Vilniaus Universitetas, Semiconductor Physics Department, Lithuania, http://www.mtmi.vu.lt/pfk/index_EN.html Functional combinations in solid states h.d. A.?ukauskas, h.d. V.Gavriu?inas http://www.mtmi.vu.lt/pfk/funkc_dariniai/diod/schottky.htm

# The Metal-Semiconductor Junction.Schottky Diode. OHMIC CONTACTS

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.

## 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.

## 2.3. Thermal equilibrium

 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)

## 3.1. 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)
 with
 (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.

## 4.1. Diffusion current

 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)
 yielding:
 (4.5)
 so that the prefactor equals the drift current at the metal-semiconductor interface, which for zero

## 4.2 Thermionic emission

 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)

## 4.3. Tunneling

 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

1. Physics of Semiconductor Devices, Second edition, S. M. Sze, Wiley & Sons, 1981, Chapter 5.
2. 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 VbiV,  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 (EEc)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 cm3 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.

04.19.2002

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