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\title{Optical and Electronic Properties of Diamonds}
\author{  Vincent J. Schmithorst\\
          \\
          Department of Physics\\
          University of Cincinnati \\
          Cincinnati, Ohio 45221\\[1em]}

\date{November 17, 1995}
\maketitle
\begin{abstract}

Diamond shows interesting optical and electronic properties. 
Absorption and emission exist at energies well below the gap energy
of around 5.6 eV even in the most perfect diamonds due to lattice
defects.  Because there is intense emission in the blue and green
regions, diamonds can be used as scintillation detectors of
ionizing radiation.  The presence of defects causes some diamonds
to be semi-conductors.  Because of their high thermal conductivity
at room temperature, such semi-conductors can be used as
thermistors.\\[1em]
\end{abstract}

\pagebreak

\def\baselinestretch{1.1}

Diamonds are interesting to study due to their optical and
electrical properties.  Perfect diamonds have a band gap of
approximately 5.6 eV; however, diamonds with defects will narrow
the band gap considerably.  The presence of these defects accounts
for optical emission and absorption at visible wavelengths, and for
much lower resistivities than would be expected for a perfect
diamond; in fact, some diamonds are like semiconductors with
resistivities on the order of a few ohm-cm.

Due to optical emission at visible wavelengths, diamonds can be
used as scintillation detectors.  Semi-conducting diamonds make
useful practical thermistors due to their very high thermal
conductivity.  There is still much research to be done in order to
effectively utilize the capabilities of diamonds; in particular, to
cost-effectively produce synthetic diamonds with the desired
properties.

DIAMOND STRUCTURE

The diamond structure consists of two interlaced fcc structures of
carbon atoms with relative positions of (0,0,0) and (1/4,1/4,1/4). 
The lattice constant $a$=3.56 $\AA$.  This arrangement yields
tetrahedral bonding between nearest neighbor atoms with a bond
length $b$=1.53 $\AA$.  The bond is a covalent bond with sp-3
hybridization.

OPTICAL PROPERTIES OF PERFECT DIAMONDS

Because the gap energy is around 6 eV for a perfect diamond, there
would be expected to be no optical absorption nor stimulated
emission for wavelengths longer than ultra-violet (about 2000
$\AA$).  In practice, there exist 'water-white' diamonds, but also
diamonds with different colors under white light, including pink,
blue, yellow, and green, which happens due to the absorption of
certain wavelengths, thus causing the diamond to appear the
corresponding complementary color.  It is evident therefore that
many diamonds have electronic band separations with energies much
less than 6 eV.

Diamonds are classified as 'type 1' and 'type 2' depending on
whether significant absorption occurs at wavelengths shorter than
approximately 3000 $\AA$ or 2200 $\AA$, respectively.  A type 2
variety corresponds better to the theoretical prediction for a
perfect diamond stated above.  Figure 1 shows the absorption
spectra of two type 2 diamonds at room temperature.  There is some
small absorption at low energies, but a very sharp increase in the
absorption coefficient at around 5.4 eV or so, which corresponds to
the theoretical prediction.

\begin{figure}[htb]

\centerline{\psfig{figure=fig1.ps,height=2.5in}}

\caption[]{\em{Absorption spectra at room temperature of
two type 2 diamonds.\cite{refB}}}
\end{figure}

Intense optical emission from diamonds may be stimulated by
ionizing radiations such as X-rays, fast electrons, or alpha-
particles.  For even the most perfect type 2 diamonds, as shown in
Figure 2, the emission is largely confined to a broad blue band,
whose peak shifts to shorter wavelengths at lower temperatures. 
These results are in sharp contrast to the theoretical prediction,
since, for perfect diamond, the only possible emission is with
energy $E_{G}$ approximately 5.6 eV (corresponding to a wavelength
of approximately 2200 $\AA$).  For the most perfect diamonds, the
concentration of defects is only around a few parts per million. 
Even so, this small concentration of defects is sufficient to
noticeably affect the emission peaks.
\begin{figure}[htb]

\centerline{\psfig{figure=fig2.ps,height=5in}}

\caption[]{\em{Cathodoluminescence spectrum at different
temperatures for a type 2 diamond.\cite{refC}}}
\end{figure}

OPTICAL PROPERTIES OF DIAMONDS WITH DEFECTS

After many specimens of diamond were examined, the distinction made
between 'type 1' and 'type 2' diamonds due to whether there was
significant absorption at wavelengths longer than 2200 $\AA$ was
shown to be inadmissible, as there was a variety of different 'cut-
off' wavelengths.  It was therefore concluded that the difference
\pagebreak
was arbitrary and simply due to the relative proportion of defects
in the diamonds.  This fits in with the prediction that for
diamonds with defects the electronic separation between levels will
be less than 5.4 eV; how much less depends on the proportion of
defects.

Figure 3 shows a variety of emission spectra for natural diamonds. 
It can be seen that the wavelengths are approximately between 4000
and 7000 $\AA$, and that the spectrum is highly specimen dependent. 
It is interesting that the range of emitted wavelengths is similar
for the most perfect diamonds and for the more imperfect ones,
although the shapes of the spectra are different.

\begin{figure}[htb]

\centerline{\psfig{figure=fig3.ps,height=2.5in}}

\caption[]{\em{Cathodoluminescence spectra for four
specimens of natural diamond.\cite{refC}}}
\end{figure}

DIAMONDS AS SCINTILLATION DETECTORS

As shown above, de-excitation transitions at defect levels causes
the emission of light at visible wavelengths, in the blue or green
part of the spectrum, and this behavior is present in all diamonds. 
The intensity of this behavior has been shown to rarely differ by
as much as a factor of 10, even among specimens with markedly
different electrical and optical properties, and this intensity is
more than sufficient to allow detection of any individual ionizing
radiation of more than a few KeV by collection with a
photomultiplier tube.  After comparing $\alpha$- and $\beta$-
particle scintillation and cathodoluminescent spectra it was found
that:

1) for $\beta$-particles the absolute magnitude was approximately
proportional to the energy, and that the shape did not change with
change in the energy;

2) the shape of the spectrum was almost completely specimen-
independent for $\alpha$-particles, whereas for $\beta$-particles
a greater relative proportion of defects produced relatively more
long-wavelength emission.\cite{refA}

ELECTRICAL PROPERTIES OF DIAMONDS WITH DEFECTS

Since diamond has the same crystal structure as germanium and
silicon, many of the electrical properties of diamond can be
predicted.  Solid state theory gives the following formula for the
number of negative carriers per unit volume, $n$:

\begin{displaymath}
n=2 \left(\frac{2 \pi m_{n} k T}{h^{2}} \right)^{\frac{3}{2}}
e^{\left(\frac{-(-E_{C}-E_{F})}{k T} \right)}
\end{displaymath}


where $m_{n}$ is the effective mass of the negative carrier, $E_{
C}$ and $E_{F}$ are the energies of the conduction level and the
Fermi level, respectively, and ($E_{C}-E_{F}$) is approximately
equal to $E_{G}$/2 where $E_{G}$ is the energy gap between the
bottom of the conduction band and the top of the valence band.  A
similar expression will hold for the number of positive carriers
(holes in the valence band) per unit volume, $p$, with $m_{n}$
replaced by $m_{p}$, the effective mass of the hole, and ($E_{C}$-
$E_{F}$) replaced by ($E_{F}$-$E_{V}$), where $E_{V}$ is the energy
level of the top of the valence band, and again, ($E_{F}$-$E_{V}$)
is approximately equal to $E_{G}$/2.  If we define the mean number
of carriers, $N$, as equal to $\sqrt{np}$, and insert $E_{G}$=5.6
eV into the above expressions, there are less then $10^{-60}$
intrinsic carriers per unit volume at room temperature.  Since the
mobilities of the carriers for diamond would be expected to be of
the same order of magnitude as for germanium, and since the
resistivity, $\rho$, is approximately inversely proportional to N,
we can estimate the resistivity of diamond by comparing with the
values for germanium: (N=2.4 X $10^{13}$,$\rho$ is on the order of
$10^{2}\Omega$-cm), and thus the resistivity of a perfect
diamond is expected to be above $10^{70}\Omega$-cm.

In practice, however, the resistivity of real diamonds rarely
exceeds $10^{18}$ $\Omega$-cm, and in fact may be as low as a few
$\Omega$-cm in semi-conducting diamonds.  This fact further
indicates that there are electronic levels in real diamonds with
separations much less than $E_{G}$ due to defects in the lattice,
and that the resistivity decreases as the relative proportion of
defects increases.

SEMI-CONDUCTING DIAMONDS

Certain aspects of a few specimens of semi-conducting diamonds have
been studied and their general characteristics are similar to those
of other impurity-activated semiconductors of low impurity
concentration.  They may show pronounced macroscopic
inhomogeneities.  They are often colored, as the lattice defects
responsible for the semiconducting properties may also result in
optical absorption at visible wavelengths, as discussed above.


By performing a Hall effect experiment, the sign of the predominant
charge carriers may be determined.  In the case of diamond, the
carriers have been found to be of $p$-type (the charge carriers are
positive holes travelling in the valence band).  From the equation
previously given for the number of charge carriers, the activation
energy can be found by plotting ln T vs. 1/T; it is found to be
0.35 eV.  The simultaneous measurement of the Hall constant and the
resistivity allows the carrier mobility to be determined as about
1550 $\frac{cm sec^{-1}}{V cm^{-1}}$.  This value agrees within
experimental error with that found in several determinations by
other methods.\cite{refD}  The mobility as a function as
temperature is approximately proportional to $T^{\frac{-3}{2}}$
which indicates that lattice scattering predominates.  At room
temperature $n_{p}$ is approximately 2 X $10^{13}$ cm$^{-3}$, which
is about the same as the intrinsic carrier density of germanium.

Semi-conducting diamonds, because of their high thermal
conductivity, show a strong coupling between the charge carriers
and the lattice vibrations.  This coupling is known as phonon drag. 
An electric field drags the electron distribution from equilibrium,
and then due to interaction between electrons and phonons, this is
communicated to the long wavelength phonon modes.  Since the
phonon-phonon interaction is very small, the long wavelength phonon
distribution is pushed far from equilibrium, creating a large heat
current.  Due to the high Debye temperature of diamond, this effect
is appreciable even at room temperature, thus making semi-
conducting diamonds useful practical thermistors.\cite{refA}

\begin{thebibliography}{99}
\bibitem{refA} F.C.\ Champion, Electronic Properties of Diamonds,
Butterworths, 1963.
\bibitem{refB} C.D.\ Clark, R.W.\ Ditchburn and H.B.\ Dyer, Proc.
roy. Soc., {\bf A234}, 363(1956).
\bibitem{refC} P.J.\ Dean, P.J.\ Kennedy and J.E.\ Ralph, Proc.
phys. Soc. Lond., {\bf 76}, 670(1960).
\bibitem{refD} A.G.\ Redfield, Phys. Rev., {\bf 94}, 526(1954).

\end{thebibliography}
\end{document}
