Materials Transactions, Vol. 48, No. 6 (2007) pp. 1292 to 1298
Special Issue on Materials Science of Bulk Metallic Glasses-VII#2007 The Japan Institute of Metals

Electronic Structure and Stability of the Pd-Ni-P Bulk Metallic Glass
Tsunehiro Takeuchi1;2;3 , Daisuke Fukamaki3 , Hidetoshi Miyazaki4 , Kazuo Soda4 ,Masashi Hasegawa5 , Hirokazu Sato6 , Uichiro Mizutani7 ,Takahiro Ito8 and Shinichi Kimura8
1

EcoTopia Science Institute, Nagoya University, Nagoya 464-8603, Japan
Department of Crystalline Materials Science, Nagoya University, Nagoya 464-8603, Japan3
Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
4
Department of Quantum Engineering, Nagoya University, Nagoya 464-8603, Japan
5
Institute of Materials Research, Tohoku University, Sendai 980-8577, Japan
6
Department of Physics, Aichi University of Education, Kariya 448-8542, Japan
7
Toyota Physical and Chemical Research Institute, Nagakute, Aichi 480-1192, Japan
8
UVSOR facility, Institute for Molecular Science, Okazaki 444-8686, Japan
2

In order to investigate the stabilization mechanism of the Pd-Ni-P bulk metallic glass (BMG), electronic structure of the (Ni,Pd)9 P trigonalclusters with a phosphorus atom in the center and that of the relevant crystals were investigated by use of ﬁrst principle calculations; discretevariational X potential (DVX) cluster calculation and full potential linearized augmented planewave (FLAPW) band calculation. Presence ofthe covalent bonds between phosphorus and nickel/palladium was conﬁrmed in the relevant crystals by observing their density of states, that ischaracterized by the narrow bandwidth and the eigen values well reproduced by the (Ni,Pd)9 P cluster calculation. We found, as a consequence ofthe theoretical calculations, that the electronic structure and the number of electrons in the trigonal cluster allow the (Ni,Pd) atoms at the vertices
and sides of the clusters to be shared by the neighboring ones, and that the connection-direction and connection-angle of the cluster do notsigniﬁcantly alter the cluster levels nor the internal energy of the cluster. Consequently the network of the cluster has large degree of ﬂexibility
with keeping the low internal energy. These characteristics lead to the highly-stable Pd-Ni-P BMG. [doi:10.2320/matertrans.MF200617](Received December 4, 2006; Accepted March 5, 2007; Published May 25, 2007)
Keywords: bulk metallic glass, electronic structure, internal energy, band calculation, cluster calculation

1.

Introduction

Bulk metallic glasses (BMGs) have attracted a great dealof interests because of their ability in practical usage, such asmechanical parts of high strength, surface coating materials
with high degree of hardness, soft-magnetic materials, etc.1)
In order to design the BMGs suited for each practical usage,
stabilization mechanism of the BMG’s has to be revealed
from microscopic view points; the local atomic arrangements
and the electronic structure. Since it is naturally consideredthat the highly-stable BMGs possess a low free-energy
together with a large activation energy for crystallization, thestabilization mechanism of BMG’s should be discussed interms of free-energy and activation energy for crystallization.
One may easily realize that the free-energy of anamorphous solid is reduced when it consists of characteristic
clusters of low internal energy and these clusters areconnected randomly without altering the internal energy of
the clusters. Silica glasses are known to possess suchcharacteristics; SiO4 tetrahedral clusters of low internalenergy construct a random network by sharing oxygen atoms
at the vertices of the tetrahedron with ﬂexibility in theirconnection-direction and connection-angle. The low internalenergy of the SiO4 clusters are brought about by the s, pelectron based covalent bonds. Presence of the relatively
large, stable, rigid SiO4 units leads to the large activationenergy for crystallization and, therefore, prevents the systemfrom easily crystallizing at a given temperature. Thus, bothnecessities for the stable amorphous solid are realized in thesilica glasses.
We also found, in our resent work,2,3) that covalent bonds

persist in the Zr-based BMGs, and lead to the signiﬁcantreduction in the internal energy and increase the activationenergy for crystallization by producing a large, stable prism
clusters. The covalent bonds in the Zr-based BMGs aremainly produced by the hybridization between Zr d-orbitals
and s; p; d-orbitals of the neighboring atoms. The Zr-based
BMGs also possess metallic bonds especially around thesimple metal and noble metal elements, and that the clustersof the metallic nature, which are considered as the Kasper
polyhedra including the well-known icosahedron, play a role
to connect the covalent clusters of low internal energy withlarge degree of freedom in the direction of the inter-clusterconnections. It is strongly argued, therefore, that (a) the
presence of the covalent bands in the local atomic arrangements to reduce the internal-energy and (b) the possession ofa mechanism to construct the random network of the covalentclusters without altering the low internal-energy of theclusters are observable both in silica glass and Zr-based
metallic glasses, and that these two conditions would be thecommon factors to stabilize the amorphous phase.
In order to further conﬁrm this scenario, we investigate, inthis study, the electronic structure of the Pd-Ni-P BMG, thatis one of the most famous, stable BMGs obtained in metalmetalloid systems.4) We employed the relevant crystals toinvestigate the stability of the Pd-Ni-P BMG in the same way
as we did in our previous study on the Zr-based BMGs.2,3) Bymaking full use of the relevant crystals with the ﬁrst principle
cluster calculation and band calculation, we demonstrate thatthe local atomic arrangements of the Pd-Ni-P BMG and its
relevant crystals are indeed characterized by the covalentbonds, and that large degree of freedom in the inter-cluster-Electronic Structure and Stability of the Pd-Ni-P Bulk Metallic Glass

P
(a)

(c)
NiP3
NiP2

PdP2

(b)
PdNiP
Pd5P2
Pd3P
Pd6P

Pd3NiP

Pd

NiP
Ni5P4
Ni2P
Ni5P2
Ni3P
PdNi2P

Pd40Ni 40P20

Ni

(d)

Pd3P, Pd3NiP

Ni2P, PdNiP

Pd 6 P

N5 P2

Ni 3P, PdNi 2P

Fig. 1 (a) Trigonal prism cluster and (b) tetragonal dodecahedron cluster
exist in the Pd-Ni-Cu-P bulk metallic glass. Ref. 7) (c) Composition
diagram in which compositions of the bulk metallic glass (the solid circle
and hatched area) are plotted with that of the relevant crystals (the open
circles, gray circles, and gray area). The relevant crystals of the graycircles possess the characteristic clusters as those in the bulk metallic
glass. Those clusters are shown in (d).

connection enhances entropy without signiﬁcantly altering
the internal energy of the clusters.
2.

First Principle Calculations

We employed, in this study, the ﬁrst principle cluster- andband-calculations of the distributed package programs,
DVX and Wien2K, respectively. The former uses the X
potential with the wave functions obtained by linear
combination of atomic orbitals, and the latter the generalized
gradient approximation (GGA) with the full potential
linearized augmented planewave (FLAPW). Details of thesecalculations were reported elsewhere.5,6)
Generally speaking, band calculations are not applicablefor the amorphous materials because of their lack of theperiodicity. We therefore used the FLAPW method to reveal
the band structure of the relevant crystals, Ni2 P, PdNiP, and
Pd3 P. The cluster calculation, on the other hand, is applicableeven for the amorphous materials provided that the local
atomic arrangements were already known. Fortunately, thelocal atomic arrangements of the BMG have already been
investigated by a large number of studies.7–10) The trigonal
prism cluster and the tetragonal dodecahedral cluster,7)
shown in Fig. 1(a) and (b), respectively, were proposed forthe structure units constructing the Pd-Ni-Cu-P metallic
glass. Notably, these clusters also exist in the relevant
crystals, such as PdNiP, Ni2 P, Pd3 P, Pd3 NiP, Pd6 P, Ni5 P2 ,Ni3 P, and PdNi2 P.

1293

The tetragonal dodecahedral cluster is obtained by slightly
deforming the trigonal prism cluster, indicating both cluster’s
possession of the similar electronic structure and consequently of the similar internal-energy. Therefore we calculated the cluster levels only of the trigonal prism cluster as astructure unit of the BMG and relevant crystals.
3.

Results

Figure 1(c) shows a composition diagram on which thecomposition of the most stable BMG, Pd40 Ni40 P20 , is plotted
together with that of the relevant crystals. Local atomic
arrangements about phosphorus atom in these relevant
crystals11,12) are shown in Fig. 1(d). One may easily realizethat the phosphorus atoms in the relevant crystals are
surrounded by Pd and Ni atoms to form the trigonal prism
cluster or tetragonal dodecahedral cluster as well as thosereported for the Pd-Ni-P BMGs.7) It is, therefore, naturally
expected that the trigonal prism structure with a phosphorus
atom in its body-center has low internal energy andcontributes to reduce the free-energy of the relevant crystals
and perhaps of the bulk metallic glass.
In the ﬁrst step, we calculate the density of states of thethree relevant crystals, Ni2 P, PdNiP, and Pd3 P, by using the
FLAPW method. The structure reported in the Peason’s
handbook11) were used for the calculations. The resulting
density of states NðEÞ is plotted in Figs. 2(a)–(c). Since therelevant crystals have almost the same local atomic arrangements about phosphorus, all relevant crystals show thesimilar characteristics in NðEÞ which are described below indetail.
The band at 15 eV < E  EF < 12 eV marked as A in
Fig. 2 consists mainly of phosphorus 3s and s of nickel/
palladium at the vertices of the trigonal prism cluster. HereEF indicates the Fermi level. This A-band has bonding nature,and the corresponding anti-bonding band is located morethan 5 eV above EF . Two groups of bands observed at
10 eV < E  EF < 5 eV marked with B and C containmainly phosphorus 3p-electrons. In addition to the phosphorus 3p electrons, the these bands has d-components frompalladium/nickel located at the vertices and sides of thetrigonal prism cluster. These bands also have bonding nature,and the bands located just above EF marked as F are assigned
as the corresponding anti-bonding bands. The role of hybridization between phosphorus and transition metal elements onstabilization mechanism of the BMG and its relevant crystals
will be discussed in more detail after showing the cluster
levels later.
These s, p-dominant bands marked as AC possess muchnarrower bandwidth than the free electron like s, p-bands inthe typical metallic phases, such as fcc aluminum shown in
Fig. 2(d). This fact indicates that covalency develops in the s,
p-orbitals in the relevant crystals of the Pd-Ni-P BMG as
those in the silica glass. The s, p-dominant bands of therelevant crystals (AC) are located at lower energy thanthose in the fcc Al. This feature would be responsible for thedevelopment of the covalent bonds rather than the metallic
bonds.
One may naturally consider that the covalent nature isbrought about by the metalloid element, phosphorus. This

1294

T. Takeuchi et al.
(a) Ni2P(b) PdNiP

30

E

Total

(c) Pd3P
40

30

E

Total

D

20

F

B

A

10

0
1.0

B

B

A

F

0
1.5

3s
3p
3d

P1 site

A

C

20

C

C
10

D E

Total

D

20

F

0
1.5

P1 site

3s
3p
3d

1.0

P site3s
3p
3d

Pd1 site5s
5p
4d

Pd2 site

5s
5p
4d

1.0

DOS, N (E) / states eV -1unit cell-1

0.5

0.0
1.0
3s
3p
3d

P2 site

0.5

0.5

0.0
1.5

0.0
3

3s
3p
3d

P2 site
1.0

2

0.5

0.0
6

4s
4p
3d

Ni1 site
4

0.5

1

0.0
6

0
3

5s
5p
4d

Ni1 site (Pd)
4

2

2

2

1

0
6

0
6

0

4s
4p
3d

Ni2 site
4

4s
4p
3d

Ni2 site
4

2

-15

-10

(d) fcc Al

-5

0

5

0

5

Energy, E-E F /eV

1.0
Total
3s
3p

2
0.5

0

-15

-10

-5

Energy, E-E F / eV

0

5

0

-15

-10
-5
0
Energy, E-E F / eV

5
0.0

-15

-10

-5

Energy, E-E F / eV

Fig. 2 Electronic density of states of the relevant crystals, (a) Ni2 P, (b) PdNiP, and (c) Pd3 P calculated by the FLAPW-GGA method(Wien2K). The total density of states is shown together with the partial density of states for each atomic sites, the name of which wasreported in Ref. 11). The structure of the PdNiP were obtained by replacing nickel atoms in Ni1 site of the Ni2 P with palladium. All
phosphorus sites in these three relevant crystals are surrounded by nickel and/or palladium atoms to form a trigonal prism cluster. The
density of states of fcc Al, that possesses an electronic structure well accounted for with the nearly free electron approximation, are shownin (d). The density of states of the relevant crystals are commonly characterized by possession of the narrow bands marked as AF.

consideration is fully consistent with the presence of thephosphorus atom in the center of the characteristic cluster(the phosphorus centered trigonal prism cluster). Indeed, thebonding bands including the phosphorus s, p-components
stay in the relatively low energy range, and the important roleof the phosphorus on the covalent bonds is well conﬁrmed.
The bands mainly consisting of the d-components of thetransition metal elements exist at 5:5 eV < E  EF < 0 eV
as marked with D and E in Fig. 2. These bands contain verysmall amount of the components form the phosphorus.
Nearly 90% d-orbitals of the transition elements havenonbonding nature and its contribution to stabilize the cluster
is of less importance. It is also easily realized that the energywidth of the d-bands becomes wider when palladium is

involved as one of the constituent elements. This palladiuminduced widening of the band is caused most likely because
the larger extent of d-orbitals in palladium than that of nickel
leads to the larger transfer integral with the neighbouring
atoms.
Fermi level is located in the p, d anti-bonding bands
marked with F in Fig. 2. The metallic electrical conduction in
the relevant crystals is caused by these bands. Note here that
it is not necessary to consider the metallic cohesive energy
for its stabilization mechanism even though a BMG and its
relevant crystals have metallic electrical conduction. Indeed,
the band structure of the relevant crystals are well accounted
for almost solely by the cohesive mechanism of covalentbonds.

Electronic Structure and Stability of the Pd-Ni-P Bulk Metallic Glass
(a)

1295

-1

DOS, N (E) / states eV cluster

-1

(b)

20
15

E

20

Total

15

Pd P

D

10
5

A

B

F'

5

0
5
4

P

4

3s
3p
3d

B

F1 F'

C

2

1

1

0
5

0
5

Ni

4

4s
4p
3d

3

P
3s
3p
3d

3

2

Ni
4s
4p
3d

3

2

2

1

1

0
5

0

4

A

0
5

3

4

P

10

F1
F2

C

Total

E

Ni

Ni

Pd

-15

-10
-5
Energy, E-EF / eV

0

5

5s
5p
4d

3
2
1
0

-15

-10
-5
Energy, E-EF / eV05Fig. 3 Electronic density of states for the (a) Pd3 Ni6 P and (b) Ni6 P trigonal prism clusters. The schematic illustration of the cluster isshown in insets. Electronic structure of these clusters shows nearly the same characteristics except for Ni6 P’s absence of peaks D and F2,
which come from palladium atoms at the sides of the prism cluster.

Figure 3(a) shows the density of states deduced fromcluster orbitals calculated by the DVX method for the
Pd3 Ni6 P trigonal cluster found in the PdNiP relevant crystals.
To obtain the density of states, each cluster level was
broadened by multiplying a gaussian-function of 0.2 eV inwidth.
The bands obtained by the FLAPW method are qualitatively reproduced by the cluster calculations as marked withAF in Fig. 3(a). It is, therefore, strongly argued that theelectronic structure of the relevant clusters and perhaps thatin the corresponding BMGs are determined solely by the
local atomic arrangements about the phosphorus atom. One
may also realize that each cluster level has slightly diﬀerent
energies from those calculated by the FLAPW method. These
diﬀerences are mainly caused by the diﬀerence in the
coulomb potential between the crystalline solid and isolated
cluster. Since the coulomb potential in the cluster is diﬀerent
form that in the solid, we do not have to seriously consider
the minor diﬀerence of the energy levels. We stress here,
however, that presence of the bonding-orbitals, which areAC, is more important for the stability of the local atomic
arrangements and their constructing phases.
By using the cluster orbitals, we gain deeper insight into

the hybridization of each atomic orbitals. The structure in theNð"Þ marked with B and C in Fig. 3 are made up with threecluster orbitals mainly containing phosphorus 3p andpalladium/nickel s, p, d. Phosphorus 3pz are more overlapped with s, p, and d orbitals of the nickel at the vertices ofthe prism, while the remaining 3px and 3py orbitals overlap
mainly with s, p, d-orbitals of the palladium atoms at thethree sides of the trigonal prism cluster. By observing
components of the each cluster levels, we safely assigned thatthe band-B is made up with 3pz of the phosphorus and s, d ofthe nickel at the vertices, while the band-C with 3px =3py ofthe phosphorus and s, p, d of the palladium at the side of thecluster. Both B- and C-bands have bonding nature, and thethe corresponding anti-bonding bands are located above EF
as marked with F1 and F2 in Fig. 3(a), respectively.
We also calculated the cluster orbitals with removing thepalladium atoms from the sides of trigonal prism cluster toinvestigate the eﬀect of the hybridization between phosphorus and transition metal elements at the side of the prism.
The resulting density of states is shown in Fig. 3(b). Despite
the minor diﬀerences, the s, p-dominant cluster-orbitalsof bonding nature (AC) stay essentially the same energy
range. Therefore presence or absence of the transition metal

1296

T. Takeuchi et al.

(a)
TM

E

Ni2P

P3p xyTMs,dP3p xyTMs,dbonding

Ps-TMsbonding

Intensity (arb. units)Ps,pP3p z-TMs

TM dbondinganti-bonding

nonbondingP3p xyTM1,2 s,d

TM2 P

-16

anti-bonding

bondingbonding

B

C

F

-10

-8

-6

-4

-2

0

2

Energy, E-EF / eV

E

Pd3P

anti-bonding

TM2 d

-12

(b)
s,p

nearlynon-bonding

-14

D

TM1 d
nealy
nonbonding

Intensity (arb. units)

P3p z
-TMs

A

TM1

energyreduction

Ps-TMsDP3p xy-TM1,2 s,dbonding

Fig. 4 Schematic illustrations of the electronic structure of the trigonalcluster. Important role of the s, p-dominant bonding band AC and aminor role of the d-bands to stabilize the structure are clearly understood.
Less important role of the palladium/nickel atoms at the side of thetrigonal cluster is also understood by comparing the electronic structures
with/without those atoms as schematically illustrated.

(A)
B C

F

-16

-14

-12

-10

-8

-6

-4

-2

0

2

Energy, E-EF / eVelements at the sides of the trigonal prism has less importantrole in stabilizing the relevant crystals and perhaps to that of
the bulk metallic glasses.
Although their presence is not of crucially importance, thetransition metal elements at the sides of the cluster still
contribute to reduce the internal energy of the cluster. This is
conﬁrmed by observing that the energy of the bonding-bands
AC were slightly increased when the palladium atoms at
the sides of the cluster are removed. Simultaneously, one of
the nonbonding d-bands assigned as D was disappeared
together with the p  d anti-bonding bands of F2. We also
found, by removing palladium atoms, that the components
from the transition metal elements in the band-C issigniﬁcantly reduced, and the energy diﬀerence between
the bands C and B becomes larger. It is, therefore, consideredthat the hybridization between phosphorus p and d of thetransition metal elements pushes the p-bands (B and C)
towards lower energy with mixing the small amount of dcomponents in the p-bands, and creates the correspondinganti-boding bands above the nonbonding d-bands. The samerole is expected for the d-orbitals of the nickel atoms at thevertices. This scenario is well understood from the detailed
assignment of the bands schematically drawn in Fig. 4.
In order to conﬁrm the reliability of the present calculations, we measured the photoemission spectra of therelevant crystals, Ni2 P and Pd3 P, with an incident energyof h ¼ 40 eV at the BL-5U of the synchrotron radiation
facility UVSOR in Okazaki, Japan. The resulting photo-Fig. 5 Photoemission spectra of the (a) Ni2 P and (b) Pd3 P relevant crystals
measured with an incident photon energy of 40 eV. The characteristics ofelectronic structure expected from band-calculation and cluster-calculation are qualitatively reproduced in the spectra. Splitting between D and F
becomes larger with Pd atoms because of their large extent of the atomic
orbitals. Large spectral weight at (A) of the Pd3 P comes from unidentiﬁed
extrinsic extrinsic, such as secondary electrons or surface contamination.

emission spectra are shown in Figs. 5(a) and (b). The detailed
structures observed in these photoemission spectra weresuccessfully assigned with the bands AF of our present
calculation on the relevant crystals and the trigonal prism
cluster. The photoemission spectra of the Pd-Ni-P BMG
reported by several diﬀerent groups13–15) also show similar
characteristics. These experimental facts indicate reliabilityof our present analysis and usefulness of relevant crystals toinvestigate the stability of the BMG.
4.

Discussion

By calculating electronic structure of the relevant crystals
and the characteristic clusters, we realized that the trigonal
prism clusters, (Pd,Ni)9 P, would be very stable because of
their possession of the s, p-dominant covalent bonds about
the phosphorus atom, and that these trigonal prism clusters
behave as structure units in the relevant crystals and perhaps
in the corresponding bulk metallic glass. However, even if it
has the stable structure units (clusters) possessing extremely

1297

40

(a)

E

30

P

Ni

20

D
10

F

B,C

A

0
-15

-10

-5

0

5

Energy, E-EF / eV

40

(b)

E

-1

DOS, N(E) / states eV cluster

100°

30
20
10

D

0
-15

F

B,C

A
-10

-5

0

5

Energy, E-EF / eV

-1

-1

DOS, N(E) / states eV cluster-1-1

DOS, N(E) / states eV cluster-1

Electronic Structure and Stability of the Pd-Ni-P Bulk Metallic Glass

40

(c)

E

30
20

D
10
0
-15

B,C

A
-10

90°

F
-5

0

5

Energy E-EF / eV

Fig. 6 Density of states of the clusters consisting of two trigonal clusters.
Two trigonal prism clusters are connected with (a) face-, (b) edge- and (c)vertex-sharings. Presence of the bands AF are persisted in the same way
as those in a single trigonal cluster.

low internal energy, the metallic glass cannot be stabilized
without a speciﬁc mechanism leading to a random network of
the clusters. It should be realized only when the inter-clusterbonds have ﬂexibility in its angle and direction. Therefore,
we discuss whether the phosphorus centered trigonal prism
clusters can be connected with a large degree of ﬂexibility intheir connection angle and direction.
It is naturally considered that the palladium/nickel atoms
in the trigonal prism cluster provide some electrons into thebonding-bands (AC) because there are four orbitals ofbonding nature which is able to contain eight electrons, while
the phosphorus atoms has ﬁve s, p-electrons. Three vacancies
in the bonding-bands are ﬁlled by the 3 electrons from thesurrounding 9 palladium/nickel atoms, and therefore eachpalladium/nickel atom provides 0.33 (¼ 3=9) electrons to thebonding orbitals. Since each palladium/nickel atoms has 10
valence-electrons, remaining 9.67 electrons mainly ﬁll thenonbonding d-bands marked as D and E. Note here that mostof the s, p-components of the palladium/nickel atoms arelifted up into the anti-bonding bands located above EF andstay unoccupied. Thus the (Pd,Ni)9 P cluster is highly stable,
because almost no electrons ﬁll the anti-bonding bands while
the bonding bands are fully occupied.
When the palladium/nickel atoms at the surface of thetrigonal prism clusters are shared, on average, by twoneighboring clusters, each palladium/nickel atom turns out toprovide 0.67 (¼ 3=ð9=2Þ) electrons to the bonding orbitals.
This condition leads to a reduction in the number of electrons
in the nonbonding d-bands, and consequently to the reduction
both of the Fermi level and total energy of the electrons. This
energy reduction must cause the condensation of the clusters.
We also realized that the presence of the bonding levels
does not signiﬁcantly altered by the angle nor the directionof the cluster-connection because the orbitals of bonding
nature are mainly composed of s and p components of theFig. 7 Schematic illustrations of possible connections of the trigonal prism
clusters. Large degree of freedom is easily expected for the inter-clusterconnection.

phosphorus at the center of the cluster. The px and py
dominant orbitals (p  d bonding orbitals) would be slightly
aﬀected by variations in the angle and direction of the intercluster connection because of the p  d hybridization, but its
eﬀect is considered not to be very serious because we alreadyconformed that the eigen values of the bonding bands havesmall variation with presence or absence of the palladium/nickel atoms at the sides of the trigonal prism cluster. (SeeFigs. 3(a) and (b)).
In order to prove this scenario, we calculated the cluster
orbitals of some diﬀerent inter-cluster-connections. Three
diﬀerent clusters consisting of two prisms and the resulting
density of states are shown in Fig. 6. Obviously the electronic
structure does not signiﬁcantly altered by the inter-cluster
connection with keeping the bonding-bands (AC) far below
the Fermi level. These characteristics must lead to ﬂexible
connections of the clusters as schematically drawn in Fig. 7.
Consequently (Pd,Ni)9 P clusters are able to construct a
random network to form the bulk metallic glass without
signiﬁcantly increasing their internal energy. The enhanced
entropy and the small internal energy result in the very low
free-energy of the BMG.
We conclude, therefore, that the Pd-Ni-P BMG is stablebecause of the random cluster network of the rigid, stable,
large structure unit (cluster) of covalent nature. As it isalready argued in Introduction, this stabilization mechanism
is applicable for the Zr-based BMG’s and even for the silica
glasses, and, therefore, commonly acceptable for highlystable amorphous solids.
Finally, we comment on the composition of the most stable
Pd-Ni-P bulk metallic glass. Since the results obtained by thepresent analysis on the electronic structure suggest that the(Pd,Ni)9 P clusters construct a random network with the1298

T. Takeuchi et al.

palladium/nickel atoms on the cluster surface being sharedby the neighbouring clusters. If all palladium/nickel atoms
are shared, on average, by two clusters, the composition of
the BMG is predicted to be (Pd,Ni)4:5 P = (Pd,Ni)81:8 P18:2 ,which has almost the same phosphorus concentration with
Pd40 Ni40 P20 where the most stable BMG is observed. Slightly
larger phosphorus concentration in the calculated value iscompensated, if there exists, on average, one vacancy in the(Pd,Ni) sites of each prism cluster. This condition leads to thecomposition of BMG as (Pd,Ni)4 P = (Pd,Ni)80 P20 . The
assumption of the vacancies sounds reliable partly because
the (Pd,Ni) atoms at the sides of the prism cluster have lessimportant role in reducing internal energy of the cluster, andpartly because the introduced vacancies naturally lead toincrease of the entropy. Since palladium and nickel stay inthe same column in the periodic table, substitution of
palladium for nickel would not cause a signiﬁcant variationin the electronic structure nor the internal-energy but in theentropy, which becomes maximum at Pd : Ni ¼ 50 : 50. It is,therefore, concluded that the minimum in the free energy of
the Pd-Ni-P should be obtained at (Pd0:5 Ni0:5 )P20 = Pd40 Ni40 P20 . This composition is indeed good consistency withthat of the most stable BMG.
5.

Conclusion

By calculating electronic structure of the relevant crystals
and that of the (Ni,Pd)9 P trigonal prism clusters with aphosphorus atom in the center, we found that the trigonal
prism cluster has covalent nature and highly-stable, and thatit persists as a structure unit in the relevant crystals andperhaps in the corresponding bulk metallic glasses. The
electronic structure of the trigonal cluster allow the palladiumand nickel atoms at the vertices to be shared by theneighboring clusters without seriously altering the eigenval-

ue of the cluster orbitals. This characteristic leads to the large
degree of ﬂexibility in the direction and angle of intercluster-connections. Consequently the network of the cluster
has large degree of ﬂexibility to build a random structure withkeeping the low internal energy of the clusters. Thismechanism must lead to the highly-stable Pd-Ni-P bulk
metallic glasses.

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