First-order pressure-induced polyamorphism in germanium
Murat Durandurdu and D. A. Drabold
Department of Physics and Astronomy, Condensed Matter and Surface Science Program, Ohio University, Athens, Ohio 45701
Received 11 March 2002; published 16 July 2002
We report on the pressure-induced phase transition in amorphous Germanium ( a -Ge using an ab initio
constant pressure-relaxation simulation. a-Ge exhibits a first-order polyamorphic phase transition at 12.75 GPa
with a discontinuous volume change of 19%. The transition pressure is also calculated from the Gibbs
free-energy curves, and it is found that the transition occurs at 5.2 GPa in agreement, with the experimental
result of 6 GPa. The pressure-induced delocalization of electronic and vibrational states is obtained.
DOI: 10.1103/PhysRevB.66.041201 PACS numbers: 64.70.Kb, 61.43.-j, 71.30.+h
Some disordered materials show an unusual response to
applied pressure. H
2
O Ref. 1 undergoes a first-order phase
change from a low-density amorphous phase to a high-
density amorphous HDA phase. The existence of such mul-
tiple disordered phases is termed ‘‘polyamorphism.’’A simi-
lar transition to that of H
2
O was reported in amorphous
silicon ( a -Si,
2
and in SiO
2
.
3,4
The general problem of dis-
order to disorder phase transitions in tetrahedrally bonded
materials is little explored with theoretical methods because
of the challenge of constructing realistic models and the lack
of the good empirical potentials.
Experiment has shown that amorphous germanium
( a -Ge undergoes a transition to a metallic amorphous phase
with a sharp drop in resistivity and the optical gap at room
temperature around 6 GPa,
5
and it appears that this transition
was first order. Minomura
6
reported that a-Ge transforms to a
disordered -Sn structure at 6–7 GPa. An amorphous to
-Sn phase transition with a 5% volume drop is seen at
room temperature near 6 GPa in an x-ray diffraction study.
7
However, the amorphous sample contains some crystalline
grains, and with the application of pressure the crystalline
parts undergo a phase change to -Sn only 25% of the
amorphous structure transforms to -Sn while the other
parts still remain amorphous, a ‘‘partial structural
transition.’’
7
On the other hand, no phase transition was ob-
served up to 8.9 GPa in an EXAFS analysis of a-Ge.
8
These
studies indicate that the different types of high-pressure
structures can form amorphous or crystal depending on the
sample preparation and loading condition.
7,8
In a theoretical investigation using the Tersoff potential, a
gradual amorphous to amorphous phase transformation was
obtained.
9
In the same study, however, a free-energy calcu-
lation predicts a first-order amorphous to amorphous phase
transition in a-Ge.
9
It is also argued that the HDA phase of
a-Ge is similar to liquid-Ge ( l -Ge.
Despite extensive experimental studies and one theoreti-
cal analysis, several issues concerning a-Ge under pressure
remain: 1 What are the microscopic changes in the struc-
ture which occur with the application of pressure? 2 Is the
transition is first order? 3 Is the transition reversible? 4
What is the nature of insulator-metal transition? In this paper,
we perform accurate ab initio simulations of the response of
a-Ge to pressure and give unambiguous answers to the issues
reviewed above.
The model used here is generated using an improved ver-
sion of the Wooten-Winer-Weaire algorithm.
10
At zero pres-
sure, the model is equilibrated and relaxed with a local-
orbital first-principles quantum molecular-dynamic method
of Sankey and Niklewski.
11
The energy difference between
diamond and the amorphous structure is found to be 150
meV/atom in agreement with 120 meV/atom from a heat
crystallization measurement.
12,13
This Hamiltonian was ap-
plied to study a first-order amorphous to amorphous phase
change in silicon,
2
a continuous amorphous to amorphous
phase transformation in GeSe
2
,
14
ZB→Cmcm →Imm 2 tran-
sitions in GaAs,
15
and a diamond to simple hexagonal phase
transition in silicon.
2
Pressure is applied via the method of
Parrinello-Rahman,
16
and it is increased in increments of 2
GPa up to 12 GPa, after which an increment of 0.25 GPa is
carried out in order to accurately estimate the transition pres-
sure. Dynamical quenching at zero temperature under con-
stant pressure is performed to fully relax the system accord-
ing to the criterion that the maximum force is smaller than
0.01 eV/Å. We use -point sampling for the supercells’
Brillouin-zone integration, which is reasonable for a 216-
atom model. A fictitious cell mass of 1610
3
amu was
found to be suitable for these simulations.
As a preliminary, we perform a simulation for crystalline
Ge ( c -Ge. At 22–24 GPa the diamond structure transforms
into a -Sn structure in excellent agreement with experi-
ments. The computed transition volume ( V
t
-Sn
/ V
diamond
) of
the -Sn is 0.65 and the axial ratio, c / a , is 0.52 at 24 GPa.
Both values, however, are less than the experimental results
of 0.69 and 0.551 Ref. 17, respectively. We calculate the
bulk modulus B and its pressure derivative ( B ' ) of dia-
mond and -Sn structure using the Birch-Murnaghan equa-
tion of state
18
and find B =80 GPa and B ' =5.19 for dia-
mond, which are consistent with the experimental values for
diamond of B =77 GPa and B ' =4.6,
19
and B =89 GPa and
B ' =3.5 for -Sn structure, in agreement with B =86 GPa
reported in a theoretical calculation using the local-density
approximation with a nonlinear core-valence interaction.
20
The details of this simulation will be discussed elsewhere.
In the rest of the paper, we will concentrate on the amor-
phous structure. The pressure-volume curve of a-Ge is given
in Fig. 1. The volume changes smoothly up to 12.75 GPa,
and at this pressure an abrupt decline of the volume is seen,
indicating a first-order phase transition. The volume drops
about 19%, which is close to the value of 19.2% obtained in
diamond to -Sn transformation of c-Ge.
17
a-Ge transforms
from a low-density amorphous phase to a metallic HDA
phase in agreement with the experiment,
5
but the predicted
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