Ab-initio calculations of the atomic and electronic structure of metallic glasses

270

Journal of Non-Crystalline Solids 106 (1988) 270 273 North-Holland, Amsterdam

AB-INITIO CALCULATIONS OF THE ATOMIC AND ELECTRONIC STRUCTUREOF METALLIC GLASSES J.HAFNER and M.TEGZE+ I n s t i t u t f u r Theoretische Physik, Technische Universit~t Wien, A 1040 Wien, Austria S.S.JASWAL Behlen Laboratory of Physics, University of Nebraska, Lincoln, Nebraska 68 588, USA A b - i n i t i o c a l c u l a t i o n s of the atomic and the e l e c t r o n i c structure of m e t a l l i c glasses have been performed, with no other input than the composition and the atomic numbers and atomic weights of the components. Pseudopotential-derived interatomic forces, molecular dynamics c a l c u l a t i o n s , and a "potential-energy-mapping" technique have been used to construct r e a l i s t i c models f or the atomic structure. The e l e c t r o n i c structure is calculated using a supercell l i n e a r i z e d muffin t i n o r b i t a l method. Results are presented f o r Mg- and Ca-based m e t a l l i c glasses. They are compared with the atomic structures and the e l e c t r o n i c densities of state of the c r y s t a l l i n e phases and the i n t e r play of the atomic and e l e c t r o n i c structures is discussed. 1. INTRODUCTION The knowledge of the valence band-structure

is not the only d i f f i c u l t y

that has to be over-

come: in a c r y s t a l l i n e s o lid the positions of

of a material is essential fo r the understanding

the atoms within the u n i t cell are known from

of many of i t s physical properties. With the ad-

d i f f r a c t i o n experiments. In an amorphous s o lid

vent of modern t h e o r e t i c a l concepts such as den-

the experiment y i e l d s only a s t a t i s t i c a l

s i t y functional theory, pseudopotentials, and

mation on the atomic structure. The atomic co-

l i n e a r i z e d band-structure techniques the elec-

ordinates that c o n s t i t u t e the input f o r a calcu-

infor-

t r o n i c theory of c r y s t a l l i n e solids has become

l a t i o n of the e l e c t r o n i c structure must be de-

a tool f o r q u a n t i t a t i v e predictions of materials

rived from an atomic structure c a l c u l a t i o n that

properties I. The adequate description of the

should be consistent with the c a l c u l a t i o n of the

e l e c t r o n i c structure of a crystal consists of

e l e c t r o n i c structure i t s e l f .

the dispersion r e l a t i o n s E(k) along the major

The i n v e s t i g a t i o n of the e l e c t r o n i c structure

symmetry d i r e c t i o n s and of the e l e c t r o n i c densi-

of amorphous m e t a l l i c alloys was a quite popular

ty of states (DOS) n(E).

subject at the end of the seventies. The i n t e r -

The f a c t that the one-electron states may be c l a s s i f i e d according to the wavenumber ~ is a

est was motivated on one hand by the conjecture of Nagel and Tauc2 that the e l e c t r o n i c structure

consequence of the t r a n s l a t i o n a l p e r i o d i c i t y of

should be an important factor determining the

the c r y s t a l . For an i s o t r o p i c amorphous s o l i d or

glass-forming a b i l i t y ,

a l i q u i d the lack of t r a n s l a t i o n a l invariance

the f i r s t results from photoelectron spectro3 scopy showing large and unexpected valence-

means not only that the dispersion r e l a t i o n is

and on the other hand by

no longer defined (whereas the concept of the

band s h i f t s in m e t a l l i c glasses compared to the

DOS is s t i l l

pure metals. However, the a c t i v i t y in this f i e l d

meaningful) but that the powerful

t h e o r e t i c a l and computational tools based on

declined r a p i d l y when i t was found to be d i f f i -

Bloch's theorem are no longer applicable. This

c u l t (both experimentally and t h e o r e t i c a l l y ) to

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J. Hafner et al. / Atomic and electronic" structure of metallic glasses

2 71

dl

g4+a(R) 6

CasAI5

CaTAt3

~d,

Ca A[ z

5

dl

3

~d, 2

"~-d~

2di

I

71

2 I

0

9jI o

5

5

& 3

~.d, I

~

2

®,

d,

1

1

9 ~ROI

0

~-~ 4

& 3

3 2"d,z. ~

2a~ Zd '

2

2

1

1

0 0

0

&

6

8

10

0

:2

6

RE~]

8

10 0

2

6

8

10

FIGURE 1 P a r t i a l pair c o r r e l a t i o n functions in amorphous Ca-A1 a l l o y s . Full l i n e - configuration average over 25 independent 800-atom c o n fi g u r a ti o n s , histogram - single 60-atom model (see t e x t ) • v e r i f y the existence of the structure-induced

•

energy mapplng

,,5,6

techniques f or the determina-

minimum in the DOS predicted by Nagel and Tauc

tion of the amorphous structures. The computer

and that s i m i l a r valence-band s h i f t s e x i s t in

experiments are performed f or large (N 800-1500

the c r y s t a l l i n e a l l o y s as w e l l , the differences

atoms) f o r the i n t e r p r e t a t i o n of the d i f f r a c t i o n

in the e l e c t r o n i c structure being very d i f f i c u l t

data, and f o r small models (N 60 atoms) which

to characterize• In t h i s paper we return to the problem of the i n t e r p l a y of the atomic and e l e c t r o n i c structures.

serve as the basis f o r the e l e c t r o n i c structure c a lc ulat ions . (c) The e l e c t r o n i c DOS is obtained from a s e l f - c o n s i s t e n t l i n e a r i z e d - m u f f i n - t i n - o r b i t a l (LMTO)7 c a l c u l a t i o n f o r the sixty-atom su-

2. THEORY Our c a l c u l a t i o n s are based on: (a) I n t e r a t o -

p e r c e l l . Technical d e t a i l s w i l l be given in forthcoming

reports 8'9'10.

mic forces derived from optimized f i r s t - p r i n c i p les pseudopotentials. The optimization serves to old back higher-order perturbation terms into

3. ATOMIC STRUCTURE The phase diagrams of the glass-forming sim-

lower order and is important to achieve a r e a l i s -

ple-metal systems show a uniform pattern: stable

t i c v a r i a t i o n of the interatomic forces with com-

congruently melting compounds with t e t r a h e d r a l l y

position 4. (b) Molecular-dynamics and " p o t e n t i a l -

close-packed structures (often a Laves-phase) at

J. Hafner et aL / Atomic and electronic structure of metallic glasses

272

a_Mg?Zn3

LEF

o

0B

t

/

l

-

I

-

,o,o, -

-

zn

o

I

Mg

o

0.8

"6

i°'

......

I

totot

- -

°,

g 0 0

d

...

0.l,

8

~

0

o

~

-8

-6

-4

( E - E F)

(E-E F) (eV)

~ -2 ~

0 0

-8

-6

(eV)

• -2

-4 ( E - EF)

7, -0

(eV)

FIGURE 2 Total and p a r t i a l e l e c t r o n i c DOS f o r c r y s t a l l i n e (C14-structure) and amorphous Mg-Zn a l l o y s . a majority concentration of the smaller atoms

4. ELECTRONIC STRUCTURE

and a deep eutectic minimum at a majority of the

In Mg-Zn a l l o y s (Fig.2) we f i n d a broad s,p-

larger atoms. Our r e s u l t s show that even in the amorphous state the atomic structure is s t i l l

band (Mg and Zn) and at i t s bottom a narrow Zn-3d band. In the crystal the e l e c t r o n i c OOS is stron-

l a r g e l y dominated by the e f f e c t s of tetrahedral

gly structured by the van-Hove s i n g u l a r i t i e s of

close-packing: the peaks in the c o r r e l a t i o n func-

the complex crystal s t r u c t u r e , in the glass the

tions coincide with the i s t ,

DOS of the s,p-band is very close to a f r e e - e l e c -

2nd and 3rd neigh-

bour distances in the Laves phase CaAl2(Fig.1 ).

tron parabola - except for a small "dip" at the

The bond-angle d i s t r i b u t i o n functions show that

Fermi l e v e l . This is the structure-induced DOS

there are many icosahedrally coordinated sites

minimum postulated by the Nagel-Tauc conjecture.

in both the c r y s t a l l i n e and the amorphous structures 6 ' 8 ' 9 . In the Ca-based alloys e l e c t r o n i c

That i t is so very weak is well confirmed by photoemission and e l e c t r o n i c s p e c i f i c heat data9:

bonding effects r e s u l t in a contraction of the

Ca-Mg and Ca-A1 a l l o y s o f f e r an i n t e r e s t i n g

Ca-Ca distances in both the c r y s t a l l i n e compounds

contrast in t h e i r e l e c t r o n i c properties: low

and in the m e t a l l i c glasses. This chemical com-

resistivity

pression of the Ca-atom (small in Ca-Mg, large

f o r Ca-Mg, high r e s i s t i v i t y

in Ca-A1, Ca-Zn) is associated with a concentra-

rature gradient f o r Ca-A1. Usually this s t r i k i n g

and p o s i t i v e temperature c o e f f i c i e n t and negative tempe-

tion-dependent v a r i a t i o n of the pseudopotential

difference is a t t r i b u t e d to the occupation of

and a r e - d i s t r i b u t i o n of the p a r t i a l e l e c t r o n i c

strong-scattering Ca-3d states in Ca-A1 glasses,

charges.

but not in Ca-Mg glasses. In agreement with re-

J. Hafner et al. / A tomic and electronic structure of metallic glasses

n(E) n(E)

COAL, Laves phose

i

E~

] n~,,(E) [ pa~t,al AL

273

E~

%6 04

0,4

d

(I

0,2 0

8

n(EZ) n,(E)12 I

6

L,

-2

0

Co,Al - glass

-8 n4~ (E)

E,

6

-4

-2

0

-8

/

nooSE) 1,2

E,

partial At

0,8

0,8

0,4

0,4

0 nE

-6

2

-4

0

FI

, -6

total

,~

---

EF

Mg

L

t

-2

-2

2

'

-6

-&

2

0

Mg]

~ [

~ -~

0

E~ j

7-.

0

0

-4

,_

_

~- ~-. c_7.s

i

Ca Mg 2- Laves phase

6

n,(E)

-2

! ,

0,8

0

~ -4

I

n(E) L - '

o

-6

parbal Ca s P d

-6

0

n~(E) I

-4

-6

-2

pQrha[ Mg

n~(E)l,0 [

C

-4

2

0

-2

0

parha[Ca

1~0 1,0 f

0,5 0

0,5

-6

4

-2

0

0

0

6

-4

-2

0

L -6

-&

BINDING ENERGY (E-E~) (eV)

FIGURE 3 Total and partial electronic DOS for c r y s t a l l i n e (C14-CaMg 2, C15-CaAI2) and amorphous Ca-based alloys. cent photoemission data our results show that

REFERENCES

this explanation cannot be correct.

1. Computer-based microscopic description of the structure and properties of materials, eds. J.Q.Broughton et al. (MRS,Pittsburg 1986).

In both cry-

s t a l l i n e and amorphous Ca-Mg alloys we find a strong onset of Ca-3d states already 2 eV below EF. The Mott factor is g=n(EF)/nfree(EF)~117. In Ca-A1 the s,p-hybridization is strongly reduced, in the glass the AI-3s dominated band is separated by a DOS-minimum from the AI-3p states i n t e r -

2. S.Nagel and J.Tauc, Phys.Rev.Lett. 35(1975)380 3. P.Oelhafen et al. Phys.Rev.Lett. 4.33(1979~i134 4. J.Hafner. From Hamiltonian~ to Phase Diagrams (Snrinner. Berlin 1987~

acting with the Ca-states, the Mott factor is on-

5. F . H . S t i l l i n o e r and T.A.Weber. Phvs.Rev. B31 (1985)5262

ly g=l.4. The narrowing of the sub-bands is con-

6. J.Hafner, J.Phys. F18(1988)153.

firmed by UPS and SXS data. The integrated DOS

7. J.Skriver, The LMTO Method (Springer, Berlin 1981).

shows that the interatomic charge transfer is small, but there is an important intra-atomic charge r e d i s t r i b u t i o n from s to d on the Ca-sites

8. S.S.Jaswal and J.Hafner, Phys.Rev.B (in print) 9. J.Hafner et a l , J.Phys.F (submitted). I0. S.K.Bose et a l , Phys.Rev. B (in print)

and from d to s on the A l - s i t e s . Evidently this is related to the compression of the Ca-atoms.

ACKNOWLEDGEMENTS

Preliminary results show that the incipient lo-

This work has been supported by the Jubil~ums-

calization of the electron states in Ca-A1, but

fonds der Usterreichischen Nationalbank under

not in Ca-Mg is responsible for the difference

project no. 6191.

in the transport properties.