|
|
Metal-to-semiconductor contacts are of great importance since they are
present in every semiconductor device. They can behave either as a Schottky
barrier or as an ohmic contact dependent on the characteristics of the
interface. This chapter contains an analysis of the electrostatics
of the M-S junction (i.e. the charge, field and potential distribution within
the device) followed by a derivation of the current voltage
characterisitics due to diffusion, thermionic emission and tunneling and a
discussion of the
non-ideal effects in Metal-Semiconductor junctions.
Structure and principle of operation
|
|
1. Structure
The structure of a metal-semiconductor junction is shown in
Figure 1. It consists of a metal contacting a piece of
semiconductor. An ideal Ohmic contact, a contact such that no
potential exists between the metal and the semiconductor, is made to
the other side of the semiconductor. The sign convention of the
applied voltage and current is also shown on Figure 1. |
|
Figure 1 : Structure and sign
convention of a metal-semiconductor junction
|
2. Flatband diagram and built-in potential
|
|
| The barrier between the metal and the semiconductor
can be identified on an energy band diagram. To construct such
diagram we first consider the energy band diagram of the metal and
the semiconductor, and align them using the same vacuum level as
shown in Figure 2 (a). As the metal and semiconductor are brought
together, the Fermi energies of the metal and the semiconductor do
not change right away. This yields the flatband diagram of Figure 2
(b). |
|
Figure 2 :
|
Energy band diagram of the
metal and the semiconductor before (a) and after (b) contact is
made. |
| The barrier height, fB,
is defined as the potential difference between the Fermi energy of
the metal and the band edge where the majority carriers reside. From
Figure 2 (b) one finds that for an n-type semiconductor the barrier
height is obtained from: |
 |
(1.1) |
| Where FM is
the work function of the metal and c is
the electron affinity. The work function of selected metals as
measured in vacuum can be found in Table 1. For
p-type material, the barrier height is given by the difference
between the valence band edge and the Fermi energy in the metal: |
 |
(1.2) |
| A metal-semiconductor junction will therefore form
a barrier for electrons and holes if the Fermi energy of the metal
as drawn on the flatband diagram is somewhere between the conduction
and valence band edge. |
| In addition, we define the built-in potential,
fI, as the difference between
the Fermi energy of the metal and that of the semiconductor. |
 |
(1.3) |
 |
(1.4) |
| The measured barrier height for selected
metal-semiconductor junctions is listed in Table 1.
These experimental barrier heights often differ from the ones
calculated using (1.1) or (1.2).
This is due to the detailed behavior of the metal-semiconductor
interface. The ideal metal-semiconductor theory assumes that both
materials are infinitely pure, that there is no interaction between
the two materials nor is there an interfacial layer. Chemical
reactions between the metal and the semiconductor alter the barrier
height as do interface states at the surface of the semiconductor
and interfacial layers. Some general trends however can still be
observed. As predicted by (1.1), the
barrier height on n-type semiconductors increases for metals with a
higher work function as can be verified for silicon. Gallium
arsenide on the other hand is known to have a large density of
surface states so that the barrier height becomes virtually
independent of the metal. Furthermore, one finds the barrier heights
reported in the literature to vary widely due to different surface
cleaning procedures. |
| Table 1: Workfunction of selected
metals and their measured barrier height on Ge, Si and GaAs. |
|
|
| The flatband diagram, shown in Figure
2 (b), is not a thermal equilibrium diagram,
since the Fermi energy in the metal differs from that in the
semiconductor. Electrons in the n-type semiconductor can lower their
energy by traversing the junction. As the electrons leave the
semiconductor, a positive charge, due to the ionized donor atoms,
stays behind. This charge creates a negative field and lowers the
band edges of the semiconductor. Electrons flow into the metal until
equilibrium is reached between the diffusion of electrons from the
semiconductor into the metal and the drift of electrons caused by
the field created by the ionized impurity atoms. This equilibrium is
characterized by a constant Fermi energy throughout the structure.
|
|
Figure 3 : |
Energy band diagram of a
metal-semiconductor contact in thermal equilibrium. |
| It is of interest to note that in thermal
equilibrium, i.e. with no external voltage applied, there is a
region in the semiconductor close to the junction ( ), which is
depleted of mobile carriers. We call this the depletion region. The
potential across the semiconductor equals the built-in potential,
fi. |
2.4. Forward and reverse bias
|
|
| Operation of a metal-semiconductor junction under
forward and reverse bias is illustrated with Figure 4. As a positive
bias is applied to the metal (Figure 4 (a)), the Fermi energy of the
metal is lowered with respect to the Fermi energy in the
semiconductor. This results in a smaller potential drop across the
semiconductor. The balance between diffusion and drift is disturbed
and more electrons will diffuse towards the metal than the number
drifting into the semiconductor. This leads to a positive current
through the junction at a voltage comparable to the built-in
potential. |
|
Figure 4 : |
Energy band diagram of a
metal-semiconductor junction under (a) forward and (b) reverse bias |
| As a negative voltage is applied (Figure 4 (b)),
the Fermi energy of the metal is raised with respect to the Fermi
energy in the semiconductor. The potential across the semiconductor
now increases, yielding a larger depletion region and a larger
electric field at the interface. The barrier, which restricts the
electrons to the metal, is unchanged so that the flow of electrons
is limited by that barrier independent of the applied voltage. The
metal-semiconductor junction with positive barrier height has
therefore a pronounced rectifying behavior. A large current exists
under forward bias, while almost no current exists under reverse
bias. |
| The potential across the semiconductor therefore
equals the built-in potential, fi,
minus the applied voltage, Va. |
 |
(2.1) |
3. Electrostatic analysis
3.1. General discussion - Poisson's equation
|
|
| The electrostatic analysis of a metal-semiconductor
junction is of interest since it provides knowledge about the charge
and field in the depletion region. It is also required to obtain the
capacitance-voltage characteristics of the diode. |
| The general analysis starts by setting up Poisson's
equation: |
 |
(3.1) |
| where the charge density, r,
is written as a function of the electron density, the hole density
and the donor and acceptor densities. To solve the equation, we have
to express the electron and hole density, n and p, as
a function of the potential, f, yielding:
|
 |
(3.2) |
 |
(3.3) |
| where the potential is chosen to be zero in the
n-type region, where x >> xn. |
| This second-order non-linear differential equation
(3.2) can not be solved analytically. Instead we will make the
simplifying assumption that the depletion region is fully depleted
and that the adjacent neutral regions contain no charge. This full
depletion approximation is the topic of section 3.2. |
3.2. Full depletion approximation
|
|
| The simple analytic model of the
metal-semiconductor junction is based on the full depletion
approximation. This approximation is obtained by assuming that the
semiconductor is fully depleted over a distance xd,
called the depletion region. While this assumption does not provide
an accurate charge distribution, it does provide very reasonable
approximate expressions for the electric field and potential
throughout the semiconductor. |
3.3. Full depletion analysis
|
|
| We now apply the full depletion approximation to an
M-S junction containing an n-type semiconductor. We define the
depletion region to be between the metal-semiconductor interface (x
= 0) and the edge of the depletion region (x = xd).
The depletion layer width, xd, is unknown at this
point but will later be expressed as a function of the applied
voltage. |
| To find the depletion layer width, we start with
the charge density in the semiconductor and calculate the electric
field and the potential across the semiconductor as a function of
the depletion layer width. We then solve for the depletion layer
width by requiring the potential across the semiconductor to equal
the difference between the built-in potential and the applied
voltage, fi - Va.
The different steps of the analysis are illustrated by Figure 3.1. |
| As the semiconductor is depleted of mobile carriers
within the depletion region, the charge density in that region is
due to the ionized donors. Outside the depletion region, the
semiconductor is assumed neutral. This yields the following
expressions for the charge density, r:
|
 |
(3.3.4) |
| where we assumed full ionization so that the
ionized donor density equals the donor density, Nd.
This charge density is shown in Figure 3.1 (a). The charge in the
semiconductor is exactly balanced by the charge in the metal, QM,
so that no electric field exists except around the
metal-semiconductor interface. |
| Figure 3.1 : |
(a) Charge density, (b)
electric field, (c) potential and (d) energy as obtained with the
full depletion analysis. |
| Using Gauss's law we obtain electric field as a
function of position, also shown in Figure 3.1 (b): |
 |
(3.5) |
| where es is
the dielectric constant of the semiconductor. We also assumed that
the electric field is zero outside the depletion region. It is
expected to be zero there since a non-zero field would cause the
mobile carriers to redistribute until there is no field. The
depletion region does not contain mobile carriers so that there can
be an electric field. The largest (absolute) value of the electric
field is obtained at the interface and is given by: |
 |
(3.6) |
| where the electric field was also related to the
total charge (per unit area), Qd, in the depletion
layer. Since the electric field is minus the gradient of the
potential, one obtains the potential by integrating the expression
for the electric field, yielding: |
 |
(3.7) |
| We now assume that the potential across the metal
can be neglected. Since the density of free carriers is very high in
a metal, the thickness of the charge layer in the metal is very
thin. Therefore, the potential across the metal is several orders of
magnitude smaller that that across the semiconductor, even though
the total amount of charge is the same in both regions. |
| The total potential difference across the
semiconductor equals the built-in potential, fi,
in thermal equilibrium and is further reduced/increased by the
applied voltage when a positive/negative voltage is applied to the
metal as described by equation (3.2.5). This boundary condition
provides the following relation between the semiconductor potential
at the surface, the applied voltage and the depletion layer width:
|
 |
(3.8) |
| Solving this expression for the depletion layer
width, xd, yields: |
 |
(3.9) |
3.4. Junction capacitance
|
|
| In addition, we can obtain the capacitance as a
function of the applied voltage by taking the derivative of the
charge with respect to the applied voltage yielding: |
 |
(3.10) |
| The last term in the equation indicates that the
expression of a parallel plate capacitor still applies. One can
understand this once one realizes that the charge added/removed from
the depletion layer as one decreases/increases the applied voltage
is added/removed only at the edge of the depletion region. While the
parallel plate capacitor expression seems to imply that the
capacitance is constant, the metal-semiconductor junction
capacitance is not constant since the depletion layer width, xd,
varies with the applied voltage. |
3.5. Schottky barrier lowering
|
|
| Image charges build up in the metal electrode of a
metal-semiconductor junction as carriers approach the
metal-semiconductor interface. The potential associated with these
charges reduces the effective barrier height. This barrier reduction
tends to be rather small compared to the barrier height itself.
Nevertheless this barrier reduction is of interest since it depends
on the applied voltage and leads to a voltage dependence of the
reverse bias current. Note that this barrier lowering is only
experienced by a carrier while approaching the interface and will
therefore not be noticeable in a capacitance-voltage measurement.
|
| An energy band diagram of an n-type silicon
Schottky barrier including the barrier lowering is shown in Figure
3.2: |
| Figure 3.2: |
Energy band diagram of a silicon Schottky barrier
with fB = 0.8 V and Nd
= 1019 cm-3. |
| Shown is the energy band diagram obtained using the
full-depletion approximation, the potential reduction experienced by
electrons, which approach the interface and the resulting conduction
band edge. A rounding of the conduction band edge can be observed at
the metal-semiconductor interface as well as a reduction of the
height of the barrier. |
| The calculation of the barrier reduction assumes
that the charge of an electron close to the metal-semiconductor
interface attracts an opposite surface charge, which exactly
balances the electron's charge so that the electric field
surrounding the electron does not penetrate beyond this surface
charge. The time to build-up the surface charge and the time to
polarize the semiconductor around the moving electron is assumed to
be much shorter than the transit time of the electron . This
scenario is based on the assumption that there are no mobile or
fixed charges around the electron as it approaches the
metal-semiconductor interface. The electron and the induced surface
charges are shown in Figure 3.3: |
| Figure 3.3: |
a) Field lines and surface
charges due to an electron in close proximity to a perfect conductor
and b) the field lines and image charge of an electron. |
| It can be shown that the electric field in the
semiconductor is identical to that of the carrier itself and another
carrier with opposite charge at equal distance but on the opposite
side of the interface. This charge is called the image charge. The
difference between the actual surface charges and the image charge
is that the fields in the metal are distinctly different. The image
charge concepts is justified on the basis that the electric field
lines are perpendicular to the surface a perfect conductor, so that,
in the case of a flat interface, the mirror image of the field lines
provides continuous field lines across the interface. |
| The barrier lowering depends on the square root of
the electric field at the interface and is calculated from: |
 |
(3.11) |
4. Schottky diode current
| The current across a metal-semiconductor junction
is mainly due to majority carriers. Three distinctly different
mechanisms exist: diffusion of carriers from the semiconductor into
the metal, thermionic emission of carriers across the Schottky
barrier and quantum-mechanical tunneling through the barrier. The
diffusion theory assumes that the driving force is distributed over
the length of the depletion layer. The thermionic emission theory on
the other hand postulates that only energetic carriers, those, which
have an energy equal to or larger than the conduction band energy at
the metal-semiconductor interface, contribute to the current flow.
Quantum-mechanical tunneling through the barrier takes into account
the wave-nature of the electrons, allowing them to penetrate through
thin barriers. In a given junction, a combination of all three
mechanisms could exist. However, typically one finds that only one
limits the current, making it the dominant current mechanism. |
| The analysis reveals that the diffusion and
thermionic emission currents can be written in the following form: |
 |
(4.1) |
| This expression states that the current is the
product of the electronic charge, q, a velocity, v,
and the density of available carriers in the semiconductor located
next to the interface. The velocity equals the mobility multiplied
with the field at the interface for the diffusion current and the
Richardson velocity (see section
3.4.2) for the thermionic emission current. The minus one term
ensures that the current is zero if no voltage is applied as in
thermal equilibrium any motion of carriers is balanced by a motion
of carriers in the opposite direction. |
| The tunneling current is of a similar form, namely: |
 |
(4.2) |
| where vR is the Richardson
velocity and n is the density of carriers in the
semiconductor. The tunneling probability term, Q,
is added since the total current depends on the carrier flux
arriving at the tunnel barrier multiplied with the probability,
Q, that they tunnel through the barrier.
|
| This analysis assumes that the depletion layer is
large compared to the mean free path, so that the concepts of drift
and diffusion are valid. The resulting current density equals: |
 |
(4.3) |
The current therefore depends exponentially on the
applied voltage, Va, and the barrier height,
fB. The prefactor can more
easily be understood if one rewrites it as a function of the
electric field at the metal-semiconductor interface,
 max: |
 |
(4.4) |
 |
(4.5) |
| so that the prefactor equals the drift current at
the metal-semiconductor interface, which for zero |
| The thermionic emission theory assumes that
electrons, which have an energy larger than the top of the barrier,
will cross the barrier provided they move towards the barrier. The
actual shape of the barrier is hereby ignored. The current can be
expressed as: |
 |
(4.6) |
where  is
the Richardson constant and fB
is the Schottky barrier height. |
| The expression for the current due to thermionic
emission can also be written as a function of the average velocity
with which the electrons at the interface approach the barrier. This
velocity is referred to as the Richardson velocity given by: |
 |
(4.7) |
| So that the current density becomes: |
 |
(4.8) |
| The tunneling current is obtained from the product
of the carrier charge, velocity and density. The velocity equals the
Richardson velocity, the velocity with which on average the carriers
approach the barrier. The carrier density equals the density of
available electrons, n, multiplied with the tunneling
probability, Q, yielding: |
 |
(4.9) |
| Where the tunneling probability is obtained from: |
 |
(4.10) |
| and the electric field equals
= fB/L.
|
| The tunneling current therefore depends
exponentially on the barrier height, fB,
to the 3/2 power. |
Bibliography
- Physics of Semiconductor Devices, Second edition, S. M. Sze,
Wiley & Sons, 1981, Chapter 5.
- Device Electronics for Integrated Circuits, Second edition,
R.S. Muller and T. I. Kamins, Wiley & Sons, 1986, Chapter 3.
SCHOTTKY DIODES
Metal-semiconductor contact at zero bias
Electrons in the
conduction band of a crystal can be viewed as sitting in a potential box formed
by the crystal boundaries (see Fig. 1). This potential box for electrons is usually
deeper in a metal than in a semiconductor. If a metal and a semiconductor are brought
together into a close proximity, some electrons from the metal will move into
the semiconductor and some electrons from the semiconductor will move into the
metal. However, since the barrier for
the electron escape from the metal is higher, more electrons will transfer from
the semiconductor into the metal than in the opposite direction. At thermal equilibrium, the metal will be
charged negatively, and the semiconductor will be charged positively, forming a
dipole layer that is very similar to that in a p+-n junction. The Fermi level will be constant
throughout the entire metal-semiconductor system, and the energy band diagram
in the semiconductor will be similar to that for an n-type semiconductor in a p+-n junction (see Fig. 2).
Fig. 1. Schematic
energy diagram for electrons in conduction bands of a metal and of a semiconductor.
Energies Fm and Fs shown in Fig. 2 are called the metal and the semiconductor work functions. The work
function is equal to the difference
between the vacuum level (which is defined as a free electron energy in vacuum) and the Fermi level. The electron affinity of the semiconductor, Cs (also shown in Fig. 2), corresponds to the energy separation
between the vacuum level and the conduction band edge of the semiconductor.
Fig. 2. Simplified energy diagram
of GaAs
metal-semiconductor barrier q fb is the barrier height (0.75 eV), Cs is the electron affinity in the semiconductor, Fs and Fm are the semiconductor and the metal work functions, and Vbi (0.591 V) is the built-in
voltage. Donor concentration in GaAs is 1015 cm–3.
A metal-semiconductor diode is called
a Schottky diode.
In the idealized picture of the Schottky
junction shown in Fig. 2, the
energy barrier between the semiconductor and the metal is
 (1)
Since Fm > Fs the metal is charged negatively. The positive net space charge in the semiconductor leads to a band
bending
 (2)
where Vbi is called the built-in
voltage, in analogy with the corresponding quantity in a p-n junction. Note that qVbi is also identical to the difference between
the Fermi levels in the metal and the semiconductor when separated by a large
distance (no exchange of charge); see Fig. 1.
However, eq.
(1) and Fig. 2 are not quite correct. In reality, a change in the metal work
function, Fm, is not equal to the corresponding change in the barrier height ,fb, as predicted by eq. (1). In actual Schottky diodes, fb increases with an
increase in Fm but only by
0.1 to 0.3 eV when Fm increases by
1 to 2 eV.
Even though a detailed and accurate understanding of Schottky
barrier formation remains a challenge, many properties of Schottky
barriers may be understood independently of the exact mechanism determining the
barrier height. In other words, we can
simply determine the effective barrier height from experimental data. Usually, as a crude and empirical rule of
thumb, we can assume that the Schottky barrier height for an n-type semiconductor is close to 1/2 and
2/3 of the energy gap.
In a Schottky
diode, the semiconductor band diagram looks very similar to that of an n-type semiconductor in a p+-n diode (compare Fig. 1a and
2). Hence, the variation of the
space charge density, r, the electric field, F,
and the potential, f, in the semiconductor near the metal-semiconductor interface can be
found using the depletion approximation:
 (3)
 (4)
 (5)
(Here x
= 0 corresponds to the metal-semiconductor
interface.) The depletion layer width, xn, at zero bias is given by
 (6)
Schottky diode under bias
Forward bias
corresponds to a positive voltage applied to the metal with respect to the semiconductor. Just as for a p+-n junction, the depletion
width under small forward bias and reverse bias may be obtained by
substituting Vbi with Vbi– V, where V is the applied voltage.
As illustrated in Fig. 3, the application of a forward bias
decreases the potential barrier for electrons moving from the semiconductor
into the metal. Hence, the
current-voltage characteristic of a Schottky diode can be described by a diode equation, similar to that for a p-n
junction diode :
 (7)
where Is is the saturation current, Rsis the series resistance, Vth = kBT/q
is the thermal voltage, and h is the ideality factor (h typically varies from 1.02 to 1.6).
 (a) (b) (c)
Fig.
3. Band diagrams for a GaAs Schottky
barrier diode at (a) zero bias, (b) 0.2 V forward bias, and (c) 5 V reverse
bias. Dashed line shows the position of
the Fermi level in the metal (x <
0) and in the semiconductor (x > 0).
Thermionic
emission.
The diode saturation
current, Is, is typically much larger for Schottky
barrier diodes than in p-n junction
diodes since the Schottky barrier height is smaller than
the barrier height in p-n junction
diodes. In a p-n junction, the height of the barrier separating electrons in the
conduction band of the n-type region
from the bottom of the conduction band in the p-region is on the order of the energy
gap. A typical Schottky
barrier height is only about two thirds of the energy gap or less, as mentioned
above. Also,
the mechanism of the electron conduction is different. One can show that the saturation current
density in a Schottky diode with a relatively low doped semiconductor is given by
 (8)
where A* is called the Richardson constant. For a conduction band
minimum with spherical surfaces of equal energy (such as the G minimum in GaAs),
 (9)
where mn is the effective mass and a is an empirical
factor on the order of unity. The Schottky diode model described by eqs. (8) and (9) is called the thermionic emission
model. For Schottky
barrier diodes of Si,
A* = 96 A/(cm2K2). For GaAs,
A* = 4.4 A/(cm2K2).
The basic assumption of the thermionic model is that electrons have to pass over the
barrier in order to cross the boundary between the metal and the
semiconductor. Hence, to find the
saturation current, we have to estimate the number of electrons passing over
the barrier and their velocities. The
number of electrons, N(E)dE, having energies between E and E + dE
is proportional to the product of the Fermi-Dirac
distribution function, f(E),
and the number of states in this energy interval, g(E)dE, where g(E) is the density of
states:
 (10)
[N(E) = dn(E)dE where n(E) is the number of
electrons in the conduction band with energies higher than E. At high
energies, the Fermi-Dirac occupation function is very
close to the Boltzmann distribution function :
 (11)
The next step should
be to multiply the number of the electrons, N(E)dE,
in the energy interval from E to E + dE by the velocity of these
electrons. We have to account for
different directions of the electron velocities and integrate over energies
higher than the barrier height in order to determine the flux of the electrons
coming from the semiconductor into the metal.
Finally, we deduct the flux of the electrons coming from the metal into
the semiconductor. The difference
between these two fluxes will be proportional to the current density predicted
by the thermionic model. However, we can take a much simpler route if
we are interested in understanding the physics of the thermionic
model. To this end, let us consider a Schottky diode under a strong reverse bias when V is negative and – V >> hkBT. Then I = – Is [see eq. (7)],
and the band diagram looks like that shown in Fig. 3c. In this case the
energy difference between the Fermi level in the semiconductor and the top of
the barrier is so large that practically no electrons are available to come
from the semiconductor into the metal.
However, the Fermi level in the metal is much closer to the top of the
barrier, and electrons still come from the metal into the semiconductor. The flux of these electrons constitutes the
saturation current. In order to estimate
this flux, we should recall that the density of states is a relatively slow
function of energy [g(E) is proportional to (E – Ec)1/2;
compared to the distribution function, which decreases by exp(1) ≈ 2.718
each time E increases by kBT.
Hence, the largest contribution into the electron flux will come from
the electrons that are a few kBT above the barrier. The number of such electrons will be
proportional to the effective density of states in the semiconductor
 (12)
and to exp(–fb/kBT). Their velocity in the
direction perpendicular to the metal semiconductor interface is proportional to
the thermal velocity
 (13)
Hence, the saturation current density is given by
 (14)
where C is a numerical constant
of the order of unity. With a proper
choice of C, this equation coincides
with eqs. (8) and (9).
Thermionic-field
emission
In relatively highly
doped semiconductors, the depletion region becomes so narrow that electrons can
tunnel through the barrier near the top (see Fig. 4b). This process is called thermionic-field emission. In order to understand thermionic-field emission, we have to recall once again
that the number of electrons with energies above a given energy E
decreases exponentially with energy as exp[–E/(kBT)].
On the other hand, the barrier transparency increases exponentially with
the decrease in the barrier width.
Hence, as the doping increases and the barrier becomes
thinner, the dominant electron tunneling path occurs at lower energies than the
top of the barrier (see Fig. 4b).
In degenerate semiconductors,
especially in semiconductors with a small electron effective mass such as GaAs, electrons can tunnel through the barrier near or at
the Fermi level, and the tunneling current is dominant. This mechanism is called field emission (see Fig. 4c).
Fig. 4. Band diagrams of Schottky barrier junctions for GaAs for doping levels Nd = 1015 cm–3 (top graph), Nd = 1017 cm–3 (middle graph), and Nd = 1018 cm–3 (bottom graph). Arrows
indicate electron transfer across the barrier under forward bias. At very low doping levels, electrons go over
the barrier closer to the top of the barrier (this process is called thermionic emission).
At moderated doping levels, electrons tunnel across the barrier closer
to the top of the barrier (this process is called thermionic-field
emission). In highly doped degenerate
semiconductors, electrons near the Fermi level tunnel across a very thin
depletion region (this process is called field emission).
The current-voltage
characteristic of a Schottky diode in the case of thermionic-field emission can be calculated using the same
approach as for the thermionic model, except that in thermionic-field emission case, we have to evaluate the product
of the tunneling transmission coefficient and the
number of electrons at a given energy as a function of energy and integrate
over the states in the conduction band. Such a calculation [see Rhoderick
and Williams (1988)] yields the following expression for the current density in
the thermionic-field emission regime under forward bias:
 (15)
where
 (16)
 (17)
(18)
In GaAs Schottky diodes, the thermionic-field emission becomes important for Nd > 1017 cm–3 at 300 K
and for Nd > 1016 cm–3 at 77 K.
In silicon, the corresponding values of Nd are
several times larger. The forward j-V
characteristics are shown in Fig. 5.
Fig.
5. Forward j-V characteristics of GaAs Schottky diodes doped at 1015, 1017, and 1018 cm–3 (curves
are marked accordingly) at T = 300 K.
The resistance of the Schottky
barrier; in the field emission regime is quite low. Therefore metal-n+ contacts are used as ohmic contacts. The specific contact resistance, rc, decreases with the increase in the doping level of the
semiconductor. (This resistance may vary
from 10-3 Ωcm2 to 10-7 Ωcm2 or even smaller depending on semiconductor material, doping level,
contact metal, and ohmic contact fabrication
technology.)
A Schottky diode is
a majority carrier device, where electron-hole recombination is usually not important. Hence, Schottky
diodes have a much faster response under forward bias conditions than p-n junction diodes. Therefore, Schottky
diodes are used in applications where the speed of a response is important, for
example, in microwave detectors, mixers, and varactors.
Prof. Dojin Kin site
OHMIC CONTACTS
In the case of a p-n diode, for example, contacts have to
be provided to both p-type and n-type regions of the device in order to
connect the diode to an external circuit.
These contacts have to be as unobtrusive as possible, so that the
current flowing through a semiconductor device and, hence, through the
contacts, leads to the smallest parasitic voltage drop possible. Whatever voltage drop does occur across the
contact has to be proportional to the current so that the contacts do not
introduce uncontrollable and unexpected nonlinear elements into the
circuit. Since such contacts satisfy
Ohm's law, they are usually called ohmic contacts.
As was discussed, a
contact between a metal and a semiconductor is typically a Schottky
barrier contact. However, if the
semiconductor is very highly doped, the Schottky
barrier depletion region becomes very
thin, as illustrated in Fig. 4. At
very high doping levels, a thin depletion layer becomes quite transparent for
electron tunneling. This suggests that a
practical way to make a good ohmic contact is to make
a very highly doped semiconductor region between the contact metal and the
semiconductor.
It may have been better to use a metal with a
work function, Fm, which is equal to or smaller than the work function of a
semiconductor, Fs. However, for most semiconductors, it is
difficult to find such a metal acceptable for practical contacts.
Current-voltage characteristics of a Schottky barrier diode and of an ohmic contact are compared in Fig.
1. As was mentioned above, a good ohmic contact should have a linear current-voltage
characteristic and a very small resistance that is negligible compared to the
resistance of the active region of the semiconductor device. An ohmic contact
with the I-V characteristic shown in
Fig. 2 does not satisfy fully these conditions since the voltage drop
across this contact is not negligibly small compared with the voltage drop
across the Schottky diode at moderate current
densities above 0.1 kA/cm2.
As was discussed, the
barrier between a metal and a semiconductor is usually smaller for
semiconductors with smaller energy gaps.Hence, another way to decrease the contact resistance is to place a
layer of a narrow gap highly doped semiconductor material between the active
region of the device and the contact metal. Some of the best ohmic contacts to date have
been made this way.
A quantitative measure of the contact
quality is the specific contact
resistance, rc, which is the contact resistance of a unit area contact. Depending on the semiconductor material and
on the contact quality, rc can vary
anywhere from 10-3 Ωcm2 to 10-7 Ωcm2 or even less.
Fig. 1. Current-voltage
characteristics of ohmic and Schottky
barrier metal-semiconductor contacts to GaAs. (Schottky contact
is to GaAs doped at 1015 cm-3.) Ohmic contact
resistance is 104 Ωcm2.
Most semiconductor devices have either
a sandwich structure or a planar structure, as illustrated in Fig.
2. The contact resistance of each
contact in a sandwich structure contact is given by
(1)
A typical current
density in a sandwich type device can be as high as 104 A/cm2. Hence, the specific contact resistance of 10-5 Ωcm2 would lead
to a voltage drop on the order of 0.1 V. This may be barely acceptable. A larger specific contact resistance of 10-4 Ωcm2 or so would
definitely lead to problems, as we can see from Fig. 1.
These estimates show that a
semiconductor material can become viable for applications in electronic devices
only when good ohmic contacts with low contact
resistances become available. Often,
poor ohmic contacts become a major stumbling
block for applications of new semiconductor materials.
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