曾在德國的我在阿亨

Tutorial website for Schottky Barrier Height Basics by Prof. Raymond T. Tung (董梓則)

http://academic.brooklyn.cuny.edu/physics/tung/Schottky/index.htm

My recent review on Schottky barrier height can be found in Applied Physics Reviews 1, 011304 (2014). 
I plan to update this entire tutorial in 2014. Please e-mail me if you have suggestions or know examples of good "tutorial" websites for me to copy. Thanks. 

One of the most interesting properties of any metal-semiconductor (MS) interface is its Schottky barrier height (SBH). The SBH is the rectifying barrier for electrical conduction across the MS junction and, therefore, is of vital importance to the successful operation of any semiconductor device. The magnitude of the SBH reflects the mismatch in the energy position of the majority carrier band edge of the semiconductor and the metal Fermi level across the MS interface. At a metal/n-type semiconductor interface, the SBH is the difference between the conduction band minimum and the Fermi level. And for a p-type interface, the SBH is the difference between the valence band maximum of the semiconductor and the metal Fermi level. The most common symbol for the SBH is F_B. Other superscripts and/or subscripts are sometimes added to indicate the type of semiconductor and whether the SBH pertains to the flat-band condition or the depletion condition. For example, the symbolF^o_B,n denotes the flat-band SBH with an n-type semiconductor. 

We usually see band diagrams drawn in this fashion to illustrate the band bendings for n-type and p-type semiconductors. (You can click on most figures in this tutorial to blow them up.) While viewing these band diagrams, however, we should keep in mind that they are drawn to show only the long-range variation of the bands in the space charge region of the semiconductor. The electronic structure and charge distribution in the immediate vicinity (~1nm) of the MS interface have been deliberately neglected. Therefore, these often-encountered band diagrams are in fact only "asymptotic band diagrams", which illustrate how the semiconductor (or metal) bands vary as the bands approach the MS interface. The next time you look at a band diagram like the above, it may not be a bad idea to imagine that there is actually a small gap, or a little black box separating the semiconductor bands and the metallic bands, as drawn below. The region inside this little black box is known as the interface specific region (ISR). This is the transitional region between the metal and the semiconductor, where the magnitude of the SBH is determined. 

"What determines the magnitude of the SBH?" is a question which has troubled scientists for decades. The first-order theory of the formation of a Schottky barrier (SB) is the view attributed to W. Schottky himself originally, and also to Sir Mott. The Schottky-Mott theory proposes that the SBH depends sensitively on the work function of the metal. However, this prediction has received little support from experiment. The SBHs measured in actual experiments often showed some dependence on the preparation of the MS interface, which indicates that the SBH depends more than just the work function of the metal. Despite some scatters in the experimental data, by and large, metals with larger work functions have been found to have systematically higher SBHs than those with lower work functions. But the actual dependence observed is much weaker than that predicted by the Schottky-Mott theory. A term, "Fermi level (FL) pinning", has often been used to describe the insensitivity of the experimental SBH to the metal work function. 

Because of the poignancy of the FL pinning phenomenon, explanation of this effect alone had occupied the attention of most SBH studies conducted before the mid 1980's. Not surprisingly, most of the SBH theories proposed during that period of time contained some features or assumptions, which would automatically make the SBH insensitive to the interface structure. A well-known example is the common assumption that the distribution of interface states is a property of only the bulk semiconductor and not the metal. Although it clearly could explain the FL pinning effect, such an ad hoc assumption is hard to rationalize from the standpoint of general physics and chemistry. When a metal and a semiconductor are joined to form a MS interface, a significant redistribution of charge is expected to take place due to the overlap of wave functions from the two sides. Old bonds are broke, and new bonds are formed. The electronic states that accommodate the charge transfer at the interface should be characteristic of the MS interface and not just the semiconductor. So quantum mechanically, one expects the SBH to depend sensitively not only on the identity of the metal but also on the interface structure, which would seem to be at odds with the experimentally observed FL pinning phenomenon. Since the mid 1980's, a few high-quality, single-crystal, MS interfaces were successfully fabricated and the SBH measured at these interfaces showed a rather dramatic dependence on the orientation/structure of the MS interface. That the SBH depended on the interface structure was very much in line with the intuitive quantum mechanics picture of interface dipole and bonding, although it seemed to disagree with the FL-pinning phenomenon widely observed at polycrystalline MS interfaces. Some of the disagreement was settled when it was pointed out a few year later that the SBHs at polycrystalline MS interfaces were often inhomogeneous. The electrical data from polycrystalline SBs had always contained clear signs of SBH inhomogeneity. However, the decipher of such evidence, as it turned out, would require some knowledge on the band bending of an inhomogeneous SB, which was lacking prior to the early 1990's. Independently, direct evidence for SBH inhomogeneity also began to mount with the arrival of spatially-resolved SBH techniques, most notably the ballistic electron emission microscopy (BEEM). The general inhomogeneity in polycrystalline SBHs was in very good agreement with the structure-dependence observed at single crystal MS interfaces. It thus appeared that a consistent view on the formation mechanism of the SBH was emerging, but there remained one more hurdle to overcome. One still had to answer the question "how can a SBH mechanism which depends so sensitively on the interface structure can lead to SBHs at polycrystalline interfaces that always average out to nearly constant values, irrespective of the metal?". In other words, how does one explain the FL-pinning phenomenon with the bonding picture? This question was answered at the turn of the century/millennium when the interface dipole associated with chemical bonding at MS interfaces was modeled using established methods borrowed from molecular physics. It was shown that FL-pinning was a natural consequence of interfacial bonding. Furthermore, a host of experimentally observed systematics which had hitherto been attributed to interface states could be directly derived within the bonding picture, through a minimization of the total energy. At the present moment, it seems that the major formation mechanism of the SBH has finally been clearly identified. It should surprise no one that the intuitive picture of the formation of the SBH should turn out to be the right one!