On the calculation of electrostriction by DFT

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.


• Electrostriction (all crystals), strain quadratic with 𝑬;

Piezoelectricity (non-centric), strain linear with 𝑬


𝑥𝑖𝑗 = 𝒅𝒊𝒋𝒎𝐸𝑚 +𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛

Strain

1.


• Electrostriction (all crystals), strain quadratic with 𝑬;

Piezoelectricity (non-centric), strain linear with 𝑬


𝑥𝑖𝑗 = 𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛

Strain

1.


Electric Field 𝑬

Strain(𝒙𝑖𝑗)

Stress(𝑋𝑖𝑗)

𝑥𝑖𝑗 = 𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛

𝑋𝑖𝑗 = 𝒎𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛

2.


Electric Field 𝑬 Polarisation Field 𝑷



𝑥𝑖𝑗 = 𝑴𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛 𝑥𝑖𝑗 = 𝑸𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛

𝑋𝑖𝑗 = 𝒎𝒊𝒋𝒎𝒏𝐸𝑚𝐸𝑛 𝑋𝑖𝑗 = 𝒒𝒊𝒋𝒎𝒏𝑃𝑚𝑃𝑛

2.







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


- W𝐡𝐲 𝐜𝐚𝐥𝐜𝐮𝐥𝐚𝐭𝐞 𝐞𝐥𝐞𝐜𝐭𝐫𝐨𝐬𝐭𝐫𝐢𝐜𝐭𝐢𝐨𝐧?- How best to calculate: Literature review

• Indirect Calculation of Electrostriction- Derivation and advantages


• Application


• 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)]

†


- Wℎ𝑦 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛- 𝑯𝒐𝒘 𝒃𝒆𝒔𝒕 𝒕𝒐 𝒄𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒆: 𝑳𝒊𝒕𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒓𝒆𝒗𝒊𝒆𝒘


- Example calculation: MgO- Comparison/validation against direct calculation

• Application


● 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


- Why calculate electrostriction?- How to calculate: Literature review



• Application


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


𝜕𝐺



𝐺 = 𝑢 − 𝑇. 𝑆 − 𝐸. 𝑃 − 𝑥. 𝑋


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


𝜕𝐺



𝐺 = 𝑢 − 𝑇. 𝑆 − 𝐸. 𝑃 − 𝑥. 𝑋


5.

New Expression







6.







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.


∆χ/∆η

∆𝑋/∆𝑥

Indirect method

Hydrostatic strain/pressure

Direct method

ꓫ Always Unidirectional field.

∆𝑥/∆𝑋

∆𝐸/∆𝑃

7.


∆χ/∆η

∆𝑋/∆𝑥

Indirect method


Direct method


ꓫ𝐸𝑔

𝑁𝑘𝑒>

𝐸𝑎

2𝜋must hold

∆𝑥/∆𝑋

∆𝐸/∆𝑃

7.


∆χ/∆η

∆𝑋/∆𝑥

Indirect method


Can investigate at closing 𝐸𝑔.

Direct method


ꓫ𝐸𝑔

𝑁𝑘𝑒>

𝐸𝑎

2𝜋must hold

∆𝑥/∆𝑋

∆𝐸/∆𝑃

7.


∆χ/∆η

∆𝑋/∆𝑥

Indirect method



Decomposition of tensor

Direct method


ꓫ𝐸𝑔

𝑁𝑘𝑒>

𝐸𝑎

2𝜋must hold

ꓫ No easy decomposition of tensor

∆𝑥/∆𝑋

∆𝐸/∆𝑃

7.


∆χ/∆η

∆𝑋/∆𝑥

Indirect method




Well established (1990): excellent infrastructure

Direct method


ꓫ𝐸𝑔

𝑁𝑘𝑒>

𝐸𝑎

2𝜋must hold


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


∆χ/∆η

∆𝑋/∆𝑥

Indirect method





Direct method


ꓫ𝐸𝑔

𝑁𝑘𝑒>

𝐸𝑎

2𝜋must hold



∆𝑥/∆𝑋

∆𝐸/∆𝑃

7.


∆χ/∆η

∆𝑋/∆𝑥

Indirect method





Direct method


ꓫ𝐸𝑔

𝐸𝑒<

𝑁𝑘𝑎

2𝜋must hold



∆𝑥/∆𝑋

∆𝐸/∆𝑃

7.

Indirect method more robust, more efficient, easier to implement.


- Wℎ𝑦 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛- 𝐻𝑜𝑤 𝑡𝑜 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒: 𝐿𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑟𝑒𝑣𝑖𝑒𝑤

• Indirect Calculation of Electrostriction- Introduction and advantages


• Application


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



• Application


ApplicationElectrostriction at ferroelectric phase transition: KTaO3

13.

• Compressive strain induced phase transition: 𝜖𝑧 diverges

• Transition driven z-polar soft mode


13.


• Transition driven by z-polar soft mode

• 𝑀33 also diverges

• Reaches 1𝑥10−15: 𝑑33𝑒𝑓𝑓

= 60 000 pm/V; compare 162 pm/V for PZT


13.


• 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.


- Wℎ𝑦 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛- 𝐻𝑜𝑤 𝑏𝑒𝑠𝑡 𝑡𝑜 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒: 𝐿𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑟𝑒𝑣𝑖𝑒𝑤



• 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%


14.

• Q33 does vary at the phase transition, but onlyby ≈4%

• Q12 changes sign


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.

Documents

On the calculation of electrostriction by DFT