Trends in the electronic structure of the polyvalent liquid metals

304

Journal of Non-Crystalline Solids 117/118 (1990) 304-307 North-Holland

TRENDS IN THE ELECTRONIC STRUCTURE OF THE POLYVALENT LIQUID METALS W. JANK and J. HAFNER Institut fiir Theoretische Physik, TU Wien, Wiedner Hauptstrat~e 8-10, A-1040 Wien We present ab-initio calculations of the atomic and electronic structure of polyvalent metals from group IIb and IV of the periodic table demonstrating the influence of relativistic effects.

1. INTRODUCTION

2.

THEORY

It is by now well established that both the crystalline

The calculation of the atomic and electronic structure of

and the liquid structures of the s,p-bonded elements follow

the molten metals is an exceedingly complex task. Only the

characteristic trends through the periodic table 1. These

main lines of our approach may be stretched here:

include (a) a change from close-packed metallic to open co-

(a) Interatomic forces (consisting of pair- and vol-

valent structures within each row as the valence increases,

ume forces) have been constructed using N F E - pertur-

(b) in group IIb an increasing distortion of the dose-packed

bation theory 1.

structure of Mg, Zn and Cd, culminating in a unique crys-

tentials yield good results but for the heavy metals it is

talline and liquid structure in Hg, (c) in group I I I a simi-

necessary to use non-local pseudopotentials derived from

lar phenomenon with a close-packed structure in A1 and a

an orthogonalized-plane-waVe ( 0 P W ) expansion of the

unique structure in Ga, but then a return to a more dense

conduction-band states. At least the core-states entering

arrangement of the atoms in In and T1 and (d) in groups

the O P W have to be calculated in a scalar relativistic ap-

IV to V I a return from open, low-coordinated structures to

proximation 3

For the light elements empty-core po-

close-packed structures with high coordination numbers as

(b) The atomic structure of the melt is calculated via a

the atomic number increases, most pronounced in the series

classical constant-energy molecular dynamics (MD) for en-

Si-Ge-Sn-Pb.

sembles with 64 to 1372 atoms in a cubic box 4. The smallest

Recent photoemission studies have shown, that these trends in the atomic structure are reflected in systematic variations in the electronic spectrum 2. 0nly for the light

ensembles are used as the basis for the calculation of the electronic structure using a supercell method. (c)

The

electronic

spectrum

is

calculated

using

third-row elements (Mg, A1, Si) the electronic density of

the linearized-muffin-tin-orbital (LMTO) method in the

States (DOS) shows the expected parabolic nearly-free-

atomic-sphere-approximation 5 for instantaneous configura-

electron (NFE) form.

For all the heavier elements pro-

tions of 64 atoms in a periodically repeated supercell. To get

nounced structures in the DOS are observed: the hybridiza-

a good energy resolution in the DOS it is necessary to aver-

tion of the s- and p-orbitals characteristic for plane-wave-

age the eigenvalues over four to ten k-points in the Brillouin-

like states is broken up, s- and p-band are separated by a

zone of the cell (for further details see s).

DOS minimum in the middle of the conduction band. These structures become more pronounced as the atomic number increases. In this paper we present the results of our investigations on the influence of relativistic effects in the observed trends in the electronic and atomic structures of the polyvalent liquid metals.

0022 3093/90/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland)

3. RELATIVISTIC EFFECTS IN THE ATOMIC STRUCTURE OF THE POLYVALENT LIQUID METALS It has been shown that the open structures of the crystalline and liquid polyvalent metals, semimetals and semiconductors may be explained in terms of the interplay of volume-and pair-forces 1--7: for Si, Ge, P, As, Se, Te . . . t h e

W. Jank, J. Hafner / Electronic structure of the polyvalent liquid metals

305

nearest-neighbour distance D ~ in a close-packed structure

treatment of the core states leads to the expected screened-

(which is determined largely from the volume forces) falls

Coulomb form of ~(P~) at the nearest-neighbour distance

on a repulsiv hump of the potential. Hence it is energeti-

and to a I-IS-like structure factor(Fig.l(b)). These relativis-

cally favourable to split the nearest-neighbour shell into two

tic effects on the pair potential and on the liquid structure

subshells roughly centered at D1,2 = D ~ 4-AF/2

where

factor are a direct consequence of the strong non-locality of

AF = 2~r/2kF is the Friedel wavelength of the oscillations

the pseudopotential resulting from relativistic effects which

in the pair potential. Note that this argument is a gener-

tend to bind the s-electrons more strongly than p - and d -

alized real-space formulation of the Pelerls-distorsion argument conventionally used to explain the crystal structures of P, As, Se, Te, e t c . .

S

(q)

(b)

For these light elements even calcula-

tions using pair forces derived from simple model potentials are quantitatively successful 3,4,7. The return to close-packed

2

(and in the liquid to more hard-sphere (HS)-like) structures in Sn, Pb, Sb, Bi is more difficult to explain. It has been argued 1,8 that it is associated with a progressive damping of the Friedel oscillations in the pair potential (I)(R), but with a simple local pseudopotential large ad-hoc adjustments of the model parameter were found necessary to fit the trend. In our extended MD studies of the heavy liquid elements we

(c)

have found that it is impossible to fit both the proper atomic

~i

size and the amplitude of the oscillations in ~(R) with a sin-

T• =o 613 • expK

gle value of the model-potential radius. A calculation with a non-relativistlc pseudopotential for Pb gave a much more realistic pair interaction (Fig.l(a)), but the calculated liquid structure factor showed a shoulder like that found in liquid Sn (Fig.l(c)). Only a calculation based on a relativistic

(R)

(a)

(Ry)

Pb

0.01

0

~ 0

, 2

I Z.

I

I 6

I

I 8

I q (,~-~)

FIGURE l(b,c) Static structure factor for liqid Pb, calculated using a nonrelativistic (c) and a relativistic-core pseudopotential (b). Full line: MD simulation, circles: neutron diffraction electrons. The amplitude of the Friedel oscillations in ~(H), which is set by the square of the pseudopotential matrix element < ~ c + ~ ' [ w [ /~ > is positive in a backward- and negative in a forward-scattering geometry. Hence the effec-

J

I

I

I

tive matrix element for q = 2kF averages out to a value very

2

z,

6

8

close to zero and this leads to the screened-Coulomb form

R(A) FIGURE l(a) Effective pairpotential for liquid Pb calculated using a local empty-core pseudopotential (dotted), a n o n relativistic OPW-pseudopotential (dashed), and an O P W pseudopotential with a relativistic core (full line)

of ~(R) stabilizing a close-packed, HIS-like structure. Similar effects have been found in liquid T1 and Bi. In liquid Hg on the other hand the damping in the Friedel oscillations leads to a structure which is still essentially HS-like, although somewhat distorted.

W. Jank, J. Harrier~Electronic structure of the polyvalent liquid metals

306

4.

RELATIVISTIC EFFECTS IN THE ELECTRONIC

For liquid Pb the calculated DOS is very similar to that of

SPECTRUM OF THE POLYVALENT LIQUID

the crystalline fcc-Pb. Note that total energy calculations for

METALS

Pb have shown that the splitting of the conduction band into

We now turn to the investigation of relativistic effects in

an s- and a p - b a n d driven by relativistic effects is essential

the electronic structure. Fig.2(a) shows the variation in the

for stabilizing the fcc lattice relative to the diamond structure

electronic DOS in the series Si-Ge-Sn-Pb, as calculated self-

9. Our results show that the same relativistic effects are also

consistently using the LMTO-supercell method. As both liq-

important for understanding the HS-like structure of liquid

uid Si and Ge axe good metals, one would expect a parabolic

Pb.

NFE-like DOS, such as it is found in the metallic high-

Ge

pressure polymorphs with the simple hexagonal structure. However, only Si confirms to this expectation. In the three

/

heavier elements we find a pronounced minimum in the DOS

t

at a binding energy of about 4eV, for liquid Pb we even predict a gap of ~ 1.2eV separating s-like bands at higher binding energy and p-like bands close to the Fermi level. The theoretical predictions are in good agreement with the

n3

measured photoemission spectra (Fig.2(b)), the analysis of

..>,

/ i

Pb

1

~

the variation of the spectra with the photon energy allows to confirm the angular-momentum character of the bands.

E

]!

~0.5

i

i

,

-10 -5 0 Binding Energy (eV}

0) 0

E

FIGURE 2(b) Measured photoemission spectra I(E) for the liquid group IV-elements (afterRef. 2). Full line: 40.8 eV, dashed line: 16.8 eV

©

~co0 . 5 U)

0

4J r~ 4-3

For liquid Ge and Sn the electronic DOS is different from

cnO.5

LQ CD

that of any of the crystalline modifications: a - G e (diamond structure) has a DOS with the characteristic structure of spa-hybrids stabilizing the tetrahedral bonds and a gap at

0

the Fermi level, fl-Ge (white-tin structure) is metallic but retains a considerable degree of sp3-hybridization and simple-

0.5

hexagonal Ge has a nearly parabolic NFE-DOS. The difference is easily understood in terms of the atomic struc"0 E

-5 E:r

~ eV)

F I G U R E 2(a) Electronic DOS for liquid Si, Oe, Sn and Pb. Full line: total DOS, dotted: s-partial, dashed: p-partial and dot-dashed: d-paxtial DOS

ture: the thermally-induced disorder is strong enough to destroy the directional sp3-hybridization of the valence states and to close the gap.

On the other hand the increase in

the coordination number Arc and in the density on melting (No = 4 in a - G e , No = 6.8 in l-Ge, p = 5.26 g/cm 3 in a -

W. Jank, J. Hafner / Electronic structure of the polyvMent liquid metMs

Ge, p = 5.51 g/cm 3 i n / - G e ) is not large enough to broaden the atomic energy levels to a free-electron band.

307

liquid Hg the situation is complicated by the presence of a d-band overlapping with the bottom of the conduction band.

To first order the observed trends in the electronic DOS

Relativistic effects lead to a larger separation of the s- and

scale with the difference in the atomic s- and p-eigenvalues.

p-states and this causes the appearence of a weak DOS-

It is interesting however that the change from Si to Ge is

minimum at the Fermi level. Again this is well confirmed

much larger than that from Ge to Sn. The mechanism re-

by photoemission studies (Fig.3(b)) (note however that self-

sponsible for the large s-p splitting in Ge is the partial pen-

energy corrections will be necessary to account for the proper

etration of the 3d-core shell by the 4s-electrons, which feel

position of the d-band). We expect the DOS-minimum to

a larger attractive potential from the core.

become more pronounced in expanded l-Hg as the coordina-

Similar relativistic effects are found in all heavy polyva-

tion number decreases.

lent metals. As an exapmle we show in Fig.3(a) the selfconsistent DOS for liquid Mg and ttg. In l-Mg the DOS

5. CONCLUSIONS We have presented the first ab-initio calculations of the

is NFE-like with large s, p and d-contributions at EF. In

atomic arL.d the electronic structure of the polyvulent liquid metals. We find that relativistic effects are essential in under-

r~~{ vo 0.SE .... HGIV-.I,_=.~~!tl 0 --4.0\. o2.8 ~n 0 (--I0.5 LQ

standing the atomic structure and the electronic spectrum of the heavy molten elements. Details of the calculations and further results will be presented in forthcoming papers 3,10

HGj ~

ACKNOWLEDGEMENTS This work has been supported by the Austrian Science Foundation under projects no. 6191 and 7192. The numerical calculatuions have been performed on the IBM 3090400E VF of the University of Vienna supported by the IBM

X8

l ~

/J

i f' ,~:' -

J"

European Supercomputer Project.

.-t--

REFERENCES

~J L,;.'L"

8

( (?",/ }

FIGUI~E 3(a) Electronic DOS for liquid Mg and Hg. Symbols see Fig.2(a)

i. J. Ha:fner, From Hamiltonians to Phase Diagrams (Springer, Berlin 1987). 2. G. Indiekofer, P.Oelh~en, I{. Lapka and H.J. Giintherodt, Z.Phys.Chemie 157,465 (1988). 3. J. Hafner and W. Jank, Phys.l{ev.B (submitted). 4. A. Arnold, N. Mauser and J. Hafner, J.Phys.-Cond.Matter 1,965 (1989). 5. H. Skriver, The LMTO Method (Springer, Berlin, 1984). 6. W. Jank and J. Hafner, Europhys.Lett. 7, 283 (1988),

E

7. J. Hafner, Phys.Rev.Lett. 62,784 (1989); J.Phys.-Cond.Matter (submitted).

-10 -5 0 BindingEnergy(eV)

FIGUI{E 3(b) Photoemission intensity I(E) for liquid ttg. Fig.2(b)

Symbols see

8.

J. Hafner and V. Heine, J.Phys.F 13, 2479 (1983).

9.

N. E. Christensen, S. Satpathy and Z. Pawlowska Phys.Rev.B 34, 5977(1986).

10. W. Jank and J. Hafner, to be published.