COMPUTATIONAL STUDY OF DIFFUSIVITIES IN DIAMOND Ge-Si ALLOYS

A variety of diffusivities in Ge-Si alloys available in the literature were critically reviewed. On the basis of the critically reviewed literature data, the diffusion parameters for self diffusivities and impurity diffusivities in diamond Ge-Si alloys were determined by considering the diffusion mechanism. A phenomenological treatment of the diffusivities in Ge-Si alloys were conducted. The finally obtained atomic mobilities can reproduce most of the diffusivities in diamond Ge-Si alloys as well as the concentration profiles of Ge-Si binary diffusion couples. In addition, the Manning modification on Darken Equation in diamond structure was also tested by using the presently obtained atomic mobilities.


Introduction
Ge-Si alloys represent a very important type of semiconductor materials in microelectronic technology, and thus widely used as semiconductors in integrated circuits for heterojunction bipolar transistors, or as high performance complementary metal oxide semiconductor (CMOS) applications [1].
Since the strained Si/SiGe heterostructures offer higher electron and hole mobilities than bulk Si, Ge-Si alloys have attracted extensive attention recently [2][3].It is well known that the space distributions of Ge, Si and possible dopants are of importance for the electrical behavior of SiGe-based device, which mainly depend on their diffusivity information.Moreover, diffusion coefficient is a determinant of the annealing time, the designated dimension of electronic device, and even the control of the charge carrier mobility and band structure of Si-Ge base electronic devices.Thus, a comprehensive knowledge of various diffusivities in solid Ge-Si alloys (diamond structure) is in urgent need if one wants to improve the electrical properties of SiGebased device or even design new ones.
In order to acquire a full description of all kinds of diffusivities over the entire temperature and composition ranges, the recently developed DICTRA (DIffusion-Controlled TRAnsformation) software in the framework of CALPHAD [4,5] (CALculation of PHAse Diagram) method is an appropriate underlying tool to handle it.The so-called atomic mobility of each component is assessed based on a variety of reliable diffusivities, and stored in DICTRA type database, from which various diffusivities can be computed over the entire temperature and composition ranges, and various phase transformation processes can be simulated.The quality of atomic mobility strongly depends on the amount and the reliability of diffusivities.Though a certain amount of experimental diffusivities are available in the literature for diamondstructured Ge-Si alloys, the experimental data from difference sources are usually not mutually consistent with each other.With the newly developed experimental techniques, such as secondary ion mass spectrometry (SIMS) [6], Raman spectroscopy [7], etc., more reliable experimental data are now available and the diffusivity measurement has been extended to low temperature.Therefore, a critical evaluation of all the available experimental diffusivities and their measurement techniques is necessary.Meanwhile, there exists only two pieces of information on assessment of atomic mobility in systems with diamond structure (only for pure Si by Zhang et al. [8] and Tang et al. [9]).Besides, whether the Darken equation or the Darken-Manning equation is applicable in systems with diamond structure still needs validation.
Consequently, the major aims of the present work are: (i) to critically review all the available diffusivities in Ge-Si alloys with diamond structure, (ii) to assess the atomic mobilities of Ge and Si in diamondstructured Ge-Si alloys based on the critically reviewed diffusivities, (iii) to validate both Darken and Darken-Manning equations in diamond Ge-Si alloys, and (iv) to verify the presently obtained atomic mobilities by comparing the simulated diffusion profiles with the experimental ones in Ge-Si thin film diffusion couples.

Models for diffusivities
Diffusion in Ge-Si alloys is mediated by the migration of intrinsic piont defects, such as vacancies and interstitials.It is generally accepted that both vacancies and interstitals take part in the diffusion process in Si [7].On the other hand, vacancy mechanism prevails over the whole temperature range for the self-diffusion in Ge [10].There also exist evidences [6,11] that the diffusion mechanism for Si diffusion in Ge follows the S.L. Cui / JMM 48 (2) B (2012) 227 -249 diffusion mechanism of Ge self-diffusion, while that for Ge diffusion in Si follows the diffusion mechanism of Si self diffusion.According to Strohm et al. [6], the diffusion in Si y Ge 1-y alloys is mediated by vacancies for 0<y<0.65,while for 0.65<y<1, either interstitial or vacancy plays a pivotal role in the diffusion process in SiGe alloys.As both vacancy and interstitial (neglecting other mechanisms, e.g. the direct exchange mechanism) may contribute to the self diffusion process, self or tracer diffusion coefficients can be written as [12]: (1) where (X=I or V) represents a contribution of the interstitialcy or vacancy to tracer diffusivities.(X=I or V) is the correlation factor for interstitialcy-or vacancy-mediated diffusion in diamond structure.equals to 0.5 [13] for vacancy mechanism according to statistical diffusion theory and 0.53 or 0.46 [14] to atomistic study.While is 0.73 (statistical diffusion theory) [15] and 0.59 or 0.69 (atomistic study) [14].and (X=I or V) are the equilibrium concentration and the diffusion coefficients of interstitialcy or vacancy, respectively.Intrinsic diffusivities correlates to tracer diffusivities by [16]: (2) and (3) where is molar fraction of A, is the activity coefficient of A and is a constant for diamond structure, and equals to 2 [16]. is thermodynamic factor and is the so called 'vacancy wind term'.Manning corrected the Darken relation which relates the tracer and the chemical diffusivities by the vacancy wind factor S [16]: (4) where S is given by (5) According to Andersson and Ågren [17], the atomic mobility of element B, , can be expressed as: (6) where is the frequency factor, the activation enthalpy, R the gas constant, T the temperature in Kelvin and a factor taking into account the effect of ferromagnetic contribution to the diffusivities.For diamond structure the ferromagnetic contribution can be neglected, and then the atomic mobility parameters in the DICTRA [18] notation, and , can be grouped into one parameter, i.e.
. The composition dependency of can be represented with the Redlich-Kister polynomial [19]: (7) where is the value of for B in pure i, while are the binary interaction parameters.
The tracer diffusivities relates to the atomic mobilities via the Einstein relation: . Neglecting Manning correction, the interdiffusion coefficients defined with n as solvent are correlated to the atomic mobilities by [17]: (8) where is the Kroneker delta ( = 1, if i = j, otherwise = 0 ) and is the chemical potential of element i.

Review of literature data
In light of the technical importance of Ge-Si alloys in semiconductor industry, a host of experimental investigations were conducted for the sake of the understanding of its kinetic properties, including diffusion mechanism, atomic transportation velocity, the contribution of each type of diffusion mechanisms towards diffusion coefficients, etc.As a result, there exist a considerable large amount of experimental diffusivities in the literature, which can be categorized into self diffusivities, impurity diffusivities, tracer diffusivities and interdiffusivities.The former group contains various contributions.Master and Fairfield [20] investigated the self diffusivities of 31 Si in single crystal Si between 1373 K and 1573 K via the determination of the concentration profile by an anodization-etching technique combined with liquid scintillation radioassay.Using evaporated 30 Si source, the self diffusivities of Si at 1451 and 1573 K were determined by Ghoshtagore [21] in pand n-type single crystals Si using chemical sectioning.Si self diffusivities in intrinsic Si were investigated by Peart [22] using the radio isotope 31 Si as tracer in the temperature range of 1473 to 1673 K. Just after that, Fairfield and Masters [23] studied the diffusion of 31 Si into Si single crystals within the range of 1373 and 1573 K and monovacancy mechanism was proposed for Si self diffusion.The loop annealing technique was applied to the study of self diffusion in silicon over a wide range of temperature from 1243 to 1343 K by Sanders and Dobson [24], who found that the diffusion coefficients decreases as the concentration of n-type dopant decreases and the concentration of p-type dopant increases.With the aid of a sputter-sectioning technique, Mayer et al. [25] determined the self-diffusion coefficients in high purity ptype Si at 1593-1873 K.The (p, γ) resonance broadening method was utilized by Hirvonen and Anttila [26] to measure the self diffusivities of Si in the temperature range of 1173 and 1373 K.At the meantime, self diffusivities of Si in intrinsic and doped Si were investigated by Hettich et al. [27] via radiotracer microsectioning techniques.Kalinowski and Seguin [28] investigated the self diffusivities of 30 Si in intrinsic Si by means of an ion-analyzer technique and obtained the temperature dependence of the diffusion coefficients in the range from 1258 to 1448 K.By using the ion implantation for preparation and (p, γ) reaction for analysis of 30 Si profiles, the Si self diffusion was studied in the temperature range of 1203 and 1473 K by Demond et al. [29].Bracht et al. [30] measured the self diffusion of silicon in highly isotopically enriched 28 Si layers between 1128 and 1661 K with the profiles of 29 Si and 30 Si determined by SIMS.Their experiments indicate that self interstitials dominate the Si self diffusion process.Ural et al. [31,32] conducted experimental investigation on the self diffusivities of Si using epitaxially grown isotopically enriched Si and SIMS technique in the temperature range between 1073 and 1373 K. Strohm et al. [6] utilized the 31 Si ions produced by the ion guide isotope separator on-line technique to measure the self diffusivities of Si single crystal.By using isotopically pure 30 Si eptiaxical layers as a diffusion source to bulk Si substrates coupled with SIMS technique, Nakabayashi et al. [33,34] obtained the Si self diffusion coefficients in intrinsic single crystal bulk Si at 1143-1343 K. Using the highly isotopically enriched 30 Si eptiaxical layers as a diffusion source to bulk and epitaxial layers Si, with the concentration profiles determined by SIMS, Aid et al. [35] determined the diffusivities of 30 Si in the temperature range of 1146-1573 K. Shimizu et al. [7,36] overcame the handicap of SIMS in measuring the diffusivities of Si at low temperature by detection of the very small diffusion length in isotope super-lattices of Si via Raman spectroscopy.Their determined self diffusion coefficients of Si at temperatures 908-1148 K show low activation energy of 3.6 eV which is lower than that of the high temperature data and in agreement with the theoretical prediction for the vacancy mediated diffusion.

Self diffusivities of Si
Another type of self diffusivities measurement is based on metal elements which diffuse in Si mainly via kick-out mechanism.Stolwijk et al. [37] investigated the in-and out-diffusion of Au in Si by means of a neutron activation analysis combined with mechanical sectioning or by the spreading resistance technique.The interstitialcy contribution to the Si selfdiffusion coefficient was determined at the temperature range of 1073-1371 K from the Au solubility and diffusion measurement.By utilizing the deep-level transient spectroscopy and diode capacitance/reverse bias measurements, the in-diffusion of Pt in n-type Si from a platinum silicide source was investigated by Mantovani et al. [38] at 973-1073 K.The result indicated that the Pt diffusion in Si via kick out mechanism, while the self diffusion of Si mainly via interstitialcy mechanism at the experimental temperature range.Perrett et al. [39] studied the interstitialcy contribution of Si self diffusion via measurement of the kick out type diffusion of Zn in Si at 1262 K. Subsequently, Grünebaum et al. [40] conducted experimental investigations of Zn diffusion in dislocation free and plastically deformed Si with the same measurement techniques.They further verified the kickout mechanism of Zn diffusion in Si and determined the interstitialcy diffusivities of Si at 1175-1473 K. Likewise, Hauber et al. [41] derived the interstitialcy component of Si self diffusivities between 1273 and 1573 K by measurement of diffusivities and solubility of Pt in dislocation-free Si using both neutron-activation analysis in combination with mechanical serial sectioning and spreading-resistance measurement.With similar approach, Bracht et al. [12] investigated the diffusion of Zn in dislocation free Si between 1143 and 1481 K and obtained the interstitialcy component of Si self diffusivities.

Self-diffusivities of Ge
In contrast to Si, Ge has more stable radioisotope, i.e. 71 Ge.The self diffusion measurement using the radioactive tracer method is much more applicable with respect to Si.As a result, the available experimental data from different sources are generally in good agreement.A brief summary of these data are listed in Table 2. Letaw et al. [42,43] reported two pieces of information on the self diffusivities in Ge using 71 Ge as tracer.Their data cover the temperature range from 1039 to 1201 K. Valenta and Ramasastry [44] determined the self diffusivities of Ge for intrinsic n-type and p-type single crystal Ge at various temperatures by utilizing 61 Ge.Widmer and Gunther-Mohr [45] measured the Ge self diffusivities in the temperature range of 993 to 1027 K by way of a residual activity technique with radioactive 71 Ge as tracer.Subsequently, Widmer [46] studied the 71 Ge diffusion in intrinsic Ge single crystals at the temperature near 1013 K using the same measurement method.Campbell [47] performed experimental investigation of Ge self diffusion by simultaneously diffusing the isotopes 77 Ge and 71 Ge into single crystal intrinsic Ge at 1173 and 1198 K with a sectioning and counting method.Using 71 Ge as radioisotope and a sputtering technique for serial sectioning, Vogel et al. [48] investigated the self diffusivities in intrinsic Ge single crystals in the temperature range of 822-1163 K.The tracer diffusion coefficients for 71 Ge were measured in Ge single crystals as a function of pressure, temperature, and doping in the temperature range of 808-1177 K by Werner and Mehrer [49] with the ion beam sputtering for microsectioning.By using isotope heterosturctures 70 Ge and 74 Ge as tracer, with SIMS measured the diffusion profiles; Fuchs et al. [50,51] determined the self diffusivities of Ge at the temperature range of 816 to 913 K. Silveira et al. [52] studied the self diffusion of Ge using the Roman scattering by optical phonons in isotopic ( 70 Ge) n ( 74 Ge) m super-lattices at 773 K.The obtained diffusion coefficient agrees well with that from the previously reported ones.Almazouzi et al. [53] determined the bulk and grain boundary diffusion of Ge in Ge using radioactive 68 Ge as tracer in conjunction with mechanical sectioning at several temperatures.Strohm et al. [6,54] conducted two pieces of experiments in the measurement of the self diffusion coefficients in single crystals Ge.The self diffusivities of 71 Ge in relaxed Ge epitaxial layers were measured at 1167 to 1536 K by means of a radioactive technique combined with ion beam sputtering.By neutron reflectometry from the decay of the first and third order Bragg peak, Hüger et al. [10] investigated the self diffusion in intrinsic a RAT=the residual activity technique, SCM=the sectioning and counting method, IBSM=the ion beam sputtering for microsectioning, SIMS=secondary ion mass spectrometry, SRT=the spreading resistance technique.b Indicates whether the data are used or not used in the atomic mobility assessment: ■, used; □, used but with low weight;+, not used but considered as reliable data.
single crystalline Ge between 702 and 869 K using 70 Ge/ nat Ge isotope multilayer structures.Single vacancies were considered as the main diffusion mechanism in Ge over the whole temperature range.Laitinen et al. [55] studied the self diffusion coefficients implanted 71 Ge in bulk Ge at the temperature range 813-1123 K by means of a modified radiotracer technique.By measurement of the diffusion profiles and the solubility of Cu in Ge via the spreading resistance technique, self diffusivities of Ge were calculated between 850 and 1200 K by Stolwijk et al. [56].

Impurity diffusivities of Ge in Si
Several groups of authors contribute to the measurement of the diffusion coefficients of Ge in Si as summarized in Table 3. Petrov et al. [57] obtained the diffusivities of 71 Ge in polycrystalline p-type Si by way of a residual activity measurement method at 1423 to 1623 K. Subsequently, McVay and DuCharme [58] measured the diffusion of Ge in single crystalline Si by utilizing the radioactive tracer 71 Ge and a thin sectioning technique at various temperatures.Impurity diffusivities of Ge in intrinsic and doped Si

Tracer diffusivities
A brief summary of the tracer diffusivities available in the literature is in Table 4.The measurement of the tracer diffusivities in Ge-Si alloys started from McVay and DuCharme [69] who determined the tracer diffusivities of 71 Ge in polycrystalline Ge-Si alloys containing 22.4 at.%Ge, 30.8 at.%Ge, 55.4 at.%Ge and 77.7 at.%Ge.Their data revealed an abrupt decrease of the activation energy as the composition of Ge increases and reaches a platform at about 40 at.%Ge.More than three decades later, Zangenberg et al. [63] determined the diffusion coefficients and activation energies for 72 Ge diffusion in strain relaxed Si 1-x Ge x ( x= 0.1, 0.2, 0.3, 0.4 and 0.5) by means of SIMS technique in the temperature range of 1123 to 1323 K.According to their study, the activation energy for Ge diffusion is 4.7 eV for Si 1-x Ge x alloys with x = 0~0.1, and 3.7 to 4.0 eV for alloys with x=0.2~0.4.Strohm et al. [6,54] also contributed to the measurement of tracer diffusivities in single crystalline Ge-Si alloys.The diffusivities of 71 Ge were determined in a wide composition range of 923-1536 K via a radioactive technique coupled with ion beam sputtering.Similarly, the diffusivities of 31 Si in Si 1-x Ge x (x = 0.2 and 0.5) alloys were measured.Their research indicated that the activation energies for 71 Ge tracer diffusivities slightly decrease from 4.6 eV to approximately 4.25 eV firstly, and then shoot up to about 5.5 eV for Ge self diffusivities.This trend is in contradiction with that reported by McVay and DuCharme [69].While the preexponential factors for 71 Ge tracer diffusivities rise considerably from 2×10 -4 to 4×10 -2 m 2 /s.Tracer diffusivities of implanted 31 Si and 71 Ge in relaxed Si 0.20 Ge 0.80 layers were investigated by Laitinen et al. [55] in the temperature range of 813-1123 K by way of a modified radioactive technique.Experiments on the diffusion of Si and Ge in Si 1-x Ge x (x = 0.05 and 0.25) isotope heterostructures between 1143 and 1543 K were conducted by Kube et al. [64] with the concentration profiles determined by SIMS.

Inter-diffusivities
Early measurement of interdiffusion coefficients in Si/Ge amorphous multilayer films was conducted by Prokes and Spaepen [70] in the temperature range of 550-630 K.The interdiffusivities were determined by measuring the intensity of the X-ray satellite arising from the modulation as a function of annealing time.
A systematically measurement of Ge-Si interdiffusivities for Ge concentration between 0.075 and 0.192 over the temperature range of 1043~1143 K was conducted by Aubertine and Mcintyre [71].Holländer et al. [72] studied the thermal interdiffusion in both asymmetrically and symmetrically strained Si/Si 1-x Ge x superlattices with Ge concentration between x=0.2 and 0.70 using Rutherford backscattering spectrometry in the temperature range between 1173 to 1398 K. Recently, the interdiffusion coefficients were determined by Gavelle et al. [73,74] using the Boltzmann-Matano method from the concentration profiles of Ge derived by SIMS measurement in the temperature range of 1023-1173 K. Xia et al. [75][76][77] derived the diffusion coefficients in epitaxial strained Si/Si 1-y Ge y /strained Si/relaxed Si 1-x0 Ge x0 heterostructures for Ge concentration between 0 and 0.56 over the temperature of 1043 to 1193 K by using the Boltzmann-Matano method.The concentration profiles were determined via SIMS technique.Meduňa et al. [78][79][80] contributed to the measurement of interdifusivities of GeSi alloys containing 0.25, 0.50, 0.70 and 0.90 at.%Ge by X-ray specular reflectivity using ex-situ and in-situ annealing experiments.Similarly, Ozguven and Mclntyre [81] investigated the interdiffusion in eptiaxial Si x Ge 1-x /Si y Ge 1-y superlattices that have an average Ge composition of 91 at.%.The interdiffusion information mentioned above is also summarized in Table 4.

Determination of diffusion parameters
Diffusion process in Ge-Si alloys is of significance for the fabrication of electronic devices.In order to understand the properties of point defect in Ge-Si alloy, it is necessary to determine the diffusion parameters for Ge-Si alloys.Here, the diffusion parameters for self diffusivities and impurity diffusivities were evaluated first.And then the tracer diffusivities were modeled by employing the model developed by Andersson and Ågren [17].During the present modeling, the thermodynamic description for Ge-Si binary system obtained by Bergman et al. [82] was utilized to calculate the thermodynamic factor.Fig. 1 presents the calculated Ge-Si phase diagram.As discussed in section 3.1, there are two types of measurements for the self diffusivities of Si: direct measurement and metal experiment.Direct self diffusion measurement in Si cannot separate the relative contribution of interstitialcy mechanism and vacancy mechanism.Fortunately, this drawback can be compensated by metal diffusion experiment [12], because information about intrinsic point defects can also be obtained by studying foreign-atom diffusion in Si when the interstitialcy and/or vacancy are involved in the diffusion process.For example, mainly substitution dissolved foreign atoms, like group III and V elements, need vacancies and interstitials as vehicles for transportation in Si. [12,39,40] , Au [37], and Pt [38,41] in Si via kick-out mechanism.In these studies, the interstitialcy component of Si self diffusivities was determined.As a result, the self diffusivities of Si can be described using a double Arrhenius equation [83].Firstly, the interstitialcy component of Si self diffusivities was determined by least square fit of experimental data obtained using metal experiment [12,[37][38][39][40][41].One thing worth addressing is that the correlation factors used by different sources are quite different, i.e Stolwijk et al. [37], Perrett et al. [39], Grünebaum et al. [40] and Hauber et al. [41] utilized 0.5, Mantovani et al. [38] used 0.99999, and Bracht et al. [12] used 0.73.For consistency, a correlation factor of 0.73 [15] was accepted and the experimental data were adjusted by this correlation factor.Then, the vacancy component was evaluated mainly using the direct measurement by SIMS from Bracht et al. [30], Ural et al. [31,32], Strohm et al. [6], Aid et al. [35], Shimizu et al. [7,36] and Nakabayashi et al. [33,34].While other data are not utilized or only with a low weight, this is due to the fact that 31 Si has a half life of only about 2.6 h which limits the self diffusion studies to a rather narrow temperature and short annealing time or due to the Si samples used were doped.In addition, the experimental data from Master and Fairfield [20], Mayer et al. [25] and Hirvonen and Anttila [26] exhibit apparent divergence with others.The finally obtained diffusion parameter for Si self diffusivity is presented as:

RT
where the first term is the interstitialcy component and second term is the vacancy component.Fig. 2 presents the Arrhenius plot of Si self diffusivities together with the literature data.The solid line refers to the Si self diffusivities, while the dotted and dashed lines are the interstitialcy component and the vacancy component, respectively.It is apparent form the figure that interstitialcy mechanism dominates the diffusion process at high temperature, while vacancy mechanism is the main mechanism at low temperature.From Eq. 9, the activation energy for interstitialcy mechanism and vacancy mechanism are 5.04 eV and 3.59 eV which agree well with the theory value [84] 4-4.9 eV and 3.07-4.9eV respectively.[6,7,12,.The symbols in the rectangle [12,[37][38][39][40][41] are interstitialcy component of Si self diffusivities.
The literature data available for self diffusivities of Ge briefly reviewed in section 3.2 are in good agreement.A single Arrhenius equation was utilized to fitting the selected diffusion coefficients measured using single crystalline Ge, as vacancy mechanism dominate the whole temperature range.The obtained equation for Ge self diffusivities is shown as:  Lockwood et al. [62], Uppal et al. [67] and Silverstri et al. [11,68] were utilized.That is because the data from Räisänen et al. [65] were measured in doped Ge samples.For the Ge diffusivities in Si, the data from Petrov et al. [57] were in polycrystalline Si and those from Lockwood et al. [62] show divergence with other data.Thus, those data [57,62] were excluded from the parameter determination procedure.Due to the fact that there are not enough low temperature data, it is impossible to evaluate the vacancy component for Ge diffusion in Si.As a result only single Arrhenius was used.The obtained single exponentials are: The comparison between the measured and the evaluated diffusivities of Si in Ge and those Ge in Si are presented in Fig. 4 and Fig. 5 respectively.The symbols are the reported diffusivities in the literature [6,27,54,[57][58][59][60][61][62][63][64], while the solid line is calculated according to the present atomic mobility parameters.The symbols are the reported diffusivities in the literature [6,11,62,[65][66][67][68], while the solid line is calculated according to the present atomic mobility parameters.
Thereafter, the interaction parameters in Eq. 7 were assessed from the tracer diffusivities of Si and Ge in the Ge-Si alloys.The tracer diffusivities of Ge measured by McVay and DuCharme [69] are not consistent with the data from others [6,54,55,63,64], and were thus excluded from the assessment.Besides, the tracer diffusivities measured by Zangenberg et al. [63] show divergence with the data measured by Strohm et al. [6,54] at the concentrations x(Ge) = 0.3, 0.4 and 0.5.In the present assessment, the authors tend to trust the more systematical investigations by Strohm et al. [6,54].While the tracer diffusivities of Si [6,55,64] generally agree with each other.The finally obtained atomic mobility parameters are listed in Table 5. Figs. 6 and  The interdiffusivities available in the literature are not consistent with each other.The data from Prokes and Spaepen [70] were measured in amorphous SiGe alloys.Whereas the interdiffusivities measured by Gavelle et al. [73,74] exhibit wrong feature at x(Si) = 0 to 0.7 and apparently disobey the   The symbols are the reported diffusivities in the literature [6,11,62,[65][66][67][68], while the solid line is calculated according to the present atomic mobility parameters.[73,74].While a comparison between the modelpredicted and the measured interdiffusion coefficients from Xia et al. [75][76][77] are shown in Fig. 12. From these figures, we can conclude that the presently obtained atomic mobilities can predict most of the reliable experimental data reasonably.

Validation of the Manning theory in Ge-Si alloys
It is well known that the Darken relations that correlate the tracer diffusivities with intrinsic and interdiffusivities diffusivities are not complete if considering irreversible thermodynamics.Previous tests [85][86][87] in fcc structures indicated that the Manning's correction to the Darken relation in intrinsic diffusion coefficients is not significant in most systems within the realm of experimental error.However, the Darken-Manning relation in diamond structure is still not tested as far as the knowledge of the present authors.So we tend to examine the Manning correction to the Darken relation in diamond structure.Here we chose 1118 K as the testing temperature.Firstly, the composition dependence of the thermodynamic factor was determined by utilizing the thermodynamic description from Bergman et al. [82].As presented in Fig. 13, the thermodynamic factor is symmetrical for the Ge-Si alloys at 1118 K and in the range of 0.8 ~ 1.0 over the whole composition range.Secondly, the 'vacancy wind term' and the vacancy wind factor were calculated by using the atomic mobility parameters obtained in section 4. It can be seen from Fig. 14 that the 'vacancy wind term' for Ge ranges from 1 to 2. According to  Eq. 2, this factor will enhance the intrinsic diffusion coefficient of Ge and even about 2 times for Ge rich alloys.It may be partially due to the fact that the M o is 2 for diamond structure which is quite small in comparison with 7.15 for fcc structure.Meanwhile, the vacancy wind term for Si changes substantially from -0.5 to 1 indicating that it will slow the intrinsic diffusivities of Si and even change its sign.The vacancy wind factor varies between 1 and 1.5 and will enhance the interidffusivities overall.Fig. 15 presents the calculated intrinsic diffusivities and interidffusivties at 1118 K by both Darken relation and Manning relation.As there is no report of intrinsic diffusivities in Ge-Si alloys, we could not conduct direct comparison for intrinsic diffusivities.Besides, even though there are experimental interdiffusivities at 1118 K, both Darken and Manning relation could not accurately predict the interdiffusivities as the data are much scatter (see the enlarged part of Fig. 15).But from the calculation result, the  Manning modification will enhance the faster component and retard the slower component during the whole diffusion process.And the Manning modification even will change the sign of Si intrinsic diffusivities.

Simulation of diffusion in Ge-Si layers
Several Ge-Si diffusion couples were simulated in order to verify the reliability of the presently obtained atomic mobility parameters.Fig. 16 is the model-predicted concentration profiles of a Si/Ge solid-solid diffusion couple at 1023 K for 10800 s, 36000 s, and 108000 s.The experimental data from Gavelle et al. [73,74] are also appended for comparison.Similarly, the simulation results at 1173 K for about 600 s and 1800 s are presented in Fig. 17 together with the corresponding experimental data [73,74].It is manifest from the results that the presently obtained atomic mobility parameter can accurately predict concentration profiles of these diffusion couples.The mobility parameters are capable of simulating the diffusion process in microscale and short annealing time.In details, there exist certain divergences between the simulated and the experimental concentration profiles.It may be due to the fact that thermodynamic parameters [82] are not accurate enough.In addition, experimental result [73,74] may also have certain inaccuracy as the concentration is measured in micro-scale diffusion couples.

Conclusions
Various diffusivities (self-, tracer, impurity and chemical diffusivities) in Ge-Si alloys available in the literature were critically reviewed.The diffusion parameters for Si self diffusivities, Ge self diffusivities, impurity diffusivities of Si in Ge and Ge in Si were evaluated based on the selected   [73,74] experimental information.For Si self diffusivities, the vacancy component and interstitialcy component are separated.
The Darken-Manning relation was tested in the diamond structure for the first time at 1118 K in the Ge-Si alloys.The calculated correlation factors are considerably large than those in fcc structure.The Manning's correction will retard the intrinsic diffusivities of Si, while enhance the intrinsic diffusivities of Ge and interdiffusivities.
The atomic mobility parameters for Ge-Si solid phase were determined by means of DICTRA assessment.The presently obtained atomic mobility parameters were then utilized to predict the concentration profiles of Ge-Si diffusion couples.The simulated results indicate that the obtained atomic mobility parameters are reliable.Thus, the presently obtained atomic mobility in diamond Ge-Si alloys can be utilized to construct atomic mobility database for semiconductors

Figure 1 .
Figure 1.Calculated Ge-Si phase diagram due to the thermodynamic parameters obtained by Bergman et al. [82].

Figure 2 .
Figure 2. Self diffusivities of Si.All the lines are from the present evaluation: the solid line is the Si self diffusivities, the dashed one the vacancy component of Si self diffusivities and the dotted one the interstitialcy component of Si self diffusivities.Symbols are the experimental data from literature[6,7,12,.The symbols in the rectangle[12,[37][38][39][40][41] are interstitialcy component of Si self diffusivities.

m 2 /
s(10) Hence, the activation energy for self diffusion of Ge is 3.22 eV due to the present evaluation.A comparison of the calculated diffusion coefficients and the measured ones is demonstrated in Fig.3.It can be seen from the figure that a single Arrhenius equation can describe most of the experimental diffusivities well.

Figure 3 .
Figure 3. Self diffusivities of Ge.The symbols are the reported diffusivities in the literature, while the solid line is calculated according to the present atomic mobility parameters[6, 10, 42- 56].Similarly, the diffusivities of Si in Ge and those of Ge in Si were evaluated by the corresponding experimental data.When evaluating the impurity diffusivities of Si in Ge, only the experimental data from Sodervall and Friesel[66], Strohm et al.[6],

7 Fig. 8 .
are the model-predicted tracer diffusivities of Ge in comparison with the corresponding experimental data from Laitinen et al.[55], Strohm et al.[6,54], Kube et al.[64] and Zangenberg et al.[63].Meanwhile, the calculated tracer diffusivities of Si are 0The experimental diffusivities from Strohm et al.[6], Laitinen et al.[55] and Kube et al.[64] are also appended for comparison.It is manifest that the presently obtained atomic mobility parameters can predict the experimental tracer diffusivities reasonably.

Figure 6 .
Figure 6.Model-predicted temperature dependence of tracer diffusivities of Ge in different Ge-Si alloys with diamond structure in comparison with the experimental data from Laitinen et al.[55] and Strohm et al.[6,54].

Figure 7 .
Figure 7. Model-predicted temperature dependence of tracer diffusivities of Ge in different Ge-Si alloys in comparison with the experimental data from Kube et al. [64] and Zangenberg et al. [63].

Figure 9 .
Figure 9. Model-predicted temperature dependence of interdiffusivities in different Ge-Si alloys in comparison with the experimental ones measured by and Ozeguven et al.[81].

Figure 8 .
Figure 8. Model-predicted temperature dependence of tracer diffusivities of Si in different Ge-Si alloys in comparison with the experimental data from Kube et al. [64], Laitinen et al. [55], and Strohm et al. [6].

Figure 10 .
Figure 10.Model-predicted composition dependence of interdiffusivities in different Ge-Si alloys in comparison with the experimentallymeasured ones by Aubertine and Mcintyre [71].

Figure 13 .Figure 14 .
Figure13.Calculated thermodynamic factor for Ge-Si alloys with diamond structure at 1118 K according to the thermodynamic parameters from Bergman et al.[82]

Figure 15 .
Figure 15.Model-predicted intrinsic diffusivities of Si and Ge as well as the modelpredicted interdiffusivities at 1118 K according to different relations.The thick lines are from Manning relation, while the thin lines from Darken relation.The dashed lines are intrinsic diffusivities of Si, the dotted lines intrinsic diffusivities of Ge, while the solid lines interdiffusivities.

Figure 16 .
Figure 16.Model-predicted concentration profiles of Ge/Si thin layer diffusion couples annealed at 1023 K in comparison with the experimental data from Gavelle et al.[73,74].

Figure 17 .
Figure 17.Model-predicted concentration profiles of Ge/Si thin layer diffusion couples annealed at 1173 K in comparison with the experimental data from Gavelle et al.[73,74]

Table 1 ,
and concisely presented as follows.

Table 1
Summary of experimentally measured self diffusivities of Si

Table 2
Summary of experimentally measured self diffusivities of Ge

Table 3
Summary of experimentally measured impurity diffusivities in Ge-Si alloys RAMM=the residual activity measurement methods, TST=the thin sectioning technique, RMT=the radiotracer microsectioning technique, IBSM=the ion beam sputtering for microsectioning, (p, γ)RBM=the (p, γ) resonance broadening method, SIMS=secondary ion mass spectrometry, IGISOT=the ion guide isotope separator on-line technique.b Indicates whether the data are used or not used in the atomic mobility assessment: ■, used; □, used but with low weight;+, not used but considered as reliable data. a

Table 4
Summary of experimentally measured tracer and inter-diffusivities in Ge-Si alloys

Table 5
The finally obtained atomic mobility parameters of diamond-structured Ge-Si alloys in the present work