232 7 Semiconductor surfaces and interfaces

Some of the named authors have spent time in establishing principles by which such surfaces can be understood. This is possible because a large data base of solved structures now exists; one can therefore discuss trends, and the reasons for such trends. In particular, Duke has enunciated ®ve principles in several articles, which can help us understand the following examples (Duke 1992, 1993, 1994, 1996). Zhang & Zunger (1996) and Kahn (1994, 1996) have looked at structural motifs which occur at III±V surfaces, regarding surfaces as special arrangements of these motifs. A useful point to note is that a Ga atom, being trivalent, would prefer sp2 bonding, which has the 120° angle, but that the pentavalent As atom prefers s2p3 bonding, with an inter-bond angle of 94°. Atoms at the surface have some freedom to move in directions which change their bond angles, and do indeed move in directions consistent with the above arguments.

7.2Case studies of reconstructed semiconductor surfaces

While studying this section, one needs to take enough time with a model or models to get as much of a three-dimensional `feel' of the structures discussed. Two-dimensional cuts of various low index unreconstructed surfaces can be found in Zangwill (1988) and Lüth (1993/5) along with the corresponding 2D Brillouin zones. Not all of you will need to know all the details referred to: I have found during teaching this material that any one of these sections is suitable for elaboration via a mini-project on `understanding surface reconstructions'.

7.2.1GaAs(110), a charge-neutral surface

In the f.c.c. III±V semiconductors, (110) is the cleavage face which is charge-neutral, the surface plane containing equal numbers of Ga and As atoms. Figure 7.2 shows the top view of the unit cell (a), and two side views, the dashed lines indicating dangling bonds. The unrelaxed surface (b) has the form of a zig-zag chain As±Ga±As, though, as seen in the top view, the atoms are not in the same plane. This structure is (131), so it does not introduce any further diVraction spots; however, LEED and other experiments have shown convincingly that the surface relaxes as in diagram (c): the As atom moves outwards and the Ga moves inwards, corresponding to a rotation of the Ga±As bond away from the surface plane. LEED I±V intensity analysis has been used to show that best ®ts are obtained with a rotation of 29 6 3°, with small shifts in the outer plane spacings, remarkably consistently across several III±V and even II±VI compounds. This large body of work has been reviewed by Chadi (1989), Duke (1992, 1993, 1994, 1996) and Kahn (1994, 1996). I do not give any details of 2-6 compound surfaces, nor of adsorbed atoms on any of these surfaces, but discussions of such structures and associated theoretical models are given by Mönch (1993) and Srivastava (1997).

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-®lled. The rotation results in a semiconducting surface, in

7.2 Case studies of reconstructed semiconductor surfaces	233

Ga

(a)

As

(b)

(c)	As
	Ga

Figure 7.2. Surface structure and bond rotation in GaAs (110), with broken bonds shown as

Å

dotted lines: (a) top view of the unit cell, with [110] vertical and [001] horizontal, the Ga and As forming a zig-zag chain; (b) side view of (a) without rearrangement; (c) with bond rotation of about 28°, so that Ga moves towards the planar sp2 and As towards the pentagonal s2p3 con®gurations (after Chadi 1989, redrawn with permission).

which electrons are transferred to the outer As atom and away from the Ga. Second, and intimately related, the ®lled As state is lower in energy, near the valence band edge, and its environment and angles are closer to the s2p3 con®guration. The un®lled Ga state moves up in energy, above the conduction band minimum, with its environment and angles closer to the sp2 con®guration. This is real cluster chemistry in action at the surface.

Finally, we can see that this means that the ®lled (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 visualizing this arrangement was ®rst achieved by STM and spectroscopy in 1987, as shown in ®gure 7.3. Tunneling from the sample into the tip showed the ®lled As atom states, whereas reversing the sample bias showed up the un®lled Ga states. Suitably colored in red and blue, this made an impressive cover for Physics Today in January 1987; tunneling spectroscopy was then used to verify these assignments in detail (Feenstra et al. 1987). This work was also correlated with extensive previous work on UPS and surface band structure, some of which is described by Lüth, Mönch and Zangwill. More images are given by Wiesendanger (1994), and an update on STM/STS for studying semiconductor surfaces and surface states is given by Feenstra (1994).

234 7 Semiconductor surfaces and interfaces

Figure 7.3. Constant current STM images of the GaAs(110) surface acquired at sample bias voltages (a) 11.9 V and (b) 21.9 V. Image (a) shows the unoccupied state images, dominated by the Ga sp2 con®guration, while (b) shows the ®lled states associated with the As s2p3 con®guration; (c) the corresponding unit cell and crystallography (after Feenstra et al. 1987, reproduced with permission, and Wiesendanger 1994).

7.2.2GaAs(111), a polar surface

There are many examples of polar semiconductor surfaces, but the archetype is GaAs (111). Viewed along the [111] direction we have layers: Ga As space Ga As space, so

Å Å Å

that along the [111] direction is not the same, it is As Ga space ¼. This results from the

Å

lack of a center of symmetry in the GaAs lattice, (43m), not m3m as the normal f.c.c., or the diamond lattice.

If now the Ga layers are somewhat positive, and the As somewhat negative, then there are indeed alternating sheets of charge, as discussed in problem 7.3. Consider a test charge moving through this material. It will undergo a net (macroscopic) change of potential energy as it goes through the crystal. In fact this change is HUGE! We calculated in section 6.1.4 that a dipole layer consisting of 1 electron/atom separated by 1 Å caused a potential change of about 36 V; 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 cannot be what happens in reality; nature does not like long range ®elds, which

7.2 Case studies of reconstructed semiconductor surfaces	235

Ga

As

Figure 7.4. Top view of the 232 vacancy reconstruction of Ga-rich GaAs(111). The six-fold ring of atoms surrounding the corners of the unit cell consist of alternating three-fold coordinated Ga and As atoms, closely resembling the zig-zag chains of the (110) surface (after Chadi 1989, redrawn with permission).

store large amounts of energy. 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 stoichiometric composition. If they don't, they will carry a surface charge density (opposite on the two faces), which will produce a compensating long range dipole and hence no long range ®eld. The most common solution is thought to be the 232 vacancy reconstruction, shown here in ®gure 7.4 for the Ga-rich surface. It is an interesting exercise to do the bond counting and show that it works out correctly (problem 7.3). You should also note the changes in bond angles, which take the Ga towards the sp2, and the As towards the s2p3 con®gurations, which these elements would like. The As moves into the vacancy and towards ®ve-fold coordination, and the Ga uses the extra space so created to move into the surface and to a more planar, threefold con®guration.

What is perhaps diYcult to comprehend is the fact that the changes in electronic energy involved are so large, that they are suYcient to create atomic structural defects such as surface vacancies. In this case, we have removed one Ga atom in four; so the cost of this has to be about three Ga±As bonds, of order 331.755.1 eV per surface unit cell, the excess Ga typically existing in the form of small (liquid) droplets on the surface. But instead of the metallic surface, we have four ®lled As-derived states, gaining of order 4Eg ,5.6 eV, where the energy gap of GaAs is Eg51.42 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 ®eld completely. There are delicate balances involved, but the result is clear. The arguments in favor of vacancy formation at II±VI surfaces in such situations are even stronger because of larger band gaps and lower bond energies (Chadi 1989).

7.2.3Si and Ge(111): why are they so different?

In section 1.4, we introduced the various reconstructions of Si(111), and the fact that the famous 737 structure was solved by a combination of STM, THEED and LEED.

2367 Semiconductor surfaces and interfaces

The crucial breakthrough was the proposal of the dimer-adatom-stacking fault (DAS) model by Takayanagi et al. (1985) which built upon the prior STM and LEED work, and a detailed analysis of THEED intensities. Since the diVraction pattern contains 49 beams, a truly quantitative analysis of the diVraction pattern was thought to be impossible. But once this model had been articulated, detailed surface X-ray diVraction and LEED I±V analyses were successful, and the re®nements lead to a very complete set of atomic positions in the structure (Robinson et al. 1988, Tong et al. 1988). The 737 structure is shown in ®gure 7.6; versions in color can be found via Appendix D.

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 at low temperatures, but when we cleave the crystals at room temperature, we get a (231) reconstruction. This has been found to have a p-bonded chain structure; it is illustrated and discussed in detail by Lüth (1993/5). On annealing this structure to around 250°C, it transforms irreversibly to the 737. The DAS structure is therefore more stable energetically; but it requires atom exchange, which is not possible at low temperatures. At 830°C, the 737 pattern disappears, to be replaced reversibly by a simple 131 pattern. But Ge(111) has a quite diVerent sequence: c(238) at room temperature, with a reversible transition to 131 at 300 °C.

What on earth is going on, you might well ask. More detective stories, good ones too; should the plot be spelled out, or should you be left to ®nd out? DiYcult question; the detailed history is a good topic for a mini-project during a course. Early work using a semi-empirical tight binding model showed that 737 was more stable than 131 by around 0.4 eV/(131) cell (Qian & Chadi 1987). But the 737 structure is only one of a family of DAS structures of the form (2n11)3 (2n11); the smallest of these is 333. The elements of the 131 and 333 structure are shown in ®gure 7.5. When ab initio theorists ®rst calculated the energy of DAS structures, they naturally started with this one (Payne 1987, Payne et al. 1989). The basic adatom unit is in a 232 arrangement, so that was another possible approach (Meade & Vanderbilt 1989).

There was then an enormous eVort to calculate the energy of the 737 ab initio, a huge task, resulting in two groups publishing back to back in Physical Review Letters volume 68: Stich et al. (1992) on page 1351 from Cambridge, England, and Brommer et al. (1992) on page 1355, from Cambridge, MA. Both these groups showed that the 737, illustrated in ®gure 7.6, indeed has a lower energy than both the 333 and 535, and also by a margin of only 0.06 eV/(131)cell than the 231, very close to the 0.04 eV previously estimated by Qian & Chadi (1987) ± especially considering the likely errors in the calculations. The values they quote for these energies are shown in table 7.2. To show that 737 is really the most stable structure, one should surely also calculate the 939 and 11311 and show that the energy goes up: yes, but one must remember that these calculations were at the limit of massively parallel supercomputer technology. At the time of writing, such a calculation is de®nitely feasible; but is it now anyone's ®rst priority to do it again, and be really careful? Probably not!

The stacking fault in the DAS structures enables dimers to form along the cell edges,

7.2 Case studies of reconstructed semiconductor surfaces	237

Figure 7.5. Simple Si(111) surface structures (a) bulk terminated, showing a 333 cell for comparison with (b) 333 DAS structure. In (a) the open and full circles show unrelaxed atoms of the ®rst layer (three-fold) and second layer (four-fold) coordinated. In (b) the second layer atoms form the dimers along the (dotted) cell edges, which is coupled to the existence of the stacking fault in the lower left hand half of the cell (small boxes). The larger shaded circles are the adatoms, which are three-fold coordinated, each replacing three dangling bonds by one (after Chadi 1989, redrawn with permission).

and the ring at the corners at the intersection of cell edges. Without the stacking fault, we simply have the adatoms, which are arranged in a 232 array. The Ge(111) structure is thought to be based simply on these adatoms; within the cell there are two local geometries, subunits of 232 and c234; together they make the larger c238 reconstruction as determined by X-ray diVraction (Feidenhans'l et al. 1988); reviews of this technique plus many structural details are given by Feidenhans'l (1989) and Robinson & Tweet (1992). In case you think it is always easy for great scientists, it isn't; for example, Takayanagi & Tanashiro (1986) generalized their Si(111) 737 model to produce a model of Ge(111)c238 based on dimer chains ± too bad, wrong choice!

The high temperature 131 structure is often written `131', 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 232 or similar con®gurations. DiVuse scattering from these adatoms has been seen for both Si and Ge(111), e.g. using RHEED (Kohmoto & Ichimiya 1989) and medium energy ion scattering (MEIS) (Denier van der Gon et al. 1991). Similar structures are expected on Ge/Si mixtures, where Ge segregates to the

238 7 Semiconductor surfaces and interfaces

Adatoms	Second layer
	and ©rest© atoms

	Third layer	Fourth layer
	including dimers

Figure 7.6. The equilibrium 737 DAS structure. All the DAS features are present, with 12 adatoms plus seven `rest atoms', three in each half of the cell plus one in the middle of the corner hole. There are thus 19 dangling bonds left out of 49 for the unreconstructed structure. This reduction in energy is obtained at the cost of substantial internal strain energy, stretching over at least four layers (after Takayanagi et al. 1985, redrawn with permission, and later authors).

surface because the lower binding energy. These details are also fascinating and are discussed in section 7.3.3.

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 Binnig et al. (1982), and subsequent work by many people showed up back bonds and other features of the electronic structure; i.e. one gets diVerent pictures as a function of bias voltage, because diVerent states are active. The most ambitious, yet relatively simple, attempt to understand the various structures is that by Vanderbilt (1987), 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 (domain wall) energy. He then draws a phase diagram, shown in ®gure 7.7. This exhibits a series of DAS structures if f is small, which have increasing (2n11) 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 c238.

This simple diagram explains how Si and Ge could be close together on such a diagram, and yet have such diVerent structures. It also explains (in the same sense) how the surface stress, quenching, or Ge addition to the surface can give rise to 535, 939, and mixed surfaces. Beautiful STM pictures illustrating all these possibilities have been published by Yang & Williams (1994); one example is shown here in ®gure 7.8. The important point to note is that these diVerent reconstructions do not have the same areal density of atoms; so the change from one structure to another requires a lot of