Ionic conductivity evolution of isotropic crystal with double strained interfaces

Solid State Ionics 303 (2017) 167–171

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Ionic conductivity evolution of isotropic crystal with double strained interfaces Chao Feng a,1, Jipeng Fei a,1, Kechun Wen a,1, Weiqiang Lv a, Zuoxiang Zhang a, Minda Zou a, Fei Yang a, Muhammad Waqas a,b, Weidong He a,b,c,d,⁎ a

School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, PR China Department of Electrical Engineering, Sukkur Institute of Business Administration, Sukkur, Pakistan Sichuan Alpha Scenery & Green Energy Co., Ltd., PR China d Chengdu Zi Zhi Yuan Co., Ltd., Sichuan, PR China b c

a r t i c l e

i n f o

Article history: Received 6 August 2016 Received in revised form 9 March 2017 Accepted 9 March 2017 Available online xxxx

a b s t r a c t Introducing lattice strain at dissimilar interfaces has been considered as an efficient approach to enhance the ionic conductivity in thin electrolyte film of solid oxide fuel cells (SOFCs). However, quantitative analysis on depth profile of double strained crystal interfaces is still lacking. In this report, the ionic conductivity evolution of isotropic crystal with double strained interfaces is evaluated quantitatively. The correlation between ionic conductivity enhancement and intrinsic parameters of double strained electrolyte materials (layer sequence, Young's modulus, Poisson's ratio) is investigated. A prototype isotropic cubic lattice crystal (YSZ) is employed to verify our model. The results show that the synergetic effects of double lattice strain on ionic conductivity are more prominent than that of the single lattice strain. Our work provides insightful guidelines to enhance the ionic conductivity in electrolyte of SOFCs by introducing the double lattice strain at the nanoscale heterostructures. © 2017 Published by Elsevier B.V.

1. Introduction Due to high power density, high energy conversion efficiency, environmental friendly, etc., solid oxide fuel cells (SOFC) are widely applied in large-scale power station and heavy industry [1,2]. However, it is difficult to generalize commercial applications of SOFCs because of its high operating temperature (~ 1000 °C) [2]. The key strategy to low the operating temperature of SOFCs is to enhance the ionic conductivities in solid electrolyte at low temperatures. Several ways to enhance the ionic conductivity have been studied in the past decades, including developing new electrolyte materials, depositing thin electrolyte films onto porous substrates (CVD/EVD, sputtering, etc.) and multi-elements doping in existing electrolytes [3]. Recently, it is found that the ionic conductivities in thin films with strained heterostructure interfaces are ~ 3.5 orders of magnitude larger than those in the same bulk materials [4]. Peters et al. reported that the total conductivity of the CSZ (ZrO2 + 8.7 mol% CaO) ⁎ Corresponding author at: School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, PR China. E-mail address: [email protected] (W. He). 1 These authors contributed equally to this work.

http://dx.doi.org/10.1016/j.ssi.2017.03.007 0167-2738/© 2017 Published by Elsevier B.V.

increases by two orders of magnitude as the thickness of the individual CSZ layers decreases from 0.78 μm to 40 nm [5]. Barriocanal et al. claimed ~ 8 orders of magnitude increase in ionic conductivity at room temperature in fully strained ultrathin (~ 1 nm) YSZ layers as compared to bulk YSZ [6]. Such a result is attributed to transformation of charge carriers and decrease of energy barriers of oxygen ions migration, indicating the significance of the strained interfaces in enhancing charge transport. Although numerous experiments and theoretical efforts have verified that strained interfaces can increase ionic conductivity, quantitative analysis of double strain effect at heterostructure interfaces is still yet insufficient. Zou et al. have proposed models to calculate the enhancement of conductance at strained interface in electrolyte, but they merely considered the influence of single strained interface on ionic conductivity enhancement [7]. In this report, we, for the first time, develop quantitative expressions between lattice mismatch leaded double tensile strains and ionic conductivity enhancement. Derivations are performed to accurately study the correlation between the strain effects and ionic conductivity in YSZ electrolyte films. Influences of internal and external factors on ionic conductivity enhancement are also assessed in depth in our work.

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2. Theory and computing methods There is a compressive or tensile stress in the crystal due to mismatch of different crystal lattices when these crystals grow epitaxially. As shown in Fig. 1(a), the lattice layers of phase 1 are expanded due to symmetrical tensile strain at the interfaces. Thus, the unit cell size of phase 1 at X-axis and Y-axis is extended. The volume changes of strained lattice at the interfaces are depicted in Fig. 1(b), where the cell parameters of phase 1 are a, b, c and the cell parameters of phase 2 are as, bs, cs. Phase 1 is subject to tensile strain owing to a b as, b b bs. Lattice parameters at the bottom plane of the kth lattice layer of phase 1 change to ak, bk, ck. There are n total layers of crystal faces that can be assumed in thickness direction in phase 1. Then, the function of strain over depth evolution in a given material can be expressed by the following exponential equation: εðzÞ ¼ ε 1 exp½−αz

ð1Þ

where ε1 is the xy plane strain at film/substrate interface and α is material constant obtained from experimental or stimulation result, denoting measure of stress relaxation with the distance [8–10]. Thus, the tensile strain of kth layer due to mismatch of the bottom lattice is expressed as: "

k−1

#

εk ¼ ε1 exp −α ∑ ci

ð2Þ

i¼1

where εk and ci denote the xy plane strain at the bottom plane of kth lattice layer and the thickness of ith lattice layer, respectively. Accordingly, the kth lattice layer generates another tensile stress due to mismatch of the top lattice. We can get Eq. (3), "

n−k

#

εn−kþ1 ¼ ε1 exp −α ∑ ci i¼1

ð3Þ

where εn – k + 1 is the xy plane strain at the top plane of (n − k + 1)th lattice layer. Thus, the kth lattice layer is actually subject to resultant force of two tensile stress, respectively, from mismatch of the bottom and the top. It can be expressed in Eq. (4), ε tot k ¼ ε k þ ε n−kþ1

ð4Þ

where εtot k is total strain of kth lattice layer. Stretching xy plane yields lattice strain along z-axis direction, and the thickness of c changing into ck can be expressed with Poisson's ratio and xy plane strain in Eq. (5) [11].   2ν εk ck ¼ c 1− 1−ν

ð5Þ

Accordingly,   2ν εn−kþ1 cn−kþ1 ¼ c 1− 1−ν

ð6Þ

we then get a real thickness of kth lattice layer as shown in Eq. (7).     2ν 2ν tot ¼ c þ c −c ¼ c 1− ð þ ε Þ ¼ c 1− ε ε ctot k n−kþ1 n−kþ1 k 1−ν k 1−ν k

ð7Þ

Volume change of the unit cell at the kth lattice layer is depicted in Eq. (8) [12]. ΔV k ¼

  2  2   i 1 2ν tot h 1 þ εtot þ 1 þ εtot þ 1 þ εtot 1 þ εtot abc 1− εk k kþ1 k kþ1 −abc 3 1−ν

ð8Þ

Fig. 1. (a) 3D schematic of double strained heterostructure interfaces with lattice mismatch, where phase 1 and 2 are electrolyte and substrate, respectively. (b) Sketch representing mismatch in unit cells, in which tensile strain is in phase 1 and compressive strain is in phase 2.

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Furthermore, ionic conductivity is expressed with volume change in Eq. (9), 

σk σ0

ln

 ¼

ð2 þ 2ν ÞEΔV K 9ð1−2ν ÞkB T

ð9Þ

where σk and σ0 are the total ionic conductivities of the kth lattice layer and the bulk ionic conductivity, respectively, and kB is the Boltzmann constant, T is temperature, E is Young's modulus [7,12]. Substituting Eq. (8) into Eq. (9) leads to Eq. (10),  ln

σk σ0

 ¼

ð2 þ 2ν ÞabcE 27ð1−2ν ÞkB T

h   2     i 2ν tot 2 tot tot 1− ðεk þ εn−kþ1 Þ 1 þ ε tot þ 1 þ ε þ 1 þ ε 1 þ ε −3 k kþ1 k kþ1 1−ν

ð10Þ

Lattice strain at coherent and dislocation-free interface is relative to elastic properties of ionic electrolyte films, and the specific expression is given in Eq. (11). ε1 ¼

ðas −aÞEs aEs þ as E

ð11Þ

Hereby, Es is the Young's modulus of substrate [13]. Of particular note is that Eq. (11) is available on the precondition of coherent and dislocation-free interface. When thickness of a thin electrolyte film goes beyond the limit of the critical value, misfit dislocation and stress relief are generated at the interface of two phases. The critical value can be denoted by a transcendental equation (Eq. 12) [12,14], dc ¼

  1 1−ν b dc pffiffiffi ln b 16 2 1 þ ν ε21

ð12Þ

where dc is the critical thickness and b denotes the Burgers vector.

3. Results and discussion Yttria-stabilized zirconia (YSZ), which keeps stabilized cubic phase at ambient temperature, is extensively applied as solid oxide electrolyte material for favorable mechanical and chemical properties as well as high oxygen ionic conductivity over ranges of temperatures and oxygen pressures in SOFCs. Therefore, we take YSZ as a material platform to discuss our derivations and model. Atomic oxygen anions diffuse through the crystal lattice in solid oxide electrolyte, so the ionic conductivity in SOFC depends on O2– migration rate [6,15,16]. Appropriate tensile strain can considerably increase ionic conductivity enhancement by expanding the volume of lattice cell. Fig. 2(a) and (b) reveal the correlation between ionic conductivity enhancement and interfacial strain ε1 on five lattice layers closest to heterostructure interfaces at 573 K and with total ten lattice layers, where the lattice constants of YSZ are a = b = 3.637 Å and c = 5.14 Å, and Young's modulus and Poisson's ratio are 204 GPa and 0.31, respectively [12]. The plots reveal that with a certain interfacial tensile strain, the ionic conductivity enhancement in the first layer far exceeds that in other layers. For instance, the ionic conductivity in the first

layer with a 3% tensile strain increases by 2.5 orders of magnitude, but the ionic conductivity in the second layer drops to a value with 0.09 orders of magnitude compared to in the first layer. This result clearly indicates the predominant effects of two first layers closest to heterostructure interfaces on the diffusion of atomic oxygen anions in SOFCs. In other words, atomic oxygen anions diffuse in the two first layers because lattice strain has a more obvious effect on volume change of unit cell closest to the interface. Meanwhile, ionic conductivity enhancement of each layer exhibits linearly with strain ranging from 1% to 5%. For 1.2 nm YSZ deposited on (100) SrTiO3 (STO) substrates, the tensile strain at the film/substrate interface is 3.46% in our calculation as the Young's modulus is 190 GPa. Accordingly, the critical thickness of YSZ is ~9.5 nm based on Eq. (12), which is larger than 1.2 nm, indicating coherent and dislocation-free film/substrate interface. As shown in Fig. 2(a), about 3 orders of magnitude enhancement in ionic conductance at first lattice layer is obtained with a 3.46% tensile strain in YSZ, which is quantitatively identical with experimental results [17,18]. Generally, the Young's modulus is sustainable and only reduces slightly with increase of temperature [19]. For YSZ material, the

Fig. 2. (a) Plots of the logarithm of ionic conductivity enhancement versus strain ε1 for the first lattice layers closest to the coherent interface at 573 K, with total ten lattice layers in YSZ. (b) Plots of the logarithm of ionic conductivity enhancement versus strain ε1 for the second to fifth lattice layers closest to the coherent interface at 573 K, with total ten lattice layers in YSZ.

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Fig. 3. (a) Plot of logarithm of ionic conductivity enhancement versus temperature for the first lattice layers closest to the coherent interface, with a tensile strain of 3% and total ten lattice layers in YSZ. (b). Plot of logarithm of ionic conductivity enhancement versus temperature for the second to fifth lattice layers closest to the coherent interface, with a tensile strain of 3% and total ten lattice layers in YSZ.

expression of Young's modulus at different temperatures can be depicted in Eq. (13), E ¼ E0 −BT 0 exp½−T 0 =T

ð13Þ

where E0 is the Young's modulus at absolute zero temperature [10]. The constants B, T0 and E0 are 0.065 GPaK−1, − 429 K, and 281.5 GPa, respectively [20]. Substituting Eq. (13) to Eq. (10), we can plot Log(σk/ σ0) versus temperature for five lattice layers closest to the coherent interfaces, as shown in Fig. 3(a) and (b). The plots depict the dependency between logarithm of ionic conductivity enhancement and temperature with certain tensile strain of 3%. According to Fig. 3(a) and (b), ionic conductivity enhancement exhibits a nonlinear correlation with temperature, indicating temperature has a tremendous influence on ionic conductivity enhancement. The plots also show ionic conductivity enhancement is leveling off as temperature rises, which suggests less dominant effects of strain effect in high temperature regions. In fact, bulk diffusion is stimulated by temperature increase, and atomic oxygen anions obtain sufficient thermal energy for bulk diffusivity to overcome energy barrier, giving rise to a limit in conductivity change [21,22]. Fig. 4 (a) and (b) shows logarithm of ionic conductivity enhancement versus Poisson's ratio on five lattice layers closest to strained interfaces, with temperature and tensile strain of 573 K and 3%, respectively. As shown in Fig. 4 (a), ionic conductivity enhancement goes down with increasing Poisson's ratio, suggesting that materials with a low Poisson's ratio are favorable for electrolyte in SOFCs. Meanwhile, within a certain range of Poisson's ratio, its effect far exceeds the tensile strain at heterostructure interfaces. In this double strained crystal interfaces model, we get a critical value of Poisson's ratio ~ 0.345, which leads to the variation tendency of ionic conductivity enhancement. When Poisson's ratio is smaller than this value, strained interface can promote

ion transport. Thus, Poisson's ratio is more dominant on ionic conductivity when it exceeds 0.345 according to the plots. Moreover, compared to single layer strain model, this critical value rises by 1.5% due to a superimposed effect [7]. We compare the effects of Young's modulus on ionic conductivity enhancement in a different model, where the Young's modulus ranges from 100 GPa to 208 GPa and with a tensile strain of 3%. As depicted in Fig. 5 (a), (b), and (c), the ionic conductivity enhancement induced by the double strain effect linearly increases with increase of Young's modulus, the correlation of which is the same as that in Ref. 7, but the ionic conductivity enhancement of the former is faster than the latter. Besides, we find the closer intermediate layer in the electrolyte film is, the more obviously the double strain effect shows. For example, with the Young's modulus of 160 GPa, the logarithm of ionic conductivity enhancement for three lattice layers closest to interface increases by 2.6%, 28.7% and 131%, respectively. The linear positive correlation between ionic conductivity enhancement and Young's modulus provides the basis for utilization of certain materials at strained crystal interface. Additionally, we analyze the impact on ionic conductivity enhancement at the 1st layer when the total lattice layer number changes in the electrolyte film. Ionic conductivity enhancement on the 1st layer sharply decreases with the number of total lattice layers, as shown in Fig. 6. Besides, when the total lattice layers exceed eight, the positive impact on ionic conductivity enhancement remains. In this double strained heterostructure interfaces model, tensile strains resulting from two substrate mismatch exhibit much more dominant effects on ionic conductivity enhancement in an ultrathin film, featured with a superimposed influence on each lattice layer, especially the 1st layer. Meanwhile, high temperature exhibits more side effect on ionic conductivity enhancement with total lattice layers smaller than three, from

Fig. 4. (a) The effect of Poisson's ratio on the ionic conductivity for the first lattice layer closest to the coherent interface, with a tensile strain of 3% and total ten lattice layers. (b) The effect of Poisson's ratio on the ionic conductivity for the second to fifth lattice layers closest to the coherent interface, with a tensile strain of 3% and total ten lattice layers.

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Fig. 5. (a), (b) and (c) Plots of the logarithm of ionic conductivity enhancement versus Young's modulus from 100 GPa to 208 GPa, with an interfacial strain of 3% and total five lattice layers. The red lines are calculated data obtained by Zou et al. [7]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

References

Fig. 6. Plot of the logarithm of the ionic conductivity enhancement on the 1st layer versus total lattice layers number. The corresponding Young's modulus of YSZ is 204 GPa, 201 GPa, 197 GPa, 187 GPa, 181 GPa. (Inset) Plots of the logarithm of the ionic conductivity enhancement on the 1st layer versus total lattice layer number from 3 layers to 10 layers.

which we grasp synergistic effects of both tensile strain and temperature in thin films. 4. Conclusion In this report, a double strained heterostructure interface model is proposed to investigate the depth profile of ion transport. Both external and internal factors are investigated systematically. Meanwhile, thin films are favorable for the electrolyte due to a superimposed effect. Our work indicates that lower temperature makes tensile strain a predominant effect on ionic conductivity enhancement, especially in the 1st layer. This model provides theoretical insight into acquiring enhancement in ionic conductivity at strained interfaces. Acknowledgment The work was supported by the Science & Technology Support Funds of Sichuan Province (grant no. 2016GZ0151), the Fundamental Research Funds for the Chinese Central Universities (grant no. ZYGX2015Z003), and the National Natural Science Foundation of China (grant nos. 21403031 and 51501030).

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