acad. Sci. Ukr. SSR, str. Vemadsky (Received
Cl5
36, Kiev-142 (U.S.S.R.)
April 29, 1978)
Summary The temperature dependence of the electrical resistivity ~(2’) and the magnetic susceptibility x(T) of ZrVs and HfVz compounds with the Cl5 structure have been studied in the temperature range 4.2 ” 300 K. At 120 K for ZrV, and 150 K for HfVs, p(T) and x(T) display some anomalies which are very sensitive to the addition of impurities and to any departure from the stoichiometric composition. The effect of magnetic and electrical fields on the temperature dependence p(T) has also been investigated. A model is considered in which the phase transition, accompanied by a partial dielectrization of carriers, results from the pairing of electrons and holes belonging to different bands but with similarly shaped Fermi surfaces of electrons and holes. The experimental results are interpreted in terms of this model.
1. Introduction Most of the high temperature superconducting compounds with a transition metal base are known to reveal lattice instability at temperatures ?‘, which are above the temperature Tk of a superconducting transition [l]. It has often been suggested that the closeness of T, to Tk is responsible for the high values of Tk, since the lattice instability causes softening of the phonon spectrum and hence the growth of the effective constant h in the electron-phonon interaction, this being the cause of the rise in Tk. There exists an opposite point of view concerning the effect of the structural transformation on Tk, according to which the high temperature phase becomes unstable when the temperature is decreased to T,, because of the large value of h. A new phase below T, has a smaller value of h and is therefore stable. The Tk value in the absence of structural transformations should have been higher than the observed value. However, investigations [2] of compounds with the Al5 structure have shown that the lattice instability seems to have no practical effect on the superconductivity. It has been suggested [ 21, on the basis of the data for the supe~onducting properties *Dedicated
to Professor
B. T. Matthias
in celebration
of his 60th birthday.
158
and dilatometry of VsSi, that the structural transition does not depend on the superconducting transition. It should be said that most inves~~ations deal with the change in phonon characteristics during structural transformations and with the effect of these changes on Tk, The value of Tk depends essentially on the density of electron states N(E) near the Fermi level; this density is also able to change during structural transformations. Indeed, according to current ideas the structuraI transfo~ation can be accomp~ied by the formation of a dielectric gap, owing to electron-hole correlation in a definite part of the Fermi surface similar to that which occurs during the Peierls transition in quasi-one-dimensional metals [3]. Such dielectrization has been observed experimentally in measurements of the temperature dependence of the electrical resistance and the magnetic susceptibility of V-Ru alloys [4]. These authors also found a drop in Tk and a suppression of the superconductivity owing to the fact that a fraction of carriers undergoes dielectrization. At the same time, together with such a suppression, there exists the problem of a possible increase in the superconductivity and in the value of Tk with an increase in the density of states on the dielectric gap edges [3]. This makes it interesting to study physical properties which give information about the change in electron properties during structural transitions. The present work concerns the temperature dependence of the electrical resistivity and the magnetic susceptibility of ZrVz and HfV, Laves phase compounds, which have been chosen for study for the following reasons. They belong to a series of superconducting compounds possessing high critical parameters of supe~onducti~ty, Tk is 8.7 f 0.2 K for ZrVz and 8.8 + 0.2 K for HfVz [ 5,6]. The upper magnetic field Hc, is 353 kOe for HfVz and 400 kOe for Zr,5Hf0.5Vz [7]. Since the Laves phase alloys are 2 - 5 times softer than the Al5 compounds, these superconductors may be of practical use. They are of interest because of structural transformations which occur at ~rnpe~tu~s well above Tk 18, 91 and because of the anomalous dependence of their physical properties [lo - 131.
2. Experimental ZrVz and HfVz alloys were prepared by melting in an arc. furnace with a water-cooled copper bottom, in an atmosphere of purified argon; they were then cast into cylindrical crucibles. The raw materials for the alloy preparation were the iodides of hafnium and zirconium, 99.9% pure vanadium and vanadium purified in a vacuum of lo-* Ton by re-melting in the electron beam. The alloys were subjected to homogenization annealing in a vacuum of 10s5 Torr, at 1100 “C for 200 h in the case of ZrVs and at 1300 “C!for 50 h in the case of HfVa. At 700 “C all the alloys were annealed for 500 h in a quartz capsule filled with helium. After thermal treatments the microstructural and X-ray study of the samples was carried out. X-ray
159
patterns from powder samples were obtained using Cr radiation. Microstructural analysis has shown that the alloys consist of a single phase while, according to X-ray data, all samples have a crystal structure of the MgCu, type (C15) with the lattice parameter a = 7.430 A for ZrVz and a = 7.380 + 0.001 A for HfVs. The electrical resistance p was measured by a d.c. fourprobe method. An R348 potentiometer with low resistance and a sensitivity of 2 X lo-’ V was the main measurement device. Samples for the study of p (2')were cut by the electric spark method and had a parallelopiped shape 10 - 15 mm long, 0.4 - 0.6 mm thick and 0.5 - 1.0 mm wide. The density of the measurement current was 4 A cm- 2. Uncertainty in the measured electrical resistance did not exceed 0.2%, while for the specific electrical resistance it was 1%. The magnetic susceptibility x(T) was measured by the Faraday relative method on an electronic microbalance with automatic compensation. The error in the measured value of x did not exceed 1%. The resistive method was used to measure the temperature of the superconducting transitions. It was found that for the cast state the value of Tk was 8.8 + 0.1 K for ZrV, and 8.9 + 0.1 K for HfV2. Annealing resulted in an increase of 0.1 K in the superconducting transition temperature.
3. Results and discussion Figure 1 shows p(T) for ZrV2 samples subjected to various thermal treatments. At T = 120 K there is an anomalous increase in the electrical resistance of the annealed samples. Let anomalies in p(T) be described as (where pmax,pmin are the p( T) values at the AP = (P,,, -pmin/p3,,,,-pmh)
Fig. 1. Relative resistance us. temperature for ZrV 2: (a) the cast sample (a); (b) the sample annealed at 1100 “C (X );(c) the sample subjected to additional low temperature annealing at 700 “C (A); (d) the sample prepared from vanadium purified by vacuum melting and annealed at 1100 “C and 700 “C!(0).
160
maximum and minimum points, respectively; psoo = p (300 K)). Then the Ap value will be 35% for samples subjected to high temperature homogenization annealing. No peculiarities in p(T) near 120 K were observed for ZrVz as-cast samples (Fig. l(a)). This is due to the fact that the ZrVz as-cast alloy had a complex phase composition; less than 10% of it consisted of ZrVz phases with the Cl5 structure, as shown by the data from the microstructural investigation. Microstructural analysis has shown that only the samples annealed for 200 h are single-phase while, for example, samples annealed for 24 h contain less than 25% of the Cl5 phase. Low-temperature annealing at 700 “C!ensured structure perfection but resulted in a more pronounced anomaly in p (T) (Fig. l(c)) with a simultaneous decrease in the temperature interval for the anomalous change in electrical resistance. The Ap value for alloys after additional low temperature annealing was 30% in the case of ZrVz. This agrees with the data from an X-ray structural study of ZrVz [8] , in which the low temperature orthorhombic phase was fixed only for ZrVz samples subjected to low temperature annealing, while as-cast samples and those annealed at high temperatures displayed only slight line broadening. Similar results as to the effect of thermal treatment on p(T) were obtained for HfVz samples in which the anomaly in p(T) was observed at T < 150 K. The Ap value for samples which had undergone high temperature annealing was 45%; after an additional low temperature annealing it had decreased to 35%. The results for the variation in p(T) agree with our measurements of the temperature dependence of magnetic susceptibility in the same samples (Figs. 2(a) and 2(b)). At temperatures below 120 K for ZrVz and below 150 K for HfVz the temperature run of x(T) is changed. The change in character of p(T) and x(T) is very sensitive to the introduction of impurities. Thus the use of vanadium purified in uacuo gave a more than twofold increase in Ap for ZrVz (Ap = 74%) under the same conditions of thermal treatment (high and low temperature annealing) (Fig. l(d)). It follows from these investigations that the size of the anomaly in p(T) and x(T) is also sensitive to the composition of ZrVz and HfVz alloys. p(T) for alloys of various compositions near the stoichiometric composition is shown in Fig. 3. As may be seen, Ap decreases gradually with increasing Zr content, while the anomaly in p(T) practically vanishes at a V content increase of 1 at.%. A similar dependence of Ap on alloy composition near the stoichiometric composition is also observed for HfVz. Thus it may be concluded that the anomalous increase in electrical resistance with decreasing temperature is observed in a narrow concentration range around the stoichiometric composition. It should be said that any change in the alloy composition is accompanied by a change in the temperature at which anomalies in p(T) and x(T) are observed (Fig. 3). When the alloy composition is changed, the absolute value of the magnetic susceptibility is also varied (Fig. 2(b)). This suggests that, for the above compounds, the model of the energy electronic spectrum which assumes that the Fermi level lies near the slope of the energetically wide maximum in the density of electron states N(E) is applicable, because x N N(E).
161
Fig. 2. Magnetic susceptibility us. temperature for (a) ZrV2; (b) HfVz (O), HfVs + 2 at,%V (*). Fig. 3. Relative resistance us. temperature for ZrVz near the stoichiometric 66.7at.%V(~),65.7at.%V(--),64.?at.%V(~),61.7at.%V(Q),68.7at.%V(~) 71.7 at.% V (0).
composition: and
The effect of magnetic and electrical fields on the temperature dependence of the electrical resistance has been studied. Figure 4 shows the results of p (T, E) measurements for various current densities (E is the electrical field in the sample). HfVz samples subjected to a full thermal treatment were used for these measurements. An increase in the current density of the sample leads to the suppression of the anomaly and to a conductivity increase at T < T, = 150 K. When the magnetoresistance in the field H= 40 kOe is measured, the portion with a negative tempe~ture coefficient in the curve p(T) = p fT,E + 0) is observed to be shifted to higher temperatures; accordingly, the T, value is increased by 5 K. (At temperatures above T, and below T’ = 50 K, i.e. wt~-n the temperature dependence of the electrical resistance is of a metallic character, the resistance is not observed to vary with the magnetic field.) It may be suggested from the analysis of the above data that the compounds under study undergo a phase transition followed by a change in crystal structure and partial dielectrization of the electron spectrum. In this case, for a certain fraction of carriers whose energy spectrum satisfies the condition e(P) = ---E@ + Q) where Q is the constant vector and the energy is measured from the Fermi level, a gap appears in the energy spectrum similar to the gap in the semiconductor spectrum. The other carriers display the continuous spectrum. Compounds with the Cl 5 structure [ 141 as well as
162
Fig. 4. Electrical resistivity vs. temperature represents the theoretical results.
at various values of the field B; the solid line
those with the Al5 structure [15,16] can undergo this partial dielectrization of the electron spectrum if the Fermi level is close to the X point on the Brillouin zone boundary where the electron spectrum is doubly degenerate and there are two branches of excitation in its vicinity (of electron and hole types with the linear law dispersion e(P) = *up, [ 151). The carriers that undergo dielectrization are carriers in the neighbourhood of the point X. In view of the above, it may be assumed that the behaviour of p(T) and x(T) in the region where the anomaly is observed depends on the freezing out of some of the free carriers. The expressions for the magnetic susceptibility x(T) and the electron conduction o(T) may each be given as the sum of two contributions: x(T) = X*(T) + XdfT) u(T) = o,(T) + ad(T) (1) where xn and a, are the contributions from those carriers which have not undergone dielect~zation ; Xd and (Id are the cont~butions from dielectrized carriers whose energy spectrum reveals a gap when T is less than T, . Jn the high temperature range (T > Z’,) the dependences Xd (T) and od (2’) coincide qualitatively with xn (2’) and u, (2’); this corresponds to the case for a normal metal. But at low temperatures (T < T,), Xd(T) and @d(T) are of semiconductor character and decrease sharply with temperature owing to the freezing-out of free carriers, while ~~(2’) and u,(T) remain of metal character. Hence it is apparent that eqn. (1) should describe qu~i~tively the experimentally observed behaviour of the magnetic susceptibility and the electrical resistivity p(T) = u-r(T). To illustrate this, let us consider the semimetal model: a phase transition followed by a partial dielectrization of carriers arises owing to the pairing of electrons and holes from different zones, the Fermi surfaces of the electrons and holes having similar shapes. The presence of a partially filled zone of non-dielectrized carriers is also assumed. This model describes qu~itatively the properties of systems with much more complex spectra [3]. For the sake of simplicity we shah assume that the concentrations and
163
effective masses of electrons and holes involved in pairing are the same and that the scattering mechanism is also the same for all the carriers. Then, if T is greater than T, o(T) = a,(T)
+ ud(T) = (a, + ad)f(T)
= oo(T)
(2)
where the coefficients CX,and (Ydare, roughly speaking, proportional to the number of normal and dielectrized carriers; f(T) is determined by the scattering of carriers on phonons. If T is less than or approximately equal to T,, electrons and holes are paired and transferred into the state of exciton dielectric. Their contribution to the conduction is governed by the effective concentration neff which decreases exponentially at low ~m~ratures 11’71 :
hff
=
2nd
fdq. KS-$&!!( [T X exp
0
11 -1
T-Ii2 + A2(T)}1/2
+ 1
(3)
Here A(T) is the dielectric gap. Its temperature dependence in the case under study is the same as in the Bardeen-Cooper-Schrieffer (BCS) theory. It should be mentioned that for a more complex band structure the dependence A(T) may differ from the gap dependence given by the BCS theory. When T is less than T, Gff -= 2nd
4?ff
-_
1-y!!; !
(4)
i
When T is less than 0.5 T, ~~ff=!!~[Kl\~/ where K is the Macdonald T< T,
-expi_--~~~{~/]
function;
rexp{-h$!?I
(5)
A(x) = ex Jw K,(x) dx. Hence, when x
o(T) = (a, + ~~~~f~~T))f(T) = aoo(T) + b&r(T)
(6)
where Q = (Y~/(cY~+ Q) and p = Qdf(Tm). For the alloys under study, eqn. (6) neglects the temperature dependence of the scattering time for dielectrized carriers because the residual resistance is large compared with the change of p with temperature and with the strong temperature dependence of neff . For the carriers that are not dielectrized it is assumed that the scattering law remains unchanged during the phase transition. Thus, if the oo(T) value is known from experiments, an attempt can be made to describe the behaviour of electrical conduction in the phase transition region T < T, using eqn. (6). In this case the coefficients (Yand p will be adjustable parameters. From the behaviour of the electrical resistance curve p e( T),,,, at high temperatures, as well as from the p;(T) value for the sample alloyed with impurities (whose electrical resistance curve displays no anomalies) it may be said that in our case the a&T) dependence is well described 1183 by
164
oo(T) = ---a +
1 (6a)
b(T/e)3J,(e/T)
Figure 4 gives the electrical resistance dependence for HfVs defined from eqns. (6) and (6a) at (Y= 0.900, P = 0.286 (a cm))l, a = 1.084 X 10m4 (a cm), b = 0.279 X 10e4 (a cm). In a similar way the freezing-out of some of the carriers leads to a strong temperature dependence of the spin paramagnetic component in the magnetic susceptibility. When T is greater than T, x(T) = x0 + 2p;{N,(E)
+ N,(E)}
+ BT2 = A + BT2
(7)
Here x0 is the Van Vleck orbital paramagnetic susceptibility which is independent of temperature; N,, d (E) is the density of states at the Fermi level for normal and dielectrized carriers, respectively; and the factor B is determined by the motion of a chemical potential level and by the particular density of states [19]. When T is less than T, in the phase of an exciton dielectric we have x(T) = A -A’
+A’&T)
+B’T2
(8)
where X,(T) =
t
Xd(T)_ = r 2piNd(E)
d5
-A2(T)}1’2
_{t2
Asymptotics in eqn. (8) yield the following When A(T)/T is much less than 1 x:(T)
1 _I.4Tch2(g/2T) equations.
= {A(T)/Tj2
and when A (T)/T is much greater than 1
Hence the x(T) dependence may be given as follows. When T is greater than T, x(T) = A + BT2 and when T is less than T, x(T) = A -A’
+ A’&(T)
+ B’T2
(9)
where A = Xo+%&%(~)
+Nd@)}
A’ = 2/@,(E) Figure 5 shows x(T) for HfV2 found from g-1; B = -0.04 x lo-10 cm3 g-1 K-2 ; B’ = 0.571 X 10m6 cm3 g-l. It may be seen that and x(T) are described quite well by eqns.
eqn. (9) at A = 4.169 X 10e6 cm3 0.23 X 10-10 cm3 g-1 K-2; A’ = the anomalies observed in p(T) (6) and (9). In the framework of
165
0
Fig. 5. Magnetic susceptibility us. temperature; the solid line represents the theoretical results. Fig. 6. The dependence of A@(E)on l/E at various temperatures; inset, the temperature dependence of the activation field Eo( T).
the proposed model an interpretation is given of the strong concentration dependence of the observed effect. The alloyed impurity affects the electron-hole pairing (because of the opposite charges of the electron and the hole in the pair) and suppresses it. The concent~tion eq~lib~um between electrons and holes is disturbed, and so the condition e(p) = --e(P + Q) is not fulfilled. The increase in the value of T, in a magnetic field can also be explained because a magnetic field leads to isotropization of the Fermi surface and enhancement of the constant h. The influence of the magnetic field on the metal-dielectric phase transition has been investigated expe~ment~ly [ 201. However, in order to draw conclusions about the appearance of a partial dielectrization during phase transitions in Cl5 compounds, further investigations are needed (e.g. tunnel or optical) which should be able to register the formation of a dielectric gap in the electron spectrum. The most interesting result is that the conduction in the temperature range 2’ < ?h, is non-ohmic in character. Figure 4 gives the results for electrical conducti~ty measured at various current densities, ie. at various values of the electric field E in the sample, The observed conductivity increase in the field is not related to thermal effects because such an increase was found throughout the temperature range T < Tm but was absent when T was greater than T,. The T, value was ~de~ndent of E. Figure 6 shows in a semilog plot Acr(Z’,E) = u( T, E) - o(T, E + 0) as a function of the field magnitude at various temperatures. It may be seen that the value of Ao(T, E) is well described by Ao(Z’,E) = a(T) exp(--Eo(T)/E}
(10)
The same figure gives the temperature dependence of E,. Similar non-linear conduction has recently been observed in a qu~i-one-dimensions metal TTF-TCNQ [21] and in NbSes [22] at temperatures below that of the dielectric phase transition. (In NbSes the dielectrization is also incomplete and affects only part of the Fermi surface.) The reason for the anomalous character of the conductivity is not clear. However, it is possible that this
166
conductivity increase in the field is an intrinsic property of systems where dielectrization of the electron spectrum occurs owing to the dynamics of the dielectric phase transition of the charge density wave (CDW) [Zl - 231. It is interesting to note that it has been suggested 1231 that this mechanism contributes to the conductivity on account of the CDW which is described by eqn. (10).
References 1 Yu. A. Izyumov and E, E. Kurmayev, Usp. Fiz, Nauk, 118 (1976) 53. 2 V. M. Pan, V. G. Prokhorov, A. D. Shevchenko, G. A. Takzei and M. Yurish, Fiz. Nizk. Temp., 4 (1978) 262. 3 Yu. V. Kopayev, Transactions of the Physical Inst. Acad. Sci. USSR, 86 (1975) 3. Yu. Y. Kopaev, in V. L, Ginzburg and D. A. Kirjnitz (eds.), The problem of high temperature superconductivity, Nauka, Moscow, 1977. 4 C. W. Chu, E. Bucher, A. S. Cooper and I. P. Maita, Phys. Rev. B, 13 (1971) 320. 5 B. T. Mattbias, V. B. Compton and E. Corenzwit, J. Phys. Chem. Solids, 19 (1961) 130. 6 V. Sadogapan, E. Pollard and H. G. Gates, Solid State Commun., 3 (1965) 97. 7 K. Inoue and K. Tachikawa, Appl. Phys. L&t., 18 (1971) 235. 8 D. E. Moncton, Solid State Commun., 13 (1973) 1779. 9 A. C. Lawson and W. II. Zachariasen, Phys. I&t. A, 38 (1972) 1. 10 A. C. Lawson, Phys. I.&t. A, 36 (1971) 8. 11 Y. K. Kimura, J. Phys. Sot. Jpn, 36 (1974) 306. 12 J. E. Doherty and D. E. Gibbons, Phys. Status Solidi B, 44 (1971) K5. 13 T. Takaschima and H. Hayasi, Jpn J. Appl. Phys., 12 (1973) 1659. 14 Yu. A. Izyumov, V. E. Naish and V. N. Syromyatnikov, Fiz. Met. Metalloved., 39 (1975) 455. 15 L. P. Gorkov and 0. N. Dorokhov, J. Low Temp. Phys., 22 (1976) 1. 16 G. Bilbro and W. L. McMillan, Phys. Rev. B, 14 (1976) 1877. 17 D. Jerome, T. M. Rice and W. Kohn, Phys. Rev., 158 (1967) 462. 18 F, Blatt, Physics of Electronic Conduction in Solids, McGraw-Hill, New York, 1968. 19 E. V. GaIoshina, Usp. Fiz. Nauk, 113 (1974) 105. 20 N. B. Brandt and S. M. Chudinov, J. Low Temp. Phys., 8 (1972) 339. 21 M. J. Cohen, P. R. Newman and A. J. Meeger, Phys. Rev. Lett., 37 (1976) 1500. 22 N. P. Ong and P. Monceau, Phys. Rev. B, 16 (1977) 3443. 23 K. Maki, Phys. Rev. L&t., 39 (1977) 46.
Journal of the Less~ommo~ Metals, 62 (1978) 157 - 166 0 Ekevier Sequoia S.A., Lausanne -- Printed in the Nethedands
DIELECTRIC PHASE TRANSITIONS COMPOUNDS ZrVs AND HfVz*
IN THE SUPERCONDUCTING
V. M. PAN, I. E. BULAKH,
and A. D. SHEVCHENKO
A. L. KASATKIN
Institute of Metul Physics, Acad. Sci. Ukr. SSR, str. Vemadsky (Received
Cl5
36, Kiev-142 (U.S.S.R.)
April 29, 1978)
Summary The temperature dependence of the electrical resistivity ~(2’) and the magnetic susceptibility x(T) of ZrVs and HfVz compounds with the Cl5 structure have been studied in the temperature range 4.2 ” 300 K. At 120 K for ZrV, and 150 K for HfVs, p(T) and x(T) display some anomalies which are very sensitive to the addition of impurities and to any departure from the stoichiometric composition. The effect of magnetic and electrical fields on the temperature dependence p(T) has also been investigated. A model is considered in which the phase transition, accompanied by a partial dielectrization of carriers, results from the pairing of electrons and holes belonging to different bands but with similarly shaped Fermi surfaces of electrons and holes. The experimental results are interpreted in terms of this model.
1. Introduction Most of the high temperature superconducting compounds with a transition metal base are known to reveal lattice instability at temperatures ?‘, which are above the temperature Tk of a superconducting transition [l]. It has often been suggested that the closeness of T, to Tk is responsible for the high values of Tk, since the lattice instability causes softening of the phonon spectrum and hence the growth of the effective constant h in the electron-phonon interaction, this being the cause of the rise in Tk. There exists an opposite point of view concerning the effect of the structural transformation on Tk, according to which the high temperature phase becomes unstable when the temperature is decreased to T,, because of the large value of h. A new phase below T, has a smaller value of h and is therefore stable. The Tk value in the absence of structural transformations should have been higher than the observed value. However, investigations [2] of compounds with the Al5 structure have shown that the lattice instability seems to have no practical effect on the superconductivity. It has been suggested [ 21, on the basis of the data for the supe~onducting properties *Dedicated
to Professor
B. T. Matthias
in celebration
of his 60th birthday.
158
and dilatometry of VsSi, that the structural transition does not depend on the superconducting transition. It should be said that most inves~~ations deal with the change in phonon characteristics during structural transformations and with the effect of these changes on Tk, The value of Tk depends essentially on the density of electron states N(E) near the Fermi level; this density is also able to change during structural transformations. Indeed, according to current ideas the structuraI transfo~ation can be accomp~ied by the formation of a dielectric gap, owing to electron-hole correlation in a definite part of the Fermi surface similar to that which occurs during the Peierls transition in quasi-one-dimensional metals [3]. Such dielectrization has been observed experimentally in measurements of the temperature dependence of the electrical resistance and the magnetic susceptibility of V-Ru alloys [4]. These authors also found a drop in Tk and a suppression of the superconductivity owing to the fact that a fraction of carriers undergoes dielectrization. At the same time, together with such a suppression, there exists the problem of a possible increase in the superconductivity and in the value of Tk with an increase in the density of states on the dielectric gap edges [3]. This makes it interesting to study physical properties which give information about the change in electron properties during structural transitions. The present work concerns the temperature dependence of the electrical resistivity and the magnetic susceptibility of ZrVz and HfV, Laves phase compounds, which have been chosen for study for the following reasons. They belong to a series of superconducting compounds possessing high critical parameters of supe~onducti~ty, Tk is 8.7 f 0.2 K for ZrVz and 8.8 + 0.2 K for HfVz [ 5,6]. The upper magnetic field Hc, is 353 kOe for HfVz and 400 kOe for Zr,5Hf0.5Vz [7]. Since the Laves phase alloys are 2 - 5 times softer than the Al5 compounds, these superconductors may be of practical use. They are of interest because of structural transformations which occur at ~rnpe~tu~s well above Tk 18, 91 and because of the anomalous dependence of their physical properties [lo - 131.
2. Experimental ZrVz and HfVz alloys were prepared by melting in an arc. furnace with a water-cooled copper bottom, in an atmosphere of purified argon; they were then cast into cylindrical crucibles. The raw materials for the alloy preparation were the iodides of hafnium and zirconium, 99.9% pure vanadium and vanadium purified in a vacuum of lo-* Ton by re-melting in the electron beam. The alloys were subjected to homogenization annealing in a vacuum of 10s5 Torr, at 1100 “C for 200 h in the case of ZrVs and at 1300 “C!for 50 h in the case of HfVa. At 700 “C all the alloys were annealed for 500 h in a quartz capsule filled with helium. After thermal treatments the microstructural and X-ray study of the samples was carried out. X-ray
159
patterns from powder samples were obtained using Cr radiation. Microstructural analysis has shown that the alloys consist of a single phase while, according to X-ray data, all samples have a crystal structure of the MgCu, type (C15) with the lattice parameter a = 7.430 A for ZrVz and a = 7.380 + 0.001 A for HfVs. The electrical resistance p was measured by a d.c. fourprobe method. An R348 potentiometer with low resistance and a sensitivity of 2 X lo-’ V was the main measurement device. Samples for the study of p (2')were cut by the electric spark method and had a parallelopiped shape 10 - 15 mm long, 0.4 - 0.6 mm thick and 0.5 - 1.0 mm wide. The density of the measurement current was 4 A cm- 2. Uncertainty in the measured electrical resistance did not exceed 0.2%, while for the specific electrical resistance it was 1%. The magnetic susceptibility x(T) was measured by the Faraday relative method on an electronic microbalance with automatic compensation. The error in the measured value of x did not exceed 1%. The resistive method was used to measure the temperature of the superconducting transitions. It was found that for the cast state the value of Tk was 8.8 + 0.1 K for ZrV, and 8.9 + 0.1 K for HfV2. Annealing resulted in an increase of 0.1 K in the superconducting transition temperature.
3. Results and discussion Figure 1 shows p(T) for ZrV2 samples subjected to various thermal treatments. At T = 120 K there is an anomalous increase in the electrical resistance of the annealed samples. Let anomalies in p(T) be described as (where pmax,pmin are the p( T) values at the AP = (P,,, -pmin/p3,,,,-pmh)
Fig. 1. Relative resistance us. temperature for ZrV 2: (a) the cast sample (a); (b) the sample annealed at 1100 “C (X );(c) the sample subjected to additional low temperature annealing at 700 “C (A); (d) the sample prepared from vanadium purified by vacuum melting and annealed at 1100 “C and 700 “C!(0).
160
maximum and minimum points, respectively; psoo = p (300 K)). Then the Ap value will be 35% for samples subjected to high temperature homogenization annealing. No peculiarities in p(T) near 120 K were observed for ZrVz as-cast samples (Fig. l(a)). This is due to the fact that the ZrVz as-cast alloy had a complex phase composition; less than 10% of it consisted of ZrVz phases with the Cl5 structure, as shown by the data from the microstructural investigation. Microstructural analysis has shown that only the samples annealed for 200 h are single-phase while, for example, samples annealed for 24 h contain less than 25% of the Cl5 phase. Low-temperature annealing at 700 “C!ensured structure perfection but resulted in a more pronounced anomaly in p (T) (Fig. l(c)) with a simultaneous decrease in the temperature interval for the anomalous change in electrical resistance. The Ap value for alloys after additional low temperature annealing was 30% in the case of ZrVz. This agrees with the data from an X-ray structural study of ZrVz [8] , in which the low temperature orthorhombic phase was fixed only for ZrVz samples subjected to low temperature annealing, while as-cast samples and those annealed at high temperatures displayed only slight line broadening. Similar results as to the effect of thermal treatment on p(T) were obtained for HfVz samples in which the anomaly in p(T) was observed at T < 150 K. The Ap value for samples which had undergone high temperature annealing was 45%; after an additional low temperature annealing it had decreased to 35%. The results for the variation in p(T) agree with our measurements of the temperature dependence of magnetic susceptibility in the same samples (Figs. 2(a) and 2(b)). At temperatures below 120 K for ZrVz and below 150 K for HfVz the temperature run of x(T) is changed. The change in character of p(T) and x(T) is very sensitive to the introduction of impurities. Thus the use of vanadium purified in uacuo gave a more than twofold increase in Ap for ZrVz (Ap = 74%) under the same conditions of thermal treatment (high and low temperature annealing) (Fig. l(d)). It follows from these investigations that the size of the anomaly in p(T) and x(T) is also sensitive to the composition of ZrVz and HfVz alloys. p(T) for alloys of various compositions near the stoichiometric composition is shown in Fig. 3. As may be seen, Ap decreases gradually with increasing Zr content, while the anomaly in p(T) practically vanishes at a V content increase of 1 at.%. A similar dependence of Ap on alloy composition near the stoichiometric composition is also observed for HfVz. Thus it may be concluded that the anomalous increase in electrical resistance with decreasing temperature is observed in a narrow concentration range around the stoichiometric composition. It should be said that any change in the alloy composition is accompanied by a change in the temperature at which anomalies in p(T) and x(T) are observed (Fig. 3). When the alloy composition is changed, the absolute value of the magnetic susceptibility is also varied (Fig. 2(b)). This suggests that, for the above compounds, the model of the energy electronic spectrum which assumes that the Fermi level lies near the slope of the energetically wide maximum in the density of electron states N(E) is applicable, because x N N(E).
161
Fig. 2. Magnetic susceptibility us. temperature for (a) ZrV2; (b) HfVz (O), HfVs + 2 at,%V (*). Fig. 3. Relative resistance us. temperature for ZrVz near the stoichiometric 66.7at.%V(~),65.7at.%V(--),64.?at.%V(~),61.7at.%V(Q),68.7at.%V(~) 71.7 at.% V (0).
composition: and
The effect of magnetic and electrical fields on the temperature dependence of the electrical resistance has been studied. Figure 4 shows the results of p (T, E) measurements for various current densities (E is the electrical field in the sample). HfVz samples subjected to a full thermal treatment were used for these measurements. An increase in the current density of the sample leads to the suppression of the anomaly and to a conductivity increase at T < T, = 150 K. When the magnetoresistance in the field H= 40 kOe is measured, the portion with a negative tempe~ture coefficient in the curve p(T) = p fT,E + 0) is observed to be shifted to higher temperatures; accordingly, the T, value is increased by 5 K. (At temperatures above T, and below T’ = 50 K, i.e. wt~-n the temperature dependence of the electrical resistance is of a metallic character, the resistance is not observed to vary with the magnetic field.) It may be suggested from the analysis of the above data that the compounds under study undergo a phase transition followed by a change in crystal structure and partial dielectrization of the electron spectrum. In this case, for a certain fraction of carriers whose energy spectrum satisfies the condition e(P) = ---E@ + Q) where Q is the constant vector and the energy is measured from the Fermi level, a gap appears in the energy spectrum similar to the gap in the semiconductor spectrum. The other carriers display the continuous spectrum. Compounds with the Cl 5 structure [ 141 as well as
162
Fig. 4. Electrical resistivity vs. temperature represents the theoretical results.
at various values of the field B; the solid line
those with the Al5 structure [15,16] can undergo this partial dielectrization of the electron spectrum if the Fermi level is close to the X point on the Brillouin zone boundary where the electron spectrum is doubly degenerate and there are two branches of excitation in its vicinity (of electron and hole types with the linear law dispersion e(P) = *up, [ 151). The carriers that undergo dielectrization are carriers in the neighbourhood of the point X. In view of the above, it may be assumed that the behaviour of p(T) and x(T) in the region where the anomaly is observed depends on the freezing out of some of the free carriers. The expressions for the magnetic susceptibility x(T) and the electron conduction o(T) may each be given as the sum of two contributions: x(T) = X*(T) + XdfT) u(T) = o,(T) + ad(T) (1) where xn and a, are the contributions from those carriers which have not undergone dielect~zation ; Xd and (Id are the cont~butions from dielectrized carriers whose energy spectrum reveals a gap when T is less than T, . Jn the high temperature range (T > Z’,) the dependences Xd (T) and od (2’) coincide qualitatively with xn (2’) and u, (2’); this corresponds to the case for a normal metal. But at low temperatures (T < T,), Xd(T) and @d(T) are of semiconductor character and decrease sharply with temperature owing to the freezing-out of free carriers, while ~~(2’) and u,(T) remain of metal character. Hence it is apparent that eqn. (1) should describe qu~i~tively the experimentally observed behaviour of the magnetic susceptibility and the electrical resistivity p(T) = u-r(T). To illustrate this, let us consider the semimetal model: a phase transition followed by a partial dielectrization of carriers arises owing to the pairing of electrons and holes from different zones, the Fermi surfaces of the electrons and holes having similar shapes. The presence of a partially filled zone of non-dielectrized carriers is also assumed. This model describes qu~itatively the properties of systems with much more complex spectra [3]. For the sake of simplicity we shah assume that the concentrations and
163
effective masses of electrons and holes involved in pairing are the same and that the scattering mechanism is also the same for all the carriers. Then, if T is greater than T, o(T) = a,(T)
+ ud(T) = (a, + ad)f(T)
= oo(T)
(2)
where the coefficients CX,and (Ydare, roughly speaking, proportional to the number of normal and dielectrized carriers; f(T) is determined by the scattering of carriers on phonons. If T is less than or approximately equal to T,, electrons and holes are paired and transferred into the state of exciton dielectric. Their contribution to the conduction is governed by the effective concentration neff which decreases exponentially at low ~m~ratures 11’71 :
hff
=
2nd
fdq. KS-$&!!( [T X exp
0
11 -1
T-Ii2 + A2(T)}1/2
+ 1
(3)
Here A(T) is the dielectric gap. Its temperature dependence in the case under study is the same as in the Bardeen-Cooper-Schrieffer (BCS) theory. It should be mentioned that for a more complex band structure the dependence A(T) may differ from the gap dependence given by the BCS theory. When T is less than T, Gff -= 2nd
4?ff
-_
1-y!!; !
(4)
i
When T is less than 0.5 T, ~~ff=!!~[Kl\~/ where K is the Macdonald T< T,
-expi_--~~~{~/]
function;
rexp{-h$!?I
(5)
A(x) = ex Jw K,(x) dx. Hence, when x
o(T) = (a, + ~~~~f~~T))f(T) = aoo(T) + b&r(T)
(6)
where Q = (Y~/(cY~+ Q) and p = Qdf(Tm). For the alloys under study, eqn. (6) neglects the temperature dependence of the scattering time for dielectrized carriers because the residual resistance is large compared with the change of p with temperature and with the strong temperature dependence of neff . For the carriers that are not dielectrized it is assumed that the scattering law remains unchanged during the phase transition. Thus, if the oo(T) value is known from experiments, an attempt can be made to describe the behaviour of electrical conduction in the phase transition region T < T, using eqn. (6). In this case the coefficients (Yand p will be adjustable parameters. From the behaviour of the electrical resistance curve p e( T),,,, at high temperatures, as well as from the p;(T) value for the sample alloyed with impurities (whose electrical resistance curve displays no anomalies) it may be said that in our case the a&T) dependence is well described 1183 by
164
oo(T) = ---a +
1 (6a)
b(T/e)3J,(e/T)
Figure 4 gives the electrical resistance dependence for HfVs defined from eqns. (6) and (6a) at (Y= 0.900, P = 0.286 (a cm))l, a = 1.084 X 10m4 (a cm), b = 0.279 X 10e4 (a cm). In a similar way the freezing-out of some of the carriers leads to a strong temperature dependence of the spin paramagnetic component in the magnetic susceptibility. When T is greater than T, x(T) = x0 + 2p;{N,(E)
+ N,(E)}
+ BT2 = A + BT2
(7)
Here x0 is the Van Vleck orbital paramagnetic susceptibility which is independent of temperature; N,, d (E) is the density of states at the Fermi level for normal and dielectrized carriers, respectively; and the factor B is determined by the motion of a chemical potential level and by the particular density of states [19]. When T is less than T, in the phase of an exciton dielectric we have x(T) = A -A’
+A’&T)
+B’T2
(8)
where X,(T) =
t
Xd(T)_ = r 2piNd(E)
d5
-A2(T)}1’2
_{t2
Asymptotics in eqn. (8) yield the following When A(T)/T is much less than 1 x:(T)
1 _I.4Tch2(g/2T) equations.
= {A(T)/Tj2
and when A (T)/T is much greater than 1
Hence the x(T) dependence may be given as follows. When T is greater than T, x(T) = A + BT2 and when T is less than T, x(T) = A -A’
+ A’&(T)
+ B’T2
(9)
where A = Xo+%&%(~)
+Nd@)}
A’ = 2/@,(E) Figure 5 shows x(T) for HfV2 found from g-1; B = -0.04 x lo-10 cm3 g-1 K-2 ; B’ = 0.571 X 10m6 cm3 g-l. It may be seen that and x(T) are described quite well by eqns.
eqn. (9) at A = 4.169 X 10e6 cm3 0.23 X 10-10 cm3 g-1 K-2; A’ = the anomalies observed in p(T) (6) and (9). In the framework of
165
0
Fig. 5. Magnetic susceptibility us. temperature; the solid line represents the theoretical results. Fig. 6. The dependence of A@(E)on l/E at various temperatures; inset, the temperature dependence of the activation field Eo( T).
the proposed model an interpretation is given of the strong concentration dependence of the observed effect. The alloyed impurity affects the electron-hole pairing (because of the opposite charges of the electron and the hole in the pair) and suppresses it. The concent~tion eq~lib~um between electrons and holes is disturbed, and so the condition e(p) = --e(P + Q) is not fulfilled. The increase in the value of T, in a magnetic field can also be explained because a magnetic field leads to isotropization of the Fermi surface and enhancement of the constant h. The influence of the magnetic field on the metal-dielectric phase transition has been investigated expe~ment~ly [ 201. However, in order to draw conclusions about the appearance of a partial dielectrization during phase transitions in Cl5 compounds, further investigations are needed (e.g. tunnel or optical) which should be able to register the formation of a dielectric gap in the electron spectrum. The most interesting result is that the conduction in the temperature range 2’ < ?h, is non-ohmic in character. Figure 4 gives the results for electrical conducti~ty measured at various current densities, ie. at various values of the electric field E in the sample, The observed conductivity increase in the field is not related to thermal effects because such an increase was found throughout the temperature range T < Tm but was absent when T was greater than T,. The T, value was ~de~ndent of E. Figure 6 shows in a semilog plot Acr(Z’,E) = u( T, E) - o(T, E + 0) as a function of the field magnitude at various temperatures. It may be seen that the value of Ao(T, E) is well described by Ao(Z’,E) = a(T) exp(--Eo(T)/E}
(10)
The same figure gives the temperature dependence of E,. Similar non-linear conduction has recently been observed in a qu~i-one-dimensions metal TTF-TCNQ [21] and in NbSes [22] at temperatures below that of the dielectric phase transition. (In NbSes the dielectrization is also incomplete and affects only part of the Fermi surface.) The reason for the anomalous character of the conductivity is not clear. However, it is possible that this
166
conductivity increase in the field is an intrinsic property of systems where dielectrization of the electron spectrum occurs owing to the dynamics of the dielectric phase transition of the charge density wave (CDW) [Zl - 231. It is interesting to note that it has been suggested 1231 that this mechanism contributes to the conductivity on account of the CDW which is described by eqn. (10).
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