How Li diffusion in spinel Li[Ni1/2Mn3/2]O4 is seen with μ
±SR

Jun Sugiyama; Kazuki Ohishi; Ola Kenji Forslund; Martin Månsson; Stephen P. Cottrell; Adrian D. Hillier; Katsuhiko Ishida

doi:10.1515/zpch-2021-3102

Article Open Access

How Li diffusion in spinel Li[Ni_1/2Mn_3/2]O₄ is seen with μ ^±SR

Jun Sugiyama , Kazuki Ohishi , Ola Kenji Forslund , Martin Månsson , Stephen P. Cottrell , Adrian D. Hillier and Katsuhiko Ishida

Published/Copyright: September 6, 2021

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From the journal Zeitschrift für Physikalische Chemie Volume 236 Issue 6-8

Abstract

The diffusive behavior in a spinel-type Li⁺ ion battery material, Li[Ni_1/2Mn_3/2]O₄, has been studied with positive and negative muon spin rotation and relaxation (μ ^±SR) measurements in the temperature range between 200 and 400 K using a powder sample. The implanted μ ⁺ locates at an interstitial site near O²⁻ ion so as to form a O–H like bond, while the implanted μ ⁻ is mainly captured by an oxygen nucleus, resulting in the formation of muonic oxygen. This means that local magnetic environments in Li[Ni_1/2Mn_3/2]O₄ were investigated from the two different sites in the lattice, i.e., one is an interstitial site for μ ⁺SR and the other is an oxygen site for μ ⁻SR. Since both μ ⁺SR and μ ⁻SR detected an increase in the fluctuation rate of a nuclear magnetic field for temperatures above 200 K, the origin of this increase is clearly confirmed as Li diffusion. Assuming a random walk process with the hopping of thermally activated Li⁺ between a regular Li site and the nearest neighboring vacant octahedral sites, a self-diffusion coefficient of Li⁺ was found to range above 10⁻¹¹ cm²/s at temperatures above 250 K with an activation energy of about 0.06 eV.

Keywords: Li diffusion; Li ion battery materials; muon spin rotation and relaxation

1 Introduction

In order to understand the operational processes of battery materials, it is crucial to understand the diffusive nature of ions in these materials, because the transport of ions is the fundamental mechanism behind their operation. Nuclear magnetic resonance (NMR) has been widely used for such a purpose [1], [2], [3], through the observation of nuclear magnetic fields in these materials. Recently, the positive muon spin rotation and relaxation (μ ⁺SR) technique [4, 5] was found to be a good tool for detecting ion diffusion in solids [6]. Unlike in NMR, where the polarization is dependent on an externally applied high magnetic field, the initial muon spin polarization is aligned antiparallel to its momentum due to parity violation of a muon creation reaction [7]. As a result, μ ⁺SR senses the internal magnetic field in materials without the need of any external magnetic field, i.e., even in zero field (ZF).

This leads to the following two important features: the first one is that μ ⁺SR distinguishes the nuclear magnetic field from the electron magnetic field by simply applying a weak longitudinal field (LF) [8]. Here, “weak” means the field is comparable to the field distribution width formed by surrounding nuclear magnetic moments at the muon site, which typically ranges between 5 and 10 Oe, and “longitudinal field” means the magnetic field is parallel to the initial muon spin polarization. Such weak LFs decouple the contribution from a nuclear magnetic field on the μ ⁺ spin, while the effect of weak LF on the contribution from the electron magnetic field is negligibly small. Consequently, μ ⁺SR provides information on ion diffusion even in materials that include paramagnetic ions [9, 10]. This is an attractive feature for studying Li⁺ [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], Na⁺ [10, 22, 23], and K⁺ [24] diffusion in cathode materials for an ion battery, because these materials always contain transition metal ions to keep charge neutrality during ion insertion and extraction reactions.

Another feature that makes μ ⁺SR an attractive probe is its sensitivity, which in principle is temperature independent, since the muon spin polarization is an intrinsic feature of the technique, whereas the effect of an external field on aligning nuclear spins is suppressed by thermal excitation, resulting in the decrease of NMR sensitivity with temperature. Therefore, μ ⁺SR is a very important tool to study ion dynamics at high temperatures.

Despite the above features, the information obtained with μ ⁺SR is sometimes ambiguous. Whereas the implanted μ ⁺ is presumed to be static at low temperatures in its interstitial site (where its electrostatic potential is in a local minimum), in any solid there is a chance that the μ ⁺ will start to diffuse above certain temperatures. This means that the observed diffusive behavior could be caused by not only ion diffusion but also μ ⁺ diffusion. In order to confirm the diffusing species, μ ⁺ or ion, we have initiated series of μ ⁻SR experiments on battery materials [25], [26], [27]. Here, μ ⁻SR is the “anti-technique” of μ ⁺SR by not using μ ⁺ but negative muons (μ ⁻) [28], [29], [30], [31].

The implanted μ ⁻ is captured by a nucleus and forms a stable muonic atom. The muonic atoms, apart from the ones based on the diffusing species, is immobile until the decomposition temperature of the material. In other words, by means of μ ^±SR, we are able to see the internal magnetic field from different viewpoints: at the atomic position in the lattice with μ ⁻SR and at the interstitial sites with μ ⁺SR. It should be noted that a μ ⁻SR spectrum requires about 36 times higher counts than that of μ ⁺SR to obtain the same statistical precision, since the spin polarization of μ ⁻ is down to about 1/6 of its initial polarization during the atomic capture process by a nucleus [32,33]. It is thus only reasonable to use the expensive and difficult μ ⁻SR technique in order to determine the diffusing species, while the details of the diffusion processes are best determined by μ ⁺SR.

A combined μ ^±SR work has confirmed Li⁺ diffusion in an olivine-type LiMnPO₄ above around 200 K [26] and in a spinel-type Li₄Ti₅O₁₂ ( = Li + [ Li 1 / 3 + Ti 5 / 3 4 + ] O 4 ) above around 250 K [34]. Here, we have studied the diffusive nature in another spinel-type battery material, Li[Ni_1/2Mn_3/2]O₄ ( = Li + [ Ni 1 / 2 2 + Mn 3 / 2 4 + ] O 4 , see Figure 1), which has a different cation distribution from that of Li₄Ti₅O₁₂, by means of μ ^±SR. This is because Li[Ni_1/2Mn_3/2]O₄ is highly attractive as a cathode material for an all solid-state battery [35], [36], [37], [38], due to the fact that the charge and discharge reactions occur at about 4.7 V (vs. Li/Li⁺) and such voltage is almost independent of the Li content [39]. Our previous μ ⁺SR study on a powder and film samples of Li[Ni_1/2Mn_3/2]O₄ revealed the occurrence of Li diffusion at temperatures above around 250 K [40].

Figure 1:

The crystal structure of cubic spinel Li[Ni_1/2Mn_3/2]O₄ drawn with VESTA [41]. Li⁺ ions occupy the tetrahedral (8a) site, while Ni and Mn ions occupy the octahedral (16d) site. O²⁻ ion is located in each corner of the tetrahedron and octahedron. Dark grey spheres represent a vacant 16c site.

2 Experimental

A powder sample of Li[Ni_1/2Mn_3/2]O₄ was purchased from Toshima Manufacturing, Ltd. The μ ⁻SR time spectra were measured on the decay muon beam line ARGUS [42, 43] at ISIS pulsed neutron and muon source of Rutherford Appleton Laboratory in UK. Approximately 22 g powder sample was placed in a copper container with a sample space of 30 mm diameter and 15 mm depth, made of 1 mm thick Cu plate [see Figure 2(a)]. A cap of the container was made of 3 mm thick Cu plate, which also works as a degrader to adjust the range of μ ⁻s. The copper container was then set onto the end of the He-flow cryostat (Ministat, Oxford Instruments). The momentum of the μ ⁻ beam was adjusted to 70 MeV/c to maximize the number of μ ⁻ stopped in the sample, as shown in Figure 2(b). The μ ⁻SR spectrum was measured at 200, 300, and 400 K with up to 50 Mevents for transverse field (TF) μ ⁻SR and 200 Mevents for zero field (ZF) and 150 Mevents for longitudinal field (LF) μ ⁻SR at a counting rate of 17 Mevents/h. Here TF [LF] means the applied magnetic field was perpendicular [parallel] to the initial μ ⁻ spin polarization.

Figure 2:

(a) A copper container filled with a black powder sample of Li[Ni_1/2Mn_3/2]O₄ for a μ ⁻SR experiment. The size of the sample space is 30 mm diameter and 15 mm depth. (b) The variation of the depth profile of the muons in the copper container filled with the sample by five different implantation momentum. The profile was simulated with SRIM [44].

The μ ⁺SR time spectra were measured on the surface muon beam line EMU [43, 45] at ISIS. Approximately 2 g powder sample from the same batch as for the μ ⁻SR measurements was packed into a standard ISIS titanium cell. The window of the cell was made of a titanium foil with 20 μm thickness. The cell was then mounted onto the Al plate of a closed cycle refrigerator-type cryo-furnace in the temperature range between 50 and 500 K. The counting rate was about 100 Mevents/h. The experimental techniques are described in more detail elsewhere [4, 5]. The obtained μ ^±SR data was analyzed with musrfit [46].

3 Results

While the μ ⁻SR study is the main focus of this paper, to put these results into context our discussion will start with the μ ⁺SR measurements that were carried out on the same system.

3.1 μ ⁺SR

Figure 3 shows the μ ⁺SR time spectra recorded in wTF with H = 20 Oe at selected temperatures in order to understand the magnetic nature of the Li[Ni_1/2Mn_3/2]O₄ sample. At 150 K, the spectrum exhibits a clear oscillation with a full asymmetry due to wTF, which means that the whole volume of the sample is in a paramagnetic state. As the temperature decreases from 150 K, the oscillating asymmetry decreases with decreasing temperature and then almost disappears below 100 K. In order to extract the μ ⁺SR parameters, the wTF-μ ⁺SR spectrum is fitted by a combination of an exponentially relaxing cosine oscillation due to wTF and an exponentially relaxing non-oscillatory signal, due to the 1/3 tail arising from the distribution of static electronic components in the powder sample when in an AF ordered state.

(1) A 0 P TF = A TF cos ( 2 π f TF t + ϕ TF ) exp ( − λ TF t ) + A tail exp ( − λ tail t ) ,

where A ₀ is the initial asymmetry determined by the wTF measurements at high temperatures and P _TF is the muon spin polarization function in wTF. A _TF, f _TF, ϕ _TF and λ _TF are the asymmetry, muon spin precession frequency, initial phase, and relaxation rate due to the applied TF. Here, f _TF = γ _μ/2π × 20 Oe = 13.554 kHz/Oe × 20 Oe ∼ 0.27 MHz. The A _tail component corresponds to the signal from the magnetic ordered phase, in which the internal AF field is parallel to the initial muon spin polarization.

Figure 3:

Weak transverse field (wTF) μ ⁺SR time spectra for Li[Ni_1/2Mn_3/2]O₄ recorded at 50, 100, 110, 120, and 150 K. Solid lines represent the best fit using Eq. (1).

Figure 4 shows the temperature dependencies of the four wTF-μ ⁺SR parameters, i.e., A _TF, A _tail, λ _TF, and λ _tail. The A _TF(T) curve exhibits a step-like change at around 105 K, which indicates the appearance of a very large internal magnetic field compared with wTF at low temperatures. From the middle point of the step-like change in the A _TF(T) curve, T _N is determined as 104.2(5) K using a Sigmoid function. A relatively wide transition (ΔT _N = 41 K) indicates a random distribution of Ni and Mn at the octahedral site in the spinel lattice [40,47], where ΔT _N = T ^90% − T ^10% with A TF ( T 90 % ) = 0.9 [ A TF ( 200 K ) − A TF ( 30 K ) ] + A TF ( 30 K ) and A TF ( T 10 % ) = 0.1 [ A TF ( 200 K ) − A TF ( 30 K ) ] + A TF ( 30 K ) . A non-zero A _TF well below T _N is caused by the muons stopped outside of a sample, such as, the Ti sample holder and/or the surrounding experimental setup. This is because A _TF is almost temperature independent below T _N. The λ _TF(T) curve, on the other hand, shows a critical behavior above T _N, as expected.

$Figure 4: The temperature dependencies of (a) the weak transverse field asymmetry (A TF) and tail asymmetry (A tail) (b) their exponential relaxation rates (λ TF and λ tail) for Li[Ni1/2Mn3/2]O4. The data were obtained by fitting the wTF-μ +SR spectrum with Eq. (1). A red solid line in (a) represents the best fit by a Sigmoid function and a blue dotted line indicates the predicted A tail = [ A TF ( 200 K ) − A TF ( 30 K ) ] / 3 ${A}_{\text{tail}}=[{A}_{\text{TF}}(200\text{\hspace{0.17em}K})-{A}_{\text{TF}}(30\text{\hspace{0.17em}K})]/3$ . The color of the symbol for the TF signal is slightly changed at temperatures below 90 K, since the wTF-μ +SR signal comes from not the sample but from the Ti sample holder (see text).$

Figure 4:

The temperature dependencies of (a) the weak transverse field asymmetry (A _TF) and tail asymmetry (A _tail) (b) their exponential relaxation rates (λ _TF and λ _tail) for Li[Ni_1/2Mn_3/2]O₄. The data were obtained by fitting the wTF-μ ⁺SR spectrum with Eq. (1). A red solid line in (a) represents the best fit by a Sigmoid function and a blue dotted line indicates the predicted A tail = [ A TF ( 200 K ) − A TF ( 30 K ) ] / 3 . The color of the symbol for the TF signal is slightly changed at temperatures below 90 K, since the wTF-μ ⁺SR signal comes from not the sample but from the Ti sample holder (see text).

As temperature decreases from 200 K, A _tail starts to increase below 130 K and approachs a predicted value for the 1/3 tail signal below 90 K, where A _tail well below T _N is predicted as [ A TF ( 200 K ) − A TF ( 30 K ) ] / 3 . Besides the vicinity of T _N, λ _tail is small and almost temperature independent. These features are consistent with those for the 1/3 tail signal.

Figure 5 shows the ZF- and two LF-μ ⁺SR spectra recorded at 150, 300, and 425 K, i.e., in a paramagnetic state. As temperature increases, the μ ⁺SR spectrum indicates a clear change from a low-temperature static behavior to a high-temperature dynamic behavior. The μ ⁺SR spectra were therefore fitted with a combination of an exponentially relaxing dynamic Kubo-Toyabe function (G ^DGKT) from the sample and a time independent background signal from the Ti sample holder:

(2) A 0 P LF ( t ) = A KT G DGKT ( t , Δ , ν , H LF ) exp ( − λ KT t ) + A BG ,

where P _LF is the muon spin polarization function in ZF and LF. A _KT and A _BG are asymmetries of the two signals, Δ is the field distribution width, ν is the field fluctuation rate, and λ _KT is the exponential relaxation rate. For ν = 0 and H _LF = 0 (i.e., ZF), G ^DGKT(t, Δ, ν, H _LF) becomes a static Gaussian Kubo-Toyabe function (G _zz ^KT) given by [48]:

(3) G z z KT ( t , Δ ) = 1 3 + 2 3 ( 1 − Δ 2 t 2 ) exp ( − 1 2 Δ 2 t 2 ) ,

which represents the time variation of the muon spin polarization due to the internal magnetic field formed by randomly oriented static nuclear dipoles with Gaussian distribution. Since the implanted muons feel both a nuclear magnetic field formed mainly by ⁶Li, ⁷Li, and ⁵⁵Mn and a magnetic field caused by localized 3d electrons of Ni²⁺ (3d ⁸) and Mn⁴⁺ (3d ³) [36, 39, 49], an exponential is multiplied with G ^DGKT [50] in Eq. (2).

Figure 5:

The temperature variation of the ZF- and two LF-μ ⁺SR spectra recorded at (a) 150 K, (b) 300 K, and (c) 425 K. The applied LF’s were 5 and 10 Oe. Green solid lines represent the best fit using Eq. (2).

In the paramagnetic state, the TF-μ ⁺SR spectrum is also fitted with an Abragam equation [51, 52] using Δ⁺ and ν ⁺ and exponential relaxation function exp ( λ KT + t ) as:

(4) A 0 P TF ( t ) = A TF cos ( 2 π f TF t + ϕ TF ) G A e − λ KT t + A BG cos ( 2 π f TF t + ϕ TF ) , G A ( t ) = exp [ − 2 Δ 2 ν 2 ( e − t ν − 1 + t ν ) ]

Combining with Eq. (2), the TF-, ZF-, and LF-μ ⁺SR spectra were fitted using common A _KT, A _BG, and A _TF at temperatures between 150 and 425 K, while Δ⁺, ν ⁺, and λ KT + are common at each temperature.

Figure 6 shows the temperature dependencies of the μ ⁺SR parameters obtained by fitting the TF-, ZF-, and two LF-μ ⁺SR spectra with Eqs. (4) and (8). Here, a superscript of each symbol, i.e., “+” means that the data obtained from μ ⁺SR measurements. The field distribution width (Δ⁺), which roughly corresponds to the spin-spin relaxation rate (1/T ₂), is found to be almost temperature independent.

$Figure 6: The temperature dependencies of the μ +SR parameters for Li[Ni1/2Mn3/2]O4: (a) the field distribution width (Δ+), (b) the field fluctuation rate (ν +) and the exponential relaxation rate ( λ KT + ${\lambda }_{\text{KT}}^{+}$ ), and (c) the three asymmetries ( A TF + ${A}_{\text{TF}}^{+}$ , A KT + ${A}_{\text{KT}}^{+}$ , and A BG + ${A}_{\text{BG}}^{+}$ ). “+” of each symbol means that the data was obtained from μ +SR measurements. Δ+, ν +, A KT + ${A}_{\text{KT}}^{+}$ , and A BG + ${A}_{\text{BG}}^{+}$ were obtained by fitting the TF-, ZF- and LF-μ +SR spectra with Eq. (2). Note that Δ+ corresponds roughly to a spin-spin relaxation rate (1/T 2), while ν + corresponds roughly to a spin-lattice relaxation rate (1/T 1). More correctly, 1/T 1 would be the relaxation rate of the nuclear moments that contribute to Δ+, not the relaxation rate of the μ + spin.$

Figure 6:

The temperature dependencies of the μ ⁺SR parameters for Li[Ni_1/2Mn_3/2]O₄: (a) the field distribution width (Δ⁺), (b) the field fluctuation rate (ν ⁺) and the exponential relaxation rate ( λ KT + ), and (c) the three asymmetries ( A TF + , A KT + , and A BG + ). “+” of each symbol means that the data was obtained from μ ⁺SR measurements. Δ⁺, ν ⁺, A KT + , and A BG + were obtained by fitting the TF-, ZF- and LF-μ ⁺SR spectra with Eq. (2). Note that Δ⁺ corresponds roughly to a spin-spin relaxation rate (1/T ₂), while ν ⁺ corresponds roughly to a spin-lattice relaxation rate (1/T ₁). More correctly, 1/T ₁ would be the relaxation rate of the nuclear moments that contribute to Δ⁺, not the relaxation rate of the μ ⁺ spin.

The field fluctuation rate, ν ⁺, which corresponds roughly to the spin-lattice relaxation rate, increases almost monotonically with increasing temperature, being consistent with Li diffusion. Further confirmation for Li diffusion is given in Section 3.2. On the contrary, λ KT + decreases with increasing temperature. Since λ KT + is caused by the fluctuation of 3d localized moments, such decrease is explained by a Curie-Weiss behavior of magnetization for Li[Ni_1/2Mn_3/2]O₄ [39].

In the above fit, each asymmetry is assumed to be temperature independent, because the fractions of muons stopped in the sample and the Ti cell are, in principle, independent of temperature.

3.2 μ ⁻SR

When considering the positive muon results, we cannot rule out the possibility that the muon may start to diffuse in the sample at some temperature. To be certain we are studying the change in the nuclear magnetic field from a “fixed view point” we have also measured the μ ⁻SR spectrum for Li[Ni_1/2Mn_3/2]O₄ using the same sample to that for μ ⁺SR. Figure 7 shows the time histogram of the TF-μ ⁻SR spectrum with TF = 50 Oe recorded at 300 K. Here, forward [backward] means upstream [downstream] detector positions with respect to the sample position and the incoming muon beam direction. Since the decay asymmetry is very small and the lifetime of the μ ⁻ depends upon the nucleus on which it is captured by, the histogram of μ ⁻SR was fitted by a combination of four different lifetimes (τ _i=1−4). Note that, for μ ⁺SR, the time histogram includes only one component, i.e., μ ⁺ with τ = 2.19703 µs:

(5) N ( t ) = ∑ i = 1 n = 4 N i e − t / τ i [ 1 + A i e − λ i t cos ( ω i t + ϕ i ) ] ,

where N _i is a normalization constant at t = 0 for the i-th muon lifetime (τ _i), i.e., muonic O (μ ⁻O) with τ ₁ = τ _O = 1.7954 μs [53], muonic Li (μ ⁻Li) with τ ₂ = τ _Li = 2.188 μs [53], muonic Mn (μ ⁻Mn) with τ ₃ = τ _Mn = 0.2325 μs [53], and an unknown component with τ ₄ = 26.8(6) μs, A _i is the muon decay asymmetry for each process, ω _i = 2πf _i is the angular frequency of the μ ⁻ spin precession caused by the applied TF, and ϕ _i is the initial phase. Since τ 4 ≫ τ μ = 2.19703 μ s , the τ ₄ component is caused by other particles, such as e ⁻, e ⁺ and/or neutrons perhaps created by a decay reaction of the muonic atoms. However, the contribution of the τ ₄ component is negligibly small for t < 10 μs.

Figure 7:

The time histogram of the TF-μ ⁻SR spectrum in Li[Ni_1/2Mn_3/2]O₄ for the (a) forward counter and (b) backward counter. Red open circles represent the experimental data, green solid lines represent the fit result using Eq. (5), and blue solid lines represent the histograms of the four muonic-atom lifetimes.

As a first approximation, A _i is non-zero only for the main component, i.e., μ ⁻O and negligible for the other elements. This is because both Li and Mn have nuclear spins and will therefore form hyperfine-coupled states with the μ ⁻, leading to a precession at a dramatically different frequencies compared with the applied value. Furthermore, A _i is less than 1/6 of A ₀ for μ ⁺SR (typically 0.24) due to the reduction of spin polarization, A i < 0.04 ( = 0.24 / 6 ) ≪ 1 . Therefore, it is reasonable to fix A _i = 0 for the other minor elements to estimate each N _i using Eq. (5). Due to a short lifetime of muonic nickel (μ ⁻Ni, τ _Ni = 0.1569 μs [53]) together with a relatively small amount of μ ⁻Ni compared with that of μ ⁻Mn, it was impossible to fit the histogram with a combination of five different lifetime components.

The relative asymmetric counting efficiency can be estimated for each element and are listed in Table 1. It is noted that both α Li = N Li B / N Li F and α 4 = N 4 B / N 4 F are smaller than α O = N O B / N O F . As a result, the asymmetry spectrum based on α _O, which is defined by Eq. (6), is naturally distorted in late time domain.

(6) A 0 P ( t ) ∼ N B ( t ) − α O N F ( t ) N B ( t ) + α O N F ( t ) .

Table 1:

Parameters of the TF-μ ⁻SR histograms for the Li[Ni_1/2Mn_3/2]O₄ sample obtained by fitting with Eq. (5), where τ _O, τ _Li, and τ _Mn were fixed to the values reported in Ref. [53]. Since it is extremely difficult to determine short τ components using data at early times because of the difficulty for determining a correct time zero (t ₀) of the pulsed μ ^± beam, the histograms were fitted only in the time domain between 0.4 and 16 μs.

n	Component	τ (μs)	N i F / N O F	N i B / N O B	α i ≡ N i B / N i F	α _i/α _O
1	O	τ _O = 1.7954 [53]	1	1	α _O = 1.062(5)	1
2	Li	τ _Li = 2.188 [53]	0.014(2)	0.010(2)	α _Li = 0.8(3)	0.7(2)
3	Mn	τ _Mn = 0.1569 [53]	1.05(2)	1.33(4)	α _Mn = 1.34(6)	1.26(6)
4	?	τ ₄ = 26.8(6)	0.00110(2)	0.00073(2)	α ₄ = 0.71(3)	0.67(3)

In fact, the TF-, ZF-, and two LF-μ ⁻SR spectra for Li[Ni_1/2Mn_3/2]O₄ exhibit a sudden decrease with time particularly below around 8 µs, regardless of external magnetic field (see Figure 8). This is well explained by the effect of the τ ₄ component on the μ ⁻SR spectra, because the τ ₄ component with α τ 4 / α O = 0.67 becomes predominant in a late time domain. In addition, each spectrum is distorted in an early time domain due to the effect from μ ⁻Mn. Besides these distortions, the ZF- and LF-μ ⁻SR spectra show a decoupling behavior, which evidences the observation of a nuclear magnetic field with μ ⁻SR.

Figure 8:

The TF-, ZF-, and two LF-μ ⁻SR spectra for Li[Ni_1/2Mn_3/2]O₄ recorded at 300 K. The applied TF was 50 Oe, and the applied LF were 20 and 50 Oe.

Therefore, as seen in Figure 9, the TF-, ZF-, and two LF-μ ⁻SR spectra were fitted in the time domain between 0.8 and 6 μs with a combination of an exponentially relaxing cosine oscillation and an exponentially relaxing non-oscillatory signal for the TF-spectrum, and with a combination of an exponentially relaxing dynamic Kubo–Toyabe signal and an exponentially relaxing non-oscillatory signal for the ZF- and LF-spectra. In addition, the following three parameters, i.e., A _TF, A _TFe, and λ _TFe, are common for the TF-, ZF-, and LF-μ ⁻SR spectra at each temperature, and Δ and ν are common for the ZF- and LF-μ ⁻SR spectra at each temperature:

A 0 P TF ( t ) = A TF cos ( 2 π f TF t + ϕ TF ) exp ( − λ TF t ) + A TFe exp ( − λ TFe t ) , A 0 P LF ( t ) = A KT G DGKT ( t , Δ , ν , H LF ) exp ( − λ KT t ) + A LFe exp ( − λ LFe t ) ,

where

(7) A TF = A KT , A TFe = A LFe , λ TFe = λ LFe .

Figure 9:

The TF-, ZF-, and two LF-μ ⁻SR spectra for Li[Ni_1/2Mn_3/2]O₄ recorded at (a) 200 K, (b) 300 K, and (c) 400 K in the time domain between 0 and 7 μs. The applied TF was 50 Oe, and the applied LF were 20 and 50 Oe. (b) is the same to Figure 8.

The exponentially relaxing non-oscillatory signal (A _TFe and A _LFe), which is common for the TF-, ZF-, and LF-μ ⁻SR spectra, is most likely originating from different lifetimes of μ ⁻ between μ ⁻Li and μ ⁻O [26], but both μ ⁻ spins are also depolarized by the localized 3d moments. The other possibility for the origin of the A _TFe signal is the formation of μ ⁻Al in the surroundings of the sample. The lifetime difference between μ ⁻Al and μ ⁻O could provide a comparable relaxation to the experiment.

Figure 10 shows the temperature dependencies of Δ⁻, ν ⁻, and λ _KT ⁻ for Li[Ni_1/2Mn_3/2]O₄. A superscript of each symbol, i.e., “-” means that the data obtained from μ ⁻SR measurements. Although the observed Δ⁻ slightly increases with temperature, Δ⁻ is approximately temperature independent and very consistent with the predicted value at the oxygen site in the Li[Ni_1/2Mn_3/2]O₄ lattice. Here, the predicted value ( Δ − , calc = 0.314 μ s − 1 ) was obtained by dipole field calculations with Dipelec [54]. Since the slight increase in Δ⁻ with temperature is difficult to explain, we have attempted to fit the spectra using a common Δ⁻ [Δ^−c=0.30(3) μs⁻¹] in the whole measured temperature range.

$Figure 10: The temperature dependencies of the μ −SR parameters for Li[Ni1/2Mn3/2]O4, (a) the field distribution width (Δ−), (b) the field fluctuation rate (ν −), and (c) the exponential relaxation rate ( λ KT − ${\lambda }_{\text{KT}}^{-}$ ). “-” of each symbol means that the data was obtained from μ −SR measurements. Δ−c, ν −c, and λ KT − c ${\lambda }_{\text{KT}}^{-\text{c}}$ are the fitting results with a common Δ− ( Δ - , c = 0 . 30 ( 3 ) μ s - 1 ${{\Delta}}^{-,\text{c}}=0.30(3)\hspace{0.17em}\mu {\text{s}}^{-1}$ ) in the whole temperature range measured. In (a), a dotted line represents Δ−,calc at the oxygen site predicted with dipole field calculations. The μ +SR results, i.e., Δ+, ν +, and λ KT + ${\lambda }_{\text{KT}}^{+}$ are also plotted for comparison.$

Figure 10:

The temperature dependencies of the μ ⁻SR parameters for Li[Ni_1/2Mn_3/2]O₄, (a) the field distribution width (Δ⁻), (b) the field fluctuation rate (ν ⁻), and (c) the exponential relaxation rate ( λ KT − ). “-” of each symbol means that the data was obtained from μ ⁻SR measurements. Δ^−c, ν ^−c, and λ KT − c are the fitting results with a common Δ⁻ ( Δ - , c = 0 . 30 ( 3 ) μ s - 1 ) in the whole temperature range measured. In (a), a dotted line represents Δ^−,calc at the oxygen site predicted with dipole field calculations. The μ ⁺SR results, i.e., Δ⁺, ν ⁺, and λ KT + are also plotted for comparison.

The value of ν ⁻ determined from fitting with Δ⁻ as a free parameter increases with temperature with a slope (dν ⁻/dT) that is steeper than that of ν ⁺. This is likely to be due to a correlation of the fitted parameters. However, when the spectra are fitted using Δ^−c instead, the increase in ν ^−c is comparable to that of ν ⁺. Finally, λ KT − decreases with temperature and the obtained values of λ KT − is almost equivalent to the ones of λ KT − c . The fact that λ KT − > λ KT + implies the effect of orbital hybridisation between 3d orbitals of Mn/Ni and 2p orbitals of O, on which the μ ⁻ is captured on. In other words, since the localized 3d moments are partially shared by the 2p orbital through such hybridisation, the μ ⁻ spin captured on O naturally feels a large fluctuation compared with the μ ⁺ spin at the interstitial site. This is also most likely the reason for the observed behavior ν − > ν + .

4 Discussion

Here, we attempt to estimate the self-diffusion coefficient of Li⁺ ( D Li J ) in Li[Ni_1/2Mn_3/2]O₄ using the presented μ ^±SR results. We should emphasize that μ ^±SR detects the thermally activated fluctuation of a nuclear magnetic field. As seen in Figure 11, the activation energy (E _a) is estimated as 0.068(5) eV for ν + and 0.043(18) eV for ν - c using the data between 200 and 400 K. The former E _a is comparable to the latter E _a within the estimation accuracy, despite a relatively low accuracy of E _a for ν - c due to a small signal in the μ ⁻SR spectrum. In overall, it is clear that ν increases with temperature for both μ ^±SR. This indicates Li⁺ diffusion in Li[Ni_1/2Mn_3/2]O₄ above 200 K.

$Figure 11: The relationship between ν and inverse temperature. Solid lines represent the best fit using a thermal activation process: ν = ν 0 exp [ − E a / ( k B T ) ] $\nu ={\nu }_{0}\mathrm{exp}[-{E}_{a}/({k}_{\text{B}}T)]$ , in the temperature range between 200 and 400 K, where E a is an activation energy and k B is the Boltzmann constant. The right vertical axis is converted into D Li J ${D}_{\text{Li}}^{\text{J}}$ using Eq. (8).$

Figure 11:

The relationship between ν and inverse temperature. Solid lines represent the best fit using a thermal activation process: ν = ν 0 exp [ − E a / ( k B T ) ] , in the temperature range between 200 and 400 K, where E _a is an activation energy and k _B is the Boltzmann constant. The right vertical axis is converted into D Li J using Eq. (8).

In order to estimate D Li J , we need to know the jump pathway of Li⁺ in the spinel lattice. The cation distribution of spinel Li[Ni_1/2Mn_3/2]O₄ is represented as ( Li + ) 8 a [ Ni 1 / 2 2 + Mn 3 / 2 4 + ] 16 d O 4 , where 8a is a tetrahedral site and 16d is the octahedral site (see Figure 1). For Li deficient spinels, Li⁺ ions are known to diffuse from the regular Li site, i.e., 8a site to a 16c site, which is the vacant octahedral site, and then to a vacant Li 8a site. The most reasonable pathway for D Li J in Li[Ni_1/2Mn_3/2]O₄ is therefore the route from the 8a site to the nearest neighboring 16c sites.

If we assume that the field fluctuation rate ν ^± corresponds to a hopping rate, i.e., the jump frequency of Li⁺ ions between the 8a and 16c sites, D Li J is given by [55];

(8) D Li J = ∑ i = 1 n 1 N i Z v , i s i 2 ν ,

where N _i is the number of Li sites in the i-th jump path, Z _v,i is the vacancy fraction, and s _i is the jump distance. Based on the above consideration, n = 1, N ₁ = 4, Z ₁ = 1, and s ₁ = 1.77 Å. Since D Li J is proportional to ν ^± in Eq. (7), the right vertical axis of Figure 11 is converted to D Li J . It is clearly found that D Li J ranges above 10 − 11 cm 2 / s at temperatures above 250 K for Li[Ni_1/2Mn_3/2]O₄. This result is consistent with the previous μ ⁺SR report on a powder and film samples of Li[Ni_1/2Mn_3/2]O₄ [40].

5 Summary and future

We have measured not only μ ⁺SR but also μ ⁻SR spectra for a spinel-type battery material, Li[Ni_1/2Mn_3/2]O₄, in order to investigate the diffusing species above around 250 K. The observed dynamic behavior both with μ ⁺SR and μ ⁻SR evidenced Li diffusion. There exists a wide range of cathode materials for a Li-ion battery, which are mainly classified within the following three groups; layered rocksalt-, spinel-, and olivine-type compounds. We have already confirmed Li diffusion with μ ^±SR in the spinel Li₄Ti₅O₁₂ [34] and Li[Ni_1/2Mn_3/2]O₄, and the olivine LiMnPO₄ [26]. Moreover, for a typical layered rocksalt-type compound, Li_xCoO₂, the estimated D Li J determined with μ ⁺SR [6] is consistent with the value predicted by first principles calculations [56], and the onset temperature of the Li-diffusion determined with μ ⁺SR is in good agreement with the Li-NMR result [57]. Therefore, μSR is safely applicable for measuring Li diffusion in Li-ion battery materials.

Corresponding author: Jun Sugiyama, Neutron Science and Technology Center, Comprehensive Research Organization for Science and Society (CROSS), Shirakata, 162-1, 319-1106 Tokai, Naka, Ibaraki, Japan, E-mail: juns@triumf.ca

Dedicated to Paul Heitjans on the occasion of his 75th birthday.

Funding source: Swedish Research Council http://dx.doi.org/10.13039/501100004359

Award Identifier / Grant number: Dnr. 2016-06955

Funding source: Japan Society for the Promotion Science http://dx.doi.org/10.13039/501100001691

Award Identifier / Grant number: JP18H01863

Award Identifier / Grant number: JP20K21149

Acknowledgments

We thank the staff of ISIS, in particular David Hunt for making the sample cell, for help with the μ ^±SR experiments (RB1720324 [58] and RB2010138 [59]).

Author contributions: J.S. conceived the experiments. K.O., O.K.F., M.M., and J.S. performed the μ ^±SR experiments in ISIS together with K.I. and the local ISIS support from S.P.C. and A.D.H. J.S. analyzed the data and created the first draft. All authors review the manuscript in several steps and have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: M.M. and O.K.F. were partly supported by the Swedish Research Council (VR) through a neutron project grant (BIFROST, Dnr. 2016-06955). This work was supported by Japan Society for the Promotion Science (JSPS) KAKENHI Grant No. JP18H01863 and JP20K21149.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2021-07-29

Accepted: 2021-08-20

Published Online: 2021-09-06

Published in Print: 2022-06-27

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/zpch-2021-3102

Keywords for this article

Li diffusion; Li ion battery materials; muon spin rotation and relaxation

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How Li diffusion in spinel Li[Ni1/2Mn3/2]O4 is seen with μ ±SR

Article

Abstract

1 Introduction

2 Experimental

3 Results

3.1 μ +SR

3.2 μ −SR

4 Discussion

5 Summary and future

Acknowledgments

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

How Li diffusion in spinel Li[Ni_1/2Mn_3/2]O₄ is seen with μ ^±SR

3.1 μ ⁺SR

3.2 μ ⁻SR