\documentstyle[aps,prb]{revtex}
\begin{document}
\draft

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\title{First-principles study of
phosphorus and nitrogen impurities in ZnSe}
\author{K. W. Kwak\cite{bykwak} and David Vanderbilt}
\address{Department of Physics and Astronomy, Rutgers University,
Piscataway, New Jersey 08855-0849}
\author{R. D. King-Smith}
\address{Biosym Technologies, Inc., 9685 Scranton Road, San Diego,
California 92121}
\date{\today}
\maketitle
\begin{abstract}
Phosphorus and nitrogen defects in ZnSe have been studied
by pseudopotential total-energy calculations. Various charge states
of the phosphorus and nitrogen impurities occupying zinc
sites (antisite defects) and interstitial sites (interstitial
defects) were considered; their structural and electronic properties
were investigated, and formation energies were
calculated as a function of Se and electronic chemical potentials.
The calculated electronic properties and formation energies were
used in the investigation of the
role of impurities in the $p$-type doping problem of ZnSe.
Our study suggests that the
difficulty with $p$-type doping of ZnSe by phosphorus
results from the compensation of the shallow
acceptors by the antisite defects ${\rm P_{Zn}}$ which act as
triple donors. In the case of nitrogen, the concentrations of the
antisite and interstitial defects
are too low to cause such compensation, which is consistent with the
successful $p$-type doping of ZnSe with nitrogen.
\end{abstract}
\pacs{71.55.Gs,71.45.Nt}
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\narrowtext

Current interest in the use of wide-bandgap II-VI semiconductors is
centered on the fabrication of blue diode lasers.\cite{neumark94}
The development
of reliable and inexpensive blue diode lasers is technologically and
commercially very important, since they will provide significant
benefit to virtually any technology that uses coherent visible light.
Among II-VI semiconductors, ZnSe has been considered for a long time to
be a very promising candidate for this purpose. With a direct bandgap
of 2.67 eV at room temperature, ZnSe has very efficient luminescence
in the blue spectral region.

However, the fabrication of the $p$-$n$ junctions essential
to light-emitting devices has been hindered in II-VI
semiconductors by doping difficulties. Most wide-bandgap II-VI
semiconductors (including ZnSe) can easily be doped $n$-type but
resist $p$-type doping, while ZnTe can easily be doped $p$-type
but not $n$-type. After two decades of effort directed towards
obtaining highly-conductive $p$-type ZnSe material,
high enough hole concentrations for the fabrication of
diode lasers ($\sim10^{18}{\rm cm^{-3}}$) have recently been
achieved in ZnSe using nitrogen free radicals as the
doping source.\cite{qiu91}
Subsquently, several groups \cite{haase91,jeon92} have
succeeded in demonstrating ZnSe-based semiconductor diodes that
lase in the blue-green region of the visible spectrum.

In spite of this success in fabricating low-resistivity
$p$-type ZnSe, the origin of the
doping difficulties remains unclear. Furthermore, even though
hole concentrations in the range of mid-$10^{17}$ to
$10^{18} {\rm cm}^{-3}$ are considered acceptable for the
manufacture of diode lasers, larger free-hole concentrations in
$p$-type ZnSe would be very
desirable for a practical and efficient diode laser,
particularly in view of electrical contact problems.\cite{neumark94}
Thus an issue of significant importance is to identify the
causes of the $p$-type doping difficulties and work towards
reducing their effects.

A number of explanations for the problem have been
suggested.\cite{henry71,jans89,neumark80,sasak91,ren89,dow91,
chadi93,neuma89,vdw93,chadi91,ren88}
The most commonly accepted explanation
has been that native defects (e.g., Zn self-interstitials)
which act as donors are activated as a
result of the doping, resulting in compensation of the
acceptor impurities. However, recently Laks
{\it et al.}\cite{laks92} employed {\it ab-initio}
total-energy calculations to investigate self-compensation
by native defects in ZnSe, and suggested that native-defect
concentrations in stoichiometric ZnSe were too low to cause
compensation. They also find that substantial concentrations
of native defects are expected for both $n$-type and $p$-type
doping in non-stoichiometric material, and are thus not able to
explain the $n$-type preference of ZnSe. Also, this mechanism
would predict the impossibility of
$p$-type doping regardless of dopant, which seems inconsistent
with the success of $p$-type doping of ZnSe using nitrogen.  Thus,
the native-defect self-compensation mechanism alone probably
cannot explain the doping problem. Therefore, a detailed study of
the role of impurities is necessary in order to understand this
problem.

We have studied phosphorus and nitrogen impurities in ZnSe,
using first-principles pseudopotential total-energy
calculations, to investigate the problems associated with $p$-type
doping. Our previous calculations \cite{kwak93a}
have shown that substitutional nitrogen and phosphorus impurities
on the Se site form shallow acceptors. Here, we report on
calculations for P and N impurities in antisite and interstitial
defect configurations.  (Some results on the phosphorus defects
have already appeared in abbreviated form.\cite{kwak94})
The formation energies and the electronic
and structural properties of the defects in their various charge
states have been calculated. The results
are used to interpret the role of
impurity defects in the $p$-type doping difficulty of ZnSe.

We find that the antisite defect ${\rm P_{Zn}}$ plays a critical
role in the $p$-type doping problem. When the Fermi level is near the
valence-band maximum (VBM), the formation energy of ${\rm P^{3+}_{Zn}}$
in phosphorus-doped stoichiometric ZnSe is comparable to
or lower than that of substitutional phosphorus ${\rm P_{Se}}$.
This suggests that, in the process of doping ZnSe $p$-type
with phosphorus, the shallow acceptors are compensated by
the antisite defects, which can act as triple donors.
In contrast, in the case of nitrogen-doped stoichiometric ZnSe,
the antisite defect ${\rm N_{Zn}}$ is predicted to have
a higher formation energy than ${\rm N_{Se}}$ for the
whole range of Fermi level, consistent with the
successful fabrication of $p$-type ZnSe material based on
nitrogen doping.

This manuscript is organized as follows. In Sec.\ II,
the theoretical methods employed in this study are described.
In Sec.\ III, the calculational results are presented and discussed.
We first summarize the stable ionic configurations of defects in their
various charge states. Details of the calculated electronic properties
and formation energies of phosphorus and nitrogen defects follow.
The results are then discussed in the context of the
doping problem of ZnSe. Finally, we summarize in Sec.\ IV.

\section{Theoretical Methods}

Our theoretical approach is reviewed in this section. We employed
a plane-wave pseudopotential approach within the local-density
approximation for studying the
energetics and the electronic properties of point defects in ZnSe.

Total-energy calculations with norm-conserving pseudopotentials
have been successfully applied to the study of various
solid-state systems.\cite{pickett89}
In the case of ZnSe, however, the highly
localized 3$d$ orbitals of Zn present a severe challenge for
the use of traditional norm-conserving pseudopotential
approaches using plane-wave basis sets. The condition of norm
conservation places a severe restriction on the form of the
pseudowavefunction, and makes it impossible to
construct a pseudowavefunction that is much smoother than the
all-electron one. Consequently, the norm-conserving
pseudowavefunctions of the Zn 3$d$ states require a large
number of plane waves in the basis set.
The Zn 3$d$ states may be included in the core shell, but this
results in a poor description of bulk ZnSe.  For example,
previous work on ZnSe and ZnS has
shown \cite{haas87,engel90,mart91}
that the inclusion of Zn 3$d$ states in the core shell can lead
to errors on the order of 10\% in the equilibrium lattice
constant.

In our calculations, the occupied Zn 3$d$ states are treated in
the {\it valence shell} using the
ultra-soft pseudopotential scheme developed by Vanderbilt.\cite{vand90}
This approach overcomes the problem of the
tightly bound states by relaxing the norm-conservation constraint
and introducing a generalized eigenvalue formalism.
Using the present scheme, we obtain good structural properties of
bulk ZnSe with a plane-wave cut-off of only 25 Ry, yet we can still
calculate forces on the atoms so as to relax atomic coordinates
efficiently for the defect structures.\cite{kwak93a,kwak93} For the
case of nitrogen impurities, the problematic N $p$-states
can be treated as easily as those of the heavier column-V
dopants within this scheme. For the exchange-correlation potential,
we used the Ceperley-Alder form as parameterized by Perdew and
Zunger.\cite{perdew81}

32-atom and 33-atom bcc supercells were used in the defect
calculations. A 32-atom supercell containing the formula unit
Zn$_{15}$Se$_{16}$X (X = N or P) per 32-atom was used for the
antisite defect calculation to represent an isolated defect.
For the interstitial defect calculation a 33-atom bcc supercell
was used with the formula unit Zn$_{16}$Se$_{16}$X per 33-atom
cell. We fixed the lattice constant of the cell to be the
experimental one.

The integration over the supercell Brillouin zone
was approximated by a special {\bf k}-point \cite{chadi73}
$\frac{2\pi}{b}$($\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$)
for systems with $T_d$ symmetry, where $b$ is the
lattice constant of the supercell (11.34 \AA).
For a system with $C_{3v}$ geometry, the equivalent set of
{\bf k}-points would consist of the two points
$\frac{2\pi}{b}$($\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$)
with weight $\frac{1}{4}$, and
$\frac{2\pi}{b}$(-$\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$)
with weight $\frac{3}{4}$. However, for the bcc cell these two
{\bf k}-points are related by a reciprocal lattice vector
$\frac{2\pi}{b}$(1 0 0); thus only one {\bf k}-point was needed
even for a system with $C_{3v}$ geometry. A similar argument holds
also for a system with $C_{2v}$ geometry.
Tests with denser {\bf k}-point sets indicate that the total energy
is reasonably converged with the {\bf k}-point set we used.\cite{kwak93a}

To obtain the minimum-energy structure for each defect in ZnSe (for
example, ${\rm P_{Zn}^0}$), we started with an arbitrary configuration
of the ions in a supercell which simulates the defect system. Using
a conjugate-gradient minimization technique,\cite{payne92,kingsmith94} the
Kohn-Sham energy functional was minimized to obtain the electronic
ground state for the fixed ionic configuration, and the Hellmann-Feynman
forces on the ions were calculated. Then the ionic coordinates in
the unit cell were updated using the forces as a guide.
This procedure was repeated until the forces were
negligible (less than 0.05 eV/\AA), using a modified Broyden
scheme.\cite{vand84}

For a given defect system, the formation energy $F(Q)$ for charge
state $Q$ is a function of the electron chemical potential (or Fermi
level) $\mu_e$;
\begin{equation}
F(Q)=E(Q)+Q(\mu_e+E_v) ,
\label{efq}
\end{equation}
where $E(Q)$ is the total energy of the supercell containing the
defect in charge state $Q$, and $E_v$ denotes the top of the valence
band in the bulk ZnSe crystal. Then the (thermal) ionization levels
are the values of the Fermi level at which energies of competing
charge states cross. The energy in the defect cell corresponding to the
top of the valence band in bulk ZnSe has been determined by aligning
the screened local potential at the atomic site farthest from
the impurity atom with the potential at the site of the same kind
in bulk ZnSe. If the cell was large enough, the local potential at
the farthest atomic site would correspond to a bulk calculation, giving
the exact value of $E_v$. When the cell used is not large enough, however,
the local potential is different from that of the bulk calculation, and
thus introduce an error in $E_v$. The estimatd error in $E_v$ in our
32-atom cell calculations is 0.1-0.2 eV.

In dealing with charged defects when using a periodic supercell
approach, the long-range Coulomb interaction has to be treated
carefully. The Coulomb interactions between the charged defects in 
the periodic array produce a divergent term in the total energy per
unit cell of the system. In order to avoid this unphysical situation,
the unit cell was implicitly neutralized by compensating the extra
or missing (relative to the neutral state) electron charge on the
defect with a rigid background charge, which is uniformly spread over
the whole unit cell.
(The background charge does not enter the expressions for
the exchange and correlation energies.)
In principle this could cause some systematic errors, but as long
as the supercell is large enough, any spurious interdefect
interactions or other effects of the background charge are
expected to be small.\cite{vdw89}

As a check of the charged-defect calculations, we performed 16-atom
and 32-atom cell calculations for the antisite defect ${\rm P_{Zn}}$.
The thermal ionization energies of the antisite defect were extracted
from the formation energy calculations as described above. The main
results of the two
supercell calculations are shown in Table~\ref{chadef}.
The results exhibit similar trends in
electronic structure. They predict that, when the Fermi level is near the
valence-band edge, 3+ is the stable charge state of the antisite defect.
They also agree in that both calculations predict that 1+, 2+, and 3+
charge states of the defect form a ``negative-U'' system. The reaction
$2{\rm P^{2+}_{Zn}} \rightarrow {\rm P^{1+}_{Zn}} + {\rm P^{3+}_{Zn}}$
is exothermic by 0.4 eV and by 0.3 eV in the 32-atom and 16-atom cell
calculations, respectively, suggesting that the 2+ charge state is a
metastable state, as discussed in Sec.\ \ref{sec:anti} below.

\section{Calculational Results and Discussion}

\subsection{Summary of defect configurations}

We now turn to the details of our calculations on defects in
ZnSe.  In this section, we summarize our results for the
stable ionic configurations of defects (P and N) in their
various charge states. For a given defect state, all ionic
coordinates in the supercell were fully relaxed, using the
calculated Hellmann-Feynman forces, to determine the stable
ionic configuration for the defect state.

The distortion induced
by the presence of a defect in ZnSe can be conceptually divided
into two components: a breathing relaxation which preserves the
tetrahedral ($T_d$) symmetry of the host material, and
a Jahn-Teller (JT) distortion which reduces the symmetry. When the
JT effect is present, the symmetry of the defect system
is reduced to $C_{3v}$ or $C_{2v}$, depending on the electronic
structure of the system. Our calculations indicate that, with
the exceptions of negatively charged states of interstitial
defects ${\rm P_{int}}$ and ${\rm N_{int}}$, the induced
displacements of next-nearest neighbors are much smaller than
those of the nearest-neighbor atoms of the impurities. Therefore,
in this section we report only the displacements of the nearest
neighbors.

When only breathing relaxations are induced, the defect system
stabilizes into a configuration with $T_d$-symmetry which we
describe in terms of the bond length $z$ between the impurity and
its nearest neighbors, the angle between any two such bonds being
109.5$^\circ$.  When a $C_{3v}$ JT distortion is also
present, we define $z$ to be the length of the unique (axial) bond
between the impurity and its neighbors, $z^{\prime}$ to be the
length of the other three equivalent (basal) bonds, and
$\theta$ to be the angle between the axial and basal bond. The
notations are illustrated in Fig~\ref{defconf}(a).

For the case of a $C_{2v}$ distortion, the impurity
atom moves in the [001] direction, making two bond lengths shorter
and two longer.  For this case, we define $z$ and $z^{\prime}$
to be the two different bond lengths, and $\theta$ to be the angle
between the bonds with different bond lengths, as shown in
Fig~\ref{defconf}(b). Ionic configurations with $C_{2v}$
symmetry are found to be relatively rare.  They
occur for neutral states of P and N interstitials,
for which threefold degenerate states (not counting spin) in the
bandgap are partially occupied by three electrons. However, the
departures from the breathing-relaxed ($T_d$) structure are very
small.

The results of our calculations for the stable configurations of
defects (P and N) in their various charge states are summarized
in Table~\ref{summ}. The choice of charge states will be explained
later in Secs.~\ref{pdef} and \ref{ndef}, where the defects are
discussed in detail.

\subsection{Phosphorus defects in ZnSe}
\label{pdef}

\subsubsection{${\rm P_{Zn}}$}
\label{sec:anti}

With tetrahedral symmetry, the neutral antisite defect
${\rm P_{Zn}}$ has a non-degenerate ($a_1$) and a
threefold-degenerate ($t_2$) level in the bandgap.
The non-degenerate level, which is near the VBM, is fully
occupied by two electrons, while the threefold-degenerate level,
which forms a
deep donor level, is partially occupied by a single electron.
Thus, we should check whether charge states ranging from 1-- to 5--
are possible in $n$-type ZnSe (acceptor behavior), and
whether charge states ranging from 1+ to 3+
are possible in $p$-type ZnSe (donor behavior).
Since we are mainly interested in the $p$-type material, we only
consider 0 (neutral) to 3+ charge states here.

For the neutral antisite defect, we found a JT
distortion as well as a symmetry-preserving breathing relaxation.
The breathing-relaxed bond length between the P impurity and
neighboring Se atoms was calculated to be 2.54 {\AA}.
Because of its partially filled degenerate level, the neutral
defect is found to undergo an additional JT distortion
by which the P atom moves about 0.2 {\AA} toward one of
the four neighboring Se atoms, lowering the symmetry of the
defect from tetrahedral ($T_d$) to trigonal ($C_{3v}$).
The total energy is lowered by 0.16 eV by the JT distortion.
For positively charged defects, the JT distortions are not
found, as expected from the fact only non-degenerate gap
levels are being depleted.
However, substantial breathing
relaxations (which preserve $T_d$ symmetry) are found with the
change of charge state: the bond lengths between the central
P and neighboring Se atoms are 2.50, 2.39, and
2.24 \AA\ for 1+, 2+, and 3+ states, respectively. As
${\rm P^{2+}_{Zn}} \rightarrow {\rm P^{3+}_{Zn}}$, the bond
length contracts by 0.15 \AA, which is large compared to the
typical breathing relaxations induced by phosphorus impurities
as the charge state changes. For instance, the bond length
between the P atom and neighboring Zn atoms contracts
by only 0.04 \AA\ as ${\rm P^{1-}_{Se}} \rightarrow {\rm P^0_{Se}}$.

The present calculation suggests that, because of these strong
relaxations, the states ${\rm P^{1+}_{Zn}}$, ${\rm P^{2+}_{Zn}}$,
and ${\rm P^{3+}_{Zn}}$ form a ``negative-U system.''\cite{anderson75}
The best-known example of a negative-U system is the vacancy in
Si,\cite{baraff80} and it also has been reported for defects in
III-V compound semiconductors.\cite{alt90,zhang91,alatalo93}
We find that the reaction
$2{\rm P^{2+}_{Zn}} \rightarrow {\rm P^{1+}_{Zn}} + {\rm P^{3+}_{Zn}}$
is exothermic by 0.4 eV, so that the 2+ charge state is a metastable
state, decaying either to the 1+ or 3+ state depending on the Fermi level.
The antisite defect cannot bind a single electron on the $a_1$ level in
the bandgap; instead, the occupancy changes directly from zero (3+ state)
to two (1+ state). Energy lowering due to the lattice relaxation more than
compensates for the interelectron Coulomb repulsion.
The calculated thermal depth for the 3+/1+ transition is about 0.71
eV relative to the VBM, as shown in the formation energy plot of
${\rm P_{Zn}}$
(Fig.~\ref{efpzn}). The ionic configurations in various charge states
are summarized in Table~\ref{summ}.

\subsubsection{${\rm P_{int}}$}

We turn now to a consideration of interstitial phosphorous defects.
Phosphorous can assume two kinds of tetrahedral interstitial
sites.  One, which we label ${\rm P_{int}}(T_{\rm Zn})$, is surrounded
by Zn atoms; the other, which we denote ${\rm P_{int}}(T_{\rm Se})$, by
Se atoms.
In the breathing-relaxed structure (that is, preserving
$T_d$ symmetry), the total energy of ${\rm P_{int}^0}(T_{\rm Zn})$
is lower by 0.66 eV compared to ${\rm P_{int}^0}(T_{\rm Se})$.
For the 1+ charge state, the total energy of ${\rm P_{int}}$ at the
$T_{\rm Zn}$ site is still lower in energy than at the $T_{\rm Se}$
site, but only by 0.41 eV.
Since ${\rm P_{int}}$ at the $T_{\rm Zn}$ site is
more stable than
that at the $T_{\rm Se}$ site, we consider only
${\rm P_{int}}(T_{\rm Zn})$. The P interstitial at the
$T_{\rm Zn}$ site has
a threefold-degenerate level in the bandgap, and
${\rm P_{int}^0}(T_{\rm Zn})$ contains three electrons on the
level.

The fact that the 2-- and 1-- states contain 5 and 4 electrons
respectively in the threefold-degenerate level in the gap
makes the system susceptible to a JT distortion
which would split the level into
a higher non-degenerate and a lower
twofold-degenerate level.
The calculated breathing-relaxed bond lengths between P atoms and
neighboring Zn atoms are 2.33 {\AA} and 2.38 {\AA} for 2-- and
1-- charge states, respectively.
For these states, we have found relatively small
JT effects.
The P atom moves by less than 0.1 {\AA} toward one of the
four neighboring Zn atoms,
lowering the symmetry of the
system to $C_{3v}$. Comparing to the small relaxation of
the nearest-neighbor Zn atoms, we have found large outward
relaxations of the six second-nearest-neighbor Se atoms, of
0.29 {\AA} and 0.22 {\AA} for 2-- and 1-- charge states, respectively.
Moving from the 1-- to the neutral state leads to a $C_{2v}$ JT
distortion, reducing the energy further by an additional splitting of
the twofold-degenerate level into a singly-occupied higher level and
a fully-occupied lower level.  The JT effect is very small, with the
relaxation of each Zn atom less than 0.01 {\AA}.  In this charge
state, the second-neighbor Se atoms move outward from the P atom by
0.17 {\AA}.

One of the interesting results of the present calculation is that, for
the positively charged states (1+ to 3+), the tetrahedral $T_{\rm Zn}$
site is unstable against the sites near the hexagonal interstitial site.
In ideal ZnSe, $T_{\rm Zn}$ and $T_{\rm Se}$ sites are separated by
2.45 {\AA} in the [111] direction, and the hexagonal site  is
midway between these two sites. The interstitial-site energies
depend on the bonding configuration of the impurity with
its neighbors, and can change
with charge state. For the 1+ state, the P atom
moves in the [111] direction by about 1.2 {\AA} from its
ideal position, almost reaching the hexagonal site. The
minimum-energy site is 1.2 eV more stable in energy
than the $T_{\rm Zn}$ site.
Thus, the P interstitial in ZnSe exhibits some similarities with
the Al and self-interstitial defects in silicon, in that the
total-energy difference between the hexagonal and tetrahedral site
is strongly charge-state dependent.\cite{baraf83} For ${\rm
Al_{int}}$, the relative total energy at the hexagonal site, which
acts as a barrier along the migration path of ${\rm Al_{int}}$
originating at the $T_d$ site, is significantly lower in the 1--
state compared to the neutral or positive state.
Thus, a specific electron-capture mechanism can enhance Al
interstitial migration, by converting the interstitial into the 1--
charge state.\cite{baraf83} In the case of the Si
self-interstitial, the estimated barrier lowering due to this
mechanism was found to be larger than the barrier height, so that
the hexagonal position became more stable than the tetrahedral
position in the neutral charge state.\cite{baraf83}

For the 2+ state, the P atom moves in the [111] direction by about
0.9 {\AA} from its ideal position, passing through the plane formed
by the three Zn nearest-neighbor atoms. The angle between the [111]
axis and the basal bond is calculated to be 95.3$^\circ$,
suggesting that the P and the three Zn's are
nearly coplanar.
As a result of the relaxation, the system lowers
its energy by 0.63 eV. The calculated thermal depth of the
2+/1+ transition is 0.28 eV above the VBM. For an optical
transition, which is so rapid that the instantaneous
ionic configuration (that of the 2+ state) is unchanged while
the transition takes place, the energy released as the ions relax
back to the stable (1+ state) configuration
is calculated to be 0.39 eV.
Thus, the optical ionization energy of the 2+/1+ transition
is found to be about 0.67 eV above the VBM.

For the 3+ state, the P atom moves in the [111] direction by about 0.8 {\AA}
from its ideal position. This site is lower in energy by 0.44 eV than the
$T_{\rm Zn}$ site. The thermal ionization level of the 3+/2+ transition
is found to be very near the valence-band edge.

\subsubsection{Defect formation energies}

Following the method suggested by Qian, Martin, and Chadi,\cite{qian88}
we have calculated the formation energies of defects in ZnSe as
a function of the Fermi level and the Se chemical potential.
The formation energy $F(D^Q)$ of a defect $D^Q$ in
charge state $Q$ is a function of both the Fermi level
$\mu_e$ and atomic chemical potentials $\mu_{\rm Zn},
\mu_{\rm Se}$, and $\mu_{\rm P}$. However, since each defect system contains
a P impurity, we can omit $\mu_{\rm P}$ and redefine the formation energy as
\begin{equation}
F(D^Q)=E(D^Q)+Q(\mu_e+E_v)-N_{\rm Zn}\mu_{\rm Zn}-N_{\rm Se}\mu_{\rm Se},
\label{one}
\end{equation}
where $E(D^Q)$ is the total energy of the supercell for the defect
system containing $N_{\rm Zn}$ zinc atoms and $N_{\rm Se}$ selenium atoms.
The Fermi level $\mu_e$ is measured relative to
$E_v$, the VBM.

The allowed values of atomic chemical potentials are limited by
thermodynamic constraints as follows.
The chemical potential $\mu_{\rm Se}$, which represents the energy
of Se atoms in an external reservoir, can never exceed that of
bulk Se (otherwise the external reservoir would be unstable against
the formation of bulk Se.)
Thus $\mu_{\rm Se} \leq \mu_{\rm Se}^{\rm bulk}$. Similarly,
the upper limit for the Zn chemical potential is
$\mu_{\rm Zn} \leq \mu_{\rm Zn}^{\rm bulk}$. Also,
under equilibrium conditions, ZnSe can exchange pairs of Zn and Se
atoms with bulk ZnSe, for which the energy is the total bulk
energy per pair, $E_{\rm ZnSe}$. This requires that
\begin{eqnarray}
\mu_{\rm Zn} + \mu_{\rm Se} = E_{\rm ZnSe}
 = \mu_{\rm Zn}^{\rm bulk} + \mu_{\rm Se}^{\rm bulk} - \Delta H_f ,
\label{muconstraint}
\end{eqnarray}
where $\Delta H_f$ is the heat of formation of ZnSe. This sets a lower
limit on $\mu_{\rm Se}$ or $\mu_{\rm Zn}$. Since $\mu_{\rm Se}$ and
$\mu_{\rm Zn}$ are not independent, it suffices to use only one of
them. We choose to use $\mu_{\rm Se}$, whose allowed range is given
by
\begin{eqnarray}
\mu_{\rm Se}^{\rm bulk} - \Delta H_f \leq \mu_{\rm Se}
\leq \mu_{\rm Se}^{\rm bulk} .
\end{eqnarray}
Our calculated value of the heat of formation of ZnSe is 2.79 eV.
Using the constraint (\ref{muconstraint}), we obtain
\begin{eqnarray}
F(D^Q)&=&E(D^Q)+Q(\mu_e+E_v) \nonumber \\
&&\quad -N_{\rm Zn}E_{\rm ZnSe} +(N_{\rm Zn}-N_{\rm Se})\mu_{\rm Se}.
\label{two}
\end{eqnarray}
Thus, when the formation energies of different types of defects
are compared, we must consider the energies as a function of the
Fermi level $\mu_e$ and the Se chemical potential $\mu_{\rm Se}$.

In this paper, we use a convention in which $\mu_{\rm Se}^{\rm bulk}$
sets the zero of $\mu_{\rm Se}$. Thus, when
$\mu_{\rm Se}$ is only slightly negative, it corresponds to a
Se-rich condition, while a value close to --2.79
eV corresponds to a Zn-rich condition.
The appropriate value of $\mu_{\rm Se}$ for stoichiometric ZnSe
ought to fall somewhere in the middle of this range.
Unfortunately, a determination of the precise
value of $\mu_{\rm Se}$ for stoichiometric ZnSe
would require exhaustive calculations on native defects in ZnSe,
which we have not undertaken.
We have adopted the expedient of choosing
$\mu_{\rm Se}$ = --1.4 eV (the middle of the allowed range)
to represent ``stoichiometric'' ZnSe in the calculations reported
below.

Our results for the total energies of formation of the
${\rm P_{Se}}$, ${\rm P_{Zn}}$, and ${\rm P_{int}}$ defects,
plotted vs.~$\mu_{\rm Se}$ over the estimated accessible range,
are shown in Fig.~\ref{eformp}.
Since different charge states are also considered,
Fig.~\ref{eformp} shows the results for three characteristic
Fermi levels in $p$-type material: 0.0 eV, 0.7 eV, and
1.4 eV relative to the VBM.
When the Fermi level is at 0.6 eV
or higher above the valence-band edge, the Se substitutional site
is most favored by P in stoichiometric ZnSe, as it has the lowest
formation energy. However, as the Fermi
level moves down and approaches the valence-band edge, the formation
energy of the antisite defect ${\rm P_{Zn}}$ in the 3+ charge
state is comparable to or lower than that of the substitutional
${\rm P_{Se}}$ in the neutral or 1-- charge state. This result
suggests that when shallow acceptors (${\rm P^0_{Se}}$) are created,
they are compensated by ${\rm P_{Zn}}$ which act as donors.

In their experimental study on the optical and
electrical properties of phosphorus-doped ZnSe, Reinberg
{\it et al.}\cite{rein71} observed that the conductivity at
elevated temperature is governed by a 0.68-eV level. We suggest that
this level is due to ${\rm P^{3+}_{Zn}}$ for two reasons. First, the
antisite defect is calculated to be the dominant defect in compensated
ZnSe; and second, the experimental ionization level agrees with the
one calculated for the 3+/1+ transition of the antisite defect.

\subsection{Nitrogen defects in ZnSe}
\label{ndef}

\subsubsection{${\rm N_{Zn}}$}
\label{sec:nzn}

With tetrahedral symmetry imposed, the neutral ${\rm N_{Zn}}$ has a
threefold-degenerate ($t_2$) level in the bandgap,
occupied by a single electron.
This is in contrast to the case of the neutral ${\rm P_{Zn}}$,
which has a deeper fully-occupied $a_1$ level in addition to the
singly occupied $t_2$ level in the bandgap. This difference
in the electronic structure can be understood roughly by the
trend in the $sp^3$ average energy of Zn and impurity atoms.
The average energy of P and N are about --6 eV and --10 eV,
respectively, relative to that of Zn. The lowest conduction
band of bulk ZnSe originates from the antibonding combination
of Zn 4$s$ and 4$p$ orbitals with surrounding Se orbitals.
Since the $sp^3$
average energy of the impurity P atom is substantially lower
than that of the Zn atom, the $a_1$ and $t_2$
defect states, which also have antibonding character
with the surrounding Se orbitals, are lowered and appear in the
bandgap. In the case of ${\rm N_{Zn}}$, whose $sp^3$ average
energy is even lower than P, only the $t_2$ state appears in the
gap, as the $a_1$ state has descended into the valence band.

The breathing-relaxed bond lengths between N and neighboring Se
atoms of the 1-- and neutral states of ${\rm N_{Zn}}$ were calculated
to be 2.31 {\AA} and 2.27 {\AA}, respectively.
For these states,
additional strong JT distortions were found, resulting
in $C_{3v}$-symmetry structures. The negatively charged defect
lowers its energy by 1.03 eV by the JT distortion, in which the
N impurity moves 0.53 {\AA} toward a nearest-neighbor Se atom,
expanding the bond lengths with the other three by 0.34 {\AA}.
For the neutral defect, it lowers its energy by 0.37 eV by
moving 0.39 {\AA}
toward a nearest-neighbor Se atom.
These distortions are favorable because the energy of the
electrons in the states in the gap is lowered by the distortions.
The JT distortion does not occur for 1+ charged state, since
the gap states are empty.

The present calculation indicates that the thermal ionization
energies for 1+/0 and 0/1-- transitions are 1.0 eV and 1.1 eV,
respectively, with respect to the VBM.

\subsubsection{${\rm N_{int}}$}

For the neutral nitrogen defect, as for the phosphorous defect, the
$T_{\rm Zn}$ site is found to be energetically more favorable than the
$T_{\rm Se}$ site. In the breathing-relaxed structure,
${\rm N_{int}^0}$ at the $T_{\rm Zn}$ site is 1.4 eV lower in total
energy than at the $T_{\rm Se}$ site. The neutral N
defects at the $T_{\rm Zn}$ site are found to have an almost
threefold-degenerate level, with a very small splitting due to
a JT distortion. According to our LDA calculation the energy
level is near the VBM, suggesting that the neutral defect should
be a shallow acceptor. Since the level is partially occupied by three
electrons, we consider 0 (neutral) to 3-- charge states.

The ionic configurations of N interstitial defects can be
characterized by strong breathing relaxations of the
nearest-neighbor Zn atoms and very minor JT distortions, as shown
in Table~\ref{summ}. For the 3-- state, the neighboring Zn atoms
relaxed by 0.54 {\AA} from their ideal positions toward the N
impurity.  JT distortion did not occur, as the gap
level is completely filled.
For 2-- and 1-- charge states we have found very weak JT
distortions of $C_{3v}$ symmetry, with very minor departures from
$T_d$ symmetry.
The neutral defect, as expected from the fact
that it has three electrons in the gap level, assumes a $C_{2v}$
structure, but the associated distortion is again very small.
We also find substantial outward relaxations of the six
second-nearest-neighbor Se atoms,
by 0.27 {\AA}, 0.23 {\AA}, 0.19 {\AA}, and 0.18 {\AA}
for 3--, 2--, 1--, and neutral states, respectively.

The calculated thermal depths relative to the VBM
for 0/1--, 1--/2--, and 2--/3-- are 0.06 eV,
0.17 eV, and 0.30 eV, respectively.

\subsubsection{Defect formation energies}

Our results for the total energies of formation of the
${\rm N_{Se}}$, ${\rm N_{Zn}}$, and ${\rm N_{int}}(T_{\rm Zn})$
defects are summarized in Fig.~\ref{eformn}.
Once again, this plot shows the formation energies as a function of
$\mu_{\rm Se}$ over the accessible range of the chemical
potential for three characteristic
Fermi levels in $p$-type material: 0.0 eV, 0.7 eV, and
1.4 eV relative to the VBM.

When the Fermi level is at the middle of the bandgap, the
formation energy of the ${\rm N_{int}^{3-}}(T_{\rm Zn})$ is comparable
to that of the ${\rm N_{Se}^{1-}}$ at $\mu_{\rm Se} \approx$
--1.4 eV, while that of ${\rm N_{Zn}^{1-}}$ is much higher.
As the Fermi level moves down in the bandgap, the interstitial
site becomes less and less favorable.

For the Fermi level near the VBM, the formation energy
of N substitutional defects is lower than for
N antisite and interstitial defects
at $\mu_{\rm Se} \approx$ --1.4 eV. This result agrees with
the recent theoretical calculation of Van de Walle {\it et al.}
on N impurities in ZnSe.\cite{vdw93} In their theoretical
study on acceptor impurities (Li, Na, and N) in ZnSe,\cite{vdw93}
Van de Walle {\it et al.} investigated ${\rm N_{Se}}$,
${\rm N_{Zn}}$, and N interstitials, and found that ${\rm N_{Zn}}$
and N interstitials are much higher in formation energy than
${\rm N_{Se}}$.

\subsection{Role of antisite defects in the doping problem}
\label{disdoping}

Our formation-energy calculation for phosphorus-doped
stoichiometric ZnSe suggests that, when the
Fermi level is near the valence-band edge as a result of
phosphorus doping, the formation energy of the antisite
defect ${\rm P_{Zn}}$ in the 3+ charge state
is comparable to or lower than that of the substitutional
${\rm P_{Se}}$ in the neutral or 1-- charge state.
Therefore, when shallow acceptors (${\rm P^0_{Se}}$) are
created, they are compensated by the donors
(${\rm P_{Zn}}$). Since, near the valence-band edge,
the stable charge state of the antisite defect is 3+,
three acceptors are compensated by a single antisite defect.
According to the calculations by Laks
{\it et al.} on native defects, the
concentration of native defects in stoichiometric ZnSe
is quite low (of order
$10^{10}$cm$^{-3}$ at $T = 600$K in material doped $p$-type
with $10^{18}$cm$^{-3}$ dopants).\cite{laks92}
Thus, the equilibrium state is most likely determined by the
concentrations of the dopant-associated defects.
In the equilibrium state for which the concentrations of
${\rm P^{3+}_{Zn}}$ and ${\rm P^{1-}_{Se}}$ are
the same order of magnitude, the calculated Fermi level
lies at 0.4--0.5 eV above the VBM.\cite{explan2}

In the equilibrium state,
${\rm P^{3+}_{Zn}}$ and ${\rm P^{1-}_{Se}}$ are the
dominant impurities, and small concentrations of
${\rm P^0_{Se}}$ and positively-charged
${\rm P_{int}}(H)$ with other intrinsic defects
such as Zn interstitials and Zn vacancies are expected.
For $E_F = 0.4$eV, we expect about $10^{16}$cm$^{-3}$
unionized acceptors (${\rm P^0_{Se}}$) at $T$ = 600 K
in a material with $10^{19}$cm$^{-3}$ dopants.
When the Se chemical potential is near its minimum allowed
value, the formation energy of ${\rm P^0_{Se}}$ becomes lower than
that of ${\rm P_{Zn}}$, indicating that the compensation of
shallow acceptors by ${\rm P_{Zn}}$ would be suppressed. However,
in this non-stoichiometric Zn-rich condition, compensation by
Zn interstitials acting as double donors is expected
instead.\cite{jans89,laks92} Thus, the $p$-type doping
would probably not improve.

On the other hand, the calculation for the nitrogen defects
indicates that substitutional nitrogen (which forms a shallow
acceptor) is lower in formation energy than nitrogen antisite
or interstitial defects. Therfore, even when the Fermi level is
positioned near the valence-band edge as a result of heavy
nitrogen doping, the concentrations of the antisite and the
interstitial defects are expected to be too small to cause a
substantial compensation of the free holes. These results suggest
that nitrogen should be a more robust acceptor than phosphorus,
which is consistent with the successful $p$-type doping of ZnSe
based on the introduction of nitrogen dopants.\cite{qiu91}

Fig.~\ref{efpn} compares the trends in the relative formation
energies of phosphorus and nitrogen impurities
in extreme $p$-type ZnSe (Fermi level at the VBM).

\section{Summary}
\label{sec:conc}

In summary, we have studied the electronic properties and relaxed
ionic structures of phosphorus and nitrogen defects in ZnSe, using
pseudopotential total-energy calculations within the
local-density approximation to density-functional theory.
Conventional norm-conserving pseudopotentials are not well
suited to the study of ZnSe and other II-VI semiconductors,
as the tightly-bound cation $d$ orbitals would
require a very large plane-wave basis set.
We employ instead an ultrasoft pseudopotential scheme, in which
the problematic Zn 3$d$ orbitals are included as valence states
with a modest plane-wave cutoff energy of 25 Ry. This approach,
combined with the conjugate-gradient technique for the minimization
of the total-energy functional, enables us to study the defected ZnSe
system efficiently and accurately.
Various charge states of P and N defects
are considered; their electronic states and ionic configurations
are studied, and their formation energies are calculated.

The calculational results indicate that, for stoichometric
ZnSe material, the difficulty
in doping ZnSe $p$-type with phosphorus dopants is caused by
compensation of the free holes by antisite defects P$_{\rm Zn}$
which act as triple donors. In non-stoichiometric Zn-rich conditions,
the compensation by P$_{\rm Zn}$ is expected to be supressed,
but the compensation of the holes by Zn
interstitials is expected instead. Thus, the
$p$-type doping would probably not improve. In the case of nitrogen
doping, the concentrations of the antisite defect ${\rm N_{Zn}}$ and
interstitial defect at the $T_{\rm Zn}$ site in stoichiometric ZnSe are
expected to be too small to cause significant compensation of the free
holes, which is consistent with the successful fabrication of $p$-type
ZnSe material based on nitrogen doping.

\acknowledgments

This work is  supported by NSF Grant No.\ DMR-91-15342.
Cray YMP time was provided by the National Center
for Supercomputing Applications under grant DMR920003N.

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\begin{figure}
\caption{(a) Definitions of quantities $z$, $z^\prime$, and $\theta$
for $C_{3v}$ defect configurations.  (b) Same for $C_{2v}$ defects.
Open and filled atoms represent impurity and neighboring host atoms,
respectively.}
\label{defconf}
\end{figure}

\begin{figure}
\caption{Formation energies of antisite ${\rm P_{Zn}}$ defects
in different charge states, relative to
${\rm P^0_{Zn}}$, as a function of the Fermi level position.}
\label{efpzn}
\end{figure}

\begin{figure}
\caption{Formation energies of phosphorus impurities, relative
to ${\rm P_{int}^0}(T_{\rm Zn})$, as a function of the Se chemical
potential for three characteristic Fermi levels: ({\rm a}) 0.0 eV,
({\rm b}) 0.7 eV, and ({\rm c}) 1.4 eV above the VBM.}
\label{eformp}
\end{figure}

\begin{figure}
\caption{Formation energies of nitrogen impurities, relative
to ${\rm N_{int}^0}(T_{\rm Zn})$, as a function of the Se chemical
potential for three characteristic Fermi levels: ({\rm a}) 0.0 eV,
({\rm b}) 0.7 eV, and ({\rm c}) 1.4 eV above the VBM.}
\label{eformn}
\end{figure}

\begin{figure}
\caption{Formation energies of ({\rm a}) phosphorus and
({\rm b}) nitrogen impurities
as a function of the Se chemical potential, for the Fermi level
at the VBM.}
\label{efpn}
\end{figure}

\newpage
\begin{table}
\caption{The thermal ionization levels (relative to the VBM)
of the antisite defect ${\rm P_{Zn}}$, and reaction energy of
$2{\rm P^{2+}_{Zn}} \rightarrow {\rm P^{1+}_{Zn}} + {\rm P^{3+}_{Zn}}$
(positive is exothermic),
calculated for supercells containing the defect in 32-atom and in
16-atom cell geometries.}
\vskip .2in
\begin{tabular}{ccc}
       & 32-atom cell       & 16-atom cell \\
\hline
3+/1+  & 0.71 eV            & 0.45 eV \\
1+/0   & 1.71 eV            & 2.01 eV \\
\hline
$2{\rm P^{2+}_{Zn}} \rightarrow {\rm P^{1+}_{Zn}} + {\rm P^{3+}_{Zn}}$
       &+0.41 eV & +0.27 eV \\
\end{tabular}
\label{chadef}
\end{table}

\mediumtext
\begin{table}
\caption{Stable configurations of defects in ZnSe in various
charge states. The definitions of $z$, $z^\prime$, and $\theta$ are
given in the text and in Fig~\ref{defconf}, and are
different for $C_{3v}$ and $C_{2v}$ configurations.
Given in the parenthesis is the ``breathing-relaxed'' bond length,
i.e., that between the impurity and its
nearest neighbors when ($T_d$) symmetry is imposed.
For reference, the bond length of bulk ZnSe is 2.45 \AA.
$E_{JT}$ represents the energy lowering from the Jahn-Teller distortion.
Results for ${\rm P_{Se}}$ and ${\rm N_{Se}}$ from
% Ref.\ \onlinecite{kwak93a}
Ref.\ 17
are included for completeness.}
\vskip .2in
\begin{tabular}{llcccrcc}
Defect &Charge &Structure &$z$ (\AA) &$z^\prime$ (\AA) &$\theta (^\circ)$ &
 (\AA) &$E_{JT}$(eV) \\
\hline
                &1-- &$T_d$     &2.29   &       &109.5  &(2.29) &     \\ 
${\rm P_{Se}}$  &0   &$C_{3v}$  &2.40   &2.30   &107.7  &(2.33) &0.04 \\
                &1+  &$C_{3v}$  &2.57   &2.31   &105.4  &(2.35) &0.15 \\
\hline
                &0   &$C_{3v}$  &2.33   &2.66   &113.7  &(2.54) &0.16 \\
${\rm P_{Zn}}$  &1+  &$T_d$     &2.50   &       &109.5  &(2.50) &     \\ 
                &2+  &$T_d$     &2.39   &       &109.5  &(2.39) &     \\ 
                &3+  &$T_d$     &2.24   &       &109.5  &(2.24) &     \\ 
\hline
                             &2-- &$C_{3v}$  &2.32  &2.33   &109.7 
&(2.33) &     \\
                             &1-- &$C_{3v}$  &2.32  &2.40   &110.9 
&(2.39) &0.03 \\
${\rm P_{int}}(T_{\rm Zn})$  &0   &$C_{2v}$  &2.45  &2.45   &109.5 
&(2.45) &     \\
                             &1+  &$C_{3v}$  &3.70  &2.55   &75.4  
&(2.54) &1.18 \\
                             &2+  &$C_{3v}$  &3.34  &2.64   &84.7  
&(2.58) &0.63 \\
                             &3+  &$C_{3v}$  &3.22  &2.66   &88.1  
&(2.65) &0.44 \\
\hline
% \hline
                &1-- &$T_d$       &1.96   &   &109.5  &(1.96) &   \\
${\rm N_{Se}}$  &0   &$\sim T_d$  &1.95   &   &109.5  &(1.95) &   \\
                &1+  &$\sim T_d$  &1.94   &   &109.5  &(1.94) &   \\
\hline
                &1-- &$C_{3v}$  &1.78   &2.65   &121.3  &(2.31) &1.03 \\
${\rm N_{Zn}}$  &0   &$C_{3v}$  &1.88   &2.46   &116.1  &(2.27) &0.37 \\
                &1+  &$T_d$     &2.21   &       &109.5  &(2.21) &     \\
\hline
                            &3-- &$T_d$     &1.91  &       &109.5  
&(1.91) &  \\
${\rm N_{int}}(T_{\rm Zn})$ &2-- &$C_{3v}$  &1.96  &1.95   &109.2  
&(1.95) &  \\
                            &1-- &$C_{3v}$  &2.02  &2.01   &109.0  
&(2.02) &  \\
                            &0   &$C_{2v}$  &2.06  &2.06   &109.5  
&(2.06) &  \\
\end{tabular}
\label{summ}
\end{table}

\end{document}

