The Elliott-Yafet theory of spin relaxation generalized for large spin-orbit coupling

We generalize the Elliott-Yafet (EY) theory of spin relaxation in metals with inversion symmetry for the case of large spin-orbit coupling (SOC). The EY theory treats the SOC to the lowest order but this approach breaks down for metals of heavy elements (such as e.g. caesium or gold), where the SOC energy is comparable to the relevant band-band separation energies. The generalized theory is presented for a four-band model system without band dispersion, where analytic formulae are attainable for arbitrary SOC for the relation between the momentum- and spin-relaxation rates. As an extended description, we also consider an empirical pseudopotential approximation where SOC is deduced from the band potential (apart from an empirical scaling constant) and the spin-relaxation rate can be obtained numerically. Both approaches recover the usual EY theory for weak SOC and give that the spin-relaxation rate approaches the momentum-relaxation rate in the limit of strong SOC. We argue that this limit is realized in gold by analyzing spin relaxation data. A calculation of the g-factor shows that the empirical Elliott-relation, which links the g-factor and spin-relaxation rate, is retained even for strong SOC.

where |↑ 〉 and |↓ 〉 are the pure (unperturbed) spin states and ↑  k , ↓  k are the perturbed Bloch states. In the first order of the SOC, the coefficients are given by the L matrix element of the SOC for the conduction and the near lying band, and the corresponding energy separation Δ as: |b k |/|a k | ≈ L/Δ.
Elliott showed with a first-order time dependent perturbation treatment 2 that the usual momentum scattering induces spin transitions for the admixed states, i.e. a spin relaxation. With Γ s = ħ/τ s and Γ = ħ/τ used for the spinand momentum-relaxation rates (τ s and τ are the corresponding relaxation times), respectively: where α 1 is a band structure dependent constant near unity. Elliott further showed that the magnetic energy of the admixed states is different from that of the pure spin-states, i.e. there is a shift in the electron g-factor: where ≈ . g 2 0023 0 is the free electron g-factor, α 2 is another band structure dependent constant near unity. Eqs (3) and (4) give the so-called Elliott relation which links three empirical measurables; Γ s , Γ , and Δg. In practice, the spin-relaxation time is obtained for metals from conduction electron spin resonance (CESR) measurements 1 as: Γ s = ħγΔB. Here ΔΒ is the homogeneous ESR line-width and γ/2π = 28.0 GHz/T is the electron gyromagnetic ratio. The CESR resonance line position yields the g-factor shift. Monod and Beuneu tested empirically 9 the validity of Eq. (3) and found that the atomic SOC induced energy splittings approximate well the appropriate matrix elements. For most elemental metals ∆ .  L/ 01 holds but for Cs, Cu, Ag, and In it is about 0.1 and for other metals 0.2 (Pb), 0.3 (Hg and Sn) with a 0.9 as the largest value in Au. Clearly, the validity of a first-order perturbation treatment of the SOC for these cases is questionable. This motivates us to revisit the EY theory for the case when the SOC is not small compared to other energy scales (kinetic energy or band separations).
Here, we discuss the most general form of the SOC Hamiltonian which is applicable for the EY model. We proceed with a simplified four-band Hamiltonian and solve the problem of spin relaxation in the first order of the scattering but exactly for the SOC. We find that the conventional EY result is recovered for weak SOC. A calculation of the g-factor yields the Elliott-relation for arbitrary strength of the SOC. A numerical calculation is also presented for the spin relaxation in the framework of a pseudopotential approximation for the conduction electrons, where SOC is obtained directly from the potential together with an empirical scaling factor. The numerical result qualitatively returns the result of the EY model calculations. The most important results are that the spin scattering time approaches the momentum scattering for strong SOC and that the Elliott relation remains valid irrespective of the SOC strength. We revisit previous experimental data on Au by Monod and Jánossy 10 and show that these effects were observed already but it avoided the attention.
The Elliott theory of spin relaxation. We consider the Elliott mechanism of spin relaxation 2 in which the conduction electron spin interacts with its motion in the electric field of the host lattice described by the periodic potential  V . This lattice induced spin-orbit coupling, given by the Hamiltonian H leads to the mixing of the originally pure spin states in the electron wave function. Thus, the conduction electron spins can now relax through ordinary momentum scattering caused by impurity atoms, for example. The spin-relaxation rate is estimated from the spin-flip transition probability ′ σ σ → ′ W k k of an electron scattering on the impurity potential V, while the momentum-relaxation rate includes also the non spin-flip transition probability ′ σ σ → W k k . In this picture we neglect the SOC caused by the electric field of the impurity atoms, and also assume that they give rise to much larger momentum scattering than the host ions.
To formulate the above described Elliott mechanism, we start from the wave functions of electrons in the periodic potential V ℓ , which are Bloch-type as where n is the band index, and the Bloch functions σ u k n , , are lattice-periodic. Each band is at least two fold degenerate due to the presence of time-reversal symmetry.
The spin-flip transition probability , , within a first-order perturbation theory, and in parallel, the non spin-flip transition element , , . If we assume that the impurity scattering potential V is slowly varying on the scale of the unit cell, the matrix elements can be approximated by the factorization 3 , which gives the spin-flip and non spin-flip transition elements as within the approximation obtained by applying the Fermi's golden rule. A rigorous derivation of the spin-relaxation rate  τ Γ = ( / ) s s which characterizes the process of a not fully polarized spin ensemble of conduction electrons becoming in equilibrium with the environment due to interactions is obtained through a rate equation after ref. 3: W k k given in Eq. (9), i.e. we consider only the change of the Bloch states due to the SOC, and hence obtain the spin-flip rate by using Eq. (11). Elliott assumed that the spin-relaxation rate, Γ s , can be well approximated with the spin-flip rate, Γ sf , which reads: We note that in the limit of small SOC, the spin-flip rate Γ sf given in Eq. (12) is the same as the spin-relaxation rate. Although it is not explicitly proven herein, we assume that Γ s ≈ Γ sf , is retained even for a large SOC. We therefore refer to the definition of the spin-flip rate as spin-relaxation rate throughout in this paper.
Finally, the momentum-relaxation rate is obtained as

Results and Discussion
The Elliott-Yafet model for strong SOC. We consider a four-band model which consists of two nearby bands each with a two-fold degeneracy with respect to the spin-up and spin-down states. The bands are separated by an energy gap Δ, and we neglect the dispersion of the bands. We found that this simplified model allows to derive analytic expression for the spin-relaxation rate for arbitrary value of the spin-orbit interaction. The most general form of the spin-orbit interaction with inversion symmetry is discussed in ref. 11: SOC within the basis | ↑〉 | ↓〉 | ↑〉 | ↓〉 { 1 , 1 , 2 , 2 } where 1 and 2 label the bands, and σσ′ L are SOC matrix elements. For inversion and time-reversal symmetries, the SOC matrix elements satisfy: . The detailed group theoretical considerations are given in the Supplementary Material. For simplicity, we keep only spin-flip SOC matrix elements which connect the states σ 1 and σ′ 2 and neglect the ones which mix the same spin directions. We also consider a small Zeeman energy, h, to handle computational difficulties due to the doubly degenerate bands but we reinsert h = 0 at the end of the calculations.
This Hamiltonian of this simplified model is then given by: which includes kinetic, SOC, and the Zeeman terms. The level structure and the relevant terms of this model are depicted in Fig. 1. We retain the k index for the SOC matrix elements in Eq. (15) to indicate that it depends on both the direction and magnitude of the wave vector, even though we consider dispersionless bands. This is required as otherwise ↑ u and ↓ u are orthogonal, which would give zero spin transition rate:

The original energy splitting, Δ, is modified by the SOC as
The Bloch functions in Eq. (7) are obtained as the eigenstates of Hamiltonian (15): where the index σ  denotes mixed spin states due to the SOC. For example, the lowest two states are obtained as which have the same form as the ones given in Eqs (1) and (2) derived by Elliott. The explicit expressions for the coefficients a k and b k in Eqs (17) and (18) for arbitrary value of the SOC are given in the Supplementary Material. In the limit of small SOC, i.e.
The spin-flip and non spin-flip transition probabilities can be calculated at each band based on the formulae (9) and (10) and the general result is given in the Supplementary Material. Analytic results are obtained when i) the Fermi surface is approximated with a sphere with radius k F which is a good approximation for (k ≈ 0), ii) the unknown overlap integrals of the orbital part of the states and the impurity potential are estimated by constants as . The latter two approximations are justified as these quantities are band structure dependent constants after the integration over the Fermi surface such as α 1 is introduced in Eq. (3).
With these simplifications the transition probabilities read: There is no spin relaxation =  Since the k-integrations in Eqs (12) and (13) cannot be performed in the simplified four-band model, we use the reasonable assumption that Γ and Γ + which will be justified by the subsequent numerical calculations. As a result, the ratio Γ s /Γ reads: In the limit of small SOC, i.e ∆  L , the ratio becomes: reproducing the perturbation result of Elliott given in Eq. (3). In this limit, the spin-relaxation rate is much smaller than the momentum-relaxation rate. In the opposite limit, i.e. ∆  L , the ratio tends to a constant as a function of L:  which means that the spin-relaxation rate approaches to the momentum-relaxation rate (apart from the constants α, β, and γ, which are near unity). Figure 2 shows the ratio Γ s /Γ as a function of the SOC matrix element in the entire L-range with different values of the band-structure dependent constants. The characteristic energy where the behavior of the ratio changes from L 2 -dependence to saturation is approximately L ≈ Δ, i.e. where L/Δ ≈ 1. Figure 2 also shows a fit with the following phenomenological formula, which was found to well approximate the result in the entire parameter range: with c 1 and c 2 being constants of order of unity. We note that the simple phenomenological formula (25) is already implied by the analytic result given in Eq. (21) after some manipulations (see the Supplementary Material). We also discuss the g-factor shift which is given by the expectation value of the orbital momentum in the Bloch state through the Zeeman term. A straigthforward calculation (detailed in the Supplementary Material) yields for ⊥ g and g , i.e. when the magnetic field is perpendicular or parallel to the z axis defined by the SOC Hamiltonian (Eq. (14)), respectively: The EY theory predicts an isotropic g-factor and the anisotropy in our results is due to the omission of the L ↑↑ and L ↓↓ terms in Eq. (15). To test the Elliott-relation we proceed with g ⊥ as: reproducing the Elliott result. Combining expression (28) with (25) gives: We solve it with the pseudopotential method which approximates the lattice potential V(r) with its first few Fourier components V(g) as In the calculations we take This model has some advantages and shortcomings when calculating spin relaxation. An advantage is that the SOC is obtained directly from the lattice potential thus it allows to perform a numerical analysis of spin relaxation. In addition, the pseuodpotential-based SOC automatically accounts for the crystal symmetry which is found in the SOC, too. However, the pseudopotential-based SOC is known to underestimate the experimental SOC as it considers the smooth potential of the valence electrons only and neglects the strongly oscillating potential near the atomic core 12 . To solve this problem, the pseudopotential-based SOC is magnified with a scaling parameter with a typical value of λ sc ~ 10 2 − 10 3 . We chose λ sc = 264 as it accounts quantitatively well for the k = 0 SOC gap in Ge 12 . In addition, we also introduce a tuning parameter, λ, for the SOC interaction in order to study the SOC dependent spin relaxation. As a result, the SOC Hamiltonian reads as . We note that more realistic pseudopotentials were used successfully to reproduce the SOC strength in Si 13 .
We use a pseudopotential parameter set which reproduces well the conduction and valence bands of Ge. The band structure is shown in Fig. 3 for λ = 1. Considering the nearby valence and conduction bands denoted as v n and |c〉 in Fig. 3, respectively, the characteristic features of metals with hybridized s-p orbitals can be discussed. Namely, the orbital parts of the electron wave functions at k = 0 (Γ point) without SOC have the symmetries |c〉 ~ Γ 1 and |v n 〉 ~ Γ 4 which mean s-and p-like orbitals, respectively. Symmetry operations acting on the spin wave functions have to be considered in the presence of SOC, which can be handled by double group irreducible representations.
At k = 0, the p-symmetric (Γ 4 ) valence band with spin splits into a four-and a two-fold degenerate state, separated by the SOC gap. These split states can be labeled according to the total angular momentum operator as j = 3/2 ~ Γ 8 and j = 1/2 ~ Γ 7 , respectively. As we move away from the k = 0 point, the four-fold degenerate valence band splits further but a two-fold degeneracy is kept due to the presence of time-reversal symmetry. In the vicinity of the k = 0 point, all of the valence and conduction bands have mixed s-p character due to the hybridization.
Using Eqs (9) and (10), we derive the non spin-flip and spin-flip transition matrix elements in the conduction band |c〉 as Scientific RepoRts | 6:22706 | DOI: 10.1038/srep22706 and in the upper valence band |v 1 〉 as that is not specified, but the k-dependence of the wave functions is obtained accurately in the pseudopotential approximation.
The momentum-and spin-relaxation rates are obtained according to Eqs (12) and (13) by averaging over the transition matrix elements numerically with respect to k and k′ on the Fermi surface with energy E k F as: , F Figure 4 shows the spin-and momentum-relaxation rates calculated in the pseudopotential approximation both for the upper valence band |v 1 〉 and conduction band |c〉 as a function of the spin-orbit coupling, λ. We took k F = 0.4 in the calculation and averaged the transition probabilities given in Eqs (34) and (35) over 30 points at the Fermi surface to obtain the relaxation rates, Γ and Γ s .
The seemingly different behavior of the relaxation rates in the valence and conduction bands can be consistently explained. As it is shown in Fig. 3, the 'relevant' band gap, i.e. the characteristic energy defined as for the conduction band, is quite different for the two cases. Namely, in the case of the upper valence state, |v 1 〉 , there is another valence state |v 2 〉 in its close vicinity, while the closest state to the conduction band |c〉 is |v 1 〉 , which is much further away, which leads to ∆ ∆  v c . Thus, by increasing the SOC in the calculations, the range λ ∆  v can be reached for the valence band and we indeed find the saturating behavior for both Γ and Γ s in this range in well agreement with the result of the four-band model. On the other hand, in the case of the conduction band, the relation λ ∆  c holds in the entire range of λ which we study. A much larger λ would lead to a rearranged band structure. Therefore, for the conduction band in our model, we observe the perturbative (or Elliott-Yafet) behavior Γ~(1)  and λ Γs 2 . We find that the numerical data for the ratio Γ s /Γ follows the approximate formula obtained in the model Hamiltonian calculation for both the valence and conductions bands. Namely, the approximate formula given in Eq. (25) fits well the ratios by taking the numerical data Δ v and Δ c for the band gap Δ(k F ) for the valence and conduction band, respectively. Figure 5 shows the fit for the ratio Γ s /Γ as a function of λ/Δ with this model.
The good agreement between the numerically obtained data and the phenomenological formula (25) obtained in the four-band model calculation supports the previous assumption that Γ and . In addition, we find it compelling that the two different approaches result in Γ s /Γ ratios Comparison with experiment. Among all the metals where experimental data on spin relaxation exists, Au is particularly suited to test the validity of the above discussion as it has the strongest SOC. In addition, gold has a single conduction electron per unit cell, i.e. its Fermi surface does not extend beyond the first Brillouin zone. It is known that description of spin relaxation is more complicated for metals with two or more conduction electrons per unit cell 14,15 (e.g. Mg or Al), where the Fermi surface crosses the Brillouin zone boundaries giving rise to the so-called spin relaxation "hot-spots". When hot-spots are present, the Elliott relation is known to break down even though the Elliott-Yafet theory remains valid 14,15 .
Monod and Beuneu found 9 the SOC strength in Au as L/Δ ≈ 0.9. It is therefore an appropriate candidate for a case where the perturbative treatment of the SOC breaks down. Monod and Jánossy reported in ref. 10 the electron spin resonance linewidth, ΔB, which is used to obtain Γ s . Γ is calculated from the resistivity, ρ, data in ref. 16 through: ε ω ρ Γ = 0 pl 2 , where ε 0 is the vacuum permittivity and ω p1 = 8.55 eV is the plasma frequency of gold 17 . For both sets of data, the residual linewidth and resistivity was subtracted as both quantities are known to obey the Matthiessen's rule 7 , i.e. that the respective residual and temperature dependent scattering rates are additive.
The comparison, shown in Fig. 6, shows that the measured spin-relaxation rate approaches the momentum-relaxation rate within an order of magnitude such as our theoretical result suggests. The difference between the calculated Γ s /Γ ≈ 1 and the observed Γ s /Γ = 0.06 ± 0.01 could be due to the band structure dependent constants near unity in Eq. (24). However the observation of a nearly equal Γ s and Γ is itself surprising as to our knowledge it has not been noted elsewhere that the mergence of spin and momentum relaxation times could be realized for a real material.
We finally test empirically the validity of the Elliott relation in Eq. (29), i.e. Γ Γ∆ ≈ g /( ) 1 s 2 in gold. Monod and Jánossy found g = 2.11 ± 0.01 which gives Γ Γ∆ ≈ g /( ) 5 s 2 , which is again in agreement with the theoretical prediction given the presence of the band structure dependent constant factors. This means that the Elliott relation remains a valid empirical tool to test whether the Elliott-Yafet theory applies for a given system.

Conclusions
We generalized the Elliott-Yafet theory of spin relaxation in metals with inversion symmetry for the case of arbitrary value of the spin-orbit coupling. We applied two different approaches, exact diagonalization of a model four-band Hamiltonian (without dispersion) and numerical calculation of the spin relaxation time for a pseudopotential approach. The two methods give a qualitatively similar result, which is summarized in Eq. (25). A calculation of the g-factor in the four-band model shows that the empirical Elliott-relation, which links the g-factor and spin-relaxation rate, is retained even for strong SOC. Our result indicates that spin and momentum relaxation times can have similar orders of magnitude. We show that this situation has been already observed in Au. Our result is an important step toward the unified theory of spin relaxation including the strength of the spin-orbit coupling.