Low electronic conductivity of ${\mathrm{Li}}_7{\mathrm{La}}_3{\mathrm{Zr}}_2{\mathrm{O}}_{12}$ solid electrolytes from first principles

Lithium-rich garnets such as ${\mathrm{Li}}_7{\mathrm{La}}_3{\mathrm{Zr}}_2{\mathrm{O}}_{12}$ (LLZO) are promising solid electrolytes with potential application in all-solid-state batteries that use lithium-metal anodes. The practical use of garnet electrolytes is limited by pervasive lithium-dendrite growth, which leads to short-circuiting and cell failure. One proposed mechanism of lithium-dendrite growth is the direct reduction of lithium ions to lithium metal within the electrolyte, and lithium garnets have been suggested to be particularly susceptible to this dendrite-growth mechanism due to high electronic conductivities relative to other solid electrolytes. The electronic conductivities of LLZO and other lithium-garnet solid electrolytes, however, are not yet well characterized. Here, we present a general scheme for calculating the intrinsic electronic conductivity of a nominally insulating material under variable synthesis conditions from first principles, and apply this to the prototypical lithium-garnet LLZO. Our model predicts that under typical battery operating conditions, electron and hole mobilities are low ($<1{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$), and bulk electron and hole carrier concentrations are negligible, irrespective of initial synthesis conditions or dopant levels. These results suggest that the bulk electronic conductivity of LLZO is not sufficiently high to cause bulk lithium-dendrite growth during cell operation, and that any non-negligible electronic conductivity in lithium garnet samples is likely due to extended defects or surface contributions.

Figure 1

Schematic showing the workflow used to calculate the electronic conductivity from first-principles inputs.

Figure 2

The electronic band structure of t-LLZO calculated using HSE06, plotted along a high symmetry path in the Brillouin zone according to the Bradley and Cracknell notation [91]. The colored points mark the band edges used to calculate the effective masses, with numeric labels indicating the corresponding entry in Table 1.

Figure 3

Chemical potential stability region of LLZO in the $\{\mathrm{\Delta }{\mu }_{\mathrm{Li}},\mathrm{\Delta }{\mu }_{\mathrm{O}}\}$ plane. The dark blue region is constricted by Eq. (7) to typical synthesis conditions ranging from $T=1000$ to 1500 K and $P_{{\mathrm{O}}_2}=1$ to $1\times {10}^{-10}\mathrm{atm}$.

Figure 4

n- and p-type carrier concentrations at six sets of chemical potentials (each set corresponds to a vertex of the estimated chemical potential stability region that LLZO can be synthesised within). The chemical potentials used to calculate defect concentrations are shown above each plot. The carrier concentrations are calculated at $1500\mathrm{K}$ initially, the concentrations of all defects other than lithium vacancies, interstitials and electron and hole concentrations are then fixed to these high temperature values for subsequent, lower temperature solutions. All carrier concentrations are given for both an undoped sample, and a sample containing 0.15 per formula unit of some dopant $M^{2+}$.

Figure 5

Effective “room temperature” ($298\mathrm{K}$) electronic conductivities for LLZO synthesised under O-rich/metal-poor (top panel) and O-poor/metal-rich (bottom panel) conditions, as a function of $\{V_{\mathrm{Li}},{\mathrm{Li}}_{\mathrm{i}}\}$ and $\{{\mathrm{e}}^{-}/{\mathrm{h}}^{•}\}$ pseudoequilibration temperature. Conductivities are calculated via Eq. (1), using the electronic carrier concentrations in Fig. 4 and the previously calculated maximum room-temperature electron and hole carrier mobilities of $0.2{\mathrm{cm}}^2{\mathrm{V}}^{-1}\mathrm{s}$. Solid lines show results for undoped LLZO, and dashed lines show results under supervalent doping with $M^{2+}$ at a concentration of 0.15 per formula unit.