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The metal-semiconductor junction is a diode just like the PN junction, i.e. it carries a very low current in reverse bias and a very large current in forward bias. It also carries many of the physical characteristics of the PN junction and in a much simpler way. It is sometimes called a Schottky diode after the German physicist Schottky who first analyzed it before the second world war. Figure 3.20 shows the simplified band diagrams of a metal piece and an N-type semiconducting piece just before touching.

An important quantity that has been added to these simplified band diagrams of the metal and the semiconductor, compared to what has been presented so far, is the work- function. If periodic boundary conditions are imposed on a crystalline solid, its eigenvalues Et(k) extend from zero energy to infinity with the band index / having no bound. However real solids do not obey periodic boundary conditions—in 3-dimensions it is

Band structure of a metal and an N type semiconductor before contact

FIGURE 3.20 Band structure of a metal and an N type semiconductor before contact.

impossible to deform a solid to make all its faces meet—and solids are terminated at surfaces. A consequence of this is that there is a maximum of energy the electrons can have to remain in the solid. If more energy is gained, by photoexcitation for example, the electrons get out of the solid.

This maximum energy is usually called the vacuum energy and denoted Emc as shown in figure 3.20. The difference W = Evac - EFm, where EFm is Fermi level of the metal, is the workfunction of a metal and it corresponds to the minimum energy required to extract an electron from the metal. The case of a semiconductor is somewhat different since the Fermi-level EFs lies usually in the energy-gap. Then a second quantity needs to be defined, called electron affinity, which is the difference % = Evac - Ec. The workfunction Ws = Emc - EFs is still defined for a semiconductor, but we normally work with the affinity X when it comes to semiconductor interfaces. All these quantities of a semiconductor are shown in figure 3.20.

Returning now to the problem of the junction, we observe that the N-type semiconductor Fermi level is above the Fermi level of the metal, so when the two materials come into contact electrons will flow from the semiconductor to the metal so as to equalize the two originally different Fermi levels as happened in the case of the PN junction. This will create a deficiency of electrons in the N-type semiconductor and a surplus in the metal, i.e. a space charge layer will develop with a built-in potential inside, again as in the PN junction. Only this time, the space charge layer will develop mainly in the semiconductor for the following reason: the metal has a much higher density of states than the semiconductor, so that it will take a very small volume of it to accommodate the electrons that have flown from the semiconductor. This is shown in figure 3.21a, where we plot the charge density p across the junction. Again, as in the PN junction, we have approximated the exact charge density p = - и) with p = Np.

As usual the electrostatics of the junction will be governed by Poisson’s equation. We choose to ignore the very small negative surface layer on the metal side so that Poisson’s equation reads on the semiconductor side Charge density (a), electric field (b), and potential (c) across a metal-semiconductor (Schottky) junction

FIGURE 3.21 Charge density (a), electric field (b), and potential (c) across a metal-semiconductor (Schottky) junction.

where d is the length (or sometimes called width) of the space charge layer and we assume complete ionization of the donors. Integrating and using the boundary condition that the electric field £ is zero at x = d we get

The plot of £ (x) is shown in figure 3.21b. The maximum of the absolute value of the electric field occurs at x = 0. If we call this maximum £max we get

Integrating £(x) and using as a boundary condition V(o+ J = 0 we get

From either the area of the triangle in figure 3.21b or from equation 3.69 above

The plot of V(x) is shown in figure 3.21c.

To construct the band bending that this potential will exert on the semiconductor bands we simply have to multiply by (—e). Remember electrostatic energy = charge x potential. Furthermore note that (see figure 3.20) the Fermi level in the metal differs by the constant energy difference W - %. from the bottom of the conduction band in the semiconductor. Therefore, the electronic energy band diagram of the metal-semiconductor junction is shown in figure 3.22a. It is left to the reader as an exercise to prove that the potential in the N-type semiconductor of the Schottky diode is the same as the potential in the N-side of a PN junction when overdoped in the P-side (usually denoted P+N). The band diagram of a metal P-type semiconductor follows the same principles and equations, only the RHS of

f eNA

the Poisson equation is equal to H--so that finally the bands in the P side are upward

£ /

bending instead of downward as is shown in figure 3.22b.

Let us now examine the electron flow with no bias on. We will restrict ourselves to the metal N-type semiconductor junction. The arguments for the metal P-type semiconductor are exactly the same. At zero bias there are two barriers to electron flow. From the metal (M) to semiconductor (S), the electrons face a barrier Фй = W-%. From S to M, on the other hand, the electrons face the barrier Vbl, see figure 3.22a. The barrier Фв is called the Schottky barrier of the diode. The current must be zero at zero bias. This current is indeed

a Band structure of a metal N-type semiconductor junction at zero bias

FIGURE 3.22a Band structure of a metal N-type semiconductor junction at zero bias.

the sum of the currents 7m->s and Is^>m in an obvious notation, each one corresponding to one of the electron flows described above.

We will assume that the relation for the electron density

which holds for a homogeneous semiconductor, also holds for every point in an inhomogeneous one. Looking at figure 3.22a, we see that the electron density at the surface is smaller by exp(-eVbi / kT), compared to the bulk, because (Ec - EF) is bigger by Vbi at the surface compared to the bulk. Therefore the electron density at the interface и,у is

This electron density will generate a current Is^>m from S—> M. An equal in magnitude current Tv;—>s will flow of course in the opposite direction.

b Band structure of a metal P-type semiconductor

FIGURE 3.22b Band structure of a metal P-type semiconductor.

c Band structure of a metal N-type semiconductor at positive bias

FIGURE 3.22c Band structure of a metal N-type semiconductor at positive bias.

Now if a positive voltage is applied to the junction (positive meaning the + electrode of the generator connected to the metal side), the barrier (eVj,) will be reduced to e(Vbl - V). The same argument we gave for the PN junction holds here as well, see figure 3.22c. In this figure EFm and EFs stand for the Fermi levels of the metal and semiconductor respectively. On the other hand the barrier Фв from M —» S will remain the same. We have obviously assumed that the applied voltage V drops to a very good approximation entirely in the semiconductor part of the junction. As a result of the previous arguments, the current will remain the same while Is^m will increase by exp(eV/kT). The total current therefore will be

where I0 = |T,m-s|. The graph of I(V) for the Schottky diode is the same as that of a PN junction, see figure 3.18.

Both the PN junctions and M-S Schottky junctions are used as rectifiers since the current in the negative voltage domain is negligible. The M-S junctions, however, are everywhere in microelectronic circuits as metal contacts because current must be thrown in and taken out of devices. This happens through metal contacts. Most of them have to be ohmic while the M-S junction is rectifying. Therefore, it is necessary to be able to transform the rectifying contact into an ohmic one. Indeed, overdoping of the semiconductor part of the M-S junction will produce an ohmic contact. How can this be done?

From equation 3.70 we have that if the applied voltage is zero

and if a voltage is applied to the Schottky diode

The built in voltage V*,, is practically constant with doping level No- To see this, we observe from figure 3.22a or 3.22c that

The barrier Фв is a material property of the interface equal to (0.8-0.9)eV whereas Ec~Ep =40-50meV, so Vj,, is practically constant. Hence overdoping the semiconductor by two more orders of magnitude than normal will reduce the barrier width d by a factor of 10. If the barrier width is of the order of 2-Ъпт, tunneling from the metal to the semiconductor can take place, see figure 3.23.

The tunneling probability can be calculated by the general WKB formula, equation 1.65 derived in chapter 1, with the vacuum mass m replaced by the effective mass m . If an electron stays within the semiconductor, it experiences a potential given by equation 3.69. But if it crosses the metal-semiconductor interface from the metal side, it experiences the barrier Фв, therefore the tunneling potential energy U(x) experienced by an electron tunneling from the metal into the semiconductor is

and the probability of tunneling from the Fermi level Ерт, see figure 3.22, is

Same as figure 3.22a but with the semiconductor layer being overdopped.


The critical reader may note that the applied voltage V does not appear explicitly in equations 3.75, 3.76. It does so indirectly through the quantity £max. From figure 3.21b we get £maxdl2 = Vt,j-V when an external voltage V is on, so the latter affects the transmission probability T.


  • 3.1 The mobility of a pure sample of GaAs is measured and found to be p^. The sample is then doped and its mobility measured again and found to be p2. Assume that the doping does not change its effective mass m*. Find an expression for the relaxation time of the electrons due to collisions to the dopant atoms.
  • 3.2 Prove that the hole current density

may be written compactly in one dimension as

3.3 Find the built-in potential Vbi and width w of the depletion region of a linearly doped PN function, that is of a PN function whose doping is

N(x) = cx where c is a constant and x lies--

  • 2 2
  • 3.4 By imposing the conditions that at equilibrium (V = 0) the currents Jn, Jp are each separately zero obtain the expressions for the built-in potential

3.5 Show that the potential variation of a metal-semiconductor junction is the same as that of an overdoped P+N junction.

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