Takeshi Nishimatsu

http://dx.doi.org/10.1103/PhysRevB.78.104104

http://loto.sourceforge.net/feram/

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- Various applications of ferroelectric thin-films:
- multilayer capacitors
- nonvolatile FeRAMs
- nanoactuators

- Down-sizing of FeRAMs (nano-capacitors of ferroelectric thin films) is highly demanded.
- Epitaxially constraint ferroelectric films made by PLD.
- Effect of imperfect screening of electrodes is NOT well understood.
- [M. Dawber et al.: J. Phys. Condes. Matter
**15**, L393 (2003)]

- [M. Dawber et al.: J. Phys. Condes. Matter
- Fatigue causes damages in polarization behaviors of ferroelectric capacitors.

**Dynamics**of ferroelectric thin-film capacitors:- Hysteresis loop
- Polarization switching
- Dynamics of domain wall

- Their
**nanosize effects**and**temperature dependences**remain poorly known.- Experimentally,
*in situ*observations are difficult. - Theoretically,
- The long-range Coulomb interaction limits the size and time of molecular-dynamics (MD) simulations.
- How to include depolarization fields caused by imperfect electrodes was unclear.

- Experimentally,

- Development of fast molecular dynamics (MD) code which can simulate ferroelectric thin-film capacitors for a realistic system size (up to 100 nm) and a realistic time span ($>$ 1 ns).
- Clarify the effect of dead layers between ferroelectrics and electrodes.
- Predict the effect of the compressive strain arising from epitaxial constraints.

$\to $ We develop "feram" code.

- Scientific features
- Molecular dynamics (MD) simulation with first-principles-based effective Hamiltonian
- $AB$O
_{3}-type perovskite ferroelectrics and relaxors - Uses supercell; ($AB$O
_{3})${}_{N}$, $N=32\times 32\times 512$ $\Rightarrow $ 2,621,440 atoms - Coarse-graining; reduction of the number of degree of freedom
- Long-range dipole-dipole interaction is treated in reciprocal-space; k-locality of the force matrix
- Applications:
- Phase transition of bulk ferroelectrics
- Capacitor,
**ferroelectric thin film**is sandwiched between short-circuited electrodes

- Technical features
- Fast
- FFTW library version 3 http://www.FFTW.org/
- Parallelized with OpenMP http://www.OpenMP.org/

- Multi-platform
- Linux PC
- HITACHI SR11000 Super Technical Server
- SONY PLAYSTATION3 (ongoing)

- Object oriented programming (OOP) in Fortran 2003
- GNU autotools http://www.gnu.org/software/autoconf/
- Free software (GPLv3) http://loto.sf.net/feram/

- Fast

$${u}_{\alpha}^{2}=\{{v}_{\alpha}^{A}{\}}^{2}+\{{v}_{\alpha}^{B}{\}}^{2}+\{{v}_{\alpha}^{{\mathrm{O}}_{\mathrm{I}}}{\}}^{2}+\{{v}_{\alpha}^{{\mathrm{O}}_{\mathrm{I}\mathrm{I}}}{\}}^{2}+\{{v}_{\alpha}^{{\mathrm{O}}_{\mathrm{I}\mathrm{I}\mathrm{I}}}{\}}^{2}$$

- Real perovskite-like system has $15N+6$ degree of freedom
- $N$ unit cells in a supercell
- 5 atoms per unit cell
- Each atom can move along 3 directions
- 6 components of strain

- Simplified model has $6N+6$ degree of freedom
- Define 1 dipole ${Z}^{*}\mathit{u}\left(\mathit{R}\right)$ per unit cell
- 1 acoustic displacement (Inhomogeneous strain) $\mathit{w}\left(\mathit{R}\right)$ per unit cell

Supercell of ($AB$O_{3})${}_{N}$ with $\left\{\mathit{u}\right(\mathit{R}\left)\right\}$ and $\left\{\mathit{w}\right(\mathit{R}\left)\right\}$:
$${H}^{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{{M}_{\mathrm{d}\mathrm{i}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}^{*}}{2}\sum _{\mathit{R},\alpha}{\stackrel{\cdot}{u}}_{\alpha}^{2}\left(\mathit{R}\right)+\frac{{M}_{\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}}^{*}}{2}\sum _{\mathit{R},\alpha}{\stackrel{\cdot}{w}}_{\alpha}^{2}\left(\mathit{R}\right)+{V}^{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}}\left(\right\{\mathit{u}\left\}\right)+{V}^{\mathrm{d}\mathrm{p}\mathrm{l}}\left(\right\{\mathit{u}\left\}\right)+{V}^{\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{t}}\left(\right\{\mathit{u}\left\}\right)+{V}^{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s},\phantom{\rule{0.167em}{0ex}}\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{o}}({\eta}_{1},...,{\eta}_{6})+{V}^{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s},\phantom{\rule{0.167em}{0ex}}\mathrm{i}\mathrm{n}\mathrm{h}\mathrm{o}}\left(\right\{\mathit{w}\left\}\right)+{V}^{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p},\phantom{\rule{0.167em}{0ex}}\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{o}}\left(\right\{\mathit{u}\},{\eta}_{1},...,{\eta}_{6})+{V}^{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p},\phantom{\rule{0.167em}{0ex}}\mathrm{i}\mathrm{n}\mathrm{h}\mathrm{o}}\left(\right\{\mathit{u}\},\{\mathit{w}\left\}\right)-{Z}^{*}\sum _{\mathit{R}}\mathcal{E}\cdot \mathit{u}\left(\mathit{R}\right)$$

$${V}^{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}}\left(\mathit{u}\right)={\kappa}_{2}{u}^{2}+\alpha {u}^{4}+\gamma ({u}_{y}^{2}{u}_{z}^{2}+{u}_{z}^{2}{u}_{x}^{2}+{u}_{x}^{2}{u}_{y}^{2})$$

where

$${u}^{2}={u}_{x}^{2}+{u}_{y}^{2}+{u}_{z}^{2}.$$

In the effective Hamiltonian, dipole-dipole interactions are separated into the long-rage term and the short-range term.

Long-range: $${V}^{\mathrm{d}\mathrm{p}\mathrm{l}}\left(\right\{\mathit{u}\left\}\right)=\frac{1}{2}\sum _{i=1}^{N}\sum _{\alpha}\sum _{j=1}^{N}\sum _{\beta}{u}_{\alpha}\left({\mathit{R}}_{i}\right){\mathrm{\Phi}}_{\alpha \beta}\left({\mathit{R}}_{ij}\right){u}_{\beta}\left({\mathit{R}}_{j}\right),$$

$${\mathrm{\Phi}}_{\alpha \beta}\left({\mathit{R}}_{ij}\right)=\frac{{Z}^{*2}}{{\epsilon}_{\mathrm{\infty}}}\sum _{\mathit{n}}{}^{\prime}\frac{{\delta}_{\alpha \beta}-3\left(\stackrel{\u02c6}{{\mathit{R}}_{ij}+\mathit{n}}{)}_{\alpha}\right(\stackrel{\u02c6}{{\mathit{R}}_{ij}+\mathit{n}}{)}_{\beta}}{|{\mathit{R}}_{ij}+\mathit{n}{|}^{3}},$$

$\mathit{n}$ is the supercell lattice vector: $${n}_{\alpha}=\cdots ,-2{L}_{\alpha}{a}_{0},-{L}_{\alpha}{a}_{0},0,{L}_{\alpha}{a}_{0},2{L}_{\alpha}{a}_{0},\cdots $$

Short-range: $${V}^{\mathrm{s}\mathrm{h}\mathrm{r}\mathrm{t}}\left(\right\{\mathit{u}\left\}\right)=\frac{1}{2}\sum _{i=1}^{N}\sum _{\alpha}\sum _{j}^{3\mathrm{n}\mathrm{n}}\sum _{\beta}{u}_{\alpha}\left({\mathit{R}}_{i}\right)\phantom{\rule{0.167em}{0ex}}{J}_{ij,\alpha \beta}\phantom{\rule{0.167em}{0ex}}{u}_{\beta}\left({\mathit{R}}_{j}\right)$$ ${J}_{ij,\alpha \beta}={J}_{k}$: Short-range interaction matrix ($k=1,\cdots ,7$)

LO-TO splitting is the ion version of the plasma frequency.

- [King-Smith and Vanderbilt: Phys. Rev. B
**49**, 5828 (1994)] - [Zhong, Vanderbilt, and Rabe: Phys. Rev. B
**52**, 6301 (1995)]

#--- Method, Temperature, and mass --------------- method = 'md' # 'md' - Molecular Dynamics with Nose-Poincare thermostat (Canonical ensemble) # 'lf' - Leap Frog (Microcanonical ensemble) # 'hl' - Hysteresis Loop # 'mc' - Monte Carlo (not implemented yet) GPa = -5.0 kelvin = 300 mass_amu = 39.0 # Required for MD Q_Nose = 1.0 #--- System geometry ----------------------------- bulk_or_film = 'bulk' L = 32 32 32 a0 = 3.94 lattice constant a0 [Angstrom] #--- Time step ----------------------------------- dt = 0.002 [ps] n_thermalize = 40000 n_average = 10000 n_coord_freq = 5000 Write a snapshot to the disk every 5000 steps #--- On-site (Polynomial of order 4) ------------- P_kappa2 = 5.502 [eV/Angstrom^2] # P_4(u) = kappa2*u^2 + alpha*u^4 P_alpha = 110.4 [eV/Angstrom^4] # + gamma*(u_y*u_z+u_z*u_x+u_x*u_y), P_gamma = -163.1 [eV/Angstrom^4] # where u^2 = u_x^2 + u_y^2 + u_z^2 #--- Inter-site ---------------------------------- j = -2.648 3.894 0.898 -0.789 0.562 0.358 0.179 j(i) [eV/Angstrom^2] #--- Elastic Constants --------------------------- B11 = 126. B12 = 44.9 B44 = 50.3 [eV] #--- Elastic Coupling ---------------------------- B1xx = -211. [eV/Angstrom^2] B1yy = -19.3 [eV/Angstrom^2] B4yz = -7.75 [eV/Angstrom^2] #--- Dipole -------------------------------------- init_dipo_avg = 0.00 0.00 0.00 [Angstrom] # Average of initial dipole displacements init_dipo_dev = 0.02 0.02 0.02 [Angstrom] # Deviation of initial dipole displacements Z_star = 9.956 epsilon_inf = 5.24

feram is a **molecular dynamics (MD)** program for bulk and thin-film ferroelectrics.

Animations of horizontal slices of heating-up and cooling-down
simulations for BaTiO_{3} thin-film capacitors with short-circuited
electrodes under 1% in-plane biaxial compressive strain.
The +z-polarized and −z-polarized sites are
denoted by red open squares and blue filled squares, respectively.
Used supercell sizes are Lx×Ly×Lz = 40×40×2(l+d) .

- (a) l=15, d=1
- (b) l=31, d=1
- (c) l=127, d=1
- (d) l=255, d=1
- heating-up (coming soon)
- cooling-down

- (e) l=32, d=0

Relaxors Pb${B}_{x}^{\prime}$${B}_{1-x}^{\prime \prime}$O${}_{3}$: Pb${}^{2+}$

- Pb(Sc${}_{1/2}$Nb${}_{1/2}$)O${}_{3}$ (PSN): Sc${}^{3+}$, Nb${}^{5+}$
- Pb(Sc${}_{1/2}$Ta${}_{1/2}$)O${}_{3}$ (PST): Sc${}^{3+}$, Ta${}^{5+}$
- Pb(Mg${}_{1/3}$Nb${}_{2/3}$)O${}_{3}$ (PMN:) Mg${}^{2+}$, Nb${}^{5+}$
- $(1-x)$Pb(Mg${}_{1/3}$Nb${}_{2/3}$)O${}_{3}$-xPbTiO${}_{3}$ (PMN-$x$PT): Mg${}^{2+}$, Nb${}^{5+}$, Ti${}^{4+}$
- Pb(Mg${}_{1/3}$Ta${}_{2/3}$)O${}_{3}$ (PMT): Mg${}^{2+}$, Ta${}^{5+}$
- Pb(Zn${}_{1/3}$Nb${}_{2/3}$)O${}_{3}$ (PZN): Zn${}^{2+}$, Nb${}^{5+}$

The averaged valence number of B-site ions ${B}_{x}^{\prime}$ and ${B}_{1-x}^{\prime \prime}$ is $4+$.

Displacement of Pb is the main source of dipole moment.

$$\epsilon \left(\omega \right)-{\epsilon}_{\mathrm{\infty}}=\frac{1}{3{\epsilon}_{0}V{k}_{B}T}\left[\u27e8{\mathit{p}}^{2}\u27e9+i\omega {\int}_{0}^{\mathrm{\infty}}dt{e}^{i\omega t}\u27e8\mathit{p}\left(t\right)\cdot \mathit{p}\left(0\right)\u27e9\right]$$ where $\mathit{p}\left(t\right)$ is the total electric dipole moment in the supercell at time $t$,

$$\mathit{p}\left(t\right)={Z}^{*}\sum _{\mathit{R}}\mathit{u}(\mathit{R};t).$$

Estimation of computational time

32x32x32 unit cells, $\mathrm{\Delta}t=2$ fs, [AMD64 1.8GHz dual core] or [SR11000 1 node = 16 cores]

1THz, $T=1\phantom{\rule{0.167em}{0ex}}$ps, 500 steps, 54 sec or 8.4 sec.

1GHz, $T=1\phantom{\rule{0.167em}{0ex}}$ns, 500,000 steps, 900 min. or 140 min.

1MHz, $T=1\mu $s, 500,000,000 steps, 620 days or 97 days

So, I am planning to calculate and compare $\epsilon \left(\omega \right)$ for $\omega /\left(2\pi \right)$ = 10M ... 10GHz.

- We developed "feram", a fast simulator for perovskite-type ferroelectric bulks and thin films
- Molecular dynamics (MD) simulation with first-principles-based effective Hamiltonian
- Phase transition of bulk ferroelectrics
- Thin-film capacitor: perfect and imperfect short-circuited electrodes
- Striped domain structure is predicted
- Hysteresis loops for epitaxially constrained and "free" BaTiO
_{3}film capacitors

- Investigations of relaxors with MD are proposed.