1. INTRODUCTION
This piece of work is the further investigation of the 3B5 - Schottky Barrier Diode (SBD) experiment. It involves a comprehensive data error analysis to both the raw data from the experiment and the parameters derived from them. An in-depth probe to the SBD is also done by examining the fundamental theory of SBD with illustrative band diagrams and reference to the thermionic emission theory. The ideal SBD model is then compared with common p-n junction and their differences are discussed. Further analysis on the non-ideal behaviours of both the SBD and p-n junction diodes are performed as well.
Evac Metal eΦm EF n-type semiconductor Evac eχsc eΦsc Ec EF Ev Figure 1 – Band diagram of metal and n-type semiconductor when they are not in contact

2. RESULTS 3. DISCUSSION
3.1 Accuracy of Measurements There are a number of techniques to determine uncertainties in data, analytically or graphically. Table 1 shows some general rules in calculating data errors when simple algorithmic operations are performed.

Table 1 – Some rules to calculate data errors in algorithmic operations

For multi-variable functions, e.g. f(x,y,z), the resultant error in the value of the function comes from the contribution of individual error in each variable, i.e. , and the relation is as follows

Graphically, to obtain the 3.2 Theory of SBD (Compared with P-N Junction) 3.2.1 SBD Theory by Band Diagram Illustration

The SBD demonstrates a rectifying effect by taking advantage of its metal-semiconductor junction. This is basically a junction that is made by contacting a metal surface with a doped semiconductor together. Cases may vary according to the type of doping for the semiconductor and for convenience, only n-type doped semiconductor would be discussed in this work, and the situation for p-type counterparts is simply the opposite. As showed in figure 1, two band diagrams for a metal and an n-type semiconductor are provided. Before they are brought together into contact, they have independent band configuration and the uppermost energy band is the one for vacuum, Evac, which essentially marks the energy level for the electrons to go beyond the material surface, emitted into the vacuum. Intuitively, the Evac for both the materials should be at the same level and the work functions Φm, Φsc and electron affinity for the semiconductor χsc has been marked. Another feature of the typical band structures is the Fermi level EF for metal is lower than that of the semiconductor, i.e., as can see from the diagram. Thus, when these two materials are in contact with each other and reach thermal equilibrium, their EF will line up, and this effectively brings down all the energy levels of the semiconductor accordingly while the energy level remains the same as before at the metalsemiconductor interface, as shown in figure 2. This also causes a transfer of electrons to occur from the semiconductor to the metal, which creates a potential difference - the built in potential of the junction V0. Thus, in equilibrium, where e is the magnitude of the electronic charge.

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Metal Evac eΦb,m = e(Φm - χsc) EF w

N-type semiconductor Evac eΦb,s = e(Φm - Φsc) Depletion Region EF Ev Ec

through this device is attributed to those two types of carriers flowing in opposite directions. The I-V characteristic equation under forward bias (Schockley equation) of the p-n junction diode can be found by considering the Law of Mass Action, and continuity equation for the electrons and holes, which is

Figure 2 – Band diagram of metal and n-type semiconductor when they are in contact

The junction capacitance can be found to be where Is is the reverse saturated current; V is the applied bias voltage; NA is the density of acceptor atoms; ND is the density of donor atoms in the conduction band; ni is the intrinsic carrier concentration. Detailed derivations of all these above equations are available in Appendix. This equation is based on the following assumptions: - The injected minority carrier concentration is much less than the majority carrier, - No recombination occurs in the depletion region, - Negligible fields outside the depletion region. These assumptions are, nonetheless, not applicable for modelling the I-V behaviour of a SBD because the SBD is a unipolar device with only electron flow being considered, and instead of simple diffusion, the electrons are injected into the metal from a higher energy level . Additionally, there is no stored carriers in the metal. Therefore, the current flow is determined by how fast the electrons can get from the semiconductor to the metal and Thermionic Emission Theory is applied to model this behaviour. This theory assumes that the mobility of the electrons is such that they can get across the barrier as fast as they are emitted, and if the mobility is too low then there would be a mobility limited process. The net current from the semiconductor to the metal is where A* = A(m*/m), and A is the Richardson constant Va is the forward voltage bias. The junction capacitance is

Figure 2 also shows at equilibrium, a potential barrier of height at the interface, which prevents the electron flow from metal to the other side. On the other hand, for the electrons to flow from the semiconductor to the metal, they also have to overcome a potential barrier of . Furthermore, due to the charge transfer from the semiconductor to the metal to reach equilibrium, a depletion region (devoid of charge carrier) is formed in the semiconductor side near the interface. It should be noted that does not vary with external bias, while does. As shown in figure 3, a positive voltage V on the metal will counteract the built in potential V0 and will cause the depletion region to decrease in width. Such a forward bias would also lower the barrier for current flow from the semiconductor to the metal. A positive voltage V on the semiconductor, i.e. reverse bias, would on the other hand bring down the band levels of the semiconductor and becomes larger. However, the barrier against the electron flow from the metal to the semiconductor is held constant regardless of the direction of bias. 3.2.2 Comparison with P-N Junctions It is natural to consider the fundamental difference in conduction principles of SBD and a p-n junction diode. A p-n junction diode is a bipolar device that has two types of carriers, namely the electrons and holes. Their relative abundance due to doping makes them behave as majority and minority carriers. Thus, the current flow

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Derivation is also shown in the Appendix. Generally, a unipolar device is preferred for high frequency applications because it has only one type of charge carrier, and hence there would be less recovery time on switching from forward to reverse bias due to electron-hole recombination. Furthermore, the capacitance of SBD is smaller than that of a p-n junction diode, which also leads to high switching speed. The SBD also has a smaller device area and this could facilitate charge carrier transition as well. 3.3 Schottky Barrier Height An ideal model of SDB band diagram suggests that the Schottky barrier height is , which is simply the difference between the work function for the mteal and the electron affinity of the

Φb

Φb

a w b

eΦ b,m e(Φb,s + V) eΦb
,m

e(Φb,s - V)

Figure 4 –Band diagram of n-type semiconductor–metal contact. Part a is a Schottky-Mott model without an bonds interface layer. electronsis a modified model with an form between Part b and atoms. Such charge interface. redistribution is consistent with the second law of thermodynamics, implying that the electrical potential should be the same at equilibrium on both sides. This then leads to a potential drop at the interface. Moreover, quantum mechanically, the interface assists the tunnelling of the carriers to the metal and result in a high current flow through the barrier. As a result of all these factors, apparent barrier height is reduced. Figure 4 shows two band diagrams that illustrate the difference in the Schottky-Mott model and the actual model including the effect of an interface. There are number of suitable ways to determine the Schottky barrier height, and one of the most effective ways is discussed in this work as follows. The reverse bias saturated current can be obtained from the y-intercept of the Irev-Vrev graph (provided in the attached lab report). From the equations

Figure 3 – Band diagram of the SBD under bias. Note that the barrier height from the metal to the semiconductor is constant regardless of direction of bias

semiconductor. This model has been put forward by Schottky and Mott but experiments on this has always shown discrepancies in the measured values of from his theory (the measured height is always smaller than predicted one). This is because the original SchottkyMott model neglects the presence of an interfacial layer between the contact surfaces of the metal and semiconductor. When the metal and semiconductor are in contact, there would be an overlap of wave function from two sides and this further leads to redistribution of charges. In the meanwhile, old bonds break and new

it is obvious that Is varies only with temperature. Thus, several sets of Is and T values can be plotted on a ln(Is) vs e/kT graph, and the gradient of the line of best fit would give the value of , which is the Schottky barrier height. 3.4 Ideality of Performance and Fabrication Technique The I-V equations for both the SBD and p-n junction diode are ideal models, which do not take into consideration material defects in fabrications and inherent complexity of the device on quantum mechanical level. Thus, an ideality factor , a numerical measure of the effectiveness of the mathematical correlation, is introduced into the equations, which then become
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where the expression of Is differs for SBD and p-n junction diodes, as discussed before. For an ideal model, , while for a non-ideal one, , and the more deviated the device from ideality, the greater the value of would be. Based upon what has been mentioned in section 3.3, non-ideality of the SBD is owing to the Schottky barrier lowering. Furthermore, other factors like contact surface imperfection and electron tunnelling and so forth also contribute to the deviation of a real SBD from an ideal one. Figure 5 shows the variation of with semiconductor doping density ND for SBD and temperature. The graph suggests an exponential increase in ideality factor with an increase in doping density at a constant temperature and for a fixed level of doping concentration, the higher the temperature is, the lower the ideality factor would be. This is because the width of the Schottky barrier is

which implies that a heavily doped semiconductor would narrow the width of the barrier, this would in turn make tunnelling of electrons through the barrier easier, causing the device to deviate from ideal performance. On the other hand, a rise in temperature will raise the thermionic emission intensity and there would be larger current due to this than that caused by tunnelling, i.e. the tunnelling effect is negligible at high temperature. Hence the ideality factor at high temperature is reduced provided a constant doping level is maintained.

Variation in ND also changes the values of the reverse saturate current Is and series resistance rc of the SBD. A high ND means there are potentially many charge carriers in the ready for the conduction, so Is would be greater and effectively, rc is smaller. This fact is a very important aspect to consider when SBD is being fabricated. In the early days, a SBD was made by pressing a wire against surface of a semiconductor. Nowadays, a SBD is built by first depositing a metal film on a clean semiconductor surface and define the specific contact pattern photo-lithographically. The candidate metals can be platinum, gold, chromium and molybdenum. The ntype semiconductor can be silicon doped with phosphorous, and GaAs. Figure 6 is a cross sectional image of a typical BAT85 SBD. A close inspection of the junction reveals four distinct blocks. Block 3 is the metal that has been deposited on the semiconductor in block 2. As analysed before, if the doping concentration for the semiconductor is high, there would be electron tunnelling, and then the device cannot perform as well as an ideal one. Therefore, a lightly doped semiconductor block is used. However, if the semiconductor is only lightly doped, the lack of the charge carriers would result in a large series resistance rc, which is unfavourable. In order to make remain rc at a low level, a higher level of doping should be applied. These contradictory requirements can be both achieved by making the semiconductor near the metal surface lightly dope and the rest part highly doped, i.e., making an inhomogeneous semiconductor. In figure 6, block 1 of a highly doped semiconductor is brought in contact to block 2 to realise this idea. On the other hand, apart from the flaws in

1

3

2

1

Figure 5 – Variation of η ND for SBD

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Figure 6 – Cross sectional scanning electron microscopy image of BAT85 SBD. Block 1 – highly doped; block 2 – lightly doped; block 3 – metal

fabrication, the non-ideal behaviour of p-n diodes is mainly due to the electron/hole recombination and generation in the depletion layer. Under reverse bias, generation of extra carriers within depletion zone takes place. This would increase with the depletion width and hence increase the reverse bias current. Under forward

bias, recombination within depletion zone leads to deviation from Shockley equation, which is reflected by ideality factor . Additionally, resistance of the regions outside the depletion layer could limit high currents flowing through the diode so forward bias current would finally be saturated.

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APPENDIX
1. Derivation of Schockley equation for p-n junction diodes The electro static potential difference between the n and p type semiconductor is (1.1) Equilibrium concentration of h in p-type = pp0. Equilibrium concentration of e in p-type = np0. Equilibrium concentration of h in n-type = pn0. Equilibrium concentration of e in p-type = nn0. pp0np0=ni2 (1.2) pn0nn0=ni2 (1.3) From (1.1), (1.4) where (1.11b) 2. Derivation of p-n junction capacitance The Poisson Equation states (2.1) Considering one dimension and applying suitable boundary conditions, expression for V in the junction is given by (2.2) where xp and xn are width of the depletion region on p and n side respectively. Considering charge neutrality, (2.3) Solving (2.3) and (2.2) would give (2.4) (2.5) The expression for capacitance is (2.6) From (2.4), the charge on the n side is (2.7) Therefore, (2.8) Assuming NA >> ND, and a bias of V is applied (1.9) where V is the applied bias and the relation in (1.2) and (1.3) is used, assuming ND = nn0 A similar derivation for electron current on the p-side gives

From the Law of Mass Action pn0ND = ni2 (1.5) Assuming all donors are ionised, and so substitute for ND in equation (1.4) (1.6) From the continuity equation, assuming steady state and no drift (1.7) where pn0 has no variation with position and is the minority carrier (hole) lifetime. Considering the concentration gradient of holes related to the current flow (1.8)

(2.9) 3. Derivation of I-V equation for SBD Thermionic Emission for electron flow from the semiconductor to the metal is given for a forward bias Va by (3.1) (3.2)

(1.10) Summing equations (1.9) and (1.10) would give the ideal diode law, or Shockley equation (1.11a)

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where Φb = Φm-χsc and A* = A(m*/m). A is the Richardson constant, given by (3.3) Summing (3.1) and (3.2) gives

(3.4) where (3.5) 4. Derivation of junction capacitance of SBD Applying Poisson equation on 1D condition in the semiconductor (4.1) where ND is the donor density in the n-type semiconductor. Boundary conditions are ε = 0 at x = 2, (4.2) Setting V = 0 at x = 0, integrating (4.2) (4.3) (4.4) (4.5)

(4.6)

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