ELECTRONIC STRUCTURE OF SEMICONDUCTOR SURFACES AND INTERFACES
Walter A. Harrison Applied Physics Department Stanford University Stanford, CA 94305 USA
1. Introduction The energy band structure of bulk semiconductors is by now understood in great detail. The electrons move as particles with effective masses (which may differ from that of the free electron and may even depend upon direction) and with a potential energy (equal to the energy of the band edge) which is independent of position. It might have turned out that this description remained true up to the crystalline surface, which then simply provided a container holding the electrons or the holes in the material. This description is quite appropriate for a metal, but it was learned early that on most semiconductors the behavior is strongly modified at the surface; the effective-mass description remains appropriate, but the potential energy ordinarily shifts near the surface, sometimes providing barriers to the carriers or channels capturing carriers in a surface layer. Similar effects occur at the interfaces between different materials. These specific effects of the surfaces and interfaces on the electronic structure are only tncompletely understood and, partly because of their technological importance, are currently the SUbject of active research. In the present discussion we shall focus on the behavior of bands near an interface and attempt to assess the current understanding of this aspect of interface electronic structure.
2. Natural band Hne-ups The simplest semiconductor interface is that between two semiconductors in a heterojunction, where the crystal structure is continuous across the interface, but the composition changes at the plane. The band gap will ordinarily be different on the two sides so there must be some otsconnnotty of the band edges; the question is what the dtscontlnutty is for the valence bands and what it is for the conduction band, as illustrated in Fig. 1. For this case we could imagine obtaining the energy bands for the two materials on the same scale and simply subtract the values. The diffiCUlty is that the zero of energy in most band calculations is arbitrary and to obtain the two on the same scale it is necessary to do the much more complicated calculation for the electronic structure of a composite system containing both semiconductors. This is avoided in t igllt-binding theory /1,2/ which is based upon free-atom states and which places all energy bands on the same scale directly. This is less accurate as a technique than many other procedures, but seems to be roughly within the
E --------j c
First semi conductor
5econd semi conductor
FiO. 1. The valence-band maximum Ev and theconduction-band minimum Ec are separated inenergy by a gap Eg which will be different indifferent semiconductors. The bandIIne-lIprefers totherelative energies of theindividual bands ata junction between thetwo semiconductors experimental uncertainty of the measured line-ups and gives immediate preductions. In particular, the valence-band maximum Ev in Universal
Parameter tight-binding theory is given by a simple formula in terms of the free-atom term values (all tabulated) and the internuclear distance in the solid, and was evaluated for all semiconductorsl 1,2/. The newer set of values is given in Table 1. A comparison with experimentally measured discontinuities in heterojunctions with the values obtained by SUbtracting these values of E v was made by Kraut 13/,who found that the earlier values 111 were in slightly better agreement With current experiments, but the difference may not be significant. The result of these evaluations was to give valence-band line-ups such as that Illustrated in Fig. I. By adding the band gap on each side, one obtains also the conduction-band line-up. These are our predictions of what we call natural band line-ups. We shall see reasons In the next section, other than errors inherent in our theory of the energy bands of individual semiconductors, why these do not agree with experiment. We give a sampling of predictions of band discontinuities and the experimental values given by Kraut 131 (in parentheses) to illustrate the accuracy: AIAs/GaAs, 0.03 (0.19); InAs/GaSb, 0.81 <0.46); Si/Ge 0.38 (0.20); ZnSe/Ge, 2.09 (1.52). The agreement is limited, but so is the agreement from one set of experiments to another. It was good enough for W. R. Frensley and H. Kroemer (private communication) to use it to correctly predict that in the InAs/GaSb junction the discontinUity is large enough that the valence band edge in GaSb lies above the conduction band edge in InAs, the only such occurance among pairs which are well matched with respect to lattice distance.
Table 1. Magnitude oftheenerw ofthevalence-band maxima ofsemiconductors. -Ev• in electron volts,onthescale ofatomic energies. /2/ It is similar (but greater by a few eV) tothe photothreshold enerw; theenergy required to remove anelectron from thesolid. C Si
AlAs A1Sb eoN GaP GaAs
1518 9.35 8.97 8.00 12.59 1593 11.56 11.00 14.67 10.22 9.67 8.76
14.59 10.21 9.64 8.67
InN InP InAs InSb
14.34 10.03 9.48 8.61
BeG Be5 Be5e BeTe
18.05 12.74 11.86 10.77
17.19 12.00 11.06 9.87
InS InSe InTe
1189 10.97 980
CuF CuCl CuBr Cui
20.32 14.05 12.67 11.20
There are several immediate general predictions of the natural band line-ups. They should be independent of the crystal orientation of the interface and of the order of deposition. These have been studied experimentally by R. W. Grant, S. P. Kowalczyk, E. A. Kraut, and J. R. Waldrop (See Ref. 3 for references), and are only approximately satisfied. Before turning to corrections to natural band line-ups, we may note that the same Univeral-Parameter tight-binding theory gives predictions of the band line-ups for semiconductor-insulator interfaces and semiconductor metal interfaces. In typical ionic insulators/ I /, such as NaCl, the valence-band maximum is typically near the anion p-state energy (-13,78 eV for (1) and the conduction-band minimum near the cation s-state energy (-4.95 eV for Na) and we may see from Table I that the natural band 1ine-up puts the semiconductor band edges near the center of the large insulator gap. This is generally the case, though the exact line-ups seem not to have been extensively studied. tn a metal the levels are occupied up to a Fermi energy in the middle of a band and the interesting question is where that Fermi energy comes in relation to the semiconductor band edges In the particular case of 5n, which is both a semiconductor and a metal, depending upon structure, the Fermi energy of the metallic phase comes near the valence-band maximum (which is essentially
the conduction band minimum since tin is a semiconductor of zero gap) of the semiconducting phase. This is just above the valence-band maximum of most semiconductors, as may be seen from Table 1. This is typically the case for experimental semiconductor-metal interfaces, but the prediction is only accidentally correct. Tin is in the fourth column of the periodic table and metals of the third, second, and first columns have successivly shallower Fermi energies. Evaluation of these for the row containing tin (unpublished theory for metals, which gives a value of -7.83 eV for metallic tin rather than the -8.00 eV of Table 1) yields -6.60 eV for In and -5.49 eV for Cd. The values for the alkalt metals and alkaline earths are even shallower, though the noble metals (-7.24 eV for Ag) are deep. Thus the natural band line-up typically puts the metallic Fermi energy well above the conduct ion-band minimum, and strongly dependent upon the metal in question, both contrary to experiment. We turn next to effects which may be responsible for this discrepancy.
3. Interface dipoles and MIG states Imagine an interface between two solids, initially with natural band line-ups, but now because of an electronegativity difference some electronic charge is transferred from one material to the other, producing an electric dipole layer at the interface. This has exactly the effect of shifting the bands on the two sides of the dipole relative to each other; that is, modifying the band line-up. (In contrast, an excess charge layer at the interface would produce a uniform field in one material, or both, and can never occur at interfaces between bulk materials. In metals the electrons flow to remove fields, and once the potential difference within a semiconductor exceeds the band gap, free carriers are generated to eliminate the field.) We have estimated the dipole layers in semiconductor interfaces with normal band line-ups, as in Fig. 1, and found them to be very small, giving, for example a shift of 0.01 eV at the silicon-germanium interface. /4/ There is no universal agreement on this question but we believe these dipoles are indeed small, except in a case such as the InAs/GaSb case discussed above which is more analogous to a metal-semiconductor interface. We have indicated that experimental metal-semiconductor interfaces have the metallic Fermi energy in the semiconductor band gap, as illustrated in Fig. 2. (The semiconductor also has a Fermi energy, ordinarily in the gap but with a position determined by the doping of the semiconductor. That energy, in the bulk semiconductor, will equal the Fermi energy in the metal, but it is accomplished by charge redistribution near the interface and associated band bending over distances of thousands of Angstroms, as illustrated in Fig. 2, which do not modify the band line-ups right at the interface; we are discussing these nere.) In this case, electrons cannot flow into the semiconductor since the empty states are in the conduction band, well above the metallic Fermi energy. However, the metallic states themselves do not stop abruptly at the interface; they decay exponentially
Fig. 2. Band line-ups are associated withthe interface and independent of
theband bending which occursdeep InthesemicondUCtor. into the semiconductor and the rate of that decay depends upon their energy relative to the semiconductor band edges. These tails of the metallic states produce a dipole, just as would electrons transferred into the semiconductor, and have been called metal-induced gap states (MIG's) though the term seems Quite inappropriate. Heine /5/ noted that they tend to shift the metallic Fermi energy toward midgap; Louie, et al, 16/ found the effects to be large and Tersoff /7/ sought the energy in the gap toward which the Fermi energy is shifted and indicated it was consistent with observed metal-semiconductor band line-ups. We remain very skeptical of this conclusion. Sokel 18/ made a rigorous calculation of the dipole shift, but for a simplified model of the semiconductor bands. He found first that the effect does not diverge as the metallic Fermi energy approaches the band edges, as onemight have expected; this conclusion doesnot depend upon the model of the bands. Application to a metal on gallium arsenide suggested that the shift could not exceed 0.02 eV, much less than the dipoles required to bring the natural metal-semiconductor band line-ups into agreement with experiment. We conclude that at an ideal planar metal-semiconductor interface the Fermi energy of most metals would lie well above the conduction-band edge of the semiconductor, electrons would flow into the semiconductor and with their image charge would produce an ohmic contact with n-type semiconductors, not the SclJottky barrier corresponding to the band line-ups shown in Fig. 2. This is contrary to most experiments and we conclude that the interfaces are not ideal; there are defects at the interface. 4. Defects at the interface. We have seen that interfaces must be neutral, but if there are equal numbers of positively charged defects on one side of an interface andnegatively
charged defects on the other, the resulting dipole layer will modify the band line-up. The shift is proportional both to the number of defects per unit area and to their spacing. If the spacing is small, of the order of a bond length d, then a large number of defects per unit area is required to produce a signficant shift. The number of atoms per square centimeter on a (111) plane of a tetrahedral semiconductor is .f3/4d2. Let the fraction with charge e be x and let the spacing be d. Then the shift due to the dipole, reduced by a factor of the dielectric constant E is .f3TTxe2/f:d, or 2.78x eV for silicon. Thus even if 1% of the atoms at the surface are defects, the shift is small. If they are somewhat further from the interface a smaller number is required. The band bending shown in Fig. 2 is a special case of this in which the defects are charged impurity atoms and it is their extremely small density which has spread the shift over thousands of Angstroms. Our conclusion that there are large dipole shifts at the metal-semiconductor interface from defects implies that they must be large in number. They must be negatively charged in the semiconductor (with the balancing positive charge an image charge in the metal) in order to raise the semiconductor bands from their natural line-up position. In silicon, the metallic Fermi energy comes always about a third of the way up in the gap, though the natural line-up would place it much higher and would vary considerably from metal to metal. It is very difficult to know what defects are present and in what numbers, but these experimental facts limit the possibilities. We therefore make speculations based upon them. The simplest defect in the semiconductor might be a metallic atom impurity. Indeed an impurity of valence one less than the atom replaced will have a negative charge and enough of them will have the desired effect. The difficulty is that the number required and their positions depends upon the metal at the interface and it is not easy to imagine this leading to a Fermi energy line-up in the gap independent of the metal. The only possibility would seem to be if the diffusion of impurities went to an equilibrium and the equilibrium determined the band line-up. This is currently being investigated by Kraut and the author. 191 A second possibility is a defect Which has multiple charge states, which might fix the Fermi energy. Such defects provide deep levels ,or energy levels well within the gap, which will be occupied if the Fermi level (in the semiconductor) lies above them but empty if it lies below. If such levels are empty when the defect is neutral, a metal at the natural band line-up will provide electrons to them, providing negative charges until the resulting dipole shifts the bands pinning the Fermi energy at this energy level. The first proposed explanation of the band bending at metal-semiconductor interfaces, given by Bardeen 1101, was in fact that there are surface states one third of the way up the
gap in silicon which pins the metallic Fermi energy there, independent of which metal is present. This remains the most plausible explanation, in our view, though the nature of the states remains in question. The question is further complicated by the fact that on the gallium arsenide surface the Fermi energy of the interface with the vacuum is not pinned until metal atoms (or atoms of other types) are deposited on the surface. This has led Spicer, et al, /11/ to suggest that the defects in this case are introduced by the added atoms and since the pinning depends little upon the added atom they may be intrinsic defects rather than impurities. A second difference in gallium arsenide is that n-type and p-type specimens are pinned at different energies, the n-type some 0.2 eV higher. Grant, et aI, /12/ have suggested that this difference is an intra-atomic coulomb repulsion U so that a second electron removed requires this additional energy; this, then, is a single-defect model With a U. There is some evidence that the defect in question is an arsenic antisite (an arsenic atom in a gallium site) 113/, though this would require a spectator defect which removes one electron from the doubly occupied state of the neutral antisite so that the combination was neutral. This remains controversial. Why then is silicon different, with only a single pinning level for n- and
p-type, and with pinning on the free surface? Clearly an anttslte does not exist in silicon so the defect must be different, perhaps a vacancy. Interestingly enough, a vacancy in si 1icon is what is called a negative-V center. /14/ Because of relaxation of the atoms, a second electron is bound more strongly than the first as if it had a negative U. Then a collection of singly-occupied vacancies will disproportionate with part doubly occupied and the rest empty. Such a system pins both n-type and p-tvpe materials at the same energy, midway between the two electronic levels, consistent with the observed pinning behavior. This case also requires a spectator defect to remove one of the electrons from the doubly-occupied neutral vacancy. These questions are at the forefront of of current surface science. We can only consider some of the possibilities and express our own opinion of them. The answers should not be long in coming.
S. Acknowledgement This work was supported by the National Science Foundation under Grant No. DMR 80-22465.
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