Simulating charging characteristics of lithium iron phosphate by electro-ionic optimization on a quantum annealer

Abstract

The rapid evolution of quantum computing hardware opens up new avenues in the simulation of energy materials. Today's quantum annealers are able to tackle complex combinatorial optimization problems. A formidable challenge of this type is posed by materials with site-occupational disorder for which atomic arrangements with a low, or lowest, energy must be found. In this article, a method is presented for the identification of the correlated ground-state distribution of both lithium ions and redox electrons in lithium iron phosphate (LFP), a widely employed cathode material in lithium-ion batteries. The point-charge Coulomb energy model employed correctly reproduces the LFP charging characteristics. As is shown, grand-canonical transformation of the energy cost function makes the combinatorial distribution problem solvable on quantum annealing (QA) hardware. The QA output statistics follow a pseudo-thermal behavior characterized by a problem-dependent effective sampling temperature, which has bearings on the estimated scaling of the QA performance with system size. This work demonstrates the potential of quantum computation for the joint optimization of electronic and ionic degrees of freedom in energy materials.

Figures (6)

• Figure 1: (a) Simulation cell of Li_xFePO4 ($1\times2\times2$ supercell) comprising a total of 16 Li sites (either occupied or vacant) and 16 Fe sites (occupied by either Fe(II) or Fe(III)). The formal ionic charges used in the Coulomb energy model are indicated. (b) Comparison of Coulomb ($E_{\mathrm{coul}}$) vs. DFT ($E_{\mathrm{DFT}}$) energies for different configurations of Li ions in the semi-lithiated $1\times4\times1$ supercell. For the Coulomb energies, the minimum energy distributions of the redox electrons across the iron sites for the given (frozen) Li configurations were determined by full enumeration.
• Figure 2: (a) Coulomb energies of minimum energy configurations for semi-lithiated LFP (Li_0.5FePO4) as a function of the size (length) of the simulation cell along different lattice directions. The energies are given per Li_0.5FePO4 formula unit (FU) on a relative scale using the ground state configuration for the $1 \times 1 \times 1$ unit cell as a reference. The simulation cells had elongated shapes of dimensions $n \times 1 \times 1$, $1 \times n \times 1$, and $1 \times 1 \times n$ with increasing multiplicities ($n$) along the $a$ (red data points), $b$ (green data points) and $c$ (blue data points) directions, respectively. The respective minimum energy configurations are shown. (b) Extended plot of the minimum energy as a function of cell size in $b$ direction. The energy of the homogeneously distributed ground-state configuration for the $1 \times 1 \times n$ cells (with even $n$) and the average energy of bulk LiFePO4 and bulk FePO4 are given as reference lines. (c) Minimum energy configuration for a $1 \times 16 \times 1$ cell with a phase separation into LiFePO4 and FePO4 domains. Color labelling for different ion species as in Fig. \ref{['fig_LFP_model']}a.
• Figure 3: (a) Variable transformation in particle numbers to define the number of charge-neutral Li^+--e^- pairs, $N = (N_{\ce{Li^+}} + N_{\ce{e^-}})/2$, and the number of net charges, $\Delta N = N_{\ce{Li^+}} - N_{\ce{e^-}}$. (b) Plot of the minimum Coulomb energy ($E_{\mathrm{coul}}^{\mathrm{min}}$) vs. number of Li^+--e^- pairs ($N$), and (c) vs. number of net charges ($\Delta N$), for the $1 \times 2 \times 2$ LFP cell, with fixed $\Delta N = 0$ in (b) and fixed $N = 8$ in (c). The chemical potentials, $\mu_{N}$ and $\mu_{\Delta}$, are determined as the slopes of the respective linear fits (orange dashed lines) around the target values for $N$ and $\Delta N$.
• Figure 4: (a)--(c) Histogram of the energies of QA-sampled configurations (golden bars) and density of states (DOS) of the Coulomb model (blue curve) as a function of the energy, relative to the minimum energy in the ground state, for the $1 \times 2 \times 2$ LFP cell. (d)--(f) Normalized sampling probability $P(E)$, plotted on a logarithmic scale, and Boltzmann fit with given effective sampling temperatures $kT$. The respective ground-state configurations are shown as insets. SOC of $N=4$ in (a) and (d), $N=8$ in (b) and (e), and $N=12$ in (c) and (f) for a total of 16 Li^+ and e^- sites.
• Figure 5: Normalized sampling probability $P(E)$ (plotted on a logarithmic scale) and Boltzmann fit with given effective sampling temperature $kT$, for (a) the $1 \times 2 \times 2$ LFP cell with $N$-unconstrained cost function ($\lambda_{N} = 0$), (b) the $1 \times 4 \times 1$ LFP cell with $N = 8$ constrained cost function ($\lambda_{N} = 1\,\mathrm{eV}$), and (c) the $1 \times 4 \times 1$ LFP cell with $N$-unconstrained cost function ($\lambda_{N} = 0$).