Magnetic susceptibility and superconductivity of cubic vanadium nitrides

J. Phys.

Chem.

Solids,1975,Vol. 36, pp. 7-15.

Pergamon

Press.

Printed

in Great

Britain

MAGNETIC SUSCEPTIBILITY AND SUPERCONDUCTIVITY OF CUBIC VANADIUM NITRIDES” F. I. AJAMI~ School of Metallurgy and MateriaIs Science, University of Pennsylvania, Philadelphia,PA 19104,U.S.A.

Philadelph~,

and

R. K. MACCRONE Materials En~ine~ringDepa~ment,Rensselaer

Polytechnic institute,Troy,NY

12181, U.S.A.

(Received 11 .&ne 1973) Abstract-The superconducting transition temperature T, and the magnetic susceptibility from 77 to 300°K have been measured on five cubic vanadium nitrides: VN, VN,.,,, VN,.,,, VN, 84and VN, 7J.The materials were carefully prepared to exclude oxygen and ferromagnetic impurities. The value of T, falls from &l”K for VN to 2.3”K for VNo.,s. The mass-susceptibility decreases from + 3.94 x 10-ne.m.u./g for VN to I.88 x lo-” em&g for VN,.,, at 300°K. All samples showed a small positive sfope for the susceptibility temperature curve. The results are discussed in terms of the rigid band model. The main features are a high density of states of d electrons, 2.4 states/atom ’ eV for VN that drops off as the nitrogen content decreases, to 0.8 states/atom . eV. Preliminary considerations indicate that many-body effects could reduce this density of states by as much as a factor of 2. Lack of experimental results on Knight shifts and low-temperature specific heats prevent a more quantitative estimate being made. 1.

IiWROlXJCTION

The vanadium-nitrogen following

three

(1) A b.c.c.

system phases [ I I 21:

phase,

consists

of bonding in these compounds. In this connection the hard matals are unique in that they exhibit high electrical and thermal conductivities, normally associated with metallic bonding, and high melting points and high hardness, associated with covalent and ionic bonding. This led to some controversy about the relative strengths of the metal-metal and metal-non metal bonds, and about the direction of charge transfer and ionicity. In recent years however, the main theoretical effort has been the calculation of band structures of individual compounds, such as TiC[4-71, TiN[4,5] and NbN[S, 91. All calculations agree on the existence of a minimum in the density of states, and on a non-zero value of the density of states at the Fermi level (thus disposing of the “strictly covalent” bond model). There is disagreement however, on the relative positions of the bands, the depth and position of the main minimum and other fine structures. One point of contact between theory and experiment is the density of states at the Fermi level which can be obtained in a more or less straightforward manner from low temperature specific heat and magnetic susceptibility measurements. Furthermore, the

of the

a solution of N in VNO.,, in the composition range

essentially

V, up to a composition (2) A hexagonal

phase

VNW-vN0.49 (3) A cubic phase, with the NaCl structure (Bl), in the range VNt1.72-VN The Bl phase, with a wide range of stoichiometry is found also in the nitrides of Ti, Zr, Hf and Nb, and in the carbides of Ti, Zr, Hf, V, Nb and Ta. This class of compounds collectively called the “refractory hard metals”, has been the subject of considerable experimental efforts during the past decade [3]. Running concurrently with the experimental effort has been the attempt to understand the type *In partial fulfillment of the requirements for a Ph. D. dissertation by F. I. A. in Metallurgy and Materials Science. tPresent address: National Center for Energy Management and Power, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. 7

F. I. AJAMI and R. K. MACCRONE

8

wide range of stoichiometry in the V-N system allows a continuous variation of the Fermi level and the density of states, by varying the metal to non-metal ratio. In this study, we report the results of magnetic susceptibility and superconductivity transition temperature T, for cubic vanadium nitrides of five compositions. From the lack of a strong temperature dependence of the magnetic susceptibility, we conclude that there are no localized spins, and that x is dominated by the Pauli spin paramagnetism and the orbital paramagnetism of the d electrons. There is thus a great similarity between the magnetic properties of vanadium nitrides and those of the “non-magnetic” transition metals. (Recently, Mekata et a/.[101 have found that hexagonal VzN also behaves similarly). We then obtain the density of states at the Fermi surface as a function of composition and find that it is quite high for VN and decreases as the nitrogen content decreases. Finally, we use the observed values of T, and a single value of low-temperature specific heat to determine the influence of manybody effects such as electron-phonon and electron-electron interactions on the density of states. We find indications of a considerable influence, and propose that a detailed study of Knight shifts and low temperature specific heat as a function of composition be carried out in order to obtain quantitative results. 2. EXPERIMENTAL

Sample preparation The vanadium nitrides gas-metal reaction

were

prepared

by the

V + l/2 N~GVN, where 0.72 < x < 1. The equilibrium nitrogen content of the solid, x, depends mainly on the partial pressure of nitrogen in the gas phase, and, to a lesser extent, on temperature[2]. The partial pressure of nitrogen was varied by mixing it with argon gas. The starting material was vanadium foil obtained from Poly Research Co. Two thicknesses of foils were used, 0.0005 and 0.012 cm; the length and width were typically 2.5 and 05 cm respectively. The purity of the vanadium samples was 99.982 per cent. The main contaminants were Fe, Ni, Si and 0, in concentrations of 30,20, 30 and 300 ppm respectively. The gases used were “research grade” argon and nitrogen, supplied by Air products and Chemicals

Co. Oxygen, water, hydrogen and hydro-carbon impurities, which are the most relevant for sample purity, were less than 5 ppm each, according to the manufacturer’s analysis. The argon was further purified by gettering through Ti-Zr chips at 900°C and the nitrogen was gettered over copper heated at 350°C. The vanadium foil was placed in a tungsten boat in a double-walled mullite tube furnace. The reaction temperature was 1200°C for stoichiometric VN, and 1450°C for other compositions. Experience indicated that at these temperatures, the diffusion of nitrogen gas in vanadium is such that the sample can be homogeneously nitrided in about an hour. The rate of flow of the reaction gas to the inner furnace tube was about 300cm’/min. Typical gas compositions were: 100 per cent NZ to yield stoichiometric VN; 56 per cent N2-44 per cent Ar to yield VN0.84; 3 per cent N,97 per cent Ar to yield VNo 75. A slight positive pressure of pure nitrogen was kept in the outer tube of the furnace to prevent the diffusion of atmospheric oxygen, through the hot mullite, into the reaction chamber. Sample characterization The resulting samples had a golden yellow colour, the colour being paler for the nonstoichiometric samples. Colour seems to be a good indicator of purity because some samples, heavily contaminated with oxygen, were dark violet to black. Quantitatively, the samples were characterized with respect to nitrogen content and minor impurities content. The nitrogen content was determined by weighing the vanadium foil before and after a nitriding experiment, and assuming that the weight gain is due to nitrogen only. Also, the lattice constant was measured, and the variation of lattice constant with composition (x in VN,) is plotted in Fig. 1. The present results are in reasonable agreement with those of Brauer and Schnell[ll]. The departure from a smooth curve may be due to different oxygen contents in the different samples. From the figure, it seems that, in this study, the parameter x in VN, cannot be determined to better than 2 per cent. The impurities of interest were iron, cobalt, nickel and oxygen. The first three, of great importance in magnetic measurements, were determined semiquantitatively be spectroscopic analysis. The impurity level (combined), in the best specimens was 50ppm, essentially unchanged from that of the starting metal.

9

Magnetic susceptibility and superconductivity

-

a 4

BRAVER

4.12 -

!i 2 z 0

4.09-

E I= 2 4.06-I

4.03

-

1

direction) is constant. The highest values of H and dH/dZ were 10.5 kGauss and 0.5 kGauss/cm respectively. The magnetic field was calibrated by a Hall probe. Also, a standard substance, HgCO (CNS),, with a mass susceptibility x = 16.44 x 10e6e.m.u./g at 25°C was used to check the calibrations [ 131. In the temperature range 4.2-77”K, the susceptibility was measured by an induction method, using a vibrating sample magnetometer [ 141 operating at 78 cps. The sensitivity of the arrangement was not sufficient to give an absolute value for the susceptibility. It was useful, however, to detect the onset of superconductivity, and to follow the variation of susceptibility with temperature. Superconductivity

0.7

0.6 0.9 COMPOSITION

I.0

Fig. 1. Lattice constant as a function of composition vanadium nitrides (cf Ref. [l 11 for “Brauer”).

in

The oxygen content was determined by vacuum fusion analysis. It was found that the method, while sensitive, gave results that were reliable only to within a factor of two. Within this limitation, the upper limit for the oxygen content was 600ppm. This measn a slight pick-up of oxygen over the original level of metal. Magnetic measurements

The VN, foils were crushed manually using an agate mortar and pestle. The high hardness of VN (- 9 on the Moh scale) precluded the use of any mechanical grinding technique involving contact with any ferromagnetic metals; some samples crushed in a cobalt-bonded tungsten carbide crucible showed measurable contamination with ferromagnetic cobalt. In the temperature range 77-300”K, the static magnetic susceptibility of encapsulated powders was measured by the Curie method. The force on the sample in a magnetic field gradient was measured by a Cahn electro-balance whose output was recorded. The force on each sample was measured as a function of position in the magnetic field. Also, the force as a function of field was measured, to separate the effect of ferromagnetic impurities; this effect was easily observable for samples with more than 100 ppm impurities. The electromagnet was fitted with special pole pieces to minimize lateral pull[12] and to increase the volume over which the product of magnetic field H and gradient dHld2 (2 being the vertical

measurements

The superconducting transition temperature was measured by an induction technique, using a modified form of vibrating sample magnetometer operating at about 1 cps[l5]. The sample oscillates vertically between the centres of the two pick-up coils. The sample and the coils are within the region of uniform field of two Helmholtz coils placed horizontally, and are placed in a variable temperature metal dewar whose temperature could be regulated within O.l”K, as measured by an AulAuFe thermocouple. By varying the magnetising current in the Helmoltz coils, and observing the e.f.m. of the pick-up coils, it is possible to record the isothermal Al/H curve (.&t being the magnetic moment) at a series of successively higher temperatures, until the magnetisation disappears. In this way, the superconducting transition temperature can be determined to within O.l”K. 3. RESULTS

results of magnetic susceptibility measurements are shown in Table 1 and Figs. 2, 3. The susceptibility is paramagnetic and increases markedly with increasing nitrogen content. The slope of the susceptibility-temperature curve, dx/dT vs T is small and positive from room temperature down to the cryogenic range. It will be assumed to be constant, i.e. x increases linearly with T in that temperature range. The susceptibility was also measured as a function of magnetic field up to a value of 10.5 kGauss. After substracting the effect of ferromagnetic impurities, no dependence on magnetic field was found. These values can be compared with only one published literature value, 130 e.m.u./g . mol[4], presumably at room temperature, and for stoichiometric VN, given by Costa and Conte. This The

F. I. AJAMI and R. K. MACCRONE

10

Table 1. Experimental magnetic susceptibility of VN Compos. x in VN,

e.m.u./g e.m.u./mol T = 300°K 1.88 1.95 2.12 247 3.94

o-75 0.84 0.87 0.91 1.o

x x 10” e.m.u./mol 77°K

116 122 134 1.57 256

e.m.u./mol O°K*

(dX/dT) x 10” (e.m.u.imd . deg)

112 113 124 145 240

113 116 127 148 244

1.4 2.9 3.2 4.0 5.3

*Extrapolated.

EXP

/

250. h

i

x

G A

200.

-

8

$

3 = 150.3 :: a m 100.t Y : 50. -

compares with the value of x for nitrogen deficient VN prepared in this experiment, and suggests that their sample was also nitrogen-deficient. The variation of the superconducting transition temperature, T,, as a function of nitrogen content is shown in Fig. 4. The value of T, decreases sharply as the nitrogen content decreases, and this agrees with the results of Toth and collaborators [ 161.

IO -

8-

0.9 0.8 COMPOYTION

r-” 6

Fig. 2. Experimental and predicted values of the magnetic susceptibility at room temperature.

4

2

L

I

1

.7

.8

.9

1.0

X

Fig. 4. Superconducting transition temperature function of composition in vanadium nitride. Q EXPERIMENTAL -PREDICTED

0.9 0.8 COMPOSITION

I .D

Fig. 3. Experimental and predicted values of the average dx/dT in the range 77-300°K for the various compositions.

as a

4. DISCUSSION

The very slight temperature dependence of susceptibility for all compositions of VN indicates the absence of any type of magnetic ordering, i.e. there are no unpaired spins on individual atoms. Thus, the magnetism of VN is similar to that of the early transition metals and their “non-magnetic” alloys. Following the generally accepted procedures for in these the interpretation of magnetism materials [173, the susceptibiIity of vanadium nitrides will be considered as the sum of contributions

Magnetic susceptibility

from the tightly bound “core” electrons, collective “band” electrons.

and the

x = k-curex ,yh,¶“d.

(1)

In two-band conductors, the “band” part is the sum of contributions from the s-band and the d-band

and superconductivity

11

where n is the number of s electrons per molecule unit, M the molecular weight, and D the density. It is independent of temperature, and varies with composition primarily through the factor n. In the case of VN, for m = 1, xf = -1-8 and for n =2, XC = 11 X IO-“e.m.u./g . mol. The Landau diamagnetism is given by z

Xbd

=

xs +

c-9

Xd.

The contributions to the s-band magnetism[lS] are the Pauli spin term xp and the Landau term x L X$ =x:+x4

(3)

while the contributions to d-band magnetism include the orbital magnetism xi, first pointed out by Kubo and Obata[19], beside the Pauli and Landau terms Xd=XZ+X$fX:. (a) Core susceptibility The core susceptibility formula

(4)

is given by the Langevin

where N is Avogadro’s number, e, m, the electronic charge and mass respectively, c the velocity of light, and < r > * is the mean square distance of the core electron from the nucleus; the summation i is over all electrons in closed shells in an atom and the summation j is over all the different atoms in the molecular unit. Electrons that contribute to x_,,~are those in the Is, 2s, 2p, 3s and 3p shells in vanadium, and those in the 1s shell in nitrogen. Using Slater’s 1201method to calculate < r > *, as modified by Angus[21], the value of ,yEOcc was found to vary from - 7.94~ 10m6for VN to -7-34x 10-Oe.m.u./mol for VN,,,?. (b) s-Band susceptibility The Pauli spin paramagnetism

x4=- i -$ > xf

(8)

where m * is the effective mass of the electron. For free electrons, m* = m, and xi = -4 xc_ Now the total susceptibility of the vanadium nitrides, which is of the order of 10-j e.m.u./g * mol, is much larger than the sum of xcore,x: and x:. Even if the Slater method of calculating < r >’ were in error, or the exact number of s electrons were not known, the combined contribution of these terms to x is negligible, and can in fact be taken as zero. The susceptibility of VN is then governed by the d electrons. (c) d-Band susceptibility The Landau diamagnetism x 2 is still given by an equation equivalent to 8 above, but the effective mass of the d electrons is much larger than the free electron mass, so x 2 can be neglected. The susceptibility of vanadium nitrides is then reduced to the orbital and spin susceptibilities only x=x:+x:. The Pauli spin paramagnetism Stoner formual[22]

xf; =2p;.Nd

(9) is given by the

I+

where X’, JY’are the first and second derivatives of the density of states at the Fermi level EF. The term in curly brackets is usually small (and equation (19) reverts to equation (6)), but gives the variation of x with T. Thus,

is given by

x: = 2cri.K

(6)

where ps is the Bohr magneton, and X, is the density of states of one spin, of the s-band, at the Fermi level. If it is accepted that s electrons behave as free electrons, N can de determined analytically, and XT is given by XT = 1.86 x IO-% “l(M/D)“/” e.m.u./mol

(7)

The orbital susceptibility is quite difficult to caiculate from first principles, because it requires a knowledge of the wave functions and the energy levels throughout the d-band in contrast to xp which requires a knowledge of energy levels

F. I.

12

AJAMI

and R. K. MACCRONE

around the Fermi level only. In the absence of such detailed information, an approximate formula, due to Marshall[23], can be used to estimate x”d x”=2p;NL(L+l)

1 (12) g

2 z

3

where nd is the number of electrons in the d-band per molecular unit, L is the orbital quantum number (2 for d electrons) and E is an average width of the d-band. It is obvious that, in order to calculate x”dand x;, it will be necessary to develop a band model for the vanadium nitrides, and to deduce from that model the values of N, SIT’,x”, nd and E in equations (lO)-(12). 5. BANDSTRUCTURE Model for VN Stoichiometric vanadium nitride has ten valence electrons: five electrons from the 2s-2p shell of nitrogen, and five electrons from the 4s-4d shells of vanadium. These will be distributed in the proposed scheme: (1) A low-lying s-band, completely full, occupied by two electrons. (2) A band with predominantly d character, originating from the vanadium atoms, and split into two sub-bands separated by a minimum. The origin of this splitting is electrostatic. The five d orbitals of atomic V, in an octahedral environment, split into three TzI orbitals and two Ez, orbitals. In VN, these orbitals are widened into two sub-bands with a minimum in between. Although no band structure calculations have been made on VN, such a minimum has been found in the density of states curves of TiC[4-71, TiN [5] and NbN [g-9]. By analogy with these calculations, the d-band is relatively narrow; the density of states is high and the electrons have a large effective mass. (3) An s-p band of free electrons, with low density of states, overlapping the two d sub-bands. Assuming that the number of electrons in the s-p band is negligible, the ten valence electrons will be distributed thus: two electrons in the s-band, six electrons in the dT sub-band and two electrons in the dE sub-band; the later, which can contain four electrons, will be half-full, and the Fermi level will straddle the dE and the partially full s-p bands. Figure 5 shows a schematic density of states

ENERGY

( ARBI.

UNITS

1

Fig. 5. Schematic band structure for vanadium nitrides; E,,, Fermi level of VN; E,,, Fermi level of VNo 7~. curve for stoichiometric VN. This model can be extended to cover the non-stoichiometric compositions also. If the rigid-band model is applicable, the only difference between VN and VN, (x < 1) will be in the degree of filling of the d-band and hence the position of the Fermi level in the density of states curve. The number of valence electrons in VN, is given by n =5+5x. Subtracting the number will be

(13)

eight electrons that fill the lower bands, of electrons in the conduction dE band

ndE = 5x - 3. At the lowest limit of stoichiometry, x = 0.72 and ndE will be 0.6 electrons. The Fermi level will shift to EF2 to accommodate the smaller number of electrons. The relative positioning of the bands was determined by the experimental susceptibility results: the high value of x for VNoa requires that EFz be far from the minimum, and the steadily rising values of x as the nitrogen content increases require that the density of states curve rise accordingly. Finally, the positive values of dx/dT require that the curve be concave upwards. It is interesting to note that a similar model has been proposed for the band structure of NbN by Geballe [24] and collaborators; however, in order to explain the very low values of the magnetic susceptibility, they postulated that the d, sub-band was

Magnetic susceptibility shifted to much higher energies, far above Fermi level, which cut the s-p band only.

the

and superconductivity

13

Numerical calculations If XdE is given in the form of an analytical function,

lected. Finally, the uncertainty in the experimental results may be quite large because of the very weak temperature dependence. According to our calculations, the Fermi level lies 16-2.2 eV above the minimum in the density of states; this gives the width of the Ez, sub-band.

NdE = BE + C exp (rE)

(15)

Obviously, there is no way to predict by this method the width of the Tza sub-band which is

within the range EFI-EFz, it is possible, given B, C and r, to calculate the spin paramagnetism xz; by differentiation dx/dT can be obtained and by integration the number of electrons, holes and the average width of the dE band, and hence the orbital susceptibility can be obtained. The procedure was carried out iteratively, using a digital computer until a reasonable overall fit was obtained. The resulting values were as follows:

totally filled. An important result is that the orbital contribution is quite important, ranging from 50 to 33 per cent of x ; also its variation with composition is slight so that, as the d band is progressively filled, its increased width E is approximately sufficient to offset the increase in the factor nd[l -(n,/lO)]. Thus the major part of the variation of x with composition is due to ~2 and accordingly, the density of states increases with increasing nitrogen content from 0.75 to 2.5 states/(atom . eV).

B = 0.5 states/atom (eV)’ C = lo-’ states&atom . eV) r = 9(eV)-‘. These values were used to obtain the theoretical curves in Figs. 2, 3 and 6. The predicted values of d,y/dT are positive, but are an order of magnitude smaller than the observed values. The discrepancy may be due to the neglect of the variation of orbital susceptibility with temperature, but this cannot be determined without a rigorous solution of the band structure on the lines of Mori’s[25] calculations. Also, the effect of thermal expansion has been neg-

k g

I.O-

4

x 0.5

-

I

0.7

i 0.8

0.9

Comparison with theory and experiment There are two independent checks on the validity of the model. First we can compare the density of states curve with those obtained from band structure calculations on similar compounds. This implies that the rigid band model holds for all carbides and nitrides; within this limitation we find that the present density of states agrees fairly well with the calculations of Em and Switendick[5] on TiN, of Conklin and Silversmith [7] on Tic, and of Bilz [26] on a rigid band structure for carbides and nitrides in general. However, our results disagree with the density of states curve for the iso-electronic compound NbN, where both theory [8,9] and experimental x [24] indicate a much lower density of states. Next, we can compare the “magnetic” density of states with the “specific heat” density of states. Unfortunately, low temperature specific heat measurements have been done only on stoichiometric VN[27], so a comparison can be made only for that composition. Hulm et al. have obtained the value 8.60 mJ/(mol. deg*) for y, the coefficient of the linear term in the low temperature specific heat (they give no details about the composition, but, from their measured T,, 8,5”K, it appears to be stoichiometric VN). The density of states is given by 3y_ Nd = 2&k2 - 1.83 states (atom . eV).

1.0

COMPOSITION

Fig. 6. Density of states at the Fermi level for various compositions of vanadium nitride.

This value is 25 per cent lower than the “magnetic” Nd. It is in qualitative agreement in that y for VN is higher than that for any other refractory carbide or

F. I. AJAMI and

14

R. K.

MACCRONE

x:(O) = x:(O) = x”d= A=

nitride, just as A for VN is higher than that of any other carbide or nitride. Furthermore, the magnetic X could be brought in line with the specific heat X if the orbital contribution were raised from 5.5 to 100 X 10m6e.m.u./g . mol. This is not totally unrealistic, given the very approximate nature of equation (12). However, the infiuence of many-body effects has hitherto been neglected. This point will now be taken up, since it offers an alternative explanation.

a: = 0.58. According written as

6. MANY-BODY EFFECTS The many-body effects include the influence of electron-phonon interactions on the specific heat, and the influence of exchange and correlation on the spin susceptibility. In the case of superconductors, the starting point in the discussion is the MC Millan [28] equation

2~&$(0) with Xd(0) = 1.2 78 X lo-” e.m.u./g . mol 55 X 10e6 e.m.u./g . mol 240 x 10e6 e.m.u./g . mol

to Jensen

and Maita[30],

a = Nd(O)V

a! can be

(19)

where V is the Coulomb interaction energy at the Fermi surface. For X&O) = 1.2, V = 0.48 eV. There is a check on the self-consistency of the procedure, as Jensen and Andres[31] have shown that, under certain approximations, the electron-electron coupling constant, p * in equation (16), can be written

1 (16) where B,, is the Debye temperature, A is the electron-phonon coupling constant and u* is the electron-electron coupling constant, commonly taken as 0.13 for transition metals and their alloys; this value will be accepted for VN. Using T, = 8.2”K (cf Fig. 4) 8, = 77029 (obtained from an average determined from the Lindemann formula and specific heat result), then

/--

*_JlrdKvV zqz0.19 3

as

compared to the assumed value of 0.13. In the previous calculation, the contribution of the orbital susceptibility xi (equation 18) was taken as 55 e.m.u./g . mol; this could very well be a low value. By taking Ad as 100 e.m.u./g . mol the value of cr is reduced to O-44; this also yields V = 0.37 eV and CL*- 0.15, in better agreement with the original value of 0.13.

A = 056. 7. CONCLUSION

The “bare” density of states X,(O), after correction for the electron-phonon enhacement, is obtained from the specific heat coefficient thus: y = 3 n2k2Nd(0)[l

+ A].

(17)

This gives sITd(0) = 1.2 states (mol. eV), compared with the uncorrected value sITd= 1.83, and with the “magnetic” value of 2.4. The difference may be explained in terms of exchange and correlation factors which increase the spin susceptibility. In this case, the total susceptibility A, can be written as

x=x!+=

x;(o)

(18)

where A z (0) is the spin susceptibility assuming the absence of many body effects, and a! represents a measure of these effects. Using:

By taking into consideration the experimental results of superconductivity and low temperature specific heat, together with estimates of many-body effects, it is seen that the density of states of stoichiometric VN is reduced to about 1.2 states/(atom . eV), as compared with about 2.4 states/(atom . eV) obtained by considerations of magnetic susceptibility alone. Strictly speaking, the calculation of Section 5 has to be repeated; taking into consideration the manybody effects. However, this is not warranted at the present time, as the many approximations involved in the various steps of the derivations make it improbable that a rigorous quantitative understanding can be achieved. It is instructive to state explicitly some of these approximations: (1) The most serious approximation is the use of equation (12) for the orbital susceptibility. The value of As may be larger by a factor of 2. The

Magnetic susceptibility and superconductivity uncertainty can be resolved in one of two ways. Either x”d can be calculated from first principles[25], if the wave functions and energy levels are known. Alternatively, xp1can be isolated by simultaneous measurements of susceptibility and Knight shift[l7,32]. (2) Much of the discussion of many-body effects is dependent on a single measurement of specific heat. The conclusions would be more valid if y was known for a number of compositions in VN,. (3) Related to (2) above is the fact that the McMillan equation (16) or any similar equation, such as Garland and Allen’s[33] or Jensen and Andres’ are more useful in correlating trends of T,, x, &, h and p* within a particular class of compounds, rather than predicting exact values for a single material. (4) We have assumed the validity of the rigid band model. While this model has been useful in explaining many variations in properties of substitutional alloys of transition metals, its validity has not been tested in compounds such as VN,, where the variation is due to the introduction of an increasing number of vacancies in the lattice, and where the disorder is more serious than in substitutional alloys. At this point however, this approximation is the least serious. In summary, we can conclude that magnetic susceptibility and superconducting transition temperature determinations are consistent with a rigid band model for the vanadium nitrides, in which the density of states is quite large [ > 1 state/ (atom. eV)] and due to (mainly) d-electrons. Acknowledgements-We would like to thank Professor C. D. Graham (University of Pennsylvania, Metallurgy and Materials Science) for his help in the superconductivity measurements, and his interest in the project. We would like to thank also Dr. S. K. Iyer (Max Planck Institute, Dussetdorf, Germany) for his help in the preparation and characterization of the samples, and Mr. P. Flanders (University of Pennsylvania, Laboratory for Research on the Structure of Matter) for his assistance in some magnetic measurements. The project was supported by the Advanced Research Projects Agency of the Department of Defence, through a grant to the Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia.

REFERENCES 1. Storms E. K., A Critical Review of Refractories(Los Alamos Scientific Laboratory, University of California. New Mexico. 1964, Report LA-2942). 2. Iyer S. K., The Kinetics ojZ&idation of Ti, Zr and V (Ph. D. Dissertation, MetaIlurgy and Materials Science, University of Pennsylvania, Philadelphia, 1971), ch. V.

15

3. Toth L. E., Tr~nsftfon Metaf Carbides and Nitrfdes. Academic Press, New York (1971). 4. Costa P. and Conte R. R., In Compounds of Interest in Nuclear Technology (Edited by J. T. Waber, P. Chiotti and W. N. Miner), p. 3. Metallurgical Society of A.I.M.E., Ann Arbor (1964). 5. Em V. and Switendick A. C., Phys. Rev. 137, 202 (1965). 6. Lye R. G. and Logothetis E. M., Phys. Rev. 147, 622 (1%). 7. Conklin J. B. and Siiversmith D. J., ht. J. Quant. Chem. 25, 243 (1%8). 8. Schwarz K., Monat. fur Chem. 102, 1400 (1972). 9. Fong C. Y. and Cohen M. L., Phys. Rev B 6, 3633 (1972). 10. Mekata M., Yoshimura H. and Takaki H., J. Phys. Sac. .Zapan 33, 62 (1972). 11. Brauer G. and Schnell W. IX. .I. fess-common Metafs 6, 326 (1964). 12. McGuire T. R. and Flanders P. J., In Magnetism and Metalfuruv (Edited bv A. El Berkowitz and E. Kneller),~Vol. I, p. 123: Academic Press, New York (1969). 13. Figgis B. N. and LewisJ., In Modern Coordination Chemfstry (Edited by J. Lewis and R. G. Wilkins), Chapter 6. Interscience, New York (1960). 14. Flanders P. J. and Doyle W. D., Rev. scient. Instrum. 33, 69 (1962).

15. McGuire T. R., J. appf. Phys. 38, 1299 (1967). 16. Toth L. E., Wang C. P. and Yen C. M., Acta Met. 14, 1403 (1%6). 17. Clogston A. M., Jaccarino V. and Yafet Y., Phys. &PI.

134, A650 (1964). 18. Seitz F.. Modern Theory of Solids. Chaa. 16. McGrawHill, New York (1940; _ _ 19. Kubo R. and Obata Y., J. Phys. Sot. Japan 11, 547 (1956). 20. Slater J. C., Phys. Rev. 36, 57 (1930). 21. Angus W. R., Proc. R. Sot. Land. A136,569(1932). 22. Stoner E. C., Proc. R. Sot. Land. A154, 656 (1936). 23. Childs B. G., Gardner W. E. and Penfold J., Phif. Mug. 8, 419 (1963). 24. Geballe T. H., Matthias B. T., Remeika J. P., Clogston A. M., Compton V. B., Maim J. P. and Williams H. J., Physics 2, 293 (1966). 25. Mori N., J. Phys. Sot. .Zapan 20, 1383, (1%5); Ibid. 26, 926 (1969); Ibid. 29, 366 (1970). 26. Bilz H., 2. Phgs. 153, 338 (1958). 27. Hulm J. K., Walker M. S. and Pessall N., In Superconductivity (Edited by F. Chihon), p. 60. North Holland, Amsterdam (1971). 28. McMillan W. L., Phys. Rev. 167, 331 (1968). 29. Ajami F., Electronic Properties of Vanadium Nitrides, (Ph. D. Dissertation, Metallurgy & Materials Science, University of Pennsylvania, Philadelphia, 1972), p. 120). 30. Jensen M. A. and Maita J. P., Phys. Rev. 149, 409 (1966). 31. Jensen M. A. and Andres K., Phys. Rev. 65, 545 (1968). 32. Clogston A. M., Gossard A. C., Yaccarino V. and Yafet Y., Phys. Rev. Lett. 9, 262 (1962). 33. Garland J. W. and Allen P. B., In Supe~onductivfty {Edited by F. Chilton), p. 669. North Holland, Amsterdam (1971).