Donor-vacancy complexes in Ge: Cluster and supercell calculations
J. Coutinho,
1
S. Öberg,
2
V. J. B. Torres,
1
M. Barroso,
1
R. Jones,
3
and P. R. Briddon
4
1
Department of Physics, University of Aveiro, Campus Santiago, 3810-193 Aveiro, Portugal
2
Department of Mathematics, Luleå University of Technology, Luleå S-97187, Sweden
3
School of Physics, University of Exeter, Exeter EX4 4QL, United Kingdom
4
School of Natural Sciences, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, United Kingdom
Received 20 March 2006; published 26 June 2006
We present a comprehensive spin-density functional modeling study of the structural and electronic prop-
erties of donor-vacancy complexes PV, AsV, SbV, and BiV in Ge crystals. Special attention is paid to
spurious results which are related to the choice of the boundary conditions supercell-cluster approach, the
resulting band-gap width, and the choice of the points to sample the Brillouin zone. The underestimated energy
gap, resulting from the periodic conditions together with the local-density approximation to the exchange-
correlation energy, leads to defect-related gap states that are strongly coupled to crystalline states within the
center of the zone. This is shown to produce a strong effect even on relative energies. Our results indicate that
in all E centers the donor atom occupies a nearly substitutional site, as opposed to the split-vacancy form
adopted by the SnV complex in Si. The E centers can occur in four charge states, from positive to double
negative, and produce occupancy levels at E0/+ = E
v
+0.1 eV, E-/0 = E
v
+0.3 eV, and E=/- = E
c
-0.3 eV.
DOI: 10.1103/PhysRevB.73.235213 PACS numbers: 61.72.Bb, 61.80.Az, 71.55.Cn, 71.70.Ej
I. INTRODUCTION
Intrinsic limitations in carrier mobility in Si, together with
the recent advances in high- dielectrics research, has led to
the resurgence of Ge as a key ingredient in a new generation
of ultrafast devices to operate in a regime of tens of
gigahertz.
1,2
Recent defect studies in Ge are scarce, particu-
larly those dealing with the atomic and electronic details of
elemental radiation induced defects. This deficiency applies
not only to experimental reports see Refs. 3, 4, and refer-
ences therein, but to modeling studies as well.
5–11
In this
context, a detailed understanding of the properties of defects
in Ge, especially those that may affect device yield and per-
formance, is highly desirable.
The class of vacancy-impurity complexes is particularly
important as we know that many substitutional centers in-
cluding dopants migrate by reacting with radiation- or ther-
mally generated vacancies.
12
This also includes the actual Ge
atoms in Ge crystals. According to
71
Ge tracer-diffusion
measurements, self-diffusion in Ge is dominated by a va-
cancy mechanism.
13,14
This is a major departure from what
we know in Si, where self-interstitial mediated self-diffusion
plays an important role.
15,16
The importance of vacancies in
Ge has been recently highlighted after ab initio calculations
predicting a formation energy for the single Ge vacancy as
low as 1.7–1.9 eV depending on its charge state.
6
This is
considerably less than the 3.55 eV formation energy of a
self-interstitial in Ge,
17
and comparably less than that of a
vacancy in Si which has been estimated as 4.4 eV.
18
It also
suggests that at 600 ° C the concentration of thermal vacan-
cies is about 10
12
cm
-3
in Ge, contrasting with much less
than one single vacancy per cm
-3
under similar conditions in
Si.
Here, we report on ab initio density functional studies of
a class of prominent radiation defects in Ge, namely donor-
vacancy DV complexes D = P,As,Sb,Bi, also known as E
centers, which are responsible for heavy compensation ef-
fects in n-Ge crystals subject to MeV irradiation. We pay
special attention to the electronic structure of such com-
plexes, as well as to the approximations involved in treating
the host crystal, i.e., by using Ge supercells or Ge clusters.
Before summarizing previous experimental and theoreti-
cal results concerning the E centers in Ge, it is instructive to
present a short description of the analogous centers in Si.
Early electron paramagnetic resonance EPR experiments
on e-irradiated n-doped Si established much of their proper-
ties, including an atomic model shown in Fig. 1a.
19,20
This
model is referred to as full-vacancy structure here, and we
discern it from the split-vacancy model shown in Fig. 1b,
which has been assigned to the tin-vacancy complex in
Si.
21,22
The E centers in Si are produced by trapping mobile
vacancies next to n-dopants. They introduce a deep acceptor
state at around E
c
-0.4 eV, and are stable up to about
170 °C.
19,23
Three valence electrons from the group-V
atom saturate Si atoms i', j ', and k', leaving two other elec-
trons in a lone pair state below the valence band maximum.
In the neutral defect, the Si radicals i, j , and k contribute with
a total of three electrons. These hybridize to form a fully
occupied a
1
↑↓
state also lying below the valence band maxi-
FIG. 1. Color online Full-vacancy a and split-vacancy b
atomic models for a donor-vacancy complex in Si and Ge. The
donor and host atoms are represented in blue and yellow, respec-
tively. Vacant sites are shown in white.
PHYSICAL REVIEW B 73, 235213 2006
1098-0121/2006/7323/23521310 ©2006 The American Physical Society 235213-1