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\title{\bf Sulfur: a potential donor in diamond}
\author{D. Saada$^{1,2}$,
Joan Adler$^{1}$ and R. Kalish$^{1,2}$}
\address{$^{1}$Department of Physics and $^{2}$Solid State Institute,
Technion-IIT, Haifa 32000, Israel.} \date{\today} \maketitle
\begin{abstract}
We performed first-principle calculations of substitutional sulfur in
diamond, in the neutral (S$^0$) and charged states. The energy levels
induced by sulfur in diamond are calculated to be 0.15 eV and 0.5 eV
from the bottom of the conduction band, for S$^0$ and the singly
ionized state S$^+$, respectively. The formation energy for the
neutral state of sulfur is found to be 7.2 eV, lower than that of
phosphorus in diamond. The most likely state of sulfur in diamond is
found to be the doubly ionized state S$^{++}$, which cannot act as a
donor. However, a small fraction of sulfur can be found in the singly
ionized state S$^+$, which can donate an electron for conduction at
reasonable temperatures.
\end{abstract}
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The utilization of many of the outstanding electronic properties of
diamond is hampered by the lack of electronic-grade n-type
diamond. Hence, much theoretical and experimental work has been
devoted to the search of a suitable donor in diamond, however, so far,
with only limited success. Till most recently only phosphorus was
unambiguously proven to have a donor level in diamond
\cite{Koizumi1,Koizumi2}. Unfortunately, the electron mobilities
($\mu$) obtainable are rather low ($\sim$ 100-200cm$^2$/V sec), and
the activation energy (E$_a$) is too high ($\sim$ 0.5 eV) to yield
reasonable conduction at room temperature, as required for most
applications. The recent report on high quality n-type conduction in
sulfur doped diamond \cite{Sakaguchi} ($\mu \sim$ 600 cm$^2$/V sec and
E$_a$ = 0.37 eV) has been proven \cite{Kalish} to be p-type due to an
unintentional boron contamination, hence they can not be attributed to
S doping. Samples of p-type diamond implanted with sulfur ions have
exhibited rectifying features \cite{Hasegawa}, however no clear n-type
could be found in Hall effect measurements, and the possibility that
the donor action is due to residual, unannealed defects \cite{Reznik},
can not be excluded. Non of these measurements, however, prove that
sulfur can not act as a donor in diamond, they only mean that the
appropriate doping method has not yet been found.
In the present work we report on the results of {\em ab initio}
computations of sulfur in diamond. We show that substitutional sulfur
in the neutral state (S$^0$) has an energy level 0.15 eV below the
conduction band minimum and an energy level at 0.5 eV when in the
positively charged state (S$^+$). Unfortunately, the computed
formation energy for the doubly ionized S$^{++}$ is found to be the
lowest, hence the likelihood of finding sulfur with a shallow donor
state (in the S$^+$ or S$^0$ states) in diamond is low. The results of
the present calculations are validated by comparison with earlier
theoretical and experimental results on the energy levels induced by
substitutional N, P, O and B in diamond. Excellent agreement is
obtained, giving confidence in the present results.
We used an {\em ab initio} model based on density functional theory
\cite{Hohenberg}. Nonlocal norm-conserving pseudopotentials were
constructed using the Troullier-Martins \cite{Troullier} procedure,
and implemented in the fully separable form of Kleinman and Bylander
\cite{Kleinman} (KB), with $s$ as the local component. With this
choice of pseudopotentials, the kinetic energy cutoff of up to 50 Ry
lead to excellent convergence with respect to the plane-wave basis.
Supercells of 64 atoms in a zinc-blende structure have been used, and
a 2 $\times$ 2 $\times$ 2 Monkhorst-Pack {\bf k}-point mesh
\cite{Monkhorst} has been employed. No symmetry was assumed for the
atomic relaxation. Both local density approximation {LDA}
\cite{Perdrew} and generalized-gradient approximation (GGA)
\cite{Perdrew2} were used for the exchange-correlation
functionals. Similar results were obtained with these two
approximations, using the same functional for the pseudopotential
generation and the plane-wave calculations. The code used is Fhi98md
\cite{Fuchs} from the Fritz-Haber-Institut.
We first checked the reliability of the code by calculating the energy
levels induced by substitutional nitrogen, phosphorus, oxygen, and
boron in diamond, and compared our results with those of previous
first-principle calculations \cite{Kajihara,Poykko,Katayama} and
experiments \cite{Farrer,Collins}. The results and parameters of the
calculations are shown in Table \ref{table}, together with the
ionization energies measured in experiments. For boron in diamond, we
calculated the ionization energy from \cite{Barraf}
\begin{equation}
E_{ion} = E^{tot} (-1) - E^{tot} (0) + E_v,
\label{ion}
\end{equation}
where $E^{tot} (-1/0)$ is the total energy for boron in diamond with a
charge state B$^{-1/0}$, and $E_v$ is the energy of the valence band
top. Our results are in good agreement with those of
Refs. \cite{Kajihara} and \cite{Katayama}, and with experiments
\cite{Farrer,Collins}, although some discrepancy exists between our
results and those of P\"{o}ykk\"{o} et. al. \cite{Poykko} for oxygen,
probably due to the different sample sizes used in the calculations.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
& Ref. [16] & Ref. [17]
& Ref. [18] & this work & exp. \\ \hline
No. of atoms & 64 & 32 & 64 & 64 & \\ \hline
{\bf k}-points & $\Gamma$ & 2 $^3$ mesh & $\Gamma$
& 2 $^3$ mesh & \\ \hline
Dopant: & & & & & \\
N (n) & 1.9 & 2.3 & 1.7 & 1.7 & 1.7 (Ref. [14]) \\
P (n) & 0.2 & & 0.53 & 0.4 & 0.5 (Ref. [2]) \\
O & & 1.8 (p) & & 1.8 (n) & \\
B (p) & & & & 0.4 & 0.37 (Ref. [15]) \\
\end{tabular}
\end{center}
\caption{Energy levels (eV) induced by nitrogen, phosphorus, oxygen
and boron in diamond, and ionization energy for boron in diamond
(calculated from equ. \ref{ion}). Some parameters of the calculations
are given. $n$ and $p$ indicate energy level calculated from the
bottom of the conduction band and the top of the valance band,
respectively.}
\label{table}
\end{table}
The level obtained from our calculations for oxygen in diamond is very
deep (1.8 eV), and is occupied by two electrons. Like oxygen in
diamond, sulfur is expected to be double donor. In our calculations,
the level induced by the presence of a neutral sulfur atom in diamond
is $\sim$ 0.15 eV below the conduction band. This level is the
shallowest level predicted until now for n-type dopants in diamond. It
should, however, be noted that such a shallow level is associated with
an extended wave function, which may introduce interaction between
dopants of neighboring supercells. The position of this level is thus
only approximate, nevertheless, the fact that it is a shallow donor
level is unambiguous. In the ionized state S$^+$, the level occupied
by one electron is 0.5 eV below the conduction band. Therefore, even
in an acceptor environment, the partial passivation of sulfur should
yield n-type conductivity in diamond, with a reasonably shallow level.
It is not sufficient for an impurity atom to have the required energy
level in the gap, it must also have a sufficiently low formation
energy to enable its existence in the crystal. The formation energy of
a substitutional dopant $X$ in a charge state $q$ can be calculated
from
\begin{equation} E^f_X (q) = E^{tot}_X (q) - E^{tot}_{bulk} -
\mu_X + q E_F, \nonumber
\end{equation}
where $E^{tot}_X (q)$ is the total energy of $X$ in the charge state
$q$ in diamond, $E^{tot}_{bulk}$ is the total energy of the bulk
supercell containing 63 atoms of carbon, $\mu_X$ is the chemical
potential for $X$, which is taken here to be the energy of a free $X$
atom. The Fermi energy $E_F$ is measured from the top of the valence
band. We have calculated the formation energies associated with
substitutional sulfur in diamond in its various charged states as a
function of the Fermi level. The results are shown in Fig. \ref{form}.
In the neutral state S$^0$, sulfur in diamond has a formation energy
of 7.2 eV, which is lower than the formation energy of neutral
phosphorus in diamond (10.4 eV \cite{Kajihara}). As can be seen from
Fig. \ref{form}, the positive ionized states of sulfur in diamond are
energetically the most stable species for the whole range of Fermi
level investigated. Furthermore, it is shown from Fig. \ref{form} that
in p-type conditions, the solubility of sulfur should be enhanced.
However, the doubly ionized state S$^{++}$, which has always the
lowest formation energy, is most likely state for sulfur in
diamond. Therefore, only a small fraction of sulfur introduced in
diamond will be in the singly ionized S$^+$ configuration, which can
donate an electron for conduction at reasonable temperatures. Since
the S$^{++}$ state does not induce any level in the band gap, the
majority of S atoms should be electrically inactivated.
Phosphorus was previously thought to be the only potentially
attractive n-type dopant, due to the relatively shallow level
induced. However, its large formation energy leads to small solubility
and complicates its incorporation into diamond. In comparison with
phosphorus in diamond, the sulfur-induced-level is even shallower, and
makes this atom a good candidate for doping in diamond. We can thus
conclude that sulfur is potentially an n-type dopant, although the
number of active centers should be very small, and we encourage
studies to obtain useful n-type diamond by sulfur doping.
\begin{figure}
\centerline{\epsfysize=8.5cm \epsfbox{039032jap.eps}}
\caption{Formation energy as a function of the Fermi level for S$^0$,
S$^+$, S$^{++}$, and S$^-$. The formation energy is relative to the
energy of a free sulfur atom.}
\label{form}
\end{figure}
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