Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 2 Part 7 docx

Accordingly, the self-diffusivity Val-of intrinsic silicon is described by [23] Measurements of the tracer self-diffusion coefficient alone are not definitive in establishing the self-interst

23.2 Germanium 401

Fig 23.3 Doping dependence of Ge self-diffusion according to Werner et al [12]

– Further information about the nature and properties of point defects volved in the diffusion process has been deduced from the effect of pressure

in-on diffusiin-on (see Chap 8) Measurements of Ge self-diffusiin-on under sure are reported in [12] The activation volumes are comparatively smalland vary from 0.24 to 0.41 atomic volumes (Ω) as the temperature in-creases from 876 to 1086 K (Fig 23.4) Values of the activation volumefor self-diffusion of gold are shown for comparison [14] These larger val-ues are typical for vacancy-mediated diffusion in close-packed metals Thelower values for Ge support the concept that self-diffusion in Ge occursvia vacancies, which are more relaxed or ‘spread-out’ than in close-packedmaterials The positive sign very likely excludes self-interstitials as the de-fects responsible for Ge self-diffusion For an interstitialcy mechanism theactivation volume should be negative (see Chap 8)

pres-The experimental activation volume increases with temperature This crease can be attributed to the fact that neutral and negatively chargedvacancies with differnet activation volumes contribute to self-diffusion.The activation volume of the neutral vacancy contribution

in-(∆V V0 = 0.56 Ω) is larger than that of the negatively charged vacancy (∆V V − = 0.28 Ω) [12] The experimental activation volume is an aver-

age value weighted with the relative contributions of the two types ofvacancies to the total self-diffusivity;

402 23 Self-diffusion in Elemental Semiconductors

Fig 23.4. Activation volumes of Ge self-diffusion according to Werner

et al [12] For comparison activation volumes of Au self-diffusion are alsoshown [14]

23.3 Silicon

Natural silicon has three stable isotopes with the following abundances:92.2 % 28Si, 4.7 % 29Si, and 3.1 % 30Si Self-diffusion studies on silicon areaggravated by the fact that the only accessible radioisotope,31Si, has a half-life of 2.6 hours This limits the time for a diffusion experiment Nevertheless,

in view of the great importance of silicon as base material for microelectronicdevices several attempts have been made to apply tracer methods using eitherthe radioisotope 31Si, the stable enriched isotope 30Si, or isotopically con-trolled heterostructures Ghoshtagore [15] evaporated enriched30Si layersonto Si wafers, performed neutron activation of the sample after diffusion an-nealing and subsequent chemical sectioning for profile analysis Peart [16]evaporated the radiotracer31Si and utilised mechanical sectioning Fairfieldand Masters[17] studied 31Si diffusion in intrinsic and doped Si and sec-tioned the samples by chemical etching Mayer et al [18] and Hettich

et al [19] sputter-deposited thin layers of neutron activated Si ing 31Si) and used sputter sectioning for profile analysis Hirvonen andAnttila[20] and Demond et al [21] implanted a layer of30Si and studiedits diffusional broadening by a proton beam utilising the resonant nuclear re-action30Si(p, γ)31Si Kalinowski and Seguin [22] evaporated layers of30Siand studied in-diffusion profiles by SIMS The study by Bracht et al [23]employed stable silicon isotope heterostructures with highly enriched 28Si

(contain-23.3 Silicon 403

Fig 23.5 Self-diffusion coefficients of Si measured by tracer methods [15–23]

layers The heterostructures were grown by chemical vapour deposition onnatural Si wafers Diffusion profiles of29Si and30Si isotopes were determined

by SIMS

The data of the various authors are shown in Fig 23.5 The results areless consistent than those on Ge self-diffusion (see Fig 23.2) and far lessconsistent than those on self-diffusion in metallic elements (see, e.g., [24]).This is surprising since in most of the above mentioned studies virtuallydislocation-free Si single crystals of high purity were used The reason forthese discrepancies are not completely clear Oxygen is one of the main im-purities in both Czochralski-grown and float-zone single crystals It is well-known that oxidation at the surface or oxide formation or dissolution causesdeviations of intrinsic point defects from their thermal equilibrium concen-trations Implantation damage may be an additional reason for discrepancies

in those experiments where 30Si was implanted prior to diffusion All earlierexperiments either suffer from the short half-life of31Si or from the naturalabundance of stable30Si, when enriched30Si was used as tracer

404 23 Self-diffusion in Elemental Semiconductors

Drawbacks of the earlier experiments have been avoided in the alreadymentioned study with isotope heterostructures by Bracht et al [23] Nei-ther a short-lived radioisotope nor the background of the natural30Si limitedthe experiment An influence of the surface can very likely be also excluded,since diffusion is observed near an interface of the isotope heterostructure.Even at the lowest temperature no doping effects could be detected Thus, it

is believed that these data reflect intrinsic behaviour of Si self-diffusion ues of the self-diffusion coefficient cover about seven orders of magnitude andthe widest temperature range of all studies Accordingly, the self-diffusivity

Val-of intrinsic silicon is described by [23]

Measurements of the tracer self-diffusion coefficient alone are not definitive

in establishing the self-interstitial and vacancy contributions to self-diffusion.Additional information is needed and can be obtained, for example, fromthe analysis of self- and foreign-atom diffusion experiments involving non-equilibrium concentrations of intrinsic point defects Below, we summarisethe main conclusions that have been drawn from such experiments concerning

Si self-diffusion [25]

The native point defect which predominantly mediates Si self-diffusion isthe self-interstitial This conclusion is consistent with experimental data for

the transport product C I eq D I deduced from diffusion of hybrid foreign

ele-ments (see Chap 25) The self-interstitial contribution, C I eq D I, is well-knownfrom diffusion studies of hybrid foreign atom diffusers (Au: [27], Zn: [28])

Therefore, one can extract the vacancy term, C V eq D V in Eq (23.7), fromthe total self-diffusivity From an analysis of Zn diffusion in Si Bracht

The agreement between D ∗ and f I C eq

I D I + f V C V eq D V (taking into accountthe correlation factors) implies that self-diffusion is mediated by both self-

interstitials and vacancies C I eq D I equals C V eq D V at about 890◦C and not

at temperatures between 1000 and 1100◦C as had been assumed earlier [4,

5] At temperatures above 890◦C the self-interstitial contribution

domi-nates, whereas at lower temperatures vacancy-mediated diffusion dominates

23.3 Silicon 405

Fig 23.6 Si self-diffusion coefficients (symbols) compared with the self-interstitial

and vacancy transport products, C I eq D I and C V eq D V, according to Bracht

H V SD = H V F + H V M = 4.14 eV and S V SD = S V F + S V M ≈ 5.5 kB. (23.18)Theoretical calculations of intrinsic point defect properties are in good agree-ment with these values Tang et al [29] obtained from their tight-binding

molecular dynamic studies H I SD = (5.18 ± 0.2) eV, H SD

V = (4.07 ± 0.2) eV, and S SD

I = 14.3 kB The first principles calculations of Bl¨ochl et al.[26]

yield S F

V = (5± 2) kB leaving for S M

V values between 1 and 2 kB

In conclusion, we can state that experiments and theory suggest that Siself-diffusion is mediated by simultaneous contributions of self-interstitials

406 23 Self-diffusion in Elemental Semiconductors

and vacancies Self-interstitials dominate at high temperatures whereas cancies take over at lower temperatures

va-The doping dependence of Si self-diffusion [5] allows the conclusion thatneutral as well as positively and negatively charged defects are involved inself-diffusion However, the data are not accurate enough to determine theinidvidual terms in Eqs (23.5) and (23.6) Since the total tracer diffusivity

as well as the transport products consist of several terms, Eqs (23.15) and(23.16) can only be approximations holding for a limited temperature range

References

1 S.M Sze, Physics of Semiconductor Devices, Wiley, New York, 1967

2 S.T Pantelides, Defect and Diffusion Forum 75, 149 (1991)

3 U G¨osele, T.Y Tan, Defect and Diffusion Forum 83, 189 (1992)

4 A Seeger, K.P Chik, Diffusion Mechanisms and Point Defects in Silicon and

Germanium, Phys Stat Sol 29, 455–542 (1968)

5 W Frank, U G¨osele, H Mehrer, A Seeger, Diffusion in Silicon and

Germa-nium, in: Diffusion in Crystalline Solids, G.E Murch, A.S Nowick (Eds.),

Academic Press, 1984

6 T.Y Tan, U G¨osele, Diffusion in Semiconductors, in: Diffusion in Condensed

Matter – Methods, Materials, Models, P Heitjans, J K¨arger (Eds.), Verlag, 2005

Springer-7 H Letaw Jr., W.M Portnoy, L Slifkin, Phys Rev 102, 636 (1956)

8 M.W Valenta, C Ramsastry, Phys Rev 106, 73 (1957)

9 H Widmer, G.R Gunter-Mohr, Helv Phys Acta 34, 635 (1961)

10 R Campbell, Phys Rev B 12, 2318 (1975)

11 G Vogel, G Hettich, H Mehrer, J Phys C 16, 6197 (1983)

12 M Werner, H Mehrer, H.D Hochheimer, Phys Rev B 32, 3930 (1985)

13 H.D Fuchs, W Walukiewicz, E.E Haller, W Dondl, R Schorer, G Abstreiter,

A.I Rudnev, A.V Tikhomirov, V.I Oshogin, Phys Rev B 51, 16817 (1995)

14 M Werner, H Mehrer, in: DIMETA-82, Diffusion in Metals and Alloys, F.J.

Kedves, D.L Beke (Eds.), Trans Tech Publications, Diffusion and Defect graph Series No 7, 392 (1983)

Mono-15 R.N Ghostagore, Phys Rev Lett 16, 890 (1966)

16 R.F Peart, Phys Status Solidi 15, K119 (1966)

17 J.M Fairfield, B.J Masters, J Appl Phys 38, 3148 (1967)

18 H.J Mayer, H Mehrer, K Maier, Inst Phys Conf Series 31, 186 (1977)

19 G Hettich, H Mehrer, K Maier, Inst Phys Conf Series 46, 500 (1979)

20 J Hirvonen, A Anttila, Appl Phys Lett 35, 703 (1979)

21 F.J Demond, S Kalbitzer, H Mannsperger, H Damjantschitsch, Phys Lett

A 93, 503 (1983)

22 L Kalinowski, R Seguin, Appl Phys Lett 35, 211 (1979); Erratum: Appl Phys Lett 36, 171 (1980)

23 H Bracht, E.E Haller, R Clark-Phelps, Phys Rev Lett 81, 393 (1998)

24 H Mehrer, N Stolica, N.A Stolwijk, Self-diffusion in Solid Metallic Elements, Chap 2 in: Diffusion in Solid Metals and Alloys, H Mehrer (Vol.Ed.), Landolt-

B¨ornstein, New Series, Group III: Crystal and Solid State Physics, Vol 26,Springer-Verlag, 1990, p.32

References 407

25 H Bracht, MRS Bulletin, June 2000, 22

26 P.E Bl¨ochl, E Smargiassi, R Car, D.B Laks, W Andreoni, S.T Pantelides,

Phys Rev Lett 70, 2435 (1993)

27 N.A Stolwijk, B Schuster, J H¨olzl, H Mehrer, W Frank, Physica

(Amster-dam) 116 B&C, 335 (1983)

28 H Bracht, N.A Stolwijk, H Mehrer, Phys Rev B 52, 16542 (1995)

29 M Tang, L Colombo, T Diaz de la Rubbia, Phys Rev B 55, 14279 (1997)

24 Foreign-Atom Diffusion

in Silicon and Germanium

Diffusion of foreign atoms in Si and Ge is very important from a technologicalpoint of view Due to its complexity it provides a challenge from a scientificpoint of view as well Group-III elements B, Al, Ga and group-V elements P,

As, Sb are a special class of foreign elements known as dopants Dopants are

easily ionised and act as donors or acceptors Their solubility is fairly highcompared to most other foreign elements except group-IV elements Diffu-sion of dopants plays a vital rˆole in diffusion-doping to create p-n junctions

of microelectronic devices Diffusion also controls the incorporation of wanted’ foreign atoms, e.g., of the metal atoms Fe, Ni, and Cu during thermalannealing treatments of device fabrication A detailed understanding of thediffusion behaviour of unwanted foreign atoms is of technological significance

‘un-to keep the contamination of electronic devices during processing ‘un-to a less state Oxygen diffusion and growth of SiO2 precipitates play a crucial

harm-rˆole in gettering processes of unwanted foreign elements Other foreign atomslike Au in Si are used for tuning the minority-carrier lifetime

Dopants are incorporated in substitutional sites of the host lattice, some

foreign elements dissolve in interstitial sites only, others are hybrid foreign

elements, which are dissolved on substitutional and on interstitial sites Thevery high mobility of the interstitial fraction can dominate diffusion of hy-brid elements In addition, the interstitial-substitutional exchange reactions –either the dissociative or the kick-out reaction – can lead to non-Fickian dif-fusion profiles of hybrid diffusers

The diffusion behaviour of foreign elements is largely determined by thetype of solution, i.e whether they are located at substitutional or intersti-tial sites, or a mixture of both In what follows, we consider first solubilities

of foreign elements and their site preference Then, we give a brief review

of the diffusion of foreign elements in Ge and Si and classify them ing to their site occupancy and diffusivity The theoretical framework of therelatively complex diffusion patterns of hybrid solutes involving interstitial-substitutional exchange reactions is postponed to Chapt 25

accord-24.1 Solubility and Site Occupancy

The solubility of a foreign element is the maximum concentration which can

be incorporated in the host solid without forming a new phase Solid

sol-410 24 Foreign-Atom Diffusion in Silicon and Germanium

ubilities are temperature-dependent as represented by the solvus or soliduslines of the phase diagram The solubility limit is defined with respect to

a second phase For most foreign elements in solid Ge or Si at high atures equilibrium is achieved with the liquid phase At lower temperaturesthe reference phase usually is the solid foreign element or a compound of theforeign element When the foreign atom is volatile, the saturated host crystal

temper-is in equilibrium with a vapour In thtemper-is case, the solubility depends not only

on the temperature but also on the vapour pressure

For solid-solid equilibria and for solid-vapour equilibria the temperaturedependence of the solubility limit is usually described by an Arrhenius rela-tion containing a solution enthalpy If a solid-liquid equilibrium is involved,the behaviour is more complex The right-hand side of Fig 24.1 shows thenormal variation of a solubility giving rise to a maximum at the eutectic tem-

perature A frequently encountered case is the so-called retrograde solubility,

illustrated on the left-hand side of Fig 24.1 This phenomenon implies thatthe maximum solubility is achieved at a temperature which lies below themelting temperature of the host crystal but above the eutectic temperature.Below this maximum temperature, the solubility can often but not always beapproximated by an Arrhenius relation

Solubility data of foreign elements have been collected for Si by Schulz [6],for Ge by Stolwijk [7] and have been updated for Ge by Stolwijk andBracht [8] and for Si by Bracht and Stolwijk [9] Depending on theforeign element, the solubility can vary over orders of magnitude: B, P, As in

Si and Al, Ga, Sn in Ge can be incorporated to several percent into the hostcrystals Other dopant elements, such as Ga, Sb, Li in Si and As and Sb in Ge

Fig 24.1 Schematic phase diagram of a semiconductor and a foreign element

(or a compound of a foreign element) illustrating the phenomenon of retrogradesolubility

24.1 Solubility and Site Occupancy 411have maximum solubilities in the range of 10−3atomic fractions Noble metal

impurities, iron group impurities, nickel group impurities, cobalt group rities, and Zn have very low solubilities in the range of 10−6atomic fractions.Generally speaking, every foreign atom A may occupy both substitutional

impu-A s and interstitial A isites in the lattice The equilibrium solubility of either

configuration, C eq

s and C i eq, is determined by the pertinent enthalpy andentropy of solution, which refers to some external phase of the foreign atom.Dealing with equilibrium solubilities one should be aware of the the followingfeatures [10]:

– For most foreign atoms one atomic configuration dominates in the well

accessible temperature range of diffusion studies between about 2/3 T m and 0.9 T m:

– Dopants are dissolved in substitutional sites and diffuse with the aid

of intrinsic point defects

– Foreign atoms with interstitial site preference (C i eq  C eq

s ) diffusevia a direct interstitial mechanism Examples are group-I and somegroup-VIII elements

– Foreign atoms with substitutional site preference (C i eq  C eq

s ), butsome minor interstitial fraction are interstitial-substitutional exchangediffusers (hybrid solutes) Examples are some noble metals and furtherelements mainly from neighbouring groups of the noble metals.– The equilibrium solubilities on substitutional and interstitial sites, C eq

s and C i eq, depend sensitively on the solute, the solvent, and on tempera-

comitant phenomenon is that the ratio C i eq /C s eq may change with

back-ground doping, because of the different electronic structure of A i and A s.Dramatic effects have been observed for 3d transition elements in Si [11,12] For example, in intrinsic Si the predominant species of cobalt, Coi, is

a donor By contrast, in heavily phosphorous doped Si acceptor-like Cosbecomes more abundant

In what follows, we confine ourselves to intrinsic conditions This plies that the intrinsic carrier concentration at the diffusion temperature,

im-n i (T ), exceeds the maximum concentration of electrically active foreign

atoms This is usually fulfilled for host crystals having doping levels nothigher than 1018 cm−3.1

1 In shallow dopant diffusion experiments on, e.g., GaAs the relatively small n

i (T )

in conjunction with the high solubility of dopants causes a shift of the Fermi level

412 24 Foreign-Atom Diffusion in Silicon and Germanium

Fig 24.2 Equilibrium solubilities of substitutional (open symbols) and interstitial

(filled symbols) Cu in Ge according to [10]

– As already mentioned, when the diffuser is supplied from a vapour phasethe solubilities depend on the vapour pressure However, according to

the mass action law the ratio C i eq /C eq

s will not change as long as both

A i and A s are dissolved as isolated atoms Pairs and larger clusters offoreign atoms are only formed upon slow cooling Despite the fixed value

of C i eq /C eq

s at any temperature, the absolute magnitude of the A i and A s

concentration may influence the diffusion behaviour This is the case fordiffusion of hybrid elements, which diffuse via the kick-out and/or via thedissociative mechanism treated in detail in Chap 25

24.2 Diffusivities and Diffusion Modes

Figures 24.3 and 24.4 provide overviews of the diffusivities of technologicallyimportant foreign elements in Ge and Si, respectively In both figures self-away from the mid of the band gap at the diffusion temperature In an extreme

case like diffusion of the acceptor type dopant Zn in GaAs (C s eq  n i (T )), the

Zn concentration equals the hole density This is called self-doping

24.2 Diffusivities and Diffusion Modes 413

Fig 24.3 Diffusion of foreign atoms in Ge compared with Ge self-diffusion

ac-cording to [3]: Solid lines represent diffusion of elements that are incorporated on substitutional sites and diffuse via the vacancy mechanism Long-dashed lines rep-

resent diffusion of hybrid elements, which are mainly dissolved on substitutionalsites; their diffusion proceeds by the dissociative mechanism via a minor fraction

in an interstitial configuration (Au, Ag, Ni, Cu) The short-dashed lines represent

diffusion of elements that diffuse via a direct interstitial mechanism (H, Li [16, 17])

The short-dashed line on top shows the diffusivity deduced for interstitial Cu

diffusion and diffusion of substitutional atoms (mainly dopants and for Sialso C) are represented by solid lines; diffusivities of foreign elements withinterstitial site preference and direct interstitial diffusion are represented byshort-dashed lines; hybrid diffusing elements are represented by long-dashedlines Outstanding features of both figures are the grouping of the diffusivities

of dopant elements in a range around or moderately higher than self-diffusionand the fact that the diffusivities of interstitial and hybrid foreign atoms areseveral orders of magnitude higher A comprehensive and critical collection