Interconnection of a thermodynamical model for point defect parameters in solids with the dynamical theory of diffusion

Solid State Ionics 335 (2019) 82–85

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Interconnection of a thermodynamical model for point defect parameters in solids with the dynamical theory of diffusion

T

Nicholas V. Sarlis*, Efthimios S. Skordas Section of Solid State Physics and Solid Earth Physics Institute, Department of Physics, National and Kapodistrian University of Athens, Greece

A R T I C LE I N FO

A B S T R A C T

Keywords: Thermodynamics of point defect parameters Dynamical theory of diffusion Superionic conductors

A multitude of recent studies of point defect parameters in solids including several defect processes in fluoride superionic conductors as well as self-diffusion and heterodiffusion in silicon have been found to obey the cBΩ thermodynamical model. This model is shown to be compatible with the dynamical theory of diffusion.

1. Introduction Silicon (Si) is the most scientifically and technologically important semiconductor material with diverse applications in nanoelectronic, sensor and photovoltaic devices [1-5]. During the last few years, diffusion properties in group IV semiconductors have been determined by recent experimental techniques, i.e., Time-of-Flight Secondary Ions Mass Spectrometry, TOF-SIMS [6-8]. The experimental data showed that the lnD vs 1/T self-diffusion plot by Kube et al. [9] and by Bracht [10] exhibit an upwards curvature at higher temperatures. Such a curvature which is usually associated with two mechanisms (vacancies and self-interstitials) has been alternatively explained by Saltas et al. [11] by a single mechanism with temperature dependent thermodynamic parameters, i.e., activation enthalpy hact and activation entropy sact obtained from the cBΩ thermodynamical model. Both hact and sact were found to increase upon increasing the temperature in accordance with expectations based on thermodynamical grounds [12]. In a recent publication on self-diffusion in crystalline Si, Südkamp and Bracht [13] found no upward curvature and concluded that their experimental results can be described by a single activation enthalpy (4.73 ± 0.02) eV down to 755 ° C. The cBΩ model, which interconnects point defect parameters with bulk elastic and expansivity properties (shortly explained in Section 2), has been also found to successfully reproduce the nickel and copper fast diffusion in silicon [14] as well as the magnesium diffusion in silicon [15]. Beyond the aforementioned applications to the case of silicon, this model has been earlier successfully employed in the study of various defect processes in a multitude of solids including alkali halides [16,17], alkaline earth fluorides [18,19], silver halides (e.g., see Ref. [20]), fluoride superionic conductors, e.g., β − PbF2 (see Ref. [21]), in a

*

variety of oxides, for example, in UO2 and ThO2 particularly important for nuclear fuel applications [22,23], in anatase TiO2 particularly important for its high chemical stability and photocatalytic properties [24], in Li5FeO4 [25] and Li2CuO2 [26] that are candidate materials as cathode in lithium ion batteries and in Na2MnSiO4 which is a promising positive electrode material in rechargeable sodium ion batteries [27] as well as for Si diffusing in silicates [28] and aluminum in MgO [29], oxygen self-diffusion in minerals [30] etc. The foundation of cBΩ model has been treated in detail in Ref. [31] (see also Ref. [32]), which at that time (i.e., in the 1980’s) did not include any application to semiconductors [33]. Since however this model has been recently found to give also successful results to diffusion processes in silicon, as mentioned above, it is of paramount importance to show that it is compatible with the dynamical theory of diffusion by considering also the following fact emerged in silicon (see Ref. [34] and references therein): Beyond nonlinearity, elastic anisotropy is known to shed light on the mechanical properties of solids as dislocation dynamics, plastic deformations or structural stability. For cubic symmetry as silicon, Zener's definition of the acoustical anisotropy A = 2C44/(C11 − C12) has been revisited [35] in order to quantify the single-crystal elastic anisotropy 6 1 from a universal point of view AU = 5 ( A − A )2 (cf. for an isotropic U crystal A = 1 but A = 0). It was experimentally found [34], that the degree of elastic anisotropy of Si is increasing upon increasing pressure (P) which could be related to the observation that the transverse phonon frequency (i.e., the Gruneisen parameter, see also below) has been found to be the unique mode that decreases with P. It is the Gruneisen constant of the mode that is mainly responsible for the defect migration process through which the dynamical theory of diffusion is interconnected with the cBΩ model as it will be discussed later in Section 3.

Corresponding author. E-mail addresses: [email protected] (N.V. Sarlis), [email protected] (E.S. Skordas).

https://doi.org/10.1016/j.ssi.2019.02.002 Received 19 December 2018; Received in revised form 31 January 2019; Accepted 4 February 2019 0167-2738/ © 2019 Elsevier B.V. All rights reserved.

Solid State Ionics 335 (2019) 82–85

N.V. Sarlis and E.S. Skordas

Thus, in short, this paper is structured as follows: In Section 2 we briefly present the cBΩ model, while its interconnection with the dynamical theory is discussed in Section 3. Finally, our main conclusion is given in Section 4.

18 16 14 (cm3/mol)

2. The cBΩ thermodynamical model

sf = − υf =

dg f dT

dg f dP

sm = −

and P

and

υm =

T

dg m dT

dg m dP

dg act dT

and

υact =

P

2 0 0

(1)

T

dg act dP

T

g i = c iBΩ

(2)

=

⎝⎝ dP

T

υi 1 dB ≈ ⎛ hi B ⎝ dP

g i dB = ⎛ B ⎝ dP

T

3

T

− 1⎟⎞ ⎠ i

(The same conclusion holds for the ratio s /h , e.g., see the example of LiF in Ref. [48].) The validity of Eq. (7) is checked for Si in Fig. 1 where we plot υi versus hi for the self diffusion activation parameters of Si and for the corresponding heterodiffusion parameters for Cu, Ni and Mg in Si. In this plot the straight line corresponds to the ratio υi/hi predicted by Eq. (7) by using the elastic data reported by Decremps et al. [34] and the expansivity data of Gu et al. [49] that have been also used by Saltas et al. [50] to apply the cBΩ model to the self-diffusion studies in Si1−xGex alloys. 3. Compatibility of dynamical theory of diffusion with cBΩ model Flynn [51] obtained for the migration (m) process the following relation for the dynamical theory of diffusion:

(4)

2γ υm = i gm B

− 1⎞⎟ ⎠

4

(8) i

and combining this equation with Eq. (3) we get:

υi

2 hact (eV)

expansivity data [21,47]. Whenever Tsi ≪ hi, Eq. (7) can be approximated by:

The symbol B stands for the isothermal bulk modulus, and Ω the mean volume per atom while ci is dimensionless which can be considered independent of temperature and pressure. By differentiating Eq. (3) in respect to pressure, we find that the volume υi ≡ (dgi/dP)T is given by:

− 1⎞⎟ ⎠

1

Fig. 1. Check of the validity of Eq. (7) of cBΩ model for Si. The green asterisk corresponds to the self-diffusion process at T1 = 923 K, while the other symbols, i.e., blue circle, red diamond and cyan triangle to the heterodiffusion of Ni, Cu and Mg in Si, respectively. The straight line has been drawn according to the slope deduced from Eq. (7) by using solely bulk elastic and expansivity data, while the values of hact and υact come from the studies described in the text and can be also found in Table 1 in Ref. [15] in which we also consider the latest Si self-diffusion experimental results by [13]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(3)

dB c iΩ ⎛⎜⎛

Ni Cu Mg Si cBΩ

4 P

In the case of the (re)orientation of electric dipoles comprising an aliovalent impurity attracting [36,37] a neighboring bound (b) vacancy or interstitial, the results are described in terms of an activation energy gact,b, associated with an activation volume υm,b ≡ (dgm,b/dP)T for the (re)orientation process. This case is of particular importance since it provides the basis for the explanation [38,39] of the generation of low frequency electric signals that are observed before earthquakes [40-43] resulting from a cooperative (re)orientation of these electric dipoles when the gradually increasing stress before an earthquake reaches a critical value [44-46]. The cBΩ model asserts that the defect Gibbs energy gi, where the superscript i refers to the defect process under consideration, i.e., i = f, m or act for the formation, migration and activation, respectively, is interconnected with the bulk properties of the solid through the relation [31,47]

υi

8 6

When both processes, i.e., formation (f) and migration (m) are operating, the experimental results are described in terms of an activation Gibbs energy gact, on the basis of which an activation entropy sact and an activation volume υact is defined:

s act = −

10

act

The variation of the pressure (P) affects the formation Gibbs energy, gf, as well as the migration Gibbs energy, gm, of defects in solids. The defect entropies for the formation process (sf) and migration process (sm), as well as defect volumes for the formation process (υf) and migration process (υm) are defined as [31]:

12

(5)

(9) ∂ ln ω γ (=− ∂ ln V )

By the same token, differentiating Eq. (3) in respect to temperature we find the entropy si = −(dgi/dT)T and inserting it into the relation hi = gi + Tsi we finally find the enthalpy hi:

for the mode ωi that where γi denotes the Gruneisen constant is mainly responsible for the migration process. For monoatomic crystals and within the Debye approximation, Slater [52] deduced for the mean Gruneisen constant γ the relation

dB hi = c iΩ ⎛⎜⎛B − TβB − T dT ⎝⎝

γ=−

⎞⎟ (6)

P⎠

Thus, the ratio of Eqs. (4) and (6) leads to:

υi dB =⎛ hi ⎝ dP

T

dB − 1⎞⎟/ ⎛⎜B − TβB − T dT ⎠ ⎝

T

(10)

which was later improved by Dugdale and MacDonald [53] as well as by Shanker et al. [54] to:

⎞⎟ P⎠

1 1 dB + 6 2 dP

(7)

γ=−

which reveals that for various defect processes in the same matrix material, the ratio υi/hi is solely governed by the bulk elastic and

1 1 dB + 2 2 dP

T

By inserting this value of γ instead of γi in Eq. (9) we find 83

(11)

Solid State Ionics 335 (2019) 82–85

N.V. Sarlis and E.S. Skordas

υm 1 dB = ⎛ gm B ⎝ dP

T

− 1⎟⎞ ⎠

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(12)

which is just Eq. (5) of the cBΩ model. Introducing in Eq. (12) the definition υm =

dg m , dP T

and considering also the relation dgm/gm = d

(BΩ)/BΩ (with T constant) -see Eq. (14.11) derived in Ref. [31]- we finally get upon integration: υm

gm =

(Ω) dB | dP T

BΩ

−1

(13)

where the quantity

υm ( ) Ω dB |T −1 dP

has to be pressure independent.

In the case of other crystals such as NaCl, CsCl, CaF2 and zincblende structures, we can derive the cBΩ model as follows: We start from the first Szigeti relation [55,56] which relates the transversal optical frequencies ωt to the dielectric constants ε0 and ε∞ (static and high frequency constants respectively):

ωt2 =

Mr0 B ⎛ ε∞ + 2 ⎞ μ ⎝ ε0 + 2 ⎠ ⎜

⎟

(14)

where r0 is the equilibrium interionic distance, μ the reduced mass and M the coordination number of the crystal, i.e., M = 6 for NaCl, 8 for CsCl and fluorite and 4 for zincblende structures. Eq. (14) is strictly valid at zero pressure and was generalized for all pressures by Barron and Batana [57]

ωt2 =

3Vu ⎛ ε∞ + 2 ⎞ 4 ⎛B − P ⎞ 3 ⎠ μr02 ⎝ ε0 + 2 ⎠ ⎝ ⎜

⎟

(15)

where Vu is the volume of the unit cell. An expression for the corre∂ ln ω sponding Gruneisen parameter γt = − ∂ ln Vt can be derived by diffentiating the relation (15) with respect to the volume (see also Ref. [58]):

γt =

Bθ ⎡ ∂ε∞/ ∂P ∂ε0/ ∂P ⎤ + 1 dB − 2 ⎢ θ (ε0 − 1) + 3 ⎥ 2 dP ⎣ θ (ε∞ − 1) + 3 ⎦

− T

5 6

(16)

where the parameter θ takes different values for the different structures. Upon following in Eq. (16) the procedure of the calculation of the pressure derivative of the static dielectric constant developed in Ref. [59], we finally get for γt an expression similar to Eq. (11) and therefrom the cBΩ model emerges as explained above. 4. Conclusion The cBΩ model explains the upwards curvature of the self diffusion plot in Si by means of a single operating mechanism and satisfactorily reproduces the Ni and Cu fast diffusion in Si, as well as Mg diffusion in Si. This thermodynamical model is shown here to be compatible with the dynamical theory of diffusion. References [1] C. Gao, X. Ma, J. Zhao, D. Yang, J. Appl. Phys. 113 (2013) 093511, https://doi.org/ 10.1063/1.4794531. [2] S. Takeuchi, Y. Shimura, O. Nakatsuka, S. Zaima, M. Ogawa, A. Sakai, Appl. Phys. Lett. 92 (2008) 231916, https://doi.org/10.1063/1.2945629. [3] A. Chroneos, C.A. Londos, E.N. Sgourou, J. Appl. Phys. 110 (2011) 093507, https:// doi.org/10.1063/1.3658261. [4] E.N. Sgourou, D. Timerkaeva, C.A. Londos, D. Aliprantis, A. Chroneos, D. Caliste, P. Pochet, J. Appl. Phys. 113 (2013) 113506, https://doi.org/10.1063/1.4795510. [5] P. Chen, X. Yu, X. Liu, X. Chen, Y. Wu, D. Yang, Appl. Phys. Lett. 102 (2013) 082107, https://doi.org/10.1063/1.4793660. [6] R. Kube, H. Bracht, A. Chroneos, M. Posselt, B. Schmidt, J. Appl. Phys. 106 (2009) 063534, https://doi.org/10.1063/1.3226860. [7] R. Kube, H. Bracht, J.L. Hansen, A.N. Larsen, E.E. Haller, S. Paul, W. Lerch, J. Appl. Phys. 107 (2010) 073520, https://doi.org/10.1063/1.3380853. [8] A. Chroneos, H. Bracht, Appl. Phys. Rev. 1 (2014) 011301, https://doi.org/10. 1063/1.4838215. [9] R. Kube, H. Bracht, E. Hüger, H. Schmidt, J.L. Hansen, A.N. Larsen, J.W. Ager, E.E. Haller, T. Geue, J. Stahn, Phys. Rev. B 88 (2013) 085206, https://doi.org/10.

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