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Basic Semiconductor Material Science and Solid State Physics

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Basic Semiconductor Material Science and Solid State Physics The Effect of Electric Field on the Semiconductor Surface Before proceeding with detailed consideration of the Si/SiO2 interface, capacitance- voltage analysis, device structures, etc., it is necessary to consider the fundamental effect of an electric field on the surface of a semiconductor. To begin, one recalls, again, the physical meaning of the Fermi level (i.e., Fermi energy) which, within Fermi-Dirac statistics is defined as the energy, EF, for which electronic population probability is exactly one half. However, Fermi energy has further thermodynamic significance as the free energy of mobile carriers. Thus, for an amorphous or crystalline solid in thermal equilibrium, irrespective of whether it is a conductor, insulator, or semiconductor, EF must have the same value everywhere within the solid. This is trivially obvious if the solid has uniform composition, however, at equilibrium, this condition must be satisfied even if the solid changes properties over some lateral dimension due to extrinsic doping or even gross changes in composition. Indeed, this behavior is fundamental to any understanding of contacts and junctions in semiconductors and metals. Thus, in addition to band gap, crystallographic parameters, various thermodynamic equilibrium constants, etc., another important basic material property is work function, which is defined as the energy (typically expressed in electron-volts) required to remove an electron from within a specific material to a state of rest in the vacuum, i.e., to the vacuum level. Work functions are also commonly quoted in terms of equivalent electrical potential, i.e., in volts. Of course, energy and potential are directly related by electrical charge, which for electrons and holes has magnitude of one fundamental unit, i.e., nominally 1.602(1019) C. Consequently, energy and electrical potential units are often carelessly treated as interchangeable; however, for consistency and to avoid confusion, work functions should be regarded as having units of energy. Physically, the work function is evidently a measure of aggregate electronic binding energy in the solid and is classically observed by applying a negative electrical potential, i.e., a voltage, to some material (presumably having a reasonably clean surface) in a vacuum and measuring resulting current flow to an unbiased, i.e., grounded, counter electrode also in the vacuum. (Alternatively, a positive potential can be applied to the counter electrode with the material grounded.) Accordingly, the observed current is not a slowly varying function of bias voltage, but exhibits a definite threshold which is characteristic of the work function of the material. (Indeed, this effect was first observed by Thomas Edison when he placed a second, biasable electrode inside a light bulb, which, as such, can be regarded as the world’s first electronic device, the vacuum diode.) Typically, this threshold is found to be a few volts; hence, the work function is a few electron-volts. This is to be expected since this energy is of the same order of magnitude as electronic binding energies of electrons in atoms. Naturally, every solid material is, in principle, characterized by a different work function but, in practice, it may be very difficult to measure (as in the case of insulators). Within this context, it is useful to consider the simple case of a surface contact between two dissimilar metals. In metals, the Fermi level generally does not fall in a band gap as it does in semiconductors or insulators. This situation can be realized physically two ways. In the most common case of a classical metal, the Fermi level falls within an occupied band, therefore, this band evidently can be only partially filled. Hence, electrons can easily make transitions between occupied and empty band states and are as a consequence, entirely delocalized, i.e., mobile. In the case of a semimetal, an empty band overlaps a completely occupied band. Obviously, the Fermi level must fall at the top edge of the occupied band. Thus, a semimetal is analogous to a semiconductor having a zero or negative band gap. Again, electrons are delocalized and, hence, mobile. In any case, electronic states in metals are generally occupied right up to the Fermi level and, as a consequence, metals are characterized both by a large density of mobile, i.e., itinerant or conduction, electrons (of the same order as the atomic density) and the absence of a band gap. (For completeness, it must be noted that mobile carriers in some metals appear to be positively charged and, hence, are more properly regarded as holes; however, this does not substantially change the basic picture of metallic conduction.) As a “thought experiment”, the following figure illustrates what happens when two dissimilar metals are brought into intimate contact. Here, Evac is the energy of the vacuum level (conventionally taken to be zero), E is the Fermi level of “metal 1”, and F1 is the associated work function. Similarly, E is the Fermi level of “metal 2” and 1 F2 2 is the associated work function: Evac = 2 1 1 2 E F1 E F EF 2 Fig. 33: Appearance of a contact potential at the interface of two dissimilar metals As is indicated by the shaded regions in the figure, in a metal at low temperature, electrons occupy all available quasi-continuous band states up to the Fermi level. Of course, the Fermi levels, and must fall below Evac since electrons are in bound states. If the two metals are widely separated in space (as indicated on the left side of the preceding figure), electronic equilibrium is not established and the Fermi levels do not necessarily coincide. Of course, in isolation, the Fermi levels differ from the vacuum level precisely by the work function; however, since the work functions are unequal, a free energy difference exists between electrons in metal 1 and metal 2. Therefore, if the two metals are brought into close proximity, i.e., into contact, a spontaneous transient current flows. Physically, this current flow transfers electrons from the metal with the smaller work function (in this case, metal 1) to the metal with the larger work function (metal 2). Naturally, current continues to flow until thermodynamic equilibrium is established for mobile carriers. Of course, the condition of equilibrium requires that the Fermi level, EF, must be the same in both metals. Thus, as a consequence of charge transfer, at equilibrium an electrical potential difference appears between the two metals. This is called contact potential and simply corresponds to the quotient of difference of the work functions, , and fundamental charge. Clearly, a contact potential exactly compensates initial disequilibrium arising from any difference in Fermi levels. However, due to the high density of mobile carriers within a metal, an electric field cannot exist within the bulk, hence, all of the contact potential difference must occur at the interface. (This is illustrated on the right side of the preceding figure.) Furthermore, since the width of the interface is, at most, on the order of a few atomic diameters, the required number of electrons transferred in order to produce a potential difference corresponding to is, in fact, very small in comparison to the density of mobile carriers. As a result, for a metal-metal contact, it is extremely difficult if not impossible to measure the contact potential directly since any attempt to do so causes additional transient current flow which disturbs the equilibrium, i.e., “shorts out” the potential. (In essence, any practical measuring equipment becomes part of the whole system and participates in the equilibrium.) The situation for contact between a semiconductor and a metal is similar, with the added feature that, because of the existence of a band gap, moderate electric fields can exist within the bulk of a semiconductor. Typically, the work function of a metal, e.g., aluminum, titanium, etc., is less than that of an intrinsic semiconductor. For elemental materials, this supposition is easily rationalized from the periodic chart since metallic behavior is characterized by decreasing ionization potentials. (However, as will become evident subsequently, in the case of extrinsically doped semiconductors this situation may become inverted.) The following figure illustrates the case of a surface contact between a metal and intrinsic semiconductor, viz., silicon: Evac Si M M Si E E C FM E F EF Si EV Fig. 34: Appearance of a contact potential at the interface of a metal and intrinsic semiconductor Of course, E is the Fermi level of the metal and is the associated work function. FM M Likewise, E is the Fermi level of the semiconductor and is its work function. Just FSi Si as in the case of two dissimilar metals, if one brings an intrinsic semiconductor and a metal into close proximity, the metal tends to lose electrons to the semiconductor simply because available energy states for electrons in the semiconductor are of lower energy, i.e., because the Fermi level is lower in the semiconductor. Transient current flows until equilibrium is established and the Fermi level becomes the same in both the metal and the semiconductor. However, in contrast to the case of two metals, the transfer of electrons to the semiconductor results in an electric field that penetrates the surface and causes the band structure of the semiconductor to “bend”. If, as has been assumed, the metal work function is smaller than the semiconductor work function, then the bands must bend “downward”. This can be understood by observing that electrons transferred to the semiconductor from the metal occupy states in the conduction band. (Of course, carrier equilibrium implies that some electrons recombine with holes; however, this is only a small fraction of electrons transferred.) Since these excess electrons are supplied from an external source, holes do not appear in the valence band.
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