Lecture notes by John A. Venables. Lecture given 16 Apr 96. Notes updated 4 Nov 96.
While studying this section, take enough time with a model or models to get as much of a three-dimensional ‘feel’ of the structures discussed. A 2D cut of various low index unreconstructed surfaces, taken from Zangwill, is shown in diagram B2, along with the corresponding 2D Brillouin zones. Luth has the equivalent diagrams on page 291. Not all of you will need to know all the details referred to: any of these sections is a suitable topic for a mini-project on ‘Understanding Surface Reconstructions’.
The rotation is important for several aspects. First, the unrelaxed surface would be metallic. This arises because the cleavage results in one dangling bond per atom; thus the surface band is half-filled. The rotation results in a semiconducting surface, in which electrons are transferred to the outer As atom and away from the Ga. Second, and intimately related, the filled As state is lower in energy, near the valence band edge, and its environment and angles are closer to the s2p3 configuration. The unfilled Ga state moves up in energy, above the conduction band minimum, with its environment and angles closer to the sp2 configuration. This is real cluster chemistry in action at the surface.
Finally, we can see that this means that the filled (valence band-like) and the empty (conduction band-like) surface states will have the same periodicity, but will be shifted in phase, to be located over the As and Ga atoms respectively. The amazing feat of visualising this arrangement was first achieved by STM and spectroscopy in 1987 (R.M. Feenstra et al., Phys. Rev. Lett. 58 (1987) 1192, see also Surface Sci 299/300 (1994) 965). Tunneling from the sample into the tip showed the filled As atom states, whereas reversing the sample bias showed up the unfilled Ga states; suitably colored in red and green, this made an impressive cover for Physics Today in April(?) 1987; tunneling spectroscopy can verify these assignments in detail. This work was also correlated with extensive previous work on UPS and surface band structure, some of which is described by Luth (p 299-305) and Zangwill (page 101-4).
If now the Ga layers are somewhat positive, and the As somewhat negative, then there are indeed alternating sheets of charge as seen in Zangwill's fig 4.45(c) on page 105. Consider a test charge moving through this material. It will undergo a nett (macroscopic) change of potential energy as it goes through the crystal. In fact this change is HUGE! We calculated in Section A1(d) that a dipole layer consisting of 1 electron/atom separated by 1 Angstrom caused a potential change of about 36V; but this case has a dipole sheet of similar magnitude on each lattice plane, and gives rise to a really large dipole- of order 1 electron/atom times the thickness of the crystal. Anyway, this can't be what happens in reality. There must be an equal and opposite dipole due to the surfaces somehow.
The two opposite faces are referred to as Ga-rich (111A) or As-rich (111B), and they may well not have the stoichometric composition. If they aren't they will carry a surface charge density (opposite on the two faces), as indicated by Zangwill, which will produce a compensating long range dipole and hence no long range field. The most common solution is thought to be the 2x2 vacancy reconstruction, shown by Zangwill on page 40 (fig 3.12), and here in diagram B4 for the Ga-surface. I spent quite a bit of time describing these structures in the lectures, and it is an interesting exercise to do the bond counting and show that it works out OK. I also concentrated on the changes in bond angles, which take the Ga towards the sp2, and the As towards the s2p3 configurations, which these elements would like. The As moves into the vacancy and towards 5-fold coordination, and the Ga uses the extra space so created to move into the surface and to a more planar, 3-fold configuration.
What is perhaps difficult to get one’s mind around, is the fact that the changes in electronic energy involved are so large, that they are sufficient to create atomic structural defects such as surface vacancies. In this particular case, we have removed 1 Ga atom in 4; so the cost of this has to be about 3 Ga-As bonds, of order 3 x 1.7 = 5.1 eV per surface unit cell. But instead of the metallic surface, we have 4 filled As-derived states, gaining of order 4Eg ~ 5.6 eV, where the energy gap of GaAs is Eg = 1.42 eV; we also have to ‘pay’ for the bond (and other forms of elastic) distortion, but against that we get rid of the long range electric field completely. There are undoubtedly delicate balances involved, but the result is very clear and unambiguous.
This is the hallmark of a really extraordinarily successful piece of science: long fought for, but worth every penny. Understanding why we get these structures, and what are the competing structures, is equally fascinating. First, Si and Ge(111) are the lowest energy surfaces of these elements, but when we cleave the crystals at room temperature, we get a (2x1) reconstruction. This has been found to have a pi-bonded chain structure; it is illustrated and discussed in detail by Luth, p 292-5. On annealing this structure to around 250 C, it transforms irreversibly to the 7x7. The DAS structure is therefore more stable energetically; but it requires atom exchange, which is not possible at RT. At 830 C, the 7x7 pattern disappears, to be replaced reversibly by a simple 1x1 pattern. But Ge(111) has a quite different sequence: c(2x8) at RT, with a reversible transition to 1x1 at 300 C. What on earth is going on, you might well ask. More detective stories, good ones too; should I spell out the plot, or leave it to you to find out? Difficult question; the detailed history is a good topic for a mini-project.
The 7x7 structure is one of a family of DAS structures of the form (2n+1)x(2n+1); the smallest of these (3x3) is shown in diagram B5, in comparison with the unreconstructed 1x1. When theorists tried to calculate the energy of DAS structures, they naturally started with this one (M.C. Payne, J. Phys. C20 (1987) L983). The basic adatom unit is in a 2x2 arrangement, so that is another possible approach, also relevant for Ge (R.D. Meade and D. Vanderbilt, Phys. Rev. B40 (1989) 3905). There was then an enormous effort to calculate the energy of the 7x7, a huge task, resulting in two groups publishing back to back in Phys. Rev. Letters (Stich et al 68 (1992) 1351 from Cambridge, England, and Brommer et al on page 1355, from Cambridge, Mass). Both these groups showed that the 7x7 indeed has a lower energy than both the 3x3 and 5x5, and also than the 2x1; the values they quote for these energies are shown in Table B2. To show that the 7x7 is really the most stable structure, one should surely also calculate the 9x9 and 11x11 and show that the energy goes up: yes, but give us a break, these calculations were at the limit of massively parallel supercomputer technology! In 1996, it is definitely feasible; but is it now anyones’ first priority? I think not.
The stacking fault in the DAS structures enable the dimers to form along the cell edges, and the ring at the corners at their intersection. Without the stacking fault, we simply have the adatoms, which are arranged in a 2x2 array. The Ge(111) structure is thought to be based simply on these adatoms; within the cell there are two local geometries, subunits of 2x2 and c2x4; together they make the larger c2x8 reconstruction as determined by X-ray diffraction (Fiedenhansl et al. Phys. Rev. B38 (1988) 9715; a review of this technique plus the structural details is given in Surf. Sci. Rep 10 (1989) 105). In case you think it is always easy for great scientists, it isn’t; for example, Takayanagi and Tanashiro (Phys. Rev B34 (1986) 1034) generalised their Si(111) 7x7 model to produce a model of Ge(111)c2x8 based on dimer chains- too bad, wrong choice!
The high temperature 1x1 structure is often written ‘1x1’, meaning ‘we know it isn’t really’; both Si and Ge are thought to form a disordered structure of mobile adatoms which may locally be in 2x2 or similar configurations. Diffuse scattering from these adatoms has been seen for both Si and Ge(111), e.g. using RHEED (S. Kohmoto and A. Ichimiya, Surf. Sci. 223 (1989) 400) and Medium Energy Ion Scattering (A.W. Denier van der Gon et al. Surf. Sci. 241 (1991) 335). Similar structures are expected on Ge/Si mixtures, where Ge segregates to the surface because the lower binding energy. These details are also fascinating, but we simply don’t have time to get into all this in class- it is an ongoing research topic.
Are there any further checks on these models, and can we make sense of them? STM has been invaluable; adatoms were seen in the original pictures by G. Binnig et al. (Phys. Rev. Lett. 50 (1983) 120), and subsequent work by many people showed up back bonds and other features of the electronic structure; i.e. one gets different pictures as a function of bias voltage, as shown in diagram B7. The most ambitious, yet relatively simple, attempt to ‘understand’ the various structures is that by D. Vanderbilt (Phys. Rev B36 (1987) 6209), where he tries to estimate the energy costs of the stacking fault (f) and of the corner holes (c), expressed as a ratio to the dimer wall energy. He then draws a ‘phase diagram’, shown in diagram B8. This exhibits a series of DAS structures if f is small, which have increasing (2n+1) periodicity as c increases. At larger values of f, the stacking fault is unfavorable, and there is a ‘transition’ to an ordered adatom structure, notionally the c2x8. This simple diagram ‘explains’ how Si and Ge could be close together on such a diagram, and yet have such different structures. It also explains (in the same sense) how the surface stress, quenching, or Ge addition to the surface can give rise to 5x5, 9x9, and mixed surfaces. Some fantastic STM pictures illustrating all these possibilities have been published by Y.N. Yang and E.D. Williams, Scanning Microscopy 8 (1994) 781.
There has been much debate as to whether the 2x1 reconstruction is symmetric, or assymetric; by now you will realise that this is the same question as whether the surface is metallic or semiconducting. A consensus has emerged that the Si dimer is assymetric, but that the energies are so close that the the dimer flips between two equivalent states- either the left-hand or the right hand atom is up at any one time. At high T, this is like having a low frequency vibrational mode; at low T, ordered arrays of up and down dimers can give superstructures, such as c2x4. There are a host of such calculations in the recent literature: one (A. Ramstad et al. Phys. Rev B51 (1995) 14504) gives the c4x2 as the lowest energy structure, and calculates by how much it is stable. The dimerization gives a large energy gain over the unreconstructed 1x1 structure, about 2 eV per dimer. The asymmetric dimer is favoured by a further 0.2 eV; ordering these dimers into either the p2x2 or c4x2 gains a further 0.02 eV, and the eventual stability of the c4x2 is a mere 0.002 eV. It is not clear to me whether we should believe this slender margin, but it is clear why it won’t be stable at finite T.
For Ge (100), and more recently Ge/Si (100) also, there has been great interest in whether these surfaces are also asymmetric, in what way, and in establishing the trends in bond angles (see e.g. Tang and Freeman, Phys. Rev B50 (1994) 10941 and refs quoted). So this complexity is there, but is it interesting or important particularly? The asymetric dimers can be thought of in the same sense as the bond rotation in GaAs (110), but with a much smaller amplitude. The finite T effects are interesting examples of anharmonic lattice dynamics. Certainly, steps are very important in growth of devices, where the growers typically use substrates miscut by 2-4 deg, in order to prevent nucleation of three dimensional islands (discussed in section E). The bonding changes during growth are extremely complex, since any specifically surface reconstruction has to be undone in order for growth to proceed; at low temperatures this leads to the possibility of creation of many, possibly unwanted, metastable structures. On the other hand if growth of complex structures, such as multiple quantum wells with different compositions, is conducted at too high temperatures then they will be degraded by surface segregation and interdiffusion. Device engineers are always treading a fine line in trying to grow crystals at the lowest practicable temperature- reducing the ‘thermal budget’.
The growth of devices based on Ge/Si(100), or GaAs on Si or Ge(100) throws up a whole host of fascinating problems of this type. For example, Ge has a larger lattice parameter than Si (4%), and Ge surface segregates because of its lower binding energy. In the first few monolayers, 2xn reconstructions, with n of order 8-12, are commonly observed when monitoring the growth of Ge/Si by RHEED. STM has shown that these structures consist of rows of dimer vacancies which both relieve and respond to surface stresses. GaAs surfaces also exhibit such higher order vacancy line structures, such as the 2x4 and 6x4 arrangements shown in diagram B10. Another example is that GaAs growth quality is poor when grown on misoriented substrates which have single height steps, because anti-site defects are formed at the step: with double height steps this problem goes away. There is a vast literature on these topics, and even more unpublished empirical knowledge in the firms who make these devices. You could start with the review articles quoted in the semiconductor paragraphs of section E.
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