On the calculation of electrostriction by DFT
D. S. P. Tanner1,2 , and P-E. Janolin2 , E. Bousquet1
1Université de Liège, Q-MAT, CESAM, Institut de Physique2Université Paris-Saclay, CentraleSupélec, CNRS, Laboratoire SPMS, 91190, Gif-sur-Yvette, France
Project ANR-19-ASTR-0024, “MEGAEM”
• Crystal Mechanical response to 𝑬 feld:
Electromechanical coupling
Strain
1.
• Crystal Mechanical response to 𝑬 feld:
• Electrostriction (all crystals), strain quadratic with 𝑬;
Piezoelectricity (non-centric), strain linear with 𝑬
Electromechanical coupling
𝑥𝑖𝑗 = 𝒅𝒊𝒋𝒎𝐸𝑚 +𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛
Strain
1.
• Crystal Mechanical response to 𝑬 feld:
• Electrostriction (all crystals), strain quadratic with 𝑬;
Piezoelectricity (non-centric), strain linear with 𝑬
Electromechanical coupling
𝑥𝑖𝑗 = 𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛
Strain
1.
Electromechanical coupling
Electric Field 𝑬
Strain(𝒙𝑖𝑗)
Stress(𝑋𝑖𝑗)
𝑥𝑖𝑗 = 𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛
𝑋𝑖𝑗 = 𝒎𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛
2.
Electromechanical coupling
Electric Field 𝑬 Polarisation Field 𝑷
Strain(𝒙𝑖𝑗)
Stress(𝑋𝑖𝑗)
𝑥𝑖𝑗 = 𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛 𝑥𝑖𝑗 = 𝑸𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛
𝑋𝑖𝑗 = 𝒎𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛 𝑋𝑖𝑗 = 𝒒𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛
2.
Electromechanical coupling
Electric Field 𝑬 Polarisation Field 𝑷
Strain(𝒙𝑖𝑗)
Stress(𝑋𝑖𝑗)
𝑥𝑖𝑗 = 𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛 𝑥𝑖𝑗 = 𝑸𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛
𝑋𝑖𝑗 = 𝒎𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛 𝑋𝑖𝑗 = 𝒒𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛
Aim: Determine the best means by which to calculate electrostrictive
properties using DFT
2.
Outline• Motivation
- Why calculate electrostriction?- How best to calculate: Literature review
• DFPT Calculation of Electrostriction- Derivation and advantages
- Validation/Comparison
• Application of method
• Summary and Outlook
Outline• Motivation
- W𝐡𝐲 𝐜𝐚𝐥𝐜𝐮𝐥𝐚𝐭𝐞 𝐞𝐥𝐞𝐜𝐭𝐫𝐨𝐬𝐭𝐫𝐢𝐜𝐭𝐢𝐨𝐧?- How best to calculate: Literature review
• Indirect Calculation of Electrostriction- Derivation and advantages
- Validation/Comparison
• Application
• Summary and Outlook
• Electromechanical coupling: transducers – actuators, smart devices
• Electrostrictive strains too small~10-8 %
Motivation: Why calculate electrostriction? 3.
• Electromechanical coupling: transducers – actuators, smart devices
• Electrostrictive strains too small~10-8 %
• Giant electrostrictors*: strains of ~0.6%
• Advantages over Piezo-transducers:- Temperature stability - Low hysteresis- Lead free
• Find Giant Electrostrictors; Find origin of Giant Electrostriction
Motivation: Why calculate electrostriction? 3.
* [R. Korobko et al. Adv. Mater. (2012), 24, 5857] ; [N. Yavo et al. Acta Mater. (2018), 144, 411]
† [Q. Li et al. Phys. Rev. Mat. (2018), 2, 041403(R)]
†
Outline• Motivation
- Wℎ𝑦 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛- 𝑯𝒐𝒘 𝒃𝒆𝒔𝒕 𝒕𝒐 𝒄𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒆: 𝑳𝒊𝒕𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒓𝒆𝒗𝒊𝒆𝒘
• Indirect Calculation of Electrostriction- Derivation and advantages
- Example calculation: MgO- Comparison/validation against direct calculation
• Application
• Summary and Outlook
● Finite E field methods
Motivation: How best to calculate electrostriction? -Literature review
𝑋𝑖𝑗 = 𝒎𝐸2
● Finite D field methods
𝑥𝑖𝑗 = 𝑸𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛
4.
● Finite E field methods
Motivation: How best to calculate electrostriction? -Literature review
𝑋𝑖𝑗 = 𝒎𝐸2
● Finite D field methods
𝑥𝑖𝑗 = 𝑸𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛
4.
• All direct Electrostritive effect
• Not all methods represented
• methodological study required
Outline• Motivation
- Why calculate electrostriction?- How to calculate: Literature review
• Indirect Calculation of Electrostriction- Derivation and advantages
- Validation/Comparison
• Application
• Summary and Outlook
Gibbs free energy
First derivatives
𝜕𝐺
𝜕𝑋𝑘𝑙= −𝑥𝑘𝑙 ; 𝑥𝑘𝑙 = 𝑀𝑘𝑙𝑖𝑗𝐸𝑖𝐸𝑗
𝜕𝐺
𝜕𝐸𝑖= −𝑃𝑖 ; 𝑃𝑖 = χ𝑖𝑗𝐸𝑗
Indirect Calculation of ElectrostrictionThermodynamical derivation for 𝑀𝑖𝑗𝑘𝑙
𝐺 = 𝑢 − 𝑇. 𝑆 − 𝐸. 𝑃 − 𝑥. 𝑋
𝑑𝐺 = −𝑆. 𝑑𝑇 − 𝑥𝑘𝑙𝑑𝑋𝑘𝑙 − 𝑃𝑖𝑑𝐸𝑖
5.
Gibbs free energy
𝜕3𝐺
𝜕𝐸𝑗𝜕𝐸𝑖𝜕𝑋𝑘𝑙= −
𝜕𝑥𝑘𝑙𝜕𝐸𝑗𝜕𝐸𝑖
= −2𝑀𝑘𝑙𝑖𝑗
𝜕3𝐺
𝜕𝑋𝑘𝑙𝜕𝐸𝑗𝜕𝐸𝑖= −
𝜕χ𝑖𝑗
𝜕𝑋𝑘𝑙
Differentiate by Ei , then Ej Differentiate by Ej , then by Xkl
First derivatives
Third derivatives
𝜕𝐺
𝜕𝑋𝑘𝑙= −𝑥𝑘𝑙 ; 𝑥𝑘𝑙 = 𝑀𝑘𝑙𝑖𝑗𝐸𝑖𝐸𝑗
𝜕𝐺
𝜕𝐸𝑖= −𝑃𝑖 ; 𝑃𝑖 = χ𝑖𝑗𝐸𝑗
Indirect Calculation of ElectrostrictionThermodynamical derivation for 𝑀𝑖𝑗𝑘𝑙
𝐺 = 𝑢 − 𝑇. 𝑆 − 𝐸. 𝑃 − 𝑥. 𝑋
𝑑𝐺 = −𝑆. 𝑑𝑇 − 𝑥𝑘𝑙𝑑𝑋𝑘𝑙 − 𝑃𝑖𝑑𝐸𝑖
5.
𝑀𝑘𝑙𝑖𝑗 =1
2
𝜕2𝑥𝑘𝑙𝜕𝐸𝑗𝐸𝑖
=1
2
𝜕χ𝑖𝑗
𝜕𝑋𝑘𝑙
Gibbs free energy
𝜕3𝐺
𝜕𝐸𝑗𝜕𝐸𝑖𝜕𝑋𝑘𝑙= −
𝜕𝑥𝑘𝑙𝜕𝐸𝑗𝜕𝐸𝑖
= −2𝑀𝑘𝑙𝑖𝑗
𝜕3𝐺
𝜕𝑋𝑘𝑙𝜕𝐸𝑗𝜕𝐸𝑖= −
𝜕χ𝑖𝑗
𝜕𝑋𝑘𝑙
Differentiate by Ei , then Ej Differentiate by Ej , then by Xkl
First derivatives
Third derivatives
𝜕𝐺
𝜕𝑋𝑘𝑙= −𝑥𝑘𝑙 ; 𝑥𝑘𝑙 = 𝑀𝑘𝑙𝑖𝑗𝐸𝑖𝐸𝑗
𝜕𝐺
𝜕𝐸𝑖= −𝑃𝑖 ; 𝑃𝑖 = χ𝑖𝑗𝐸𝑗
Indirect Calculation of ElectrostrictionThermodynamical derivation for 𝑀𝑖𝑗𝑘𝑙
𝐺 = 𝑢 − 𝑇. 𝑆 − 𝐸. 𝑃 − 𝑥. 𝑋
𝑑𝐺 = −𝑆. 𝑑𝑇 − 𝑥𝑘𝑙𝑑𝑋𝑘𝑙 − 𝑃𝑖𝑑𝐸𝑖
5.
New Expression
Electromechanical coupling
Electric Field 𝑬 Polarisation Field 𝑷
Strain(𝒙𝑖𝑗)
Stress(𝑋𝑖𝑗)
𝑥𝑖𝑗 = 𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛 𝑥𝑖𝑗 = 𝑸𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛
𝑋𝑖𝑗 = 𝒎𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛 𝑋𝑖𝑗 = 𝒒𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛
6.
Electromechanical coupling
Electric Field 𝑬 Polarisation Field 𝑷
Strain(𝒙𝑖𝑗)
Stress(𝑋𝑖𝑗)
𝑥𝑖𝑗 = 𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛 𝑥𝑖𝑗 = 𝑸𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛
𝑋𝑖𝑗 = 𝒎𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛 𝑋𝑖𝑗 = 𝒒𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛
𝑴𝒊𝒋𝒎𝒏 =1
2
𝜕2𝑥𝑖𝑗
𝜕𝐸𝑚𝐸𝑛=𝟏
𝟐
𝝏χ𝒊𝒋
𝝏𝑿𝒎𝒏
𝒎𝒊𝒋𝒎𝒏 =1
2
𝜕2𝑋𝑖𝑗
𝜕𝐸𝑚𝐸𝑛=𝟏
𝟐
𝝏χ𝒊𝒋
𝝏𝒙𝒎𝒏
𝑸𝒊𝒋𝒎𝒏 =1
2
𝜕2𝑥𝑖𝑗
𝜕𝑃𝑚𝑃𝑛= −
𝟏
𝟐
𝝏𝜼𝒊𝒋
𝝏𝑿𝒎𝒏
𝒒𝒊𝒋𝒎𝒏 =1
2
𝜕2𝑋𝑖𝑗
𝜕𝑃𝑚𝑃𝑛= −
𝟏
𝟐
𝝏𝜼𝒊𝒋
𝝏𝒙𝒎𝒏
6.
A-Priori Methodological Advantages
∆χ/∆η
∆𝑋/∆𝑥
Indirect method Direct method
∆𝑥/∆𝑋
∆𝐸/∆𝑃
7.
A-Priori Methodological Advantages
∆χ/∆η
∆𝑋/∆𝑥
Indirect method
Hydrostatic strain/pressure
Direct method
ꓫ Always Unidirectional field.
∆𝑥/∆𝑋
∆𝐸/∆𝑃
7.
A-Priori Methodological Advantages
∆χ/∆η
∆𝑋/∆𝑥
Indirect method
Hydrostatic strain/pressure
Direct method
ꓫ Always Unidirectional field.
ꓫ𝐸𝑔
𝑁𝑘𝑒>
𝐸𝑎
2𝜋must hold
∆𝑥/∆𝑋
∆𝐸/∆𝑃
7.
A-Priori Methodological Advantages
∆χ/∆η
∆𝑋/∆𝑥
Indirect method
Hydrostatic strain/pressure
Can investigate at closing 𝐸𝑔.
Direct method
ꓫ Always Unidirectional field.
ꓫ𝐸𝑔
𝑁𝑘𝑒>
𝐸𝑎
2𝜋must hold
∆𝑥/∆𝑋
∆𝐸/∆𝑃
7.
A-Priori Methodological Advantages
∆χ/∆η
∆𝑋/∆𝑥
Indirect method
Hydrostatic strain/pressure
Can investigate at closing 𝐸𝑔.
Decomposition of tensor
Direct method
ꓫ Always Unidirectional field.
ꓫ𝐸𝑔
𝑁𝑘𝑒>
𝐸𝑎
2𝜋must hold
ꓫ No easy decomposition of tensor
∆𝑥/∆𝑋
∆𝐸/∆𝑃
7.
A-Priori Methodological Advantages
∆χ/∆η
∆𝑋/∆𝑥
Indirect method
Hydrostatic strain/pressure
Can investigate at closing 𝐸𝑔.
Decomposition of tensor
Well established (1990): excellent infrastructure
Direct method
ꓫ Always Unidirectional field.
ꓫ𝐸𝑔
𝑁𝑘𝑒>
𝐸𝑎
2𝜋must hold
ꓫ No easy decomposition of tensor
ꓫ New (2009): - No parallelism for D- PAW erroneous results
∆𝑥/∆𝑋
∆𝐸/∆𝑃
7.
Finite Field and PAW in ABINIT 7.
• P Vs E by DFPT and finite field with both PAW and NCPP
• PAW finite field: incorrect behaviour, permitivity
• => Incorrect electrsotrictive coefficients
• DFPT works with PAW – can use current method with PAW for large unit cells/supercells
A-Priori Methodological Advantages
∆χ/∆η
∆𝑋/∆𝑥
Indirect method
Hydrostatic strain/pressure
Can investigate at closing 𝐸𝑔.
Decomposition of tensor
Well established (1990): excellent infrastructure
Direct method
ꓫ Always Unidirectional field.
ꓫ𝐸𝑔
𝑁𝑘𝑒>
𝐸𝑎
2𝜋must hold
ꓫ No easy decomposition of tensor
ꓫ New (2009): - No parallelism for D- PAW erroneous results
∆𝑥/∆𝑋
∆𝐸/∆𝑃
7.
A-Priori Methodological Advantages
∆χ/∆η
∆𝑋/∆𝑥
Indirect method
Hydrostatic strain/pressure
Can investigate at closing 𝐸𝑔.
Decomposition of tensor
Well established (1990): excellent infrastructure
Direct method
ꓫ Always Unidirectional field.
ꓫ𝐸𝑔
𝐸𝑒<
𝑁𝑘𝑎
2𝜋must hold
ꓫ No easy decomposition of tensor
ꓫ New (2009): - No parallelism for D- PAW erroneous results
∆𝑥/∆𝑋
∆𝐸/∆𝑃
7.
Indirect method more robust, more efficient, easier to implement.
Outline• Motivation
- Wℎ𝑦 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛- 𝐻𝑜𝑤 𝑡𝑜 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒: 𝐿𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑟𝑒𝑣𝑖𝑒𝑤
• Indirect Calculation of Electrostriction- Introduction and advantages
- Validation/Comparison
• Application
• Summary and Outlook
Indirect Calculation of ElectrostrictionSimulation Details
• DFPT ABINIT;
• Ecut=50 Ha;
• 8x8x8 monkorst pack-grid;
• Pseudodojo NCPP;
• PBEsol;
• Rocksalt MgO
8.
• Strain (xij) between ±0.5%; calculate stress 𝑋𝑖𝑗,
susceptibility 𝜒, and inverse susceptibility 𝜂.
• Fit 𝜒 and 𝜂 vs 𝑋𝑖𝑗/𝑥𝑖𝑗 for electrostrictive
coefficients.
• Qh, Mh, mh and qh obtained in same calculation
Indirect Calculation of ElectrostrictionPressure derivative of χ𝑖𝑗and η𝑖𝑗 in MgO
9.
Direct Calculation of ElectrostrictionElectric/Displacement Field Response - Strain
• Full energy minimisation under fixed 𝐸𝑧, 𝐷𝑧
• Longitudinal expansion; Transverse contraction.
10.
Direct Calculation of ElectrostrictionElectric/Displacement Field Response - Strain
• Full energy minimisation under fixed 𝐸𝑧, 𝐷𝑧
• Longitudinal expansion; Transverse contraction.
10.
• Stress coefficients m, q: Relax only internal positions under fixed 𝐸𝑧, 𝐷𝑧; calculate stress.
Comparison between Direct and Indirect methods
• Agreement between methods and exp: Coefficientsnormalised to Exp 1 in bar chart.
11.
Comparison between Direct and Indirect methods
• Agreement between methods and exp: Coefficientsnormalised to Exp 1 in bar chart.
• Faster convergence with k-points and cut-off energy.
• 8 times faster for given k-point density, cutoff energy.
12.
Outline
• Motivation- Wℎ𝑦 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛- 𝐻𝑜𝑤 𝑏𝑒𝑠𝑡 𝑡𝑜 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒: 𝐿𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑟𝑒𝑣𝑖𝑒𝑤
• Indirect Calculation of Electrostriction- Introduction and advantages
- Validation/Comparison
• Application
• Summary and Outlook
ApplicationElectrostriction at ferroelectric phase transition: KTaO3
13.
• Compressive strain induced phase transition: 𝜖𝑧 diverges
• Transition driven z-polar soft mode
ApplicationElectrostriction at ferroelectric phase transition: KTaO3
13.
• Compressive strain induced phase transition: 𝜖𝑧 diverges
• Transition driven by z-polar soft mode
• 𝑀33 also diverges
• Reaches 1𝑥10−15: 𝑑33𝑒𝑓𝑓
= 60 000 pm/V; compare 162 pm/V for PZT
ApplicationElectrostriction at ferroelectric phase transition: KTaO3
13.
• Compressive strain induced phase transition: 𝜖𝑧 diverges
• Transition driven by z-polar soft mode
• 𝑀33 also diverges
• Reaches 1𝑥10−15: 𝑑33𝑒𝑓𝑓
= 60 000 pm/V; compare 162 pm/V for PZT
• decomposition shows soft polar mode responsable
⟹ Large strain in response to E field does not mean large strain in response to P field
Materials with large Qh do not necessarily have large Mh
ApplicationCorrelation between Mh and Qh
14.
Outline• Motivation
- Wℎ𝑦 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛- 𝐻𝑜𝑤 𝑏𝑒𝑠𝑡 𝑡𝑜 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒: 𝐿𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑟𝑒𝑣𝑖𝑒𝑤
• Indirect Calculation of Electrostriction- Introduction and advantages
- Validation/Comparison
• Application
• Summary
• Stress/strain dependence of permitivity best way to computeElectrostriction.
• Validated method against finite field calculation and experiment
• Advantages: Fewer calculations; Faster convergence; 8times faster computation; Decomposition of tensors
• Applications:
- Analysed electrostriction at ferroelectric phase transition
- Demonstrated 𝑀ℎ and 𝑄ℎ are uncorrelated
Summary 15.
Project ANR-19-ASTR-0024, “MEGAEM”
Electrostriction at ferroelectric phase transition of KTaO3Q coefficients
14.
• Q33 does vary at the phase transition, but onlyby ≈4%
Electrostriction at ferroelectric phase transition of KTaO3Q coefficients
14.
• Q33 does vary at the phase transition, but onlyby ≈4%
• Q12 changes sign
Electrostriction at ferroelectric phase transition of KTaO3Q coefficients
14.
• q33 less complicated behaviour
ApplicationDecomposition of BaZrO3 Mh
• Permitivity has electronic contribution, but electrostriction does not
• Softest polar mode, with largest polarity contributes most
• 4 coefficients 𝑚ℎ, 𝑀ℎ, 𝑞ℎ , and 𝑄ℎ have same composition
14.
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