Electronic structure of γ phase zirconium hydride

Journal of the Less-Common

Metals,

103 (1984)

309 - 315

309

ELECTRONIC STRUCTURE OF 7 PHASE ZIRCONIUM HYDRIDE* A. C. SWITENDICK Sandia National

Laboratories,

Albuquerque,

NM 87185

(U.S.A.)

(Received April 9, 1984)

Summary We have calculated the electronic energy band structure of y phase ZrH. This can be related to the cubic phase by the ordered removal of four hydrogen atoms from the large cubic unit cell. Very non-rigid band behavior is obtained. New structure is induced in the density of states which is not present in the original phase. A peak occurs at the Fermi energy which may be responsible for the tetragonal distortion but is insufficient to account for the observed magnetic susceptibility.

1. Introduction The group IV transition metal dihydrides TiH,, ZrH, and HfH, show a unique behavior as the hydrogen concentration approaches the stoichiometric value (X = 2). The cubic fluorite structure distorts to form a tetragonal structure with c/a < 1.0. Earlier calculations [l] verified the suggestion based on the experimental work of Ducastelle et al. [2] as to the origin of this distortion. The Fermi level for these systems falls on a peak in the density of states which arises from orbitally degenerate states. This degeneracy is split by the distortion and results in a lowering of the Fermi level and the value of the density of states at the Fermi energy. By alloying with neighbors of the group V period, we can trace out this density of states curve by a variety of experimental techniques: electronic specific heat [2, 31, magnetic susceptibility [3 - 51, Knight shift [6, 71 and nuclear magnetic relaxation parameters [6 - 81. Further detailed calculations [9] were able to achieve good agreement with most of these results; the most notable failure was in the case of the magnetic susceptibility and was due to the lack of inclusion of orbital contributions. This agreement was obtained by minor shifts of the Fermi energy and the assumption of the addition of 0.5 - 0.75 electrons at the Fermi level per added hydrogen atom. Self-consistent calculations [lo] and coherent potential approximation (CPA) calculations for the nonstoichiometric systems are being performed [ll] to verify further the *Paper presented at the International Symposium on the Properties and Applications of Metal Hydrides IV, Eilat, Israel, April 9 - 13,1984. 0022-5088/84/$3.00

@ Elsevier Sequoia/Printed in The Netherlands

310

detailed correctness of this model. This paper is a companion study which treats the electronic effects and the non-stoichiometry in the unique case afforded by the Zr-H system. For hydrogen compositions below 1.50 the Zr-H system exhibits a mixed-phase behavior of hydrogen in CYhexagonal zirconium, y phase tetragonal ZrH and a cubic 6 phase ZrH,.,. Pure samples of the y phase have not been obtained, but X-ray diffraction and metallography studies [12,13] show two crystalline lattices, of which one is cubic and the other tetragonal (c/a > 1.0). Refinements based on these measurements yielded the crystal structure P4,/n with the composition ZrH. The lack of any substantial variation in c(y) and a(r) with composition in this phase indicates that its stoichiometry is probably constant and that only the proportions of CY,y and 6 (ZrH,.,) vary. A measurement of the magnetic susceptibility [5] in this composition range indicates a value higher than that in either the (Yphase or the 6 phase, implying a fairly high value for the y phase component, while the nuclear magnetic relaxation rates indicate a fairly low constant value of the density of states. In this paper we attempt to look at the electronic properties (origins) of this y phase.

2. Results and discussion The crystal structure of the y phase is shown in the inset of Fig. 1. It is derived from the ideal dihydride CaFz structure by removing the (110) 5.0

5

y

4.0

8 LT N L’

3.0

,

,

,

,

FCC

ZRH2 ( /

, c

,

=, E 2

2.0

ifi Ff E

1.0

A

d 0.0 -5.0

0.0

5.0

ENERGYCELECTRON

10.0

J

1= _i.1

VOLTS1

Fig. 1. Density of states for f.c.c. ZrH 2. The arrow shows the position of the Fermi level for 5 electrons per zirconium atom (ZrH). Inset shows structures of ZrHz and ZrH as explained in text.

311

planes of hydrogen atoms. There are four zirconium and four hydrogen atoms in the unit cell which are shown by the large open circles and the large full circles respectively. The small full circles represent the hydrogen atoms present in ZrH, which have been removed. The lattice also distorts from its cubic value a = 0.4785 nm to a = 0.4586 nm and c = 0.4949 nm. These are the values used in our calculation. Because of the obvious relation to the fluorite structure we used the same potential as adopted in earlier calculations [ 1,9] to compare directly the differences induced by the structural changes, and to relate our results to the dihydride where possible. This can be done by starting with the large cubic unit cell containing four zirconium and eight hydrogen atoms which represents the fluorite structure and then removing four of the hydrogen atoms and distorting to the correct lattice constants. We have followed this procedure in our calculations to study the effect of each of these changes. The density of states of f.c.c. ZrH, is shown in Fig. 1. The Fermi level shown is determined by filling with six electrons for ZrH,. The structure between -1.0 and 5.0 eV is associated with H-H and M-H (M f metal) bonds. The structure above 5.0 eV is largely associated with the metal d states. There is a peak near the Fermi level. This peak splits in the tetragonal e phase and we were able to obtain qualitative agreement with the specific heat data by assuming that each hydrogen atom removed subtracts 0.50 electrons from the Fermi level. Also shown in Fig. 1 is the Fermi level corresponding to five electrons (ZrH). The density of states is fairly low and smooth and there are no peaks in this region. The value of the density of states is 40% of the value for ZrH, and one-third of the peak value. There would appear to be no electronic origin for the y phase nor for the high susceptibility observed by Venturini [5]. We have therefore calculated the electronic energy bands for y phase ZrH. Because the unit cell contains four zirconium and four hydrogen atoms the Fermi level will be determined by 20 electrons or 5 electrons per ZrH unit. The band structure will be determined by folding the f.c.c. band structure back into the smaller cubic Brillouin zone. This procedure has been illustrated earlier in our calculations for Y4Hs, Y,H,, Y4Hll and Y,H12 [14]. The band structure for Zr4H8 is shown in Fig. 2(a). The old zone boundary points X(2n/u, 0, 0), X(0, 27r/a, 0) and X(O,O,27r/a) are folded back to the zone center. The eight states rl, X3, X4,, and r2, are associated with the (now) eight hydrogen atoms in the unit cell and the Fermi level is determined by 24 electrons (filling 12 bands on average). The Fermi level for 20 electrons (denoted by 5 - - - 5) is at 0.527 rydbergs, which is a drop of over 1.85 eV. Because of the closeness of the band levels and the (near) degeneracies along A and at X, it was necessary to employ the linearized augmented plane wave (LAPW) technique 1151, which was very efficient and gave very good agreement for Zr,Hs with our earlier calculations for ZrH*. The energy bands for 7 phase ZrH for r, X, R, A(n/a, 0,O) and A(n/a, n/a, n/a) are shown in Fig. 2(b). The points are connected by straight lines to show the presumed connectivity. Several changes occur when the four

312

\

\

,, 4- 72 , \ ‘\ \ ,A; t %. / \ t 3, ‘\ ’ \ I’ \\ I’ ‘\: ‘\\ .I j. _ - 2‘-_ I. E f - ,tJ

-9 A1 ( ,' \ ,' \ I sj __l\_._____._____,i___ 5 .' - '\ 0.5 \\ ? ‘\

W2’. ‘--2.

I

X (a)

\\

-SW_, ‘4 x-J: / / I

1” \

x

\

I

A

r-

I

x

R

X

r

R

(b)

Fig. 2. Schematic representations of (a) folded energy bands for Zr4H8 and (b) energy bands for Zr&. Degeneracies are illustrated by the numeral on the line.

hydrogen atoms are removed from the unit cell. The structure is still pseudocubic with c = a, but ~(71, 0,O) = ~(0, ‘I), 0) f e(0, 0,q). We now have only four low-lying bands. The old I’zS-l’l splitting has decreased and two of the states associated with X3 have disappeared as have two of the states associated with X4,. If this were all that had happened, then the Fermi level would have to rise relative to ZrH2 since we have lost four bands capable of holding eight electrons but have lost only four electrons (hydrogen atoms). However, on closer inspection we see that some of the states lost to the hydrogen manifold can still be occupied (below the Fermi level). Just above the states associated with the Xi is a new band. Similarly, new states appear above the (old) lr2, and between rzsP and I’is. These are metal d-p states which were previously involved in bonding with the hydrogen and are now free to bond to each other. When we fill these states with 20 electrons we find the Fermi level shown in Fig. Z(b). Instead of rising, it actually drops by about 0.70 eV - only 40% of the rigid band prediction. This corresponds to a rigid band removal of only 0.40 electrons from ZrH,. The density of states derived from our eigenvalues for 75 points in the irreducible sector of the Brillouin zone is shown in Fig. 3. A comparison

3x3 CUB iC

ZR4H4

ENERGYlELECTRON

VOLTS)

Fig. 3. Density of states for pseudocubic ZrH (Zr4H4).

with Fig. 1 indicates a decrease in the width and value of the hydrogenassociated states. A very large peak is found at the bottom of the d states. The presence of this new feature at first concerned us since there is no evidence of it in ZrH, and it might be an artifact of the “no band crossing” procedure adopted in the tetrahedron method [16,17] used to derive our density of states. We therefore calculated Zr,Hs by identical procedures, since this problem should be even worse in this case, and we found eigenvalues and densities of states indistinguishable from those in Figs. 1 and 2(a). There is also considerable new structure near the Fermi level with a peak of value 1.46 electron states eV_’ (Zr atom)-’ which is diminished from the ZrH, value of 1.76 electron states eV- ’ (Zr atom)-‘. The value at the Fermi level is 1.12 electron states eV_’ (Zr atom)-“ compared with 1.48 electron states eV- ’ (Zr atom)-’ for ZrH, and is significantly larger than the rigid band value of 0.59 electron states eV_’ (Zr atom)-‘. The peak above the Fermi level appears to be associated with the atmost degenerate states at R. Finally we allow the tetragonal distortion to occur and calculate the energy bands. The density of states derived from this calculation is shown in Fig. 4. As in ZrHz the states at the Fermi level are pushed up and down away from the Fermi level and the density of states falls. The Fermi level actually rises (by 0.05 eV), but this is not significant, whereas the value of the density of states falls to 0.88 electron states eV_’ (Zr atom)-‘, which is larger than the rigid band value derivable from Fig. 1 and about 80% of the peak value expected for b-ZrH1_,. Thus the large value of the magnetic susceptibility attributable to 7 phase zirconium hydride is in part derivable from the modest increase in the density of states. The numerical value and the experimentally observed low and constant density of states implies a

314 GAMMA

5.0

PtiASE

r.,

ZR4H4

,

,

E

1

P 5. 0 ENERGYCELECTRON

VOLTS)

Fig. 4. Density of states for tetragonal y phase ZrH (ZraH4). large orbital contribution to the magnetic susceptibility. This could be contributed by a Van Vleck orbital type of mechanism between the states in the peak below I& and the states in the peak above Ef . The peak separation, which occurs in the denominator, is only 0.61 eV.

3. Summarizing

remarks

We have calculated evolution of the states

the electronic energy bands of y phase ZrH. The with the removal of hydrogen atoms shows little

TABLE 1 Density of states N(E) for electron counts corresponding to the Fermi level, rigid band filling or peak positions na

4.82 5.00 5.17 5.47 6.00 6.07

Zr4H4 (cubic)

ZrHz (f. CC.)

Zr4H4 (tetragonal)

E (rydbergs)

N(E)b

E (rydbergs)

N(E)b

0.527

0.58

0.616C 0.626

1.13 1.46

0.665’ 0.669

1.46 1.75

aIn electrons per zirconium atom. bin states of both spins per electronvolt per zirconium atom. CElectron counts corresponding to the Fermi level.

E (rydbergs)

N(E) b

0.603 0.61gc

1.38 0.88

0.651

1.62

315

evidence of rigid band behavior, and the results do not agree with the CPA result [ 111 which shows only a blurring of structure in the density of states and very little movement of the Fermi level. States are lost, states are shifted and the densities of states are drastically modified. The results are summarized in Table 1. A new peak appears at the Fermi level which is split and lowered by the tetragonal distortion. These results are in qualitative agreement with the limited experimental results for the y phase if we assume a large orbital contribution to the susceptibility. Such an assumption is necessary for other phases of the Zr-H and Ti-H systems [9]. The new large peak at the bottom of the d bands and the quantitative change in the hydrogen-derived states should be looked for in photoemission spectra.

Acknowledgments The author wishes to thank Dr. E. L. Venturini for conveying his experimental results prior to publication and for helpful discussions. This work was supported by the U.S. Department of Energy under Contract DE-AC0476DP00789.

References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

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