6.1 Introduction: Module-6: Schottky barrier capacitance-impurity concentration The electric current flowing across a metal semiconductor interface is generally non-linear with respect to the applied bias voltage, as the result of a discontinuity in the energy scale of the electronic states responsible for conduction in the two materials. In Figure 1, a band diagram of the interface is sketched to show this discontinuity. Delocalized electronic states around the Fermi level (FL) are responsible for electrical conduction in the metal, drawn on the left, but these states are not coupled to any delocalized electronic states in the semiconductor drawn on the right. The set of electronic states responsible for electrical conduction in the semiconductor depends on the type of dopingof the semiconductor. Figure 1 Band diagram at a metal-semiconductor interface. The electrostatic potential energy, - ev(r), is drawn as curve lines. Dotted lines indicate average electrostatic potential. In case of n-type semiconductors, the electrons present close to the conduction band minimum (CBM) contribute to the electrical conduction, whereas in p-type semiconductors, holes available near valence band maximum (VBM) contribute in the electrical conduction. Because of the presence of the fundamental band gap, the lowest-lying states for n-type semiconductor that can communicate with electrons in the metal are now at an energy ɸ 0 B,nabove the FL, as shown in the figure. When metalsemiconductor interface is formed, this energy offset (ɸ 0 B,n), also known as n-type Schottky barrier height, causes a potential energy barrier which acts as a rectifier, such that electrons can readily flow from semiconductor to metal, while the electron flow in opposite direction is difficult. When the semiconductor in contact with the metal is doped p-type, the energy gap between the FL and the VBM, marked as ɸ 0 B,pbecomes the energy barrier, i.e., the p-type Schottky barrier height, that governs the hole transport across themetal-semiconductor interface. Since theflow of current across ametal-semiconductor interface exponentially depends upon the magnitude of the Schottky barrier heightat commonapplied voltages, the Schottky barrier heightis a vital characteristic of ametal-semiconductor interface, that decides its electrical characteristics. 6.2 Physics of formation
6.2.1 Metal semiconductor (or M-S)junction A junction made up of the close contact of a metal and a semiconductor is termed as the metalsemiconductor junction, and is the earliest semiconductor device which was practically used. Both rectifying and non-rectifying type of contacts can be prepared froma metal-semiconductor junction. Rectifying type M-S junction comprises a Schottky barrier, and the resulting device is termed as a Schottky diode. On the other hand, a non-rectifying type junction is termed as ohmic contact. Another type of contact is a rectifying type semiconductor-semiconductor (S-S) junction, which is extensively used semiconductor device. It is also termed as p-n junction. M-S junction is integral to the operation of all the semiconductor devices. Ohmic contact is desirable when electric charge is required to be easily transported between the active region of a transistor and the external circuit. In contrast, Schottky contact is required in Schottky diodes, transistors, M-S field effect transistors, etc. 6.2.2 The critical parameter: Schottky barrier height The Schottky barrier height(φ B) of a junction decides whether a particular M-S junction is ohmic or Schottky. When Schottky barrier height is high enough to cause Φ B much larger than thermal energy kt, the semiconductor gets depleted near the metal and behaves as a Schottky barrier. In contrast, when Schottky barrier height is low, there is no depletion of the semiconductor and an ohmic contact is formed between the metal and the semiconductor. For both p-type and n-type semiconductors, Schottky barrier height is described differently, such that for p-type semiconductor, it is measured from the valence band edge, whereas for n-type, it is measured from the conduction band edge. Near the junction, the alignment of bands of the semiconductor does not depend upon the doping level of the semiconductor. Thus, heights of n- and p-type Schottky barriers are interrelated by the following relation:..... (1) here Eg represents the bandgap of the semiconductor. Figure 2 The band structure of M-S junction under equilibrium (or zero bias). Generally, the height of Schottky barrier is not constant across the interface, rather it varies atits surface.
6.3 Schottky Mott rule and Fermi level pinning: Schottky-Mott rule is an approximation which can be used to predict the height of the Schottky barrier. According to this rule, the height of the Schottky barrier formed at the M-S junction is approximately equal to the difference in the metal work function and electron affinity of the semiconductor, where both the quantities are measured in vacuum, i.e.:.... (2) The idea behind this model considers bringing together two materials in vacuum, and is similar to the formation of semiconductor-semiconductor junctionsvia Anderson s rule. Schottky-Mott rule is obeyed by different semiconductors to varying degrees. Even though Schottky-Mott model could correctly predict the band bending phenomenon in a semiconductor, experimental results demonstrated that it could not yield Schottky barrier height precisely. It suffered a the mechanism termed as Fermi level pinning wherein some points of the bandgap (having DoS) locked (or pinned) to the Fermi level, thereby making the height of the Schottky barrier nearly independent of the work function of metal:....... (3) here E bandgap represents the semiconductor bandgap. John Bardeen observed in 1947 that Fermi level pinning occurs naturally in a semiconductor if it has chargeable states near the interface, having energies in the energy gap of the semiconductor. Such chargeable states are either created as a result of direct chemical bonding between the metal and the semiconductor (and are termed as metal induced gap states), or present at the surface of semiconductorvacuum interface (these are termed as the surface states). These extremely dense surface states trap most of the charge donated by the metal, thereby shielding the semiconductor from the metal. Consequently, the alignment of the bands in the semiconductor corresponds to a location relative to these surface states, which are already pinned to the Fermi level (because of their high densities), and becomes insensitive to the metal. Many commercial semiconductors suffer from Fermi pinning phenomenon, causing difficulties in device fabrication. Examples of these semiconductors include silicon, germanium, gallium arsenide, etc. Since the valence band in p-type germanium is pinned to the Fermi level of the metal, most of the metals form ohmic contacts with p-type germanium, and Schottky barrier with n-type germanium. This problem can be alleviated by including additional processing steps, e.g., addition of an insulating intermediate layer to unpin the bands. Germanium nitride insulation is usually used in case of germanium. Schottky Mott rule is schematically shown in Figure 3 (a). As the two different materials (silver and n- type silicon in this case) approach each other, the semiconductor s bands bend so as to match the work function of silicon with that of silver. This bending of the bands is retained even after contact. This model envisages a very small Schottky barrier for silver and n-type silicon junction, resulting in an ohmic contact between them.
Figure 3(a) & (b) Schematic depiction of junction formation between silver and n-type silicon. Figure 3(b)shows Fermi level pinning phenomenon due to the metal induced gap states. The surface states cause band bending in silicon. Before making the contact, the bands again bend such that the work function of both the materials is matched. However, on contact formation, the bending alters depending upon the nature of the chemical bonding between silver and n-type silicon. 6.4 Rectifying properties All M-S junctions do not produce a rectifying Schottky barrier. And a M-S junction can conduct current in both directions with equal ease, thereby forming an ohmic contact. This can be attributed to the Schottky barrier being too low at the M-S junction. When the Schottky barrier is considerably high, a depletion region is formed in semiconductor, at the M-S interface. This results in a high resistance at the interface when applied bias voltage is low. However, when large voltage is applied across the junction, laws of thermionic emission and the fixed Schottky barrier (relative to the Fermi level of metal) govern the flow of electric current across the junction. 6.4.1 Forward bias Under forward bias, there are many thermally excited electrons in the semiconductor which cancross over the barrier. The passage of electrons over the barrier (without any electrons returning back) results in the flow of an electric current in the opposite direction. When bias is increased, this current also increases.however, at very high bias voltages,the currentis limited by the series resistance of the semiconductor.
Figure 4 Forward bias, thermally excited electrons are able to spill into the metal. 6.4.2 Reverse bias Figure 5 Reverse bias, the barrier is too high for thermally excited electrons to enter the conduction band from the metal. Under reverse bias, very less number of thermally excited electronsin the metal have enough energy to cross the barrier, thus a small leakage current flows. This leakage current is not constant, but gradually increases with reverse bias voltage owing to a weak barrier lowering (same as the vacuum Schottky effect). However, when reverse biased is increased to higher values, the depletion region breaks down. 6.4.3 Minority carrier injection When Shocttky barrier is very high such that it forms a considerable fraction of the semiconductor s bandgap, the forward bias current may instead be carried underneath the Schottky barrier, as minority carriers in the semiconductor.
Figure 6Band diagram for very high Schottky barriers. However, when Schottky barrier is increased such that it is same as the bandgap of the semiconductor, the forward bias current is carried by minority carrier injection. This is shown in Figure 6 with the white arrow indicating the injection of an electron hole in the valence band of the semiconductor. 6.5 Calculation of Schottky barrier height: Schottky barrier height (SBH) isgenerally calculated from the transport measurements of Schottky barrier diodes processed or patterned on non-degenerate semiconductors. Current-voltage (I-V) and capacitancevoltage (C-V) are important methodsof determining SBH. Both these techniques presume that the barrier heightbeingstudied is uniform. In I-V method, junction current is recordedwith varying applied voltage. From logarithmic plot of the forward-bias current, the "saturation current" and "ideality factor" are obtained. By thermionic emission theory, diode saturation current and SBH are linked as: I sat = AA*T 2 exp[- F o B,n / (k BT)]...(4) Here A* = Richardson's constant for a semiconductor, anda is diode area. Anaccurateinformation about the total diode area is prerequisitefordeterminingschottky barrier height by I-V measurements. For diodes withunknown area, the schottky barrier height may still be known by examining the temperature dependence of junction current wherejunction current ismeasured at varying temperaturesat a fixed bias voltage (lying in the linear portion of constant temperature I-V plot). Schematic of Silicide Schottky diode fabricated Temperature dependent "saturation current" is the most preferred investigation since this current is the "forward" component of the total current under zero bias conditions. The logarithm of (I*T -2 ) can be plotted in a "Richardson plot" as a function of inverse temperature (T -1 ). The schottky barrier height is
then the sum of calculated activation energy from the plot and the fixed applied bias used in the experiment. In C-V measurements, high frequency (>500 khz) capacitance of uniformly-doped Schottky diodes is recorded with varying bias voltage. This experiment is conducted in the reverse bias to minimize the effects of in-phase current on the measurements. By plotting the inverse of the junction capacitance against the applied bias, "flatband" voltage can be determined by linear extrapolation of voltage axis. For instance, for n-type semiconductor, the barrier height can be obtained as F o B,n = ev bi + ev n + k BT. (5) Here ev bi is built-in voltage and ev n is difference between the Fermi level and the conduction band minimum for a neutral semiconductor. 6.6 Impurity concentration The properties of an intrinsic semiconductor may be modified by introduction of impurities (doping), the resulting material is called the extrinsic semiconductor. The amount of the dopants or impuritiesinfluencesa material's charge carrier concentration. Under thermal equilibrium, the number of electrons and holes are equal in an intrinsic semiconductor, i.e., If extrinsic semiconductor (low doping) the above thermal equilibrium relation changes to: Where n 0 and p 0 are the respective concentrations of electrons and holes, and n i = intrinsic carrier concentration of the semiconductor and it varies across different materials and also depends on temperature. Increase in the doping increases the carrier concentration, thereby enhancing the conductivity.excessively doped (or degenerate) semiconductors exhibit conductivitiescomparable to the metals and find applications in integrated circuits as replacements for metals. Superscript plus and minus signsrepresent the relative doping concentrations in semiconductors, i.e. n + showsa heavily doped n-type semiconductor and p denotes very lightly doped p-type semiconductor. 6.6.1 Effect on Band structure Electron donating impurities generate additional states close to conduction band whereas the electron deficient or acceptor impurities generate states close to valence band. The gap between these energy states and closest energy band is termed as dopant-site bonding energy or E B and is usually small. Due to small values of E B, room temperature thermally ionizes almost all the dopant atoms and creates free charge carriers in conduction or valence bands. Dopants can further cause shift in energy bands with respect to Fermi level. The energy band of a dopant havinghighest concentration lies closest to Fermi level. As Fermi level is constant for a system in thermal equilibrium, piling multiple layers of materials havingvarying properties results innovel propertiescaused by band bending. Most common example is the p-n junction which has unique properties generated by band bending whichis a consequence of joining the bands in contact regions of p- and n-type materials.
Review your learning: 1. What are the different ways of determining the Schottky barrier height? Explain C-V measurement of Schottky diode. 2. Why impurities are added to the semiconductors? Explain the effect of doping on the Fermi level in both the N type and P type semiconductor. Reference [1] https://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_2_2.html. [2] D.Dale, Kleppinger Impurity concentration dependence of the density of states in semiconductors, Solid-State Electronics Volume 14, Issue 3, March 1971, Pages 199-206. [13] https://en.wikipedia.org/wiki/doping_(semiconductor).