HTML-Faksimile aus: Annual Review of Material Science 18 (1988):
von U. Gösele
Elements diffusing in semiconductors are frequently classified in terms of "slow" and "fast" diffusors (1). Slow diffusors show diffusion coefficients reasonably close to that of selfdiffusion, whereas fast diffusors exhibit diffusion coefficients which are many orders of magnitude larger for a given temperature. In Figure 1 this classification is exemplified for the diffusion of various elements in crystalline silicon. The large difference in the diffusion rate between fast and slow diffusors derives mainly from their different diffusion mechanisms which in turn are closely related to their incorporation in the semiconductor lattice. Slow diffusors, such as the common group III and group V dopants, are substitutionally dissolved and require intrinsic point defects (vacancies and/or self-interstitials) for their diffusion process, whereas fast diffusors such as Cu, Li, H, or Fe are predominantly interstitially dissolved and move by jumping from interstitial site to interstitial site (Figure 2) without requiring the presence of intrinsic point defects. Since most interstitially dissolved elements are only weakly bound to the lattice, the jump process itself does not involve breaking bonds as in the case of substitutionally dissolved elements. There are also elements with diffusion coefficients between the extreme fast and slow diffusors, such as oxygen or gold in silicon. Oxygen is interstitially dissolved, but forms fairly strong bonds to the two neighboring silicon atoms which have to be broken during the jump from interstitial site to interstitial site (Figure 3) which slows down the diffusion process (2-4). Gold in silicon is an example of an element which is predominantly substitutionally dissolved but diffuses interstitially (5-6). The corresponding substitutional-interstitial diffusion mechanism (7,8) plays also an important role in the diffusion of dopants and impurities in III-V compounds and is considered in the category of fast diffusion in this article.
Figure 1 Diffusivities of various elements in silicon as a function of inverse absolute temperature (partly from (19)). Au_{S}^{(1)} and Au_{S} ^{(2)} refer to effective diffusivities of substitutional gold in dislocation-free and highly dislocated silicon, respectively. |
An enormous amount of information and data on slow and fast diffusors has been compiled in the literature (1,9-21) especially for the technologically important crystalline semiconductors silicon (1,9,11-19), germanium (1,11,18) and the III-V compounds (10,11,14,20,21). These papers also contain ample information on the experimental techniques for measuring diffusion profiles. It is not the purpose of this article to give a summary of all this easily accessible information but rather to concentrate on specific examples in the area of fast diffusion in crystalline semiconductors in which either significant progress has been made or obvious open questions have emerged over the last few years. Specifically discussed examples will mainly be limited to silicon, germanium and the III-V compounds GaAs and InP although many of the concepts discussed may -in appropriately modified form- be applied to other semiconductors, such as the II-VI compounds.
Figure
2 Direct interstitial diffusion mechanism, schematically; Foreign interstitial atom (o) jumps from interstitial site 1 to 2, from 2 to 3, etc. (19). |
In the elemental semiconductors silicon and germanium the main technological interest concerning diffusion is focused on the diffusion of group III and group V elements which are used as p- and n-type dopants in the fabrication of electronic devices. Fast diffusors were and are investigated mainly because of their role as deep-level contaminants which enter the semiconductor during device processing. They can either directly reduce the minority carrier lifetime or precipitate in the electrically active device region, and hence influence the device behavior negatively and often catastrophically. Only a few fast diffusors have been used purposely for fabricating or improving devices. Examples are lithium-drifted detectors, the use of interstitial oxygen for inducing gettering centers in silicon and of hydrogen for passivating electrically active defects. Gold and platinum act as efficient electron-hole recombination centers in silicon and have therefore been diffused into devices for improving their frequency behavior.
In III-V semiconductors, not only typical impurities such as Cr or Fe in GaAs diffuse via a substitutional-interstitial mechanism but also some of the most common dopants such as zinc (10,22,23) or beryllium (24,25). All of them have to be considered in the category of fast diffusors. Instead of discussing the rather well understood direct interstitial diffusion of Figure 2 (11) this article will first elaborate on the controversial role of interstitial molecules in the diffusion of oxygen and hydrogen in silicon and then concentrate on fast diffusion via substitutional-interstitial mechanisms in silicon, germanium, and some III-V compounds.
Oxygen precipitates are frequently used in Czochralski-grown silicon to induce gettering sites in the bulk region of wafers in order to getter away undesirable metallic contaminants from the electrically active device area near the surface ("intrinsic gettering") (26-28). Oxygen is dissolved interstitially in silicon and is electrically inactive in this position (Figure 3). Its diffusivity D_{i} under thermal equilibrium conditions is dominated by the process of jumping from interstitial site to interstitial site. The corresponding diffusion coefficient, given in Figure 1 and Figure 4 from the melting point down to 250°C (4,29-32), spans one of the largest ranges in the diffusivity in the area of solid state diffusion. The diffusivity at the low temperatures were measured via the change in stress-induced dichroism due to the re-orientation associated with single diffusion jumps (4,30).
Figure 3 Bond-centered configuration of the oxygen interstitial in the silicon lattice. The points labeled 1-6 mark equivalent positions of the O atom (19). |
Around 450°C interstitial oxygen forms electrically active agglomerates, which have been termed "thermal donors" (33-35). It has been noticed frequently that the diffusivity D_{i} is much too low to account for the agglomeration process (36-39). Based on thermal donor measurements at the surface of silicon wafers, Gaworzewski and Ritter (40) concluded that the diffusivity governing long-range transport of oxygen is several order of magnitude higher than D_{i} derived from single re-orientation/diffusion jumps as indicated in Figure 4. Precipitate growth experiments also required a strongly enhanced long-range oxygen diffusivity (41). Two types of diffusion models have been put forward for explaining the fast long-range diffusion of oxygen. In the first one fast diffusing complexes of intrinsic point defects (vacancies and/or self-interstitial) and interstitial oxygen are responsible for the high effective diffusivity (e.g., 38,42).
Figure 4 Diffusivity of oxygen in silicon as a function of inverse absolute temperature for isolated oxygen interstitials O_{i} . The cross indicates the high effective diffusivity found for out-diffusion (40,47). |
Specifically designed experiments could not confirm this suggestion although it cannot be ruled out conclusively (39,43). The second model suggests the formation of O_{2} molecules via the reaction (37)
1. | O_{i} + O_{t } <_{kd}->^{ka} O_{2} |
where O_{i} stands for interstitial oxygen and ka and kd for the appropriate association and dissociation constants, respectively. The O_{2} molecules are assumed to diffuse extremely fast since they are not directly bound to the silicon lattice atoms as O_{i} is. The effective long-range diffusivity of oxygen is then expected to be
2. | Deff = (D_{i}C_{i} + 2D_{2}C_{2})/(C_{ i} + 2C_{2}) |
where D_{i} is the O_{i} diffusivity, C_{i} its concentration and D_{2} and C_{2} the corresponding quantities for the O_{2} molecules. Equation 2 reduces for dynamical equilibrium according to equation 1 and dominating O_{2} contribution to 2D_{2}C_{2}/C_{i}. If the association reaction is diffusion-controlled, as indicated by experimental results on the loss of interstitial oxygen during thermal donor formation (44-46) D_{eff} may be expressed as
3. | D_{eff} 16 p D_{2}D_{i }r_{i }C_{i }/k_{d} , |
where k_{a} has been assumed as 8 pD_{i }r_{i} , with r_{i} being the reaction radius for the reaction l.
Recent work by Secondary Ion Mass Spectroscopy (SIMS) by Lee and Fellinger (47) confirmed the existence of a fast long-range diffusion process of oxygen in silicon. An example of oxygen out-diffusion from a silicon wafer at 500°C is shown in Figure 5. The calculated effective diffusivity of about 3x10^{-14}cm^{2}/s falls in the range determined by Gaworzewski and Ritter (40).
Figure
5 Oxygen concentration profile in silicon annealed at 500°C for 17 days. The solid curve is the best fit to data using a constant effective diffusivity (47). |
Lee and Fellinger (47) also measured the diffusion of implanted oxygen isotopes and observed an unexpected diffusion tail showing strongly enhanced diffusion in the temperature range of thermal donor formation (Figure 6). The depth of the tail increased with decreasing temperature. The tail is due to the dissociation of a fast diffusing oxygen-containing species.
Figure 6 Oxygen-18 concentration profile in silicon: (a) as implanted, and annealed for 67 hours at (b) 525°C, (c) 480°C, and (d) 425°C (47). |
In terms of the O_{2} model, the O_{2} diffusion may be described by (47)
4. | dC_{2}/dt = D_{2}d^{2} C_{2}/dx^{2} – k_{d}C_{2} + k_{a} C_{i}^{2} . |
If outside the implanted region the association term is neglected and steady state is assumed,
5. | C_{i} = 2tC_{2}°k_{d }exp[-(k_{d}/dD_{2})^{ 0.5} x] |
holds, where C_{i} is the O_{2} concentration in the implanted region. An equation analogous to equation 5 may be derived for any fast diffusing oxygen-containing dissociating complex. The experimental data are well described by equation 5. The measured k_{d}/D_{2} values may be used to calculate D_{eff} according to equation 3 and lead to a good agreement with the D_{eff} values measured in the actual out-diffusion experiments (47).
Equation 3 predicts that D_{eff} should linearly increase with the interstitial oxygen concentration C_{i}. Out-diffusion experiments from silicon wafers with different oxygen contents resulted in a rather unsystematic scatter of D_{eff} valves with no noticeable correlation to the oxygen concentration (48, S.-T. Lee, private communication). Since during the out-diffusion experiments oxygen precipitation is occurring, which strongly depends on the thermal history of the sample, it might well be that the precipitation process influences D_{eff} by the absorption and incorporation of O_{2} molecules into oxygen precipitates.
The presence of an effective long-range diffusivity of oxygen in silicon which is much higher than the diffusivity of single O_{i} is now well established (39,47). An explanation in terms of fast diffusing oxygen molecules is tempting, but further experiments are required to prove or disprove that O_{2} model. This model might also hold for oxygen in germanium where similar thermal donor phenomena occur (44). Nitrogen diffusion in silicon has also been suggested to proceed via molecules (50) but as in the case of oxygen molecules no clear-cut experimental proof of their existence is available.
Hydrogen can be introduced into semiconductors during crystal growth, by direct implantation, by exposure to a hydrogen-containing plasma or by chemical reaction at the surface (51). The principal interest in hydrogen in crystalline semiconductors results from its ability to passivate the electrical activity of dangling or defective bonds, although it can also deactivate shallow and deep donor and acceptor levels in elemental and compound semiconductors. Excellent review articles are available covering hydrogen in semiconductors including its diffusion behavior (48,52,53). Therefore we will only shortly comment on the diffusion of hydrogen in silicon, since it appears to involve also molecules but in a manner just opposite to that of oxygen in silicon.
Hydrogen in silicon is assumed to diffuse in the form of unbound atomic hydrogen which may be present in neutral or positively charged form. The diffusivity of hydrogen in silicon has been measured by Van Wieringen and Warmoltz (54) in the temperature range of 970-1200ºC. The results are included in Figure l. At temperatures between room temperature and 700ºC much lower hydrogen diffusivities have been measured than extrapolated from the high temperature data. Corbett and co-workers (52,55,56) rationalized this observation by suggesting that atomic hydrogen H_{i} forms interstitially dissolved hydrogen molecules H_{2} according to a reaction analogous to 1. Based on quantum-mechanical calculations they assume that H_{2} is essentially immobile, which is just opposite to what has been assumed for O_{2} molecules (37). The resulting effective diffusivity of hydrogen is D_{i}C_{i }/C_{T} where D_{i} is the diffusivity of interstitial atomic hydrogen, C_{i} its concentration and C_{T} the total hydrogen concentration, which at low temperatures may essentially consist of immobile hydrogen molecules and also includes hydrogen trapped at various trapping centers. Analogous considerations hold for hydrogen diffusion in germanium (51,52,57). Similarly as in the case of oxygen, the existence of hydrogen molecules in silicon and germanium has not been proven experimentally.
A number of elements in semiconductors are dissolved predominantly substitutionally (A_{S}), but their movement is accomplished by the fast diffusion of A-atoms in interstitial form (A_{i} ) (1,10,22). The change-over of A-atoms from interstitial sites to substitutional sites requires either vacancies (V) or self-interstitials (I). If all species are uncharged then the change-over reaction involving vacancies is described by
6. | A_{i} + V <-> A_{S} |
and termed Frank-Turnbull mechanism (1,7,58). The analogous reaction involving self-interstitials,
7. | A_{i} <-> A_{S} + I |
has been termed kick-out mechanism (8). Both mechanisms are schematically indicated for an elemental semiconductor in Figure 7. In the case of silicon the recognition that self-interstitials can be involved in a substitutional-interstitial diffusion mechanism has led to a major advance in our understanding of the diffusion of these elements and also of the mechanism of self- and dopant diffusion (19).
The diffusion via mechanism 6 may be described by three appropriate diffusion equations for A_{i}, A_{s}, and V. If necessary, the diffusion equation for vacancies should also include a term for the generation or absorption of vacancies at dislocations (8). Analogous differential equations hold for the kick-out mechanism (8,15). Instead of discussing in detail the solution of these equations (19) I will follow the treatment in (59) and derive in a simplified manner the expressions for describing the diffusion behavior of elements migrating by the Frank-Turnbull or the kick-out mechanism and apply this knowledge later on to the diffusion of Au, Pt, Ni, Zn, S, and C in silicon and Cu in germanium. For this purpose the most simple example of in-diffusion from a source of A at the surface into a dislocation-free crystal is considered.
The actual movement of the element A is accomplished by A_{i} moving with the diffusivity D_{i}. The diffusivity of A_{S} via a direct exchange with vacancies or self-interstitials is neglected. Within the framework of the Frank-Turnbull mechanism the incorporation of an A atom as A_{S} is associated with the consumption of a vacancy. If local equilibrium prevails then the concentrations of the various species are related by
8. | C_{i }C_{V}/C_{S} = C_{i}^{eq} C_{v}^{eq }/C_{S} ^{eq} |
where C_{i}, C_{V}, and C_{S} refer to the actual concentrations of A_{i}, V, and A_{S} and C_{i}^{eq}, C_{V}^{eq}, and C_{S}^{eq} to their respective thermal equilibrium concentrations. Possibly occurring activity coefficients deviating from unity are not considered in this overview (23). The incorporation rate of A_{S} and consequently the effective diffusivity D_{eff} of A_{S} is determined by the slower process of either supplying A_{i} from the surface or establishing the thermal equilibrium concentration of vacancies. If the supply of A_{i} from the surface is the slower process, then the vacancy concentration C_{V} is close to its equilibrium value C_{V} ^{eq}. Based on equation 8 and with C_{i}^{eq} « C_{S} ^{eq} the effective A_{S} diffusivity is then approximately given by (5,7,12)
9. | D_{eff}^{(i)} ~ D_{i }C_{i}^{eff} /C_{S} ^{eff} |
When the supply of vacancies from the surface (in dislocation-free material) limits the incorporation of A_{S}, then the in-diffusion of A_{S} with an effective diffusivity D_{eff}^{(V)} is governed by the flux balance equation
10. | D_{eff}^{(V)} dC_{S }/dx = D_{V }dC_{V }/dx |
where D_{V} is the vacancy diffusivity (59). With C_{i} = C_{i}^{eq} and equation 10 the effective diffusivity turns out to be (5,7, 19)
11. | D_{eff}^{(V)} = D_{V} C_{v}^{eq }/C_{S}^{eq} |
Both effective diffusion coefficients do not depend on the concentration C_{S} itself. Therefore in both cases a complementary error function (erfc) type profile is expected for constant surface concentration of A (case D = const in Figure 7). The parameter R_{V} which determines which of the two diffusion coefficients is applicable reads D_{V} C_{v}^{eq }/(D_{i }C_{i}^{eq }). For R_{V} »1 equation 9 holds and for R_{V }«1 equation ll. For known C_{S}^{eq} the quantity D_{eff}^{(V)} C_{S}^{eq} allows to determine which of the two cases is realized by comparing this quantity with the self-diffusion coefficient D_{V }C_{V}^{eq} measured by other means (19). If it coincides with D_{V }C_{V}^{eq} then the case of equation 11 is fulfilled. For equation 9 to hold D_{eff}^{(V)} C_{S}^{eq} has to be smaller than D_{V}C_{V}^{eq} in dislocation-free material. In highly dislocated material C_{V} = C_{V}^{eq} and therefore equation 9 holds even if R_{V }« 1. Intermediate dislocation densities may lead to more complicated concentration profiles (60).
Figure 7 Frank-Turnbull and kick-out mechanism, according to (59). |
In the case of the kick-out mechanism, the incorporation of an A atom as A_{S} is associated with the generation of a self-interstitial. The equation analogous to 8 reads now
12. | C_{i }/(C_{S}C_{I}) = C_{i}^{eq}/(C_{S} ^{eq}C_{I}^{eq}) |
and the parameter R_{I} analogous to R_{V} is D_{I}C_{I}^{eq} /(D_{I }C_{i}^{eq }), where D_{I} denotes the diffusivity of self-interstitials (8,19). For R_{I }» l the in-diffusion of A_{i} is much slower than the out-diffusion of the generated self-interstitials to the surface (in dislocation-free material) and equation 9 holds also for this case. For R_{I }« 1 the incorporation of A_{S} is limited by the out-diffusion of self-interstitials to the surface. The corresponding effective diffusivity D_{eff}^{(I)} of A_{S} follows from the flux balance equation
13. | D_{eff}^{(I)} dC_{S} /dx = -D_{I} dC_{I})/dx |
and equation 12 as
14. | D_{eff}^{(I)} = [D_{I} C_{I}^{eq} /C_{S}^{eq}] (C_{s}^{eq }/C_{S})^{2} , |
which, contrary to D_{eff}^{(V)}, strongly depends on the local A_{S} concentration C (8,19). If this case is fulfilled in nature it can easily be recognized by the specific profile shape, examples of which will be shown later for the case of Au and Pt in silicon (case D =_{eq}C^{-2} in Figure 8). Again, for high dislocation densities local intrinsic point defect equilibrium (C_{I} = C_{I}^{eq}) is established and equation 9 holds. For intermediate dislocation densities more complicated concentration profiles result. In this case the increase of C_{S} with time is different for the Frank-Turnbull and the kick-out mechanism and also allows to distinguish between the two mechanisms (8,19).
If both vacancies and self-interstitials are present as in silicon and if the Frank-Turnbull and the kick-out mechanism are operating simultaneously, then the effective diffusivity 9 holds for (R_{I}+R_{V}) » 1, whereas for the opposite condition an effective A_{S} diffusivity combined from 11 and 14,
15. | D_{eff}^{ (I,V)} = D_{eff}^{(I)} + D_{eff}^{(V)} |
can be derived (19,61).
Figure 8 Normalized diffusion profiles for different concentration dependencies of the effective diffusivity D. C_{S} and D_{S} refer to the concentrations and diffusivities at the surface, respectively (59). |
For III-V Compounds it has to be assumed in many cases that the species involved in a substitutional-interstitial diffusion mechanism are charged (10,22,23). In general, the Frank-Turnbull mechanism may then be written as
16. | A_{ i}^{j+} + V^{k-} <-> A_{S}^{m-} + (m + j - k)h , |
where j, k, and m are integers characterizing the charge state of the species and h stands for holes. The vacancy is assumed to be in the same sublattice in which A_{S} is substitutionally dissolved, e.g., a gallium vacancy in the case of zinc acceptors substitutionally dissolved on the gallium sublattice in GaAs (10,22,23). The first extension of the Frank-Turnbull mechanism to charged species has been given by Longing (22) for Zn in GaAs. In the literature on III-V compounds substitutional-interstitial mechanisms of this form are therefore frequently termed Longing mechanism. The corresponding extension of the kick-out mechanism may be written as (62)
17. | A_{i}^{j+} <-> A_{S}^{m-} + I^{k+} (m + j - k)h |
In general, the intrinsic point defects as well as the interstitials A; may occur in more than one charge state. For the generalized Frank-Turnbull mechanism the mass-action law for local equilibrium between the different species reads
18. | C_{i}C_{V}/(C_{S}p^{m+j-k}) = const(T) |
where p is the hole concentration. For completely ionized substitutional acceptor impurities (m>0) of sufficiently high concentration (above the intrinsic electron concentration n_{i}) the hole concentration p may be replaced by mC_{S}. For donor impurities (m<0) analogously the electron concentration n is given by |m|C_{S}. For dislocation-free material considerations similar to those for uncharged species lead to
19. | D_{eff}^{(i)} = (|m|+1)[D_{i}C_{i}^{ eq} (C_{S}^{eq})/C_{S}^{eq}](C_{S}/C_{S}^{eq} )^{|m| ± j} |
if the supply of A_{i}^{j+} limits the incorporation rate (22,23,60). The positive sign in the exponent holds for substitutional acceptors and the negative sign for substitutional donors. The factor |m| + 1 accounts for the electric field set up by the concentration gradient of the charged substitutional atoms. Equation 19 holds independent of the charge state k of the vacancies and also for the kick-out mechanism when the supply of A_{i}^{j+} limits the incorporation rate of A_{S}.
When the supply of vacancies from the surface limits the incorporation rate of A_{S}, then the effective diffusion coefficient for the A_{S} atoms is given by
20. | D_{eff}^{(V)} = (|m| + 1)[D_{V}C_{V} (C_{S}^{eq} )/C_{S}^{eq}] (C_{S}^{eq}/C_{S})^{ ±k-|m|} |
where the same sign convention holds as for equation 14. When the supply of self-interstitials from the surface limits the incorporation rate then
21. | D_{eff}^{(I)} = (|m| + 1)[D_{I}C_{I} (C_{S}^{eq} )/C_{S}^{eq}] (C_{S}/C_{S}^{eq})^{ ±k-|m|-2} |
holds. For the derivation of equation 20 and 21 it has been taken into account that C_{I}^{eq} of charged interstitials depends on the local electron or hole concentration, and is not spatially constant as had erroneously been assumed in the literature (60,62,63). Equations 19-21 have not been derived for uncharged substitutional atoms A_{S}(m=0) but hold also accidentally in this case provided all species are uncharged and then reduce to equations 9,11, and 14, respectively.
The quantities D_{V}C_{V}^{eq}(C_{S}^{eq} ) and D_{I}C_{I}^{eq}(C_{S}^{eq}) refer to the self-diffusion transport coefficients of V^{k-} and I^{k+} under the doping conditions of C_{S} = C_{s}^{eq} and not necessarily to the intrinsic self-diffusion coefficient. It is worthwhile to mention that even for charged species constant effective diffusivities may be obtained. E.g., for singly charged acceptor dopants (m=1) V^{-}(k=1) or I^{3+}(k=3) lead to constant effective diffusivities D_{eff}^{(V)} or D_{eff}^{ (I)}, respectively, with corresponding simple diffusion profiles. Since the applicable effective diffusion coefficient may change with the depth of the profile, complicated concentration profiles may result, especially if p-n junctions are present which due to the large electric fields involved may lead to locally strongly enhanced diffusion coefficients (64,65).
In-diffusion profiles of Au at and above 800°C (19,66,68), of Pt (69,70) at all temperatures investigated (T >700°C) as well as of Zn (71) and sulfur (H. Mehrer and N. Stolwijk, private communication) are dominated by the kick-out mechanism and their concentration profiles may be described by the effective concentration-dependent diffusivity 14. A typical gold concentration profile is shown in Figure 9 from the work of Stolwijk et al. (67,68). The platinum concentration profiles in Figure 10 are from the work of Mantovani et al. (70). From the known C_{S} ^{eq} values of Au and Pt the self-interstitial contribution D_{I} C_{I}^{eq} to the self-diffusion coefficient D^{SD} has been determined. The concentration-independent part of D_{eff}^{(I)} for C_{S} = C_{S}^{eq} is plotted as Au_{S}^{(1)} in Figure 1.
Figure 9 Gold concentration profile in dislocation-free silicon, according to (67,68). |
In heavily dislocated silicon the dislocations act as efficient sinks for self-interstitials and keep C_{I} close to C_{I} ^{eq} so that equation 9 should hold. This has been shown for gold and zinc in silicon (68,71). An example is the upper zinc diffusion profile in Figure 11. The D_{i}C_{i}^{eq} value determined for Au in silicon is actually much larger than D_{I}C_{I}^{eq}, as required for the kick-out mechanism to yield the concentration-dependent diffusivity 14 (see Au_{S} ^{(2)} in Figure 1). In the case of zinc D_{i} C_{i}^{eq} is not much larger than D_{I}C_{I} ^{eq} so that even in dislocation-free silicon only the profile close to the surface is governed by D_{eff}^{(I)} of 14, which strongly increases with depth. For larger penetration depths D_{eff}^{(I)} finally exceeds D_{eff} ^{(i)} and a constant effective diffusivity begins to determine the concentration profile, as shown in Figure 11.
Figure 10 Platinum concentration profiles in dislocation-free silicon (70). |
The diffusion of Au in silicon is very sensitive to the presence of dislocations, since dislocations may act as sinks for self-interstitials and therefore enhance the local incorporation rate of Au_{S}. Even in dislocation-free silicon the self-interstitials created in supersaturation by the in-diffusion of Au may agglomerate and form interstitial type dislocation loops which further absorb self-interstitials and lead to W-shaped (instead of the usual U-shaped (67,68)) concentration profiles in gold-diffused silicon wafers (72).
A detailed analysis of gold profiles at 1000°C showed the presence of a small but noticeable vacancy contribution, which is consistent with the conclusion from dopant diffusion experiments that both vacancies and self-interstitials are present under thermal equilibrium conditions. At 700°C the Au concentration profiles are characterized by a constant diffusivity (5) which indicates that at this temperature the kick-out mechanism for Au is kinetically hampered whereas the Frank-Turnbull mechanism still operates. This appears to be also the case for the incorporation of substitutional nickel in silicon (19,61).
The diffusion of substitutional carbon in silicon (see Figure 1) is an example of a case in which the diffusion is accomplished by a fast diffusing self-interstitial-carbon complex with D_{i}C_{i}^{eq} < D_{I} C_{I}^{eq} so that a "normal" constant effective diffusivity described by equation 9 is observed (59,73). Contrary to the case of Au, Pt, Zn, and S, for carbon D_{i} has been measured independently (74).
Figure 11 Concentration profiles of zinc in silicon. Solid line: erfc-profile for constant diffusivity; dotted line: kick-out type profile (71). |
A key experiment concerning diffusion mechanisms in germanium has been performed by Stolwijk et al. (75) who investigated the diffusion of copper in germanium. Copper diffuses in germanium via a substitutional-interstitial mechanism (1). In analogy to the case of gold in silicon, its diffusion behavior may be used to check diffusion profiles for any indication of a self-interstitial contribution via the kick-out mechanism. No such contribution has been found. A concentration profile of copper in a germanium wafer is shown in Figure 11. The dashed U-shaped profile, which is typical for the kick-out mechanism does not fit the experimental data, whereas the data may be well described by the constant diffusivity D_{eff}^{(V)} following from equation 11. The agreement between D_{V}C_{V}^{eq} determined from Cu diffusion profiles with corresponding tracer measurement of self-diffusion in germanium is excellent (95). This shows that self-diffusion in germanium is carried by vacancies.
Figure 12 Concentration profiles of Cu into a dislocation-free germanium wafer after diffusion for 15 minutes at 878°C (75). The solid line holds for the Frank-Turnbull and the dashed line for the kick-out mechanism. |
Substitutional-interstitial diffusion mechanisms play a much larger role in III-V compounds than in silicon or germanium, since some of the most important dopants such as Zn or Be diffuse via this type of mechanisms (10,22). Extensive investigations have been performed, especially for GaAs and InP, which are the most important base materials for optoelectronic applications and fast integrated circuits. Inspite of all these investigations the quantitative and detailed understanding of diffusion processes have not reached the level which is now typical for silicon. The reasons are manifold: Since typically donor or acceptor dopants at high concentrations are involved, charge state and concentration profile-induced electric field effects have to be considered (64,65). Comparison of effective diffusion coefficients with self-diffusion coefficients for getting more detailed information on the diffusion mechanism is difficult due to a lack of reliable self-diffusion data (21) and a possible strong dependence of self-diffusion on the doping level. It goes without saying that the presence of two sublattices also complicates the situation. This is for example the case when the climb of dislocations is considered which involves point defects from both sublattices (10). In the analysis of substitutional-interstitial mechanisms most investigators (e.g. 60,76-80) completely neglected the role of self-interstitials, which add an additional freedom on the diffusion behavior not accessible when only vacancies are considered. Because of the strong doping dependence of the effective diffusion coefficients the diffusion behavior may change from intrinsic point defect limited to interstitial-diffusion limited within a concentration profile. This leads to complex profile shapes similar to those shown in Figure 11 for the diffusion of zinc in silicon. Additional complications arise due to pn-junction formation (65) and the dependence of effective diffusion coefficients on the pressure of the more volatile group V elements (22,65). In the following, we will only deal with a very limited number of technologically relevant diffusion system such as Cr in GaAs and Zn in GaAs and InP, although many experimental investigations have recently been performed for other systems such as Cd, Mg, Mn, Be, Si and Ag in GaAs, InP and other binary and ternary group III-V semiconductors (24,25,71,77-84).
Chromium acts as a deep acceptor when substitutionally dissolved on gallium sites and is extensively used for fabricating semi-insulating GaAs (90). As long as spatially uniform doping conditions are considered no charge state effects have to be taken into account and the substitutional-interstitial diffusion mechanism of Cr may be described in terms of the Frank-Turnbull mechanism 6 and/or the kick-out mechanism 7. In-diffusion profiles are fairly complex, with a profile shape near the surface resembling a "kick-out" profile (76) and a deep profile part characterized by a larger and constant diffusion coefficient (76,91). Out-diffusion profiles may be characterized by a much slower and also constant diffusivity (91,92-95).
Tentatively, the diffusion behavior may be rationalized by assuming the coexistence of both gallium vacancies and self-interstitials and that D_{eff} of substitutional chromium is given by equation 15. Contrary to the case of gold in silicon here C_{i}^{eq} and C_{S}^{eq} depend on the chromium vapor pressure. Let us assume that for the gallium sublattice D_{I}C_{I}^{ eq} is larger than or in the same order of magnitude as D_{V}C_{V}^{eq} and slower than D_{i}C_{i}^{eq}. Then during in diffusion near the surface D_{eff}^{(I,V)} is lower than D_{eff}^{(i)} so that a kick-out profile results. Because of the (C_{S}^{eq}/C_{S} )^{ 2} term D_{eff}^{(I,V)} increases with depth and finally the in-diffusion of Cr_{S} will be limited by the in-diffusion of Cr_{i} which leads to a constant diffusivity (96)
22. | D_{eff}^{(i)} = (D_{i}C_{i}^{eq} /C_{s}^{eq} ) C_{I}/C_{Í}^{eq} . |
Equation 22 contains the effect of a supersaturation of self-interstit1als which have diffused into the interior. In addition, there will be an increase of Cr_{S} due to the action of dislocations. Contrary to the case of gold in silicon, the chromium plateau concentration in the bulk does not reach C_{s}^{eq} but stays far below this value (76,91). The most likely reason for this behavior lies in the dislocation-climb induced generation of non-equilibrium concentration of intrinsic point defects in the arsenic sublattice which finally limits the attainable C_{S} concentration by dislocation-climb processes. This complication makes it also difficult to distinguish the Frank-Turnbull from the kick-out mechanism by considering the time dependence of the concentration in the bulk, which appears to favor the Frank-Turnbull mechanism (91). For the out-diffusion profiles the prevailing chromium vapor pressure is much lower so that now D_{eff}^{(i)} controls the out-diffusion process since it is likely to be lower than the D_{eff}^{(I,V)} which increases with decreasing chromium vapor pressure.
Zinc diffusion in GaAs is probably the best investigated diffusion process in III-V compounds (10, 22, 23, 62, 63, 71, 97, 100). Zinc acts as a single shallow acceptor in most III-V compounds and can be used for doping purposes. As first suggested by Longini (22) it diffuses via a substitutional-interstitial mechanism. The strong doping dependence of D_{eff} is explained in terms of appropriately charged Zn interstitials in reaction 16 or 17. Isoconcentration diffusion experiments (zinc diffusion in sufficiently highly zinc-doped material) have shown that in reaction 16 or 17 j=1 is fulfilled, which leads to |m|+j = 2 in equation 19.
For normal in-diffusion experiments complicated diffusion profiles have been obtained for high surface concentrations which partly show kink and tail features as indicated in Figure 13 for some early profiles of radioactive zinc (98). Zinc diffusion is extremely fast at high zinc concentrations, due to a large increases in the concentration of highly mobile positively charged Zn interstitals with increasing Zn concentration. Comparison of D_{eff}C_{S}^{eq} with self-diffusion D^{SD} = D_{V}C_{V}^{eq} (and/or D_{I}C_{I}^{eq}) in undoped GaAs shows that generally D_{eff}C_{S}^{eq} is much larger. Based on our earlier discussion on substitutional-interstitial mechanisms this observation may have two explanations:
i) In the first explanation it is assumed that self-diffusion does not depend on the Fermi level. Then such a result may be obtained only in highly dislocated material, since then D_{i}C_{ i}^{eq} » D^{SD} holds, but nevertheless the D_{eff} ^{ (i)} case is fulfilled because the intrinsic point defect concentration is hold close to its equilibrium value due to dislocation climb processes. Since the in-diffusion of zinc is almost independent of the initial dislocation content (71,97) this explanation requires that the in-diffusion of zinc created such a high supersaturation of I_{Ga} or undersaturation of V_{Ga} that interstitial-type dislocation-loops are formed which dominate the generation and/or absorption of intrinsic point defects and make the in-diffusion process almost independent of the original dislocation content (71). This interpretation is in line with various observations of dislocation formation during in-diffusion of Zn in GaAs (97,101).
ii) The second explanation assumes that self-diffusion is carried by charged point defects and is therefore strongly dependent on the Fermi level. Therefore, for determining the diffusion mechanism via the effective diffusion coefficient, D_{eff}C_{ S}^{eq} has to be compared with the self-diffusion coefficient et the doping level of C_{S}^{eq}. This quantity is generally not known but may be much larger than D(n_{i}) provided positively, charged point defects play a role.
It is likely that for high concentration diffusion both non-equilibrium as well as charged po1nt defects have to be taken into account. The best quantitative attempt in doing that has been the early work of Winteler (47) who was the first to investigate the role of charged gallium vacancies and self-interstitials in the diffusion of zinc in GaAs.
Figure 13 Set of radiotracer profiles for zinc in GaAs at 1000°C for the following diffusion times; A:10 min, B:30 min, C:90 min, 0:9h, E:30 h (98). |
Zinc diffusion in InP is governed by a similar concentration dependence as zinc diffusion in GaAs and many other III-V compounds (10,22). An effective diffusion coefficient proportional to the square of the substitutional zinc concentration C_{S} describes the concentration profiles ' reasonably well. Recently it has been found that a solid zinc source leads to profiles characterized by a constant effective diffusivity (102) for about the same surface concentration. It is likely that due to the solid source the intrinsic point defect concentration is decreased so that the intrinsic point defect diffusion governs D_{eff}. The two possible cases leading to a constant D_{eff} have been discussed earlier and are related either to V^{-}_{In} or to I_{In}^{3+}.
The diffusion of zinc into n-type InP leads to abrupt diffusion fronts which are steeper than described by the usual C_{S}^{2} dependence (103,104). The reason for this abrupt profile is simple. The usual C_{S}^{2} concentration dependence is due to
23. | D_{eff}^{(i)} =_{ca.} p^{2} |
where p is given by C_{S}. Equation 23 still holds, but beyond the pn-junction p is dominated by the electron concentration n coming from the n-type background dopant,
24. | p =_{ca.} n_{i}^{2} /n |
which leads to a drastic decrease of D_{eff}^{(i)} near the pn-junction (104). Zinc diffusion into lightly-doped n-type InP gives rise to a second diffusion front (105) which way be characterized by a constant diffusivity and attributed either to a second independent diffusion process (106) or more likely to electric field effects (65).
Beryllium is presently the main acceptor dopant in III-V compounds for integrated circuit applications because of its comparatively high solubility but much lower vapor pressure than zinc (90). Beryllium diffusion in GaAs depends on its concentration and on the arsenic vapor pressure in a similar way as zinc diffusion (24,25). It has therefore been concluded that it diffuses by a substitutional-interstitial mechanism and both the Frank-Turnbull and the kick-out mechanism have been considered (24,25). Widely varying beryllium diffusion coefficients have been reported for comparable beryllium concentrations and temperatures (24,25,107). The reason for this apparent difference lies in the different supply conditions for the beryllium. If there is an external source of beryllium as, e.g., during liquid phase epitaxy growth of beryllium-doped GaAs (107), C_{i} in equation 18 is arch larger than in the case when the beryllium interstitials have to be supplied from substitutionally incorporated beryllium, as in the case of annealing of beryllium-doped GaAs grown by molecular beam epitaxy (25). A similar drastic reduction in the effective diffusivity was observed for zinc in GaAs after the external source of zinc had been removed (10). Based on the effect of beryllium and zinc on the disordering of GaAs/GaAlAs quantum-well-structures it has been concluded that both beryllium and zinc diffuse by the kick-out mechanism (108).
Let me finally mention that the dependence of D_{eff} on the vapor pressure of the more volatile group V component has often been used to obtain more information on diffusion mechanisms in III-V compounds (10,23). As yet it has not been sufficiently recognized that for a D_{eff} dominated by the diffusion of charged point defects the pressure dependence generally is different from that of D_{eff}^{(i)}. E.g., in the case of a constant D_{eff}^{(V)} or D_{eff}^{(I)} no dependence on the vapor pressure is expected.
Fast diffusion in semiconductors spans a wide range of different mechanisms and areas of applications. The present article has dealt with the possible role of interstitial molecules, such as oxygen and hydrogen in silicon. In both cases the presence of such molecules is likely but has not yet been proven experimentally. The Frank-Turnbull and the kick-out mechanism are the two substitutional-interstitial diffusion mechanisms which govern the movement of many predominantly substitutionally dissolved fast diffusing elements in silicon, germanium, the III-V compounds and probably also in the II-VI compounds. Depending on the values of the parameters involved specific concentration profiles may result which allow to distinguish between these mechanisms and also to draw conclusions on the defects, vacancies or self-interstitials, governing self-diffusion. In the case of silicon, the elements Au, Pt, Zn, and S diffuse via the kick-out mechanisms and self-interstitials dominate self-diffusion. Copper in germanium diffuses via the Frank-Turnbul1 mechanism and self-diffusion is governed by vacancies. The profiles in III-V compounds, especially for fast diffusing dopants, are more complex and presently do not allow a quantitative understanding of the prevailing diffusion mechanisms, although it is likely that both types of substitutional-interstitial mechanisms and the co-existence of both vacancies and self-interstitials on both sublatt1ces have to be taken into account.
The author appreciates financial support by WESTINGHOUSE EDUCATIONAL FOUNDATION, MOBIL FOUNDATION, and the DUKE ENDOWMENT during the preparation of this article. Fruitful collaboration with W. Frank, S.-T. Lee, B. Marioton, H. Mehrer, F. Morehead, A. Seeger, N. Stolwijk, T. Y. Tan and numerous other colleagues on the subject of this paper is also acknowledged.
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