Li-ion diffusion in Li intercalated graphite C₆Li and C₁₂Li probed by μ⁺SR

Abstract

In order to study a diffusive behavior of Li⁺ in Li intercalated graphites, we have measured muon spin relaxation (μ⁺SR) spectra for C₆Li and C₁₂Li synthesized with an electrochemical reaction between Li and graphite in a Li-ion battery. For both compounds, it was found that Li⁺ ions start to diffuse above 230 K and the diffusive behavior obeys a thermal activation process. The activation energy (E_a) for C₆Li is obtained as 270(5) meV, while E_a = 170(20) meV for C₁₂Li. Assuming a jump diffusion of Li⁺ in the Li layer of C₆Li and C₁₂Li, a self-diffusion coefficient D_Li at 310 K was estimated as 7.6(3) × 10⁻¹¹ (cm² s⁻¹) in C₆Li and 14.6(4) × 10⁻¹¹ (cm² s⁻¹) in C₁₂Li.

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1 Introduction

1.1 Li intercalated graphites

Graphite¹ is the most common material for Li-ion batteries,² because of its high rechargeable capacity, low voltage, high cycle performance, and relatively low cost. During a charge (discharge) reaction in the Li-ion batteries, Li⁺ ions are intercalated into (deintercalated from) graphite.

Fig. 1 illustrates schematic structures of three Li intercalated graphites (LIG), namely, graphite, C₁₂Li, and C₆Li, where C₆Li corresponds to the structure for a fully charged state. The structure of LIG is an alternating stack along the c-axis with a graphene layer (G) and Li layer (L). That is, C₁₂Li is represented as GGLGGL, whereas C₆Li as LGLG. Such a structure is usually called a stage structure and the stage number (n) is defined as the number of graphene layers between two adjacent Li layers.

Fig. 1 The crystal structure of (a) and (b) graphite, (c) and (d) C₁₂Li, and (e) and (f) C₆Li. (b, d, and f) Show a c-axis view of graphite, C₁₂Li, and C₆Li, in which Li locates at the center of six-member rings of carbon. The structures were drawn with VESTA.³

In the c plane of C₁₂Li, and C₆Li, one third of the centers of six-member carbon rings are occupied by Li ions, leading to the formation of a super lattice structure [Fig. 1(d) and (f)]. The structural parameters of graphite, C₁₂Li, and C₆Li are summarized in Table 1. Note that, although the c-axis length changes with n, the a-axis length is independent of n for C₁₂Li, and C₆Li.

Table 1 Structural parameters of Li-intercalated graphites

Compound	Stage	Li order	a (Å)	c (Å)
Graphite^4	—	—	2.46	6.71
C12Li^5	2		4.29	7.02
C6Li^5	1		4.29	3.737

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1.2 Motivation for measuring Li diffusion

Li diffusion in solids is one of the most important parameters to determine the charge and discharge rate of the Li-ion batteries. Particularly, we have the following two motivations to measure a diffusion coefficient of Li⁺ (D_Li) in LIG with muon spin relaxation (μ⁺SR).

• Intrinsic D_Li

Since Li⁺ ions are thought to diffuse very rapidly in the LIG lattice due to the layered structure, many researchers have believed that D_Li in LIG is larger than that in the positive electrode materials, such as, Li_xCoO₂,⁶ Li_xNiO₂,⁷ Li_xFePO₄,^8,9 and (Li_xCo_1/3Ni_1/3Mn_1/3)O₂.¹⁰ However, according to recent Li-NMR measurements on C₆Li, D_Li was reported as 3.8 × 10⁻¹¹ cm² s⁻¹ (= 3.8 × 10⁻⁴ m² s⁻¹) at 314 K,¹¹ which ranges roughly in the same order of D_Li in Li_xCoO₂. Although there are many reports on D_Li in LIG with electrochemical techniques, which provide property in a device or setup, not property intrinsic for LIG itself. It is very difficult to determine the intrinsic D_Li with such techniques due to the lack of information on the correct reaction surface area of the LIG electrode in a liquid electrolyte.¹⁰

On the contrary, μ⁺SR is known to provide a very reasonable D_Li for several positive electrode materials.^6–10,12,13 This is because μ⁺SR “sees” the change in local magnetic environments due to Li diffusion. As a result, we obtain intrinsic D_Li in battery materials. Therefore, we have attempted to investigate a diffusive behavior in LIG with μ⁺SR.

• In operando measurements in the near future

For the present Li-NMR and μ⁺SR measurements, battery materials have to be removed from a battery. However, it is highly preferable to observe the diffusive behavior both in positive electrode and anode as a function of the state of charge (SOC). In other words, we need to measure D_Li of battery materials in a living battery in an in operando manner.^14,15

Fortunately, the range of the surface muons, which is the most common muon beam in the muon facilities in the world, is about 150 mg cm⁻² (= 1.5 kg m⁻²). Based on a simple Montecarlo simulation,¹⁶ such surface muons stop in a Li_xCoO₂ electrode with 50 μm thickness through an aluminum plate with 630 μm thickness. This means that we can measure D_Li of electrode materials in functioning Li-ion batteries in the near future. For such a purpose, we also have to initiate measurement of D_Li in negative electrode materials with μ⁺SR.

Here, we report the μ⁺SR results on LIG with two different stage structures, that is, C₁₂Li, and C₆Li. Particularly, in order to apply to in operando measurements in the near future, we have not used pure LIG material but an negative electrode sheet, for which the mixture of LIG and binder (PVdF) is coated on the Cu foil.

2 Experimental

2.1 Fabrication of test cell

In order to prepare C₆Li (C₁₂Li), a graphite half (pouch) cell was fabricated with graphite and Li metal as electrodes. At first, a graphite electrode sheet was prepared by casting the slurry, which consists of 95% of artificial graphite (OMAC, Osaka Gas Chemicals Co., Ltd) and 5 wt% of polyvinylidene fluoride in the N-methyl 2-pyrrolidone solvent, onto a Cu foil with 10 μm thickness. Then, the electrode sheet was dried and pressed so as to reduce the thickness down to 82 μm and increase the areal density up to 9.89 mg cm⁻². And then, the pouch cell was assembled in an Ar-filled glove box. The electrolyte was 1 M LiPF₆ dissolved in ethylene carbonate (EC) and diethyl carbonate (DEC) (3/7 v/v) solution.

2.2 Electrochemical preparation

The pouch cell was charged and discharged under a constant current condition at 0.04 C in the voltage (E) range between 0 and 1500 mV. Note that 1 C means the charge (discharge) rate from empty (full) to full (empty) with one hour. The current and the current density were 3 mA and 0.2 mA cm⁻². Electrochemical measurements were performed at 25 °C by an electrochemical analyzer (ACD-01, Aska Electronic co., Ltd) to determine the initial capacity. Fig. 2(a) shows initial charge and discharge cycles, where Li⁺ ions are intercalated into the graphite when E decreases, while Li⁺ ions are deintercalated from the graphite when E increases. After the initial two cycles, the pouch cell was discharged down to E = 88 mV [Fig. 2(b)], which is estimated as E for C₁₂Li in Open Circuit Voltage measurements. Then, E was kept at 88 mV to obtain a homogeneous C₁₂Li sheet.

Fig. 2 The variation of voltage (E) of a graphite||Li half pouch cell. (a) Li⁺ ions are intercalated into (deintercalated from) graphite as E decreases (increases). After the initial two cycles, (b) the cell was discharged down to E = 88 mV, and then kept at E = 88 mV for 10 hours to obtain a homogeneous sample.

For C₆Li, after the initial one cycle between 1500 mV and 5 mV (Fig. 3), the test cell was discharged to E = 5 mV and wrapped with an aluminum foil so as to achieve E = 0 V and kept for a week. The color of the obtained C₆Li was typical golden yellow [Fig. 4(a)] and E was lower than 1 mV. From now on, we call the charged electrode C₆Li and/or C₁₂Li sheet. The composition of C₆Li and C₁₂Li was confirmed by the ICP-OES technique (CIROS 120EOP, Rigaku).

Fig. 3 The variation of voltage (E) of a graphite||Li half pouch cell. Li⁺ ions are intercalated into (deintercalated from) graphite as E decreases (increases). After the initial cycle, the cell was discharged down to E = 5 mV, and then wrapped with aluminum foils for a week to obtain a homogeneous sample.

Fig. 4 Photographs of (a) the C₆Li sheets in the Ti cell and (b) the Ti cell with Ti foil window. Immediately before pumping the sample space, small holes were made on the Ti window so as to avoid an increase in the pressure in the cell.

2.3 μ⁺SR

The C₆Li and C₁₂Li sheets were removed from the pouch cells, rinsed with dimethyl carbonate (DMC) solvent, and then dried for 20 minutes. Then, eight sheets were packed in a gold O-ring sealed titanium (Ti) cell (see Fig. 4), because LIG is very sensitive to moisture in the air. The surface muons penetrate a Ti window with 50 μm thickness, but stop into the sheets with 82 × 8 μm thickness.

The μ⁺SR experiments were performed on the EMU¹⁷ surface muon beam line at ISIS of Rutherford Appleton Laboratory in U.K. In order to know a small change in an internal nuclear magnetic field due to Li-diffusion, the μ⁺SR spectra were measured in zero field (ZF), a weak longitudinal field (LF), and a weak transverse field (wTF) in the temperature (T) range between 50 and 500 K using a cryo-oven. The sample in the Ti-cell was mounted on the Cu holder of the cryo-oven, which is cooled down to around 10 K with a closed cycle refrigerator and heated up to around 600 K without setup change.

μ⁺SR is a common technique for investigating magnetic properties in solids through a direct measurement of an internal magnetic field (H_int), particularly in condensed matter physics.^18,19 Recently, μ⁺SR also becomes popular in materials science,^6,20,21 because ion diffusion in solids was detected with μ⁺SR from a small deviation of H_int caused by a dynamic behavior of ions.^{6,10,12,22–25} More correctly, H_int is observed through the muon spin depolarization function of time [P(t)]. Here, the initial muon spin at t = 0 is perfectly antiparallel to its momentum due to parity violation of weak interaction, i.e. P(0) = 1. P(t) is obtained by counting a number of decay positrons (e⁺) using scintillation counters, because e⁺ is emitted preferably along the direction of the muon spin, when the muon decays. The experimental technique of μ⁺SR is described elsewhere.^18,19 The detailed explanation how to detect ion diffusion in solids with μ⁺SR is given elsewhere.^6,10,12

The implanted muons naturally stop both in LIG and the Cu collector in our arrangement mentioned above. Since the fraction of the implanted muons stopped in each phase of a sample is proportional to a volume fraction of each phase, a μ⁺SR spectrum provides mainly two components corresponding to LIG and the Cu collector. Thus, the μ⁺SR spectra for the Cu foil were also measured using the same setup as that for the LIG sheet, while twelve Cu foil sheets were placed into the Ti-cell.

3 Results

3.1 C₆Li

Fig. 5(a) shows the ZF- and LF-μ⁺SR time spectra for the C₆Li sheets recorded at 50 K. The spectra clearly indicate that the internal magnetic field (H_int) detected by μ⁺ is static. Since the applied LF = 10 Oe (= 10 000/4π A m⁻¹) significantly suppresses a relaxation due to H_int,²⁶ the magnitude of H_int ranges around 10 Oe and matches the nuclear magnetic field mainly caused by the nuclear moments of ⁶Li, ⁷Li, ⁶³Cu, and ⁶⁵Cu. However, the ZF-μ⁺SR spectrum becomes dynamic with increasing temperature and finally exhibits a small relaxation at 370 K [see Fig. 5(b)]. This means that the distribution of H_int becomes homogeneous at high temperatures due to Li-diffusion, as in the case for the other electrode materials.

Fig. 5 (a) ZF- and LF-μ⁺SR time spectra for C₆Li measured at 50 K. The magnitude of LF was 5 and 10 Oe. (b) The temperature variation of the ZF-μ⁺SR spectrum from 50 to 370 K.

In order to extract H_int in LIG, the ZF- and LF-μ⁺SR spectra of the sheet were fitted by a combination of two dynamic Gaussian Kubo-Toyabe (KT) functions (G^DGKT)²⁷ and a time-independent offset signal. Here, the two KT components correspond to the signals from C₆Li and the Cu foil, and the offset component comes from the muons stopped in the Ti-cell;

A₀P(t) = A_C6LiG^DGKT(Δ_C6Li,ν_C6Li,t) + A_CuG^DGKT(Δ_Cu,ν_Cu,t) + A_BG. (1)

Here, A₀ is the initial (t = 0) asymmetry, A_C6Li, A_Cu, and A_BG are the asymmetries for the muons stopped in C₆Li, Cu foil, and the powder cell. Δ is the static width of the local field distribution at the disordered sites, ν is the field fluctuation rate.

Fig. 6 shows the temperature dependences of Δ_Cu and ν_Cu estimated by fitting the ZF- and LF-μ⁺SR spectra for the Cu foil with eqn (1) but A_C6Li = 0. Both the Δ_Cu(T) and ν_Cu(T) curves exhibit a typical μ⁺ diffusion behavior in a metal, in which μ⁺ does not make a stable bond with negative ions, above around 125 K. That is, as temperature increases from 50 K, Δ_Cu, which corresponds to a spin–spin relaxation rate (1/T₂), is temperature independent up to around 125 K, and then starts to decrease with further increasing temperature, and finally becomes almost 0 above around 300 K. Such behavior is well known as a motional narrowing. Also, ν_Cu, which corresponds to a spin–lattice relaxation rate (1/T₁), increases with increasing temperature, and reaches a maximum at around 200 K, then rapidly decreases with further increasing temperature, and becomes 0 above 350 K. Here, ν_Cu indicates the hopping rate of μ⁺, and above 200 K, the hopping rate is too fast to be detected by μ⁺SR. Although this muon diffusion can be also described by the behavior of the fixed Δ and increasing ν,⁴⁹ we dared not to fix Δ in order to exhibit that the internal magnetic field due to the nuclear magnetic moments of Cu seems to decrease with increasing temperature from diffusing muons.

Fig. 6 The temperature dependences of field distribution width (Δ_Cu) (left) and field fluctuation rate (ν_Cu) (right) of the Cu foil. The data were obtained by fitting the μ⁺SR spectra for the Cu foil with eqn (1) but A_C6Li = 0.

Using these values for the A_Cu signal in the C₆Li sheet, both Δ_C6Li and ν_C6Li were successfully determined at each temperature with common A_C6Li, A_Cu, and A_BG, namely, A_C6Li = 0.111(2), A_Cu = 0.068(2), and A_BG = 0.041(1). Since A_i is roughly proportional to the volume fraction (V_F) of i-th phase, V_F of C₆Li is estimated as 58% and that of Cu as 42% in the anode sheet. Such estimation is comparable with V_F calculated from the thickness of the graphite layer and Cu foil in the sheet, that is, 53% and 47%.

Fig. 7(a) and (b) show the temperature dependences of Δ_C6Li and ν_C6Li, which also exhibit a motional narrowing behavior above ∼230 K. The origin of such behavior is naturally thought to be either μ⁺ diffusion or Li⁺ diffusion. Note that very similar behavior was also reported with the past ZF-μ⁺SR measurements on C₆Li.²⁸ Furthermore, since Li-NMR shows a motional narrowing behavior above ∼240 K,¹¹ the diffusive behavior detected with μ⁺SR is most likely caused by Li-diffusion. In fact, first principles calculations predicts that the implanted μ⁺ makes a bond with C, and as a result, μ⁺ is more stable than Li⁺ in the C₆Li lattice, as described later.

Fig. 7 The temperature dependences of the μ⁺SR parameters for C₆Li and C₁₂Li; (a) the field distribution width for C₆Li (Δ_C6Li), (b) field fluctuation rate for C₆Li (ν_C6Li), (c) field distribution widths for C₁₂Li , and (d) field fluctuation rate for C₁₂Li (ν_C12Li). The data were obtained by fitting the μ⁺SR spectra for C₆Li with eqn (1) [for C₁₂Li with eqn (2)].

3.2 C₁₂Li

Fig. 8(a) shows the ZF- and LF-μ⁺SR spectra for the C₁₂Li sheets recorded at 50 K. The spectra also indicate a KT type relaxation caused by the nuclear moments of Li and Cu, as in the case for C₆Li. The temperature variation of the ZF-μ⁺SR spectrum is similar to C₆Li (Fig. 5) too. However, in order to fit these spectra, we need an additional KT signal, i.e. where and (i = 1, 2) are asymmetries and field distribution width for the two KT signals. Note that the ν_C12Li is common for the two A_C12Li signals. The fit provided that = 0.0399(4), = 0.0539(7), A_Cu = 0.0657(7), and A_BG = 0.0645, respectively. Thus, V_Fs of the two KT signals from C₁₂Li were estimated as 25% and 34%, and that of Cu foil as 49% in the C₁₂Li sheet. These values are also comparable with the calculated V_F, that is, 53% for graphite and 47% for Cu.

Fig. 8 (a) The ZF- and LF-μ⁺SR time spectra for C₁₂Li measured at 50 K. The magnitude of LF was 5 and 10 Oe. (b) The temperature variation of the ZF-μ⁺SR spectrum from 50 to 370 K.

Fig. 7(c) and (d) show the temperature dependences of (i = 1, 2) and ν_C12Li for C₁₂Li. As temperature increases from 50 K, both (i = 1, 2) are almost temperature independent (or slightly increases) up to 230 K, then they start to decrease above 250 K, and decrease monotonically with temperature. On the other hand, ν_C12Li is eventually zero up to ∼230 K, but ν_C12Li starts to increase further with increasing temperature, reaches a maximum around 350 K, and then decreases with further increasing temperature. This behavior is very similar to that of C₆Li, implying of the occurrence of Li diffusion above 230 K. Since (50 K) ∼ 3 × (50 K), the second muon site in the C₁₂Li lattice is most to be likely far away from the Li layer.

It should be noted that (50 K) is almost equivalent to Δ_C6Li (50 K). This suggests that the local environment at μ⁺ in C₆Li is very similar to that in C₁₂Li. It should be also emphasized that the number of muons implanted into the C₆Li (C₁₂Li) sample is so small (10⁵ s⁻¹)²⁹ that we can ignore the interaction between μ⁺ s.

4 Discussion

4.1 Muon site

Prior to the discussion of the Experimental data, we need to know the muon site(s) and situation in the C₆Li and C₁₂Li lattices. First principles calculations based on density functional theory^30,31 revealed that the implanted μ⁺ forms a stable bond with C in the graphene layer like a C–H bond with sp³ orbital hybridization [Fig. 10(d)] and, as a result, sits at = (0.35, 0.37, 0.08). The same calculations for C₁₂Li indicated that there are two stable muon sites, namely, at = (0.37, 0.37, 0.04) in the Li layer and at = (0.67, 0.66, 0.47) in the graphene layer. μ⁺ s at both sites also form a stable C–μ bond [Fig. 10(e) and (f)], as in C₆Li.

Fig. 9 The relationship between ν and inverse temperature for C₆Li and C₁₂Li. The solid lines represent the best fit using eqn (3) with E_a = 270 ± 5 meV for C₆Li in the temperature range between 270 and 330 K and E_a = 170 ± 20 meV for C₁₂Li in the temperature range between 230 and 290 K.

Fig. 10 Illustrations of the structures optimized by the calculations of (a) μ⁺ in C₆Li, (b) μ⁺ in the graphene layer and (c) in the Li layer of C₁₂Li. (d–f) The charge contour plots in the (110) plane corresponding to these optimized structures (a–c). The contour spacing is 0.02 e Bohr⁻³. The lines for the densities higher than 0.3 e Bohr⁻³ are omitted.

Details of the calculations are described in Appendix. Consequently, these results demonstrate that μ⁺ is more stable than Li⁺ in both C₆Li and C₁₂Li lattice and support that the extracted ν from the μ⁺SR data corresponds to the hopping rate of Li⁺.

The predicted muon sites and Δ, which was given by dipole field calculations with DipElec,³² are summarized in Table 2. It is found that the calculated results are qualitatively consistent with the experimental results. Note that, since the dipole field calculations provide Δ^cal at 0 K without thermal vibration, Δ^cal is usually larger than Δ^mea.³³ The fact that Δ^cal ∼ 2/3Δ^mea only for would imply either an insufficient accuracy of the DFT calculations, a limited time window of μ⁺SR, or the presence of imperfection in the samples. For the second factor, if Δ < 0.65 Oe, we need to measure the μ⁺SR spectrum until 25 μs to estimate Δ. However, such measurements are eventually impossible due to a lifetime of μ (∼2.2 μs).

Table 2 The muon site in the Li-intercalated graphites predicted by first principles calculations and the field distribution width (Δ) at the muon site estimated by dipole field calculations and measured at 50 K

Compound	Site name	Coordinate	d μ−Li	Calculated Δ		Measured Δ at 50 K
		(x, y, z)	(Å)	(Oe)	(μs^−1)	(Oe)	(μs^−1)
C6Li		(0.35, 0.35, 0.08)	1.51	4.16	0.354	2.61(9)	0.223(2)
C12Li		(0.37, 0.37, 0.04)	3.96	4.14	0.353	3.17(1)	0.270(4)
		(0.67, 0.66, 0.47)	1.61	0.65	0.055	1.01(0)	0.086(2)
			6.38			3.15

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4.2 Temperature dependence of ν

In order to compare ν for C₆Li with that for C₁₂Li, Fig. 9 shows ν_C6Li and ν_C12Li as a function of inverse temperature (1/T). When ν is caused by Li-diffusion, a thermal activation process is naturally applicable for explaining the temperature dependence, i.e.

ν = ν₀ exp(−E_a/k_BT) + ν_BG, (3)

where ν₀ is the frequency factor, E_a is the activation energy, k_B is the Boltzmann's constant, and ν_BG is the temperature-independent ν.

From the linear relationship between log(ν) and 1/T, E_a is estimated as 270(5) meV for C₆Li in the T range between 270 and 310 K and 170(20) meV for C₁₂Li in the T range between 230 and 290 K. Such a difference in E_a is probably caused by the strength of an interlayer interaction with Li⁺ ions in the adjacent Li layers.

4.3 Temperature dependence of Δ

As described in Sections 3.1 and 3.2, Δ_C6Li and Δ_C12Li are roughly temperature independent below 230 K, at which Li⁺ ions are static. More correctly, as temperature increases from 50 to 200 K, both Δ_C6Li and increase by about 15%, for reasons currently unknown. Note that graphite^34,35 and alkali-metal intercalated graphites³⁶ exhibit negative thermal expansion along the ab-plane (α_a) below room temperature and α_a ranges between −1 and −4 × 10⁻⁶ K⁻¹. The reduction of the distance between μ⁺ and Li⁺ is naturally expected to increase Δ. However, the effect of such negative thermal expansion on Δ was estimated as 0.1% over 200 K, which is too small to explain the experimental result (∼15%).

In order to further discuss the temperature dependence of Δ below 230 K, we need to know the detailed displacement of the intercalated Li with temperature.

4.4 Diffusion coefficient of Li⁺

Assuming that Li⁺ diffuses through interstitial sites in the Li layer, the self diffusion coefficient (D_Li) is given by³⁷where N_i, s_i, and Z_v,i are the number of Li sites in the i-th diffusive path, the distance of the path, and vacancy fraction, respectively. Since one Li⁺ ion in the Li layer is surrounded six vacant Li sites for a path to jump to the next vacant sites, N₁ = 6, Å (= 2.48 × 10⁻¹⁰ m), and Z_v,i = 1. Therefore, we obtain that D^μSR_Li = 7.6 × 10⁻¹¹ (cm² s⁻¹) for C₆Li and D^μSR_Li = 1.5 × 10⁻¹⁰ (cm² s⁻¹) for C₁₂Li at 310 K (Fig. 11).

Fig. 11 The temperature dependence of the diffusion coefficient (D_Li) for C₆Li and C₁₂Li. Solid lines represent the estimate from eqn (3) and (4).

Then, we discuss the reliability of D_Li estimated with μ⁺SR. Table 3 lists several D_Li for C₆Li and C₁₂Li reported with different techniques. The present μ⁺SR showed that D_Li for C₆Li and C₁₂Li ranges in the order of 10⁻¹¹–10⁻¹⁰ cm² s⁻¹ and D_Li for C₆Li is slightly smaller than D_Li for C₁₂Li. This is very consistent with D_Li(C₆Li) determined by the recent Li-NMR measurements, i.e. D_Li^NMR = 3.8 × 10⁻¹¹ (cm² s⁻¹) at 314 K.¹¹ Also, D^μSR_Li is comparable to D_Li predicted by first principles calculations. Furthermore, the past NMR¹¹ and electrochemical measurements^38–40 are likely to support the present result, i.e. D_Li(C₆Li) < D_Li(C₁₂Li), although the latter provided not D_Li but a chemical diffusion coefficient (D̃_Li).

Table 3 The diffusive coefficient D_Li around room temperature and E_a in Li intercalated graphite estimated with different techniques. D_Li represents a self diffusion coefficient, i.e. a jump diffusion coefficient, while D̃_Li represents a chemical diffusion coefficient

	C6Li
	D Li (cm^2 s^−1)	D̃ Li (cm^2 s^−1)	E a (meV)
μ^+SR (at 310 K)	7.6(3) × 10^−11	—	270(5)
Li-NMR^11 (at 314 K)	3.8 × 10^−11	—	550
Li-NMR^44 (at 373 K)	10^−8	—	—
Electrochemical impedance^38 (at 298 K)	10^−8–10^−7	—	—
First principles calculations^42 (at 300 K)	0.9 × 10^−11	—	283
First principles calculations^45 (at 300 K)	—	1–10 × 10^−11	510

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	C12Li
	D Li (cm^2 s^−1)	D̃ Li (cm^2 s^−1)	E a (meV)
μ^+SR (at 310 K)	14.6(4) × 10^−11	—	170(20)
Li-NMR^44 (at 373 K)	2.4 × 10^−7	—	—
Electrochemical impedance^38 (at 298 K)	5.3 × 10^−7	—	—
PITT^39 (at 328 K)	—	0.01 × 10^−11	—
Current pulse relaxation^40 (at 300 K)	—	0.8 × 10^−11	—
Potential step chronoamperometry^40 (at 300 K)	—	0.1 × 10^−11	—
First principles calculations^42 (at 300 K)	0.9 × 10^−11	—	297

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In a highly oriented graphite (HOPG), D_Li for C₆Li and C₁₂Li in the direction parallel/perpendicular to the graphene layer is also reported as D̃^//_Li ∼ 10⁻⁷–10⁻⁶ cm² s⁻¹ and D̃^⊥_Li ∼ 10⁻¹¹ cm² s⁻¹, respectively.⁴¹ If we assume the relation D̃_Li = ΘD_Li, where thermodynamic factor Θ(x) = 10³–10⁴ for C₆Li and C₁₂Li,⁴²D^//_Li is similar to D^μSR_Li. This shows that μ⁺SR “sees” Li diffusion in graphene layer. Such anisotropic diffusion in layered materials could be detected separately with μ⁺SR in the future, based on a theoretical prediction of H_int in a low dimensional system.⁴³

Concerning the thermal activation energy (E_a), the present μ⁺SR result is comparable to the NMR result and very consistent with the prediction from the calculations. In overall, μ⁺SR is found to provide reasonable D_Li and to confirm that D_Li in LIG ranges in the same order to those for positive electrode materials.

5 Conclusions

We have observed Li diffusion in C₆Li and C₁₂Li by μ⁺SR using the discharged negative electrode sheet, which was prepared by coating a graphite layer onto a Cu foil. Li diffusion was clearly detected with μ⁺SR through a small deviation of H_int caused by Li diffusion. A diffusion coefficient of Li (D_Li) in C₆Li and C₁₂Li was found to range in the same order of D_Li in positive electrode materials, that is, 10⁻¹⁰ (cm² s⁻¹) at room temperature. The thermal activation energy (E_a) of D_Li was estimated as 270(5) meV for C₆Li and 170(20) meV for C₁₂Li. These values are comparable with those obtained by Li-NMR¹¹ and predicted by first principles calculations.⁴² The difference of E_a between C₆Li and C₁₂Li is probably due to an interlayer electrostatic interaction from the adjacent Li layers. Since the Li diffusion in the anode sheet has been clarified by μ⁺SR, it is ready to measure Li diffusion in a Li-ion battery in an in operando manner.

Appendix: muon sites

The first-principles calculations based on density functional theory^30,31 have been performed to determine the muon site in C₆Li and C₁₂Li. The ultrasoft pseudopotential method^46,47 is used and the generalized gradient approximation⁴⁸ is adopted for the exchange–correlation energy. The position of μ⁺ is simulated using a hydrogen pseudopotential, where a deficient electron is compensated by the uniform background charge. The Li-1s state is treated as the valence. The calculations are carried out with the 2 × 2 × 2 supercell for C₆Li and 2 × 2 × 1 for C₁₂Li. The 4 × 4 × 4 k-point mesh is used for the Brillouin zone integration. For C₁₂Li, both cases of μ⁺ being in the graphene layer and in the Li layer are considered. In the structural optimization process, the lattice constants are fixed at the experimental values and the atom positions are fully relaxed without any symmetry restrictions. The initial μ⁺ position is set to be about 1.4 Å above the center of the C–C bond. The results of the structural optimization are summarized in Fig. 10. In all cases, the μ⁺ positions are the on-top site on the C layer as shown in Fig. 10(a)–(c). The charge contour plots given in Fig. 10(d)–(f) indicate clear evidence of the C–μ⁺ bond formation which pulls up the C atom from the layer. The C atom is most likely stabilized by constructing sp³ hybrids. When the μ⁺ is located in the Li layer, the C–μ⁺ bond is slightly tilted due to an adjacent Li atom. The C–μ⁺ bond lengths are 1.13–1.14 Å which are close to typical C–H ones of ca. 1.1 Å. Because of the existence of the C–μ⁺ bond, the μ⁺ is expected to be immobile in C₆Li and C₁₂Li.

Acknowledgements

We appreciate the staff at ISIS, Rutherford Appleton Laboratory, U.K. for their supports during μ⁺SR experiments and Mr T. Inoue and Mr Y. Kishida at Toyota Central R & D Labs., Inc. for their contribution to the examination prior to the μ⁺SR experiments. This work was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, KAKENHI Grant no. JP23108003 and Japan Society for the Promotion Science (JSPS) KAKENHI Grant No. JP26286084.

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