calculations of the electronic properties of...

PHYSICAL REVIEW B VOLUME 32, NUMBER 2 15 JULY 1985

Calculations of the electronic properties of substoichiometric Ti-Fe hydride

D. A. PapaconstantopoulosNaval Research Laboratory, 8'ashington, D.C. 20375-5000

A. C. SwitendickSandia Laboratories, Albuquerque, New Mexico 87185

(Received 16 August 1984)

We have calculated the electronic structure of TiFeH„using the tight-binding coherent-potential-approximation method. The tight-binding parameters were determined by Slater-Koster fits toaugmented-plane-wave calculations of TiFe and TiFeHl o. We have computed the densities of states(DOS) for the hydrogen concentrations x =0.1, x =0.8, and x =0.9, and obtained the angular-momentum and site-decomposed DOS. We have found that the Fermi level EF is very nearly in-

dependent of x, but the DOS values at EF increase rapidly with x in agreement with experiment.We discuss various features of the DOS relating to the effects of hydrogenation and disorder.

I. INTRODUCTION

Electronic-structure calculations for binary-metal hy-drides have clarified the change in the electronic structureof the (sometimes fictitious) host lattice upon the additionof hydrogen to form the hydride. ' In the transitionseries these changes have been seen to be of three types:(1) the lowering of previously occupied states, (2) the in-troduction of new low-lying states, and (3) the lowering ofstates from above the Fermi level to below. All thesemodified states have s-like character about the hydrogensites. Since the hydrogen-to-transition-metal ratios arefrom 0—3, this means that typically —, of the d states as-sociated with canonical band structure of the transition-metal lattice are largely unaffected. The relative amountsof each of these changes vary from one host-crystal struc-ture to another and even within the same or similar struc-tures. Thus, it is unadvisable to generalize from limitedresults.

The simplest question one can ask is whether the Fermilevel goes up or down, i.e., what are the relative propor-tions of distribution of the added hydrogen electrons be-tween the latter two types of states mentioned above? Oneof the present authors, ' based on comparisons of the bandstructures of the unhydrided host and the hydride, has es-timated that the number of electrons per hydrogen addedto the canonical host d states ranges from 0.4 for TiHz to1.0 for CrH. Recent comparison of experimental data anddetailed density of states (DOS) for TiH„and ZrH„(1.6 & x & 2.0) has yielded a value in the range of0.50—0.75 electrons based on the dihydride band struc-tures. Similar estimates have been made by Chan andLouie and by Temmerman and co-workers.

In the search for hydrides for practical applications,cost and weight factors have ruled out any large-scale useof binary hydrides. Thus, one is led to consider binary-and ternary-based metal alloy systems for hydrogenstorage. Here the understanding is complicated by the in-troduction of the new component(s) which interacts bothwith the other component(s) and the hydrogen. In these

TABLE I. Slater-Koster parameters for TiFe.

E„(000)E„„(000)E (000)Ed 2,d 2(100)E„(200)E,„(200)E, d 2(002)E„„(200)E (200)Ex,xy (020)E, d2(002)E„„(200)E„„(002)Ed2 d2(002)Ed l d l (002)

1.58361.39160.77270.7873

—0.0225—0.0683—0.0648

0.1507—0.0598—0.0114—0.0262

0.0185—0.0130—0.0483

0.0062

Fe-Fe

1.12222.10640.62680.6018

—0.03300.21160.04480.33210.04370.0648

—0.07090.01520.0043

—0.0264—0.0002

E„(111)E,„(111)E, „y(111)Ex,x(111E„y(111}E„„y(111)E„~{111)Ex,d l(111)E „„(111)E y „,(111)E„„d,(111)Ed2 d2(111)

Ti-Fe

0.1218—0.0950

0.01380.03440.0603

—0.0198—0.0153—0.0085

0.01630.02000.0125

—0.0207

Fe-Ti

0.1218—0.0490—0.0373

0.03440.06030.01510.0426

—0.07140.01630.02000.0202

—0.0207

cases, one has very little understanding of even the hostband structure, and the situation becomes more compli-cated in that one may have disorder in the host or missingatoms on the hydrogen lattice (which exists even forbinary hydrides).

Due to the complexity of the problem, investigators

32 1289 1985 The American Physical Society

1290 D. A. PAPACONSTANTOPOULOS AND A. C. SWITENDICK 32

have resorted to quasichemical or empirical ap-proaches. " However, due to the development of thecoherent-potential approximation' (CPA) more realisticcalculations are now possible. Fortunately, two state-of-the-art calculations exist for one of the leading candidatesfor hydrogen storage, Ti-Fe. In an earlier work, ' one ofus calculated the electronic band structure of crystallineTi-Fe in the cesium chloride structure, and Czupta, ' in apioneering calculation for Ti-Fe-H, has provided a bench-mark for these types of ternary systems. We shall attemptto explore the physics for intermediate compositions usinga tight-binding formalism of the CPA. ' ' This work isan improvement, extension, and more complete discussionof an earlier work. '

II. SLATER-KOSTER INTERPOLATION FOR TiFe

We have performed a Slater-Koster' (SK) fit to theaugmented-plane-wave (APW) calculations for TiFe in theCsCl structure. ' This involves an 18&18 Hamiltonianconsisting of the s-, p-, and d-like orbitals of both the Ti

and Fe sites. To avoid incorrect assignments of states weused group theory to reduce the 18& 18 matrix to smallermatrices at high-symmetry points or lines in the Brillouinzone. ' We utilized 48 three-center-integral parametersthat included up to second-neighbor interactions. In ourearlier calculations, ' we had included the third-neighborinteractions, which increased the number of parameters to82. However, we found that 48 parameters fit the bandsjust as well as 82 parameters. In fact, by improving ourleast-squares computer code we have now obtained abetter fit with 48 parameters than we did before with 82parameters. We performed a fit to the APW results at 16k points in the irreducible Brillouin zone, while we havetaken a rms error at 35 k points. The rms fitting errorwas less than 10 mRy for 11 bands. The SK parametersare listed in Table I.

III. SLATER-KOSTER INTERPOLATION FOR TiFeH

The compound TiFeH& o has the orthorhombic struc-ture that results from the doubling of the CsC1 unit cell of

TABLE II. Matrix elements involving hydrogen interactions with Ti and Fe. h&

and hz indicate thetwo hydrogen atoms in the 38&38 matrix and the indices 1 and 2 on the left denote Ti and Fe, respec-tively. The notation is that of Ref. 18.

(h, fh, )

{sj+fh))

(xi+fhi)

(z~+fh, )

(xy~+ fh, )

(yz g+f

h g ) = (zx g+fh, )

[«'—y'}i'f hi]

[(3z —rz))+fh(]

(sz+fht)

(xz+fh()=(yz+

fh))

(zz+fhi)

(xy z+f

h ~ ) = (yz z+f

h ~ ) = (zx z+ /h ~ )

[(3z —rz)z+fh, ]

(s~+ fh, )

(xi+ fhp)

(y~fhz)

(zi+fhp)

(xy~f

hz)={yz+&fhz)

(zx g+fhz)

[(x'—y')i+ Ihz]

[(3z'—r'},+fhz]

(sz+fhz}

(xz+fhz)={zz+

fhz)

{xyzIhz} (yzz I

h»={zxzIhz}

[{x'—y'}z'Ihz]

[(3z'—r'),+fh, ]

Eq z(000)+2' q(200)cos(2x)2E» (011)cos(z —y)2V 2EI, ,(110)cosx cosy

2V 2E& (110)sinx cosy

2V 2Eq „(110)cosxsiny

0—2&2EI, „«(110)sinx siny

002/2E z z(110))cosx cosy

7

V 2EI, ,(001}cosz0V 2EI, ,(001)sinz

00V 2E„, z z(001)cosz

2{/2Eq, ,(110)cosx cosz

2~2Eq „(110)sinx cosz

02V 2' „(110)cosxsinz

0—2V 2EI, „«(110)sinx sinz—V 6E z z(1 10)cosx cosz

t

V2E z z(110)cosx cosz—

V'2E„,(001 )cosy

0V 2E&,(001)siny0

'{/3/2E z z(001)co—sy

—&1/2E„z z(001)cosy

32 1291

TABLE III. Slater-Koster parameters for TiFeH~ 0.

Ti-FeFe-Fe Fe-Ti

0.91402.09880.62130.6114

—0.04270.18640.04150.37520.04490.0654

—0.02480.0142

—0.0038—0.0028

0.0115

—0.0231—0.0011

0.02310.01770.0057

—0.0302

0.0399—0.0700

0.02310.01770.0159

—0.0302

E„,(111)E.„,(»1)E.,„„(»1)E y „,(111)E~y d2(1 1 1)

Ed 2 d 2(111)

1.91441.28620.84860.77200.1575

—0.1180—0.0805

0.2122—0.0205

0.03460.02460.0140

—0.0333—0.0538—0.0109

E„(000)E „(000)E„y„y(000)

E, ,(200)E,„(200)E g2(002)E (200)Ey y (200)E„„„(020)

E„y„y(200)E„y„y(002)

Ed ) d )(002)

H-H

Eh h(200)Eh h (011)

1.6662—0.0315

0.2193

0.17710.02060.13630.1230

Eh, (110)Eh (110)Eh „y(110)Eh d P(110)

T1-Fe Fe-Ti

0.1030—0.0368—0.0425—0.0326

0.02670.0222

E„(111)E,„(111)E, y(»1)

(111)E„(111)E„„y(111)

0.1030—0.1002—0.0120—0.0326

0.0267—0.0261

H-Fe

E, ,(001)Eh, (001)Eh, d 2(001)

0.59740.14000.4155

can retain the original TiFe cesium chloride SK matrix bytaking the appropriate linear combinations of the (now)two types of titanium and iron atoms in the unit cell

TiFe followed by the tetragonal and orthorhombic distor-tions. The band structure of TiFeH& o has been calculatedby Gupta' using the APW method. In order to fit thiscalculation, one must first double the size of the TiFebasis set to reflect the four transition-metal atoms in thenew unit cell and add two hydrogen basis functions andtheir interactions with the transition metals so that thematrix becomes 38&38. Although one could derive theSK matrix for the orthorhombic structure, it is easier andmore instructive to cast the problem in terms of the origi-nal SK matrix for cubic TiFe. ' This is equivalent to tak-ing b=c=v 2ao where ao is the TiFe lattice constant.Thus, the matrix elements of the transition metals retaintheir x, y, and z cubiclike symmetry, but the interactionswith (and between) the hydrogens are of true orthorhom-bic character. This pseudocubic structure differs at most

'by 8%, in a cell dimension, from the observed one. One

sz; ——[sr;(O,a/2, 0)+sz;(0, —a/2, 0)]/V 2, . . . ,

and

sp, ——[sp, (0,0,a/2) —sp, (0,0, —a/2)]/v 2, . . . .

These linear combinations also greatly simplify thetransition-metal —hydrogen interactions.

We also note that by choosing the appropriate phases(+i,+1) of the basis functions one can write the SK ma-trix without any complex elements by dropping the i inthe (all imaginary) complex elements. ' Using the sym-bolic notations + ( —) for the entire set of symmetric(antisymmetric) basis functions we have

mr+;~h)

~Fr I "i

~p,(h)

hg (hg

m~+ [n,~p+, fh,

m~, fh,

Mp, [h2

hq fh2

0 0

~g; ~~p,0 0

0 ~p, ~~p; Mp, ~Mp,0

hg (~p,h, [mp+,

h, [mr+

"2 I ~~ah, [mp,"2 l ~~a

There are no matrix elements between functions of dif-ferent parity. Furthermore, the M+

~

~+ blocks are justthe original SK matrix for TiFe (Ref. 19), while the

blocks are the SK matrix for

k =ko+(2n/a, vr/a, vr/a) .

This corresponds to folding (translating) the states at thecubic M point back to the I point as noted by Gupta. '

CALCULATIONS OF THE ELECTRONIC PROPERTIES OF. . .


Thus, as ko ranges over the orthorhombic zone the origi-nal cubic zone is sampled. The value k~+2m/a is neces-sary, for although the Ti

~Ti and Fe

~

Fe blocks areperiodic in k„of cubic (orthorhombic) period 2'/a, theTi

~

Fe and Fe~

Ti blocks are not, so in order to maintainthe correct phases for the second atom in the unit cellwith regard to the first, this choice is needed. Thus, onecan form the matrix corresponding to the second atom(s)in the unit cell from the first by substituting

k =ko+(2m/a, ~/a, ~/a) '.One can easily derive the matrix elements in the

h~ ~~~;, h~ ~~Fe, h2 ~~~;, and hq ~~F+, blocks. Itagain turns out that

h;~ ~J (k) =h;

~

~J+. (k+2m/a, m/a, rf/a),

but because of the locations of the hydrogens one cannotachieve any further simplifications (at arbitrary k) by tak-ing linear combinations hI+h2. The hydrogens mix thecubic states of k and k =(2m/a, m /a, n/a). . This matrixnow contains an additional ten parameters which describethe H-H, H-Fe, and H-Ti interactions, for a total of 58parameters. The matrix elements which involve these in-teractions are given in Table II. In our previous calcula-tions' we had made the simplifying decision to freeze themetal-metal parameters to their TiFe values and deter-mine the ten new parameters by fitting them to the threelowest bands of the Gupta's calculations. ' We have now

la

NLal ~-

mm$

TIFe TOTAL DOS

\g

w \g

E tucGY(R. )

IIr

r

NMe~is

TiEeH CPA-DOS x=0.1IIIIIII

IIIIIIIIII

IIIII

ENERGY(Ry)

TiFeH CPA-DOS x=0.9 T iFeH TOTAL DOS

iX

M ~c'-LdI—

~g-MV)Ld

MI—04oI—49Lal c-

-L25 0~ 0~ 0.75ENERGY(Ry)

I

Vl

Lab mCl Mas 0.00

yv ~ e

0~ 0~ %75ENERGY(Ry)

FIG. 1. Total densities of states for TiFeH for x=0.0, 0.1, 0.9, and 1.0.

CALCULATIONS OF THE ELECTRONIC PROPERTIES OF. . . 1293

removed this approximation and have determined all the58 TiFeH~ 0 parameters by least-squares fitting of 20bands from Gupta's results. The rms fitting error wasless than 20 mRy for 20 bands. The TiFeH& 0 SK param-eters are listed in Table III. These parameters uniformlyshift Gupta's energy bands by 0.137 Ry which align thelower I &z levels of the TiFe and TiFeH APW calculationsand put the two calculations in a common energy scale.

IV. CPA THEORY

Hamiltonian is a 38)& 38 matrix which has a 36&(36 blockidentical to that of the periodic material. In the remain-

ing 2X2 block the on-site SK parameter E~ t, (000) is re-

placed by the self-energy Xz which is determined from theCPA condition:

xt„+( I x—)t„=0,where x is the hydrogen concentration, tH and t„ thescattering matrices for hydrogen and vacancy given by thefollowing expressions:

The SK Hamiltonian that we have constructed forTiFeH~ o is used in an extension of Faulkner's CPAtheory. ' In this theory it is assumed that the metal sub-lattice is perfectly periodic while the hydrogen sublatticehas random vacant sites. So in our case the effective

tH=[E& z(000) —Xq]

&& I 1 —[Eh z(000) —Xq]Gh(z, XI, )I

t. = —[GI (»&h)]

(2)

TIFe TI 8 -L I KE

I—Ng-LLC)

TiFeK CPA-Ti-I x=0.1IIIIIIIII

IIIII

II

IIIII

III

'~ I~a aaaENERGY(Ry~

00LLl ~u M~I LOO ~~~ d'J5O L'75 COO

ENERGY(Ry)

T iFeH CPA-Ti-s x=Q.9 T iFeH T I-e

lAtak.I—e-l-40MLLI

I—

I

M

~c-vis 0.00 0~ 0~ 0.75

ENERGY(Ry)

I

Ll ~-OZI

'Loa

'Om Om

ENERGY(Ry}

FICx. 2. Titanium s —like component densities of states for compositions as in Fig. 1.


d kG(z, XI, )= f (4)

where z is the complex energy and H(k, X&) is the 38X38effective Hamiltonian in which only diagonal disorder isincorporated through the self-energy Xl, .

The numerical procedure consists of solving Eqs. (1)and (4) iterating self-consistently to determine X~ andG(z, Xh). The next step is to calculate the DOS 'from theformula

X~(E)= ——lim Im TrG~(E, Xq),1

7T ~~E+(5)

The Green's function G~ (z, X~ ) is the & X 2 block of the38X38 Green's function G(z, X~ ) that is found by the fol-lowing integration over the Brillouin zone:

where the index I refers to the s, p, t2g, and eg symmetriesof the Ti, Fe, and hydrogen components. The evaluationof the integral of Eq. (4) requires the inversion of 38X38matrices at a given energy E, for a large number of kpoints. We have found that satisfactory convergence oc-curred using the following procedure: We performed theintegration for 175 k points in the entire energy spectrumusing an energy interval of 0.01 Ry. This approach pro-vides a reliable determination of the integrated DOS andhence of the Fermi level EF. However, the values ofX(EF) are far from converged when we use 175 k's. Inorder to obtain reliable values for X(EF) we performedthe integrations around EF for a mesh of 4641 k points.

%'e performed three CPA calculations at x=0.1 usingfor the metal sites the SK parameters of TiFe, and atx =0.8 and 0.9 using the SII parameters of TiFeH] 0.

TIFe Fe SWlKE

v g v r w I e

h.~ENggGY(R )

l

I

I

tII

)C

W w1 I

~%A

40

&e~~a

TiFeH CPA-F- =e x=b.)IIIIIIIIIIIIIIIII

IIII

II

(„II

0 O'Pl & ( g "4,

A 9A LsENERGY(Ry)

MLLI

I

40LLII

La

7iFeH CPA-Fe-e x=0.9II

IIIIIIIIIIIIII

V)LJ

V)

MLd

T iEeH Fe-s

IlA

~2$ 8 0 IO.oa o-&~ om

ENERGY(Ry)

I e v w

I—

M

~ o-0.25

I ' ' '/

' ' &

I -» ~l ~ a

0.00 Q.25 0.50 0.75 1.00ENERGY(R )

FIG. 3. Iron s —like component densities of states for given compositions.

32 CALCULATIONS OF THE ELECTRONIC PROPERTIES OF. . . 1295

V. RESULTS AND DISCUSSION

In Figs. 1—11 we present the DOS of two CPA calcula-tions for hydrogen contents x=0.1 and 0.9 and comparethem with the DOS of TiFe and TiFeH~ o. Figure 1shows the total DOS where we note, for the hydrogen-richcases (x=0.9 and 1.0), that the low-lying hydrogen-induced states are centered at zero of the energy scale.The bonding and antibonding transition-metal states areseparated by a minimum in the DOS and the Fermi levelEJ; lies just above this minimum. The value of thisminimum DOS as well as the DOS at EI; increase fromTiFe to TiFeH~ o by approximately a factor of 2. In ourpreliminary paper, ' although we stressed the nonrigid

band characteristics of the hydrogen-metal bonding states,we suggested that Ez increases with increasing x. In thepresent, more accurate set of calculations, we have foundthat E~ is very nearly independent of x, near each end ofthe compositional spectrum, i.e., for x=0.0 and 0.1,Ez 0.67——Ry and for x=0.8, 0.9, and 1.0, Ez ——0.69 Ry.Our conclusion on this point is similar to that of Tem-merman and Pindor in their study of the Pd-Ag-H sys-tem. The low-lying hydrogen-induced states are largely oftype 1 (i.e., previously occupied) and grow at the expenseof the states of mostly iron character. States of type 3(previously unoccupied) are pulled down below E~ to justaccommodate the added electrons with very little readjust-ment of the Fermi level, but with a concomitant increaseof the value of the density of states at the Fermi level.

IV

lK

LLI

09

k-

V) g-

T)Fe T't P-LIKE

4t

ItIfIt

ttt

I

I

MI- e-

4040 ~-

Vl ~-4

TIFeH CPA-T I-p x=Q.)IIll

Il

tIlI

II

I

I

II

I

I

Ld ~~~a $ W T W I W % ~ t V % V $% & V

tl RA O 'tIE¹vGY(Ry)

Q ~(i IIPi

LOO e.&a n.agENERGY(R }

Q-IV

C gy

lA

+e-LLII

I—lA

TIFeH CPA-Ti-p x=0.9Q

V

CK ~y-

T IFeH T I-

4tO

I

-0.25l ~

ENERGY(R )-4~5 O.ea

'~

ENERGY(Ry)

FIG. 4. Titanium p —like component densities of states for stated compositions.


We should point out that the variations of the latticeconstant with x are included in our calculations only viathe parameters at the two ends of the compositional spec-trum. However, we have neglected the small variation ofthe lattice constant between x=1.0 and 0.8, and also be-tween x=0.0 and 0.1. In Fig. 2 the s-like DOS for Ti siteis shown. The effect of hydrogenation on the Ti s statesis that it pushes a small number of states to lower energiesbut decreases substantially the number of bonding statesaround 0.3 Ry.

In Fig. 3 the Fes —like DOS are presented. The effectof hydrogenation now is to split the bonding states intotwo distinct peaks, one of which is now centered at 0.0Ry.

In Fig. 4 the Ti p —like DOS are shown. The introduc-

tion of large amounts of hydrogen creates again states atlow energies and in addition induces a pronounced peak at0.5 Ry.

The Fe p —like DOS shown in Fig. 5 display again 1ow-

lying states induced by the hydrogen presence in the lat-tice. Also the double peak in the bonding states forx=0.0 and 0.1 is replaced by a single broader peak forx=0.9 and 1.0.

The Ti d —like DOS are shown in Fig. 6. Besides theappearance of low-lying states for the hydrogen-rich casesthe effects of hydrogenation are less pronounced here.

The Fed —like DOS shown in Fig. 7 indicate a muchstronger presence of low-lying states than in other cases.Having decomposed the d DOS into t2g and e~ sym-metries we have found that these low-lying states have ex-

Q

V

LLI

NNLal

T iFe Fe p-L I KEIIIIIIIIIIIIIIIIIIIIIIII

NNLLl

f-M

7iFeH CPA-Fe~ xM.)IIIIIIIIIIIIII

IIIIIIIII

M4l ~al &125

940

lX ~y-

VlLdI—fgggg IO ~49V)LLI

I—lA

4o

Loo ~~ A Atl LpENERGY(Ry)

7iFeH CPA-Fe p x=0.9I

v r I e v v I r

~&A giRtl

ENERGY(R )

T IFeH Fe-p

), ( j'i)

L7O %00

I

MLal c-

-ais 0.00 0~ 0.5b 0.75

I

Vl

-e~~ om OMENERGY(Ry) ENERGY(R )

FICE. 5. Iron p —like component densities of states for stated compositions.


elusively ez symmetry and that they describe the H—Febonding. As in the Ti d DOS the rest of the d DOS spec-trum remains relatively unchanged.

In Fig. 8 the H-site DOS are shown for x=0.1, 0.8, 0.9,and 1.0. %'e note that the H contribution near EI; is smallwhile the strong hydrogen participation occurs at thelow-energy end of the spectrum.

Figure 9 shows the integrated DOS for the low "impur-ity band" which ends at approximately 0.1 Ry. Lookingat the total number of electrons we note a more rapid de-pletion of states than what a bound state model wouldpredict. For example, at x=0.8 our graph shows 1.3 elec-trons while such predictions give 1.6 electrons. %'e alsonote from Fig. 9 that this impurity band mainly consistsof Fe s, Fe eg, and H s, electrons. This implies that H

bonds strongly with Fe atoms rather than Ti as might beexpected because of its positions between Fe atoms in thelattice. These results also suggest that in the formation ofthe hydride charge transfer occurs from the H to the Fesites.

In Fig. 10 the integrated DOS up to E~ is shown. Con-sistent with our observation that the d states undergominor modifications upon hydrogenation, we note that thenumber of d electrons varies slowly with H content. Onthe other hand, the number of s and p electrons shows amore drastic variation.

The variation of the DOS at Fz, X(F~), is presented inFig. 11. The total X(FF) increases from TiFe to TiFeH~ 0by approximately a factor of 3. This is consistent withthe measurements of Hempelmann et al. who reported a

QV~S-

IR

PgJ R

I~ O

Lal

Vl

LL e4C3

T iFe T i Q-LIKEIII

IIII

IIII

IIIII

II

LL

O

TiFaH CPA-T I-4 xM.'i

IIIII

II

II

I

I- R-4f)

LLI ~r1 e~5 ti 4 OygENERGY(R )

&au &&5 O.M'I ) e ~

gj QjL A Rti

ENERGY(Ry)

I

I

L75 %00

I~Q-

CL

Vl gI—I- c,

Vllal

V)

4C3

T iFeH CPA-T i-d x=0.9 T & FeH T't-d

~Q-40hdc- '

-0.25V Y

g'~ T

0-0d 0~ 0.5dENERGY(Ry)

0.7S

4I)

QA ca-.-OZI O.OO

ENERGY(R&)b.75

FIG. 6. Titanium d —like component densities of states for stated compositions.


VQ-

lX

V) gN—

Vl ~-

4A

M

taC3

T iFe Fe d-L I KE TiFeH CPA-Fe-3t

x=0.$lIII

IIIII

IIIIII

N- Q-M

Lal ~cl Mzs L25 0 L75 UNENERGY(Ry)

LOOr

y e r vy

e

A~ @Ah

ENERGY(Ry)

w vy

r e

OV~Q-lX

l/l g

C

4J

694 e4"

~Q™Miaaf c=

-ms

T iFeH CPA-Fe-d x=O.Q

IIII

I

0m Lsa O.VS

ENERGY(Ry)

OV

CL

Mg

M4kI gI—40

LL ee

TiFeH Fe-d

ENERGY(Ry)

N- Q-

-L25 0.00 0.75

FIG. 7. Iron d —like component densities of states for stated compositions.

rapid increase of the specific heat upon hydrogenation. Itis interesting to note from Fig. 11 that while all the othercomponents of X(Ez) increase as a function of x.the Feeg component decreases.

VI. CONCLUSIONS

Since our calculation at x=1.0 is fitted to Gupta'sAPW results, it is consistent with her conclusions regard-ing the lowering of states and the introduction of newones, apart from some quantitative differences due to

small errors in fitting and the different ways of obtainingthe DOS. We remind the reader that we have assumed inthe hydrogen sublattice a random substitution of hydro-gen atoms by vacant sites. So in our model the variationof the amount of hydrogen is coupled with the effect ofvacancy disorder.

Our main conclusions are that the position of EF is in-sensitive to the variation of x but N(Ez) increases rapidlywith increasing x. The general shape of the DOS is af-fected, by changing x, only at low energies where thehydrogen-metal bonding states are located. The effect ofdisorder can be seen by noting the broadening of the H-site DOS at x=0.8. It will be very interesting to examine


T1FeH CPA-H-e x=0.1 TiFeH CPA-H-e x=0.8

IlhN se)-LLII-

M~-LLC3

ilhM~ s ~ ~ I v

~Rtl

ENERGY(R )

IIIIIIIII

I

I1IIII'IIIIIIIIII

IIIIII

~e-lX

MLali- e-IMVl e~-Lal

I—EA ~

Vl

Lal—-0.25

ENERGY(Ry)

IIIII

I

IIIIIIIIIIIIII

III

IIIIII'I

%00

lK

VlLaS

M0 eo-LLII

M ~-

)ge ys ~

I

40

LLl c-0.25 0.00

I TiFeH CPA-H-e x=0.9IIIIIII

II

IIIIIIIIIIIIIIIIIIIIIIII

0.50 0.75

-V

lX

VlLLI

I49Vl ~4AI

Vl g

M.25 0.00 0.25

T i FeH H-DOS

0.50

IIIII

IIIIIIIII

I

IIIIIIIIIIIIIIII

I

0-75 1-00

ENERGY(Ry) ENERGY(Ry)

FIG. 8. Hydrogen s —like component densities of states for stated compositions.

2. 00 5.00I 1 I

Fe—tg g

4.00

1.20

1.00

1.50Impurity BarId

1.25z0 1.00I-U

o. V5ttj

&.50

0.25

0.00 I I I

0-50 0. 60 0. 70 0. 80 O. BO 1.00HYDROGEN CONTENT

+ 3.ooOK

Uw 2-OO

w

1.00

Fe—e g

HYDROGEN CONTENT

Ti —eq

0.00 Q, 20 Q. 40 Q. 60 Q. BO 1.0

p. 6oUiZ0& 0. 60I-

w 0.40

P. 2O

Tl —S0.00 I I

o. oo Q. ro Q, 4o 0.6o Qao 1.o

HYDROGEN CONTENT

FIG. 9. Character of electrons in the low-lying hydrogen-induced band for the major components.

FIG. 10. Variation of the components of the total electroncharacter with composition for the major contributors —leftpanel, and for the minor contributors —right panel.

D. A. PAPACONSTANTOPOULOS AND A. C. SWITENDICK

30.0

—20. 0

K 15.0tOLU

1 10.0I-tA

5.0

I I

N(E:~)

O

1.0W

g Q. 5

the effects of disorder by introducing into our model ran-dom substitutions of Ti and Fe atoms in the metal subjat-tice.

ACKNQ%LEDGMENTS

0.0 j0.0 O. 2

I & 2o 0.0 ~ I I

0. 4 0.6 0.6 1.0 0.0 0.2 0.4 O. 6 0.6HYDROGEN CONTENT HYDROGEN CONTENT

FIG. 11. Variation in the value of the total and componentdensities of states at the Fermi level with composition: leftpanel —major contributions, right panel —minor contributions.

We are indebted and grateful to Dr. Michelle Gupta forcopies of unpublished work and numerous helpful discus-sions. The work performed at Sandia Laboratories wassupported by the U.S. Department of Energy under Con-tract No. DE-AC04-DP00789.

~A. C. Switendick, in Hydrogen in Metals I, edited by G. Alefeldand J. Volkl [Top. Appl. Phys. 28, 101 [19781].

D. A. Papaconstantopoulos, in Metal Hydrides, No. 76 ofXA TO Advanced Study Institute Series 8, edited by G. Bam-bakidis (Plenum, New York, 1981), pp. 215-242.

M. Gupta, Metal Hydrides, No. 76 of NATO Advanced StudyInstitute Series 8, edited by G. Bambakidis (Plenum, NewYork, 1981),pp. 255—272.

C. D. Gelatt, Jr., H. Ehrenreich, and J. Weiss, Phys. Rev. 817, 1940 (1978).

5A. C. Switendick, J. Less-Common Met. 101, 191 (1984).6C. T. Chan and S. G. Louie, Phys. Rev. B 27, 3325 (1983).7W. M. Temmerman, and A. J. Pindor, J. Phys. F 13, 1627

(1983); A. J. Pindor, W. M. Temmerman, and B. L. Gyorffy,ibid. , F 13, 1869 (1983).

8J. J. Reilly, Jr., Z. Phys. Chem. Neue Folge 117, 155 (1979).A. R. Miedema, J. Less-Common. Met. 32, 117 (1973); P. C.

Bonten and A. R. Miedema, J. Less-Common Met. 71, 147(1980).

~OC. E. Lundin, F. E. Lynch, and C. B. McGee, J. Less-

Cominon Met. 56, 19 (1977).' D. G. Westlake, in Metal Hydrides, No. 76 of ERATO Ad-

vanced Study Institute Series 8, edited by G. Bambakidis (Ple-num, New York, 1981),pp. 145—176.

~2P. Soven, Phys. Rev. 156, 809 (1967).' D. A. Papaconstantopoulos, Phys. Rev. 8 11, 4801 (1975}."M. Gupta, J. Phys. F 12, L57 (1982); Phys. Lett. 88A, 469

(1982).5J. S. Faulkner, Phys. Rev. B 13, 2391 (1976).

' D. A. Papaconstantopoulos, B. M. Klein, J. S. Faulkner, andL. L. Boyer, Phys. Rev. 8 18, 2784 (1978).D. A. Papaconstantopoulos and A. C. Switendick, J. Less-Common. Met. 88, 273 (1982).

'8J. C. Slater and G. F. Koster, Phys. Rev. 94„1498 (1954).J. D. Shore and D. A. Papaconstantopoulos, J. Phys. Chem.Solids 45, 439 (1984).

2oR. Hempelmann, D. Qhlendorf, and E. Wicke, in Hydrides forEnergy Storage, proceedings of an International Symposium,Geilo, Norway, edited by A. F. Andresen and A. J. Maeland(Pergamon, New York, 1977), p. 407.

calculations of the electronic properties of...

Documents