Abstract
Designing new quantum materials with longlived electron spin states urgently requires a general theoretical formalism and computational technique to reliably predict intrinsic spin relaxation times. We present a new, accurate and universal firstprinciples methodology based on Lindbladian dynamics of density matrices to calculate spinphonon relaxation time of solids with arbitrary spin mixing and crystal symmetry. This method describes contributions of ElliottYafet and D’yakonovPerel’ mechanisms to spin relaxation for systems with and without inversion symmetry on an equal footing. We show that intrinsic spin and momentum relaxation times both decrease with increasing temperature; however, for the D’yakonovPerel’ mechanism, spin relaxation time varies inversely with extrinsic scattering time. We predict large anisotropy of spin lifetime in transition metal dichalcogenides. The excellent agreement with experiments for a broad range of materials underscores the predictive capability of our method for properties critical to quantum information science.
Introduction
The manipulation of electron spins is of increasing interest in a widerange of emerging technologies. The rapidly growing field of spintronics seeks to control spin as the unit of information instead of charge in devices such as spin transistors^{1}. Quantum information technologies seek to utilize localized spin states in materials both as singlephoton emitters^{2,3,4} and as spinqubits for future integrated quantum computers^{5}. Both spintronics and quantum information applications therefore demand a quantitative understanding of spin dynamics and transport in metals and semiconductors. Recent advances in circularly polarized pumpprobe spectroscopy^{6}, spin injection, and detection techniques^{7} have enabled increasingly detailed experimental measurement of spin dynamics in solidstate systems^{8,9}. However, a universal firstprinciples theoretical approach to predict spin dynamics, quantitatively interpret these experiments and design new materials has remained out of reach.
A key metric of useful spin dynamics is the spin relaxation time τ_{s}^{1}. For example, spinbased quantum information applications require τ_{s} exceeding milliseconds for reliable qubit operation. Consequently, accurate prediction of τ_{s} in general materials is an important milestone for firstprinciples design of quantum materials. Spin–spin^{10}, spin–phonon^{11}, and spinimpurity scatterings, all contribute to spin relaxation, but spin–phonon scattering sets the intrinsic material limitation and is typically the dominant mechanism at room temperature^{1}. Further, spin–phonon relaxation arises from a combination of spin–orbit coupling (SOC) and electron–phonon scattering, and is traditionally described by two mechanisms. First, the Elliott–Yafet (EY) mechanism involves spin–flip transitions between pairs of Kramersdegenerate states due to SOCbased spinmixing of these states^{12,13}. Second, the D’yakonov–Perel’ (DP) mechanism in systems with broken inversion symmetry involves electron spins precessing between scattering events due to the SOCinduced internal effective magnetic field^{14}.
Previous theoretical approaches have extensively investigated these two distinct mechanisms of intrinsic spin–phonon relaxation using model Hamiltonians in various materials^{15}. These methods require parametrization for each specific material, which needs extensive prior information about the material and specialized computational techniques, and often only studies one mechanism at a time. Furthermore, most of these approaches require the use of simplified formulae^{12,14} and make approximations to the electronic structure (e.g. low spinmixing) or electron–phonon matrix elements^{15}. This limits the generality and reliability of these approaches for complex materials, particularly for the DP mechanism, where various empirical relations are widely employed to estimate τ_{s}^{1}. Sophisticated methods based on spin susceptibility^{16} and time evolution of density matrix^{17} also rely on suitably chosen model Hamiltonians with empirical scattering matrix elements. Therefore, while these methods provide some mechanistic insight, they do not serve as predictive tools of spin relaxation time for the design of new materials.
A general firstprinciples technique to predict spin–phonon relaxation in arbitrary materials is therefore urgently needed. Previous firstprinciples studies have addressed the EY mechanism in centrosymmetric semiconductors^{18,19} and metals^{20}. These methods^{18,20} rely on defining a pseudospin that allows the use of Fermi’s golden rule (FGR) with only spin–flip transitions^{13}. However, this is only welldefined for cases with weak spinmixing such that eigenstates within each Kramersdegenerate pair can be chosen to have small spinminority components, precluding the study of spin relaxation of states with strong spinmixing, e.g. holes in silicon and noble metals. Firstprinciples calculations have not yet addressed systems with such complex degeneracy structures, where the simple picture of spin–flip matrix elements in a FGR breaks down, or systems without inversion symmetry that do not exhibit Kramers degeneracy. Therefore, a more general firstprinciples technique without the materialspecific simplifying assumptions of these previous approaches is now necessary.
In this work, we establish a new, accurate and unified firstprinciples technique for predicting spin relaxation time based on perturbative treatment of the Lindbladian dynamics of density matrices^{21}. Importantly, by covering previously disparate mechanisms (e.g. EY and DP) in a unified framework, this technique is applicable to all materials regardless of dimensionality, symmetry (especially inversion) and strength of spinmixing, which is critical for new material design. All SOC effects are included selfconsistently (and nonperturbatively) in the groundstate eigensystem at the density functional theory (DFT) level, and we predict τ_{s} through a universal rate expression without the need to invoke realtime dynamics. In this article, we first introduce our theoretical framework based on firstprinciples densitymatrix dynamics, and then show prototypical examples of τ_{s} for the broad range of systems, including three with inversion symmetry—silicon, iron, and graphene, and three without inversion symmetry—monolayer MoS_{2}, monolayer MoSe_{2}, and bulk GaN, in excellent agreement with available experimental data. By doing so, we establish the foundation for quantum dynamics of open systems from firstprinciples to facilitate the design of quantum materials.
Results
Theory
The key to treating arbitrary state degeneracy and spinmixing for spin relaxation is to switch to an ab initio densitymatrix formalism, which goes beyond specific cases such as Kramers degeneracy or Rashbasplit model Hamiltonians. Specifically, we seek to work with density matrices of electrons alone, treating its interactions with an environment consisting of a thermal bath of phonons. In general, tracing out the environmental degrees of freedom in a full quantum Liouville equation of the densitymatrix results in a quantum Lindblad equation. Specifically, for electron–phonon coupling^{21} based on the standard Born–Markov approximation^{22} that neglects memory effects in the environment, the Lindbladian dynamics in interaction picture reduces to
where α is a combined index labeling electron wavevector k and band index n, λ is mode index and ± corresponds to \(q=\mp \left(kk^{\prime} \right)\). \({n}_{q\lambda }^{\pm }\equiv {n}_{q\lambda }+0.5\pm 0.5\) and n_{qλ} is phonon occupation. \({G}_{\alpha \alpha ^{\prime} }^{q\lambda \!{\pm} }={g}_{\alpha \alpha ^{\prime} }^{q\lambda \!{\pm} }{\delta }^{1/2}({\varepsilon }_{\alpha }{\varepsilon }_{\alpha }^{\prime}\pm {\omega }_{q\lambda })\) is the electron–phonon matrix element including energy conservation, where ω_{qλ} is the phonon frequency.
This specific form of the Lindbladian dynamics preserves positive definiteness of the density matrix which is critical for numerical stability^{21}. In addition, the energyconserving δfunction above is regularized by a Gaussian with a width γ, which corresponds physically to the collision time. In some cases, the results depend on γ and γ → 0 is not the relevant limit^{23}. Here, the Lindblad master equation with finite smearing parameters corresponding to the collision time can be regarded as the best Markovian approximation to the exact dynamics^{23}. In the case of spin relaxation, this is particularly important for systems that exhibit the DP mechanism, as we show below. Consequently, we consistently determine the smearing parameters from ab initio electron–phonon linewidth calculations throughout^{24,25}.
The densitymatrix formalism allows the computation of any observable such as number and spin density of carriers, and the inclusion of different relaxation mechanisms at time scales spanning femtoseconds to microseconds, which forms the foundation of the general relaxation time approach we discuss below. Given an exponentially relaxing measured quantity \(O={\rm{Tr}}(o\rho )\), where o and ρ are the observable operator and the density matrix, respectively, we can define the relaxation rate Γ_{o} and relaxation time \({\tau }_{\mathrm{o}}={\Gamma }_{\mathrm{o}}^{1}\) of quantity O as
where eq corresponds to the final equilibrium state. We note that even when the observables have additional \(\cos (\omega t)\) oscillation factors, such as due to spin precession with periodicity of ω, the above equation remains an appropriate definition of the overall relaxation rate. For example, for a precessing and relaxing spin system with \(S(t)={S}_{0}\exp (t/\tau )\cos (\omega t)\), the initial relaxation rate is \(\dot{S}(0)={S}_{0}/\tau\), which is exactly the same as that of a pure exponential relaxation.
The equilibrium density matrix in band space is \({({\rho }^{{\rm{eq}}})}_{nn^{\prime} }^{k}={f}_{kn}{\delta }_{nn^{\prime} }\), where f_{kn} are the Fermi occupation factors of electrons in equilibrium. Writing the initial density matrix ρ = ρ^{eq} + δρ, assuming a small perturbation ∣∣δρ∣∣ ≪ ∣∣ρ^{eq}∣∣ and kdiagonal o and δρ, the Lindblad dynamics expression (Eq. (1)) and the definition (Eq. (2)) yield
Here, the G is exactly as defined above in Eq. (1), but separating the wavevector indices (k, \(k^{\prime}\)) and writing it as a matrix in the space of band indices (n, \(n^{\prime}\)) alone. Similarly, o and δρ are also matrices in the band space, Tr_{n} and †_{n} are trace and Hermitian conjugate in band space, and \({[o,G]}_{kk^{\prime} }\equiv {o}_{k}{G}_{kk^{\prime} }{G}_{kk^{\prime} }{o}_{k^{\prime} }\), written using matrices in band space.
Given an initial perturbation δρ and an observable o, Eq. (3) can now compute the relaxation of expectation value O from its initial value. Even for a specific observable like spin, several choices are possible for the initial perturbation corresponding directly to the experimental measurement scheme. Specifically for spin relaxation rate Γ_{s,i}, the observable is the spin matrix S_{i} labeled by Cartesian directions i = x, y, z, and the initial perturbed state should contain a deviation of spin expectation value from equilibrium. The most general (experimentagnostic) choice for preparing a spin polarization is to assume that all other degrees of freedom are in thermal equilibrium, which can be implemented using a test magnetic field B_{i} as a Lagrange multiplier for implementing a spin polarization constraint. With a corresponding initial perturbation Hamiltonian of H_{1} = −2μ_{B}B_{i}S_{i}/ℏ, where μ_{B} is the Bohr magneton, perturbation theory yields
In some cases, S_{i,k,mn} ≈ 0 when ϵ_{km} ≠ ϵ_{kn}. For these cases, \(\delta \rho \approx (2{\mu }_{\mathrm{B}}{B}_{i}/\hslash )(\partial f/\partial \epsilon ){S}_{i}^{\mathrm{deg} }\), where \({({S}_{i}^{\mathrm{deg} })}_{knn^{\prime} }\equiv {\left({S}_{i}\right)}_{knn^{\prime} }{\delta }_{{\varepsilon }_{kn}{\varepsilon }_{kn^{\prime} }}\) is the degeneratesubspace projection of S_{i}. In such cases, we can further simplify Eq. (3) to the Fermi Golden rulelike expression,
where \({\chi }_{s,i}=T{r}_{n}[{S}_{i}(\partial f/\partial \epsilon ){S}_{i}^{\mathrm{deg} }]/{N}_{k}\). Note that the test field B_{i} etc. drops out of the final expression and only serves to select the direction of the perturbation in the highdimensional space of density matrices.
Without SOC, \({S}_{i}^{\mathrm{deg} }={S}_{i}\) commutes with g, leading to Γ_{s,i} = 0 as expected. If \({S}_{i}^{\mathrm{deg} }\) is diagonal, \({[{S}_{i}^{\mathrm{deg} },{g}^{q\lambda }]}_{knk^{\prime} n^{\prime} }\) reduces to \(\Delta {s}_{i,knk^{\prime} n^{\prime} }{g}_{knk^{\prime} n^{\prime} }^{q\lambda }\), where \(\Delta {s}_{i,knk^{\prime} n^{\prime} }\equiv {s}_{i,kn}{s}_{i,k^{\prime} n^{\prime} }\) is the change in (diagonal) spin expectation value for a pair of states. Therefore, in this limit, Eq. (5) reduces to transitions between pairs of states, each contributing proportionally to the square of the corresponding spin change.
See Supplementary Note 1 and 2 for detailed derivations of the above equations. As we show in Supplementary Note 1 and 2, the above equations can be reduced to previous formulae with spin–flip matrix elements in Kramersdegenerate subspaces for systems with inversion symmetry and weak spinmixing, such as conduction electron spin relaxation in bulk Si, similar to ref. ^{18}. However, Eq. (3) is much more general, applicable for systems with arbitrary degeneracy and crystal symmetry, and we therefore use it throughout for all results presented below. In addition, the overall framework can also be extended to other observables and can be made to correspond to specific measurement techniques that prepare a different initial density matrix e.g. a circularly polarized pump pulse.
Finally, note that in our firstprinciples method, all SOCinduced effects (such as the Rashba/Dresselhaus effects) are selfconsistently included in the groundstate eigensystem or the unperturbed Hamiltonian H_{0}. This is essential to allow us to simulate τ_{s} by a single rate calculation when there is broken inversion symmetry. On the other hand, if SOC does not enter into H_{0}, as in previous work with model Hamiltonians, it must be treated as a separate term that provides an internal effective magnetic field. Consequently, those approaches require a coherent part of the time evolution to describe the fast spin precession induced by this effective magnetic field, which require explicit realtime dynamics simulations even to capture spin relaxation, going beyond a simple exponential decay as in Eq. (2). Using fully selfconsistent SOC in a firstprinciples method is therefore critical to avoid this systemspecific complexity and arrive at the universal approach outlined above.
Systems with inversion symmetry: Si and Fe
We first present results for systems with inversion symmetry traditionally described by a Elliot–Yafet spin–flip mechanism. Figure 1a shows that our predictions of electron spin relaxation time (τ_{s}) of Si as a function of temperature are in excellent agreement with experimental measurements^{26,27}. Note that previous firstprinciples calculations^{18} approximated spin–flip electron–phonon matrix elements from pseudospin wavefunction overlap and spinconserving electron–phonon matrix element, effectively assuming that the scattering potential varies slowly on the scale of a unit cell; we make no such approximation in our direct firstprinciples approach. Importantly, this allows us to go beyond the doubly degenerate Kramersdegenerate case of conduction electrons in Si. In contrast, holes in Si exhibit strong spinmixing with spin2/3 character and spin expectation values no longer close to ℏ/2. Figure 1b shows our predictions for the hole–spin relaxation time, which is much shorter than the electron case as a result of the strong mixing (450 fs for holes compared to 7 ns for electrons at 300 K) and is much closer to the momentum relaxation time. In addition, Fig. 1d shows that the change in spin expectation values (Δs) per scattering event (evaluated using Eq. (5)) has a broad distribution for holes in Si, indicating that they cannot be described purely by spin–flip transitions, while conduction electrons in Si predominantly exhibit spin–flip transitions with Δs = 1.
We next consider an example of a ferromagnetic metal, iron, which exhibits a complex band structure not amenable for model Hamiltonian approaches. Previous firstprinciples calculations for ferromagnets employ empirical Elliott relation^{28} or FGR formulae with spin–flip matrix elements specifically developed for metals or ferromagnets^{20}. Here, we apply exactly the same technique used for the silicon calculations above and predict spin relaxation times in iron in good agreement with experimental measurements (Fig. 1c)^{29,30,31}. Our Wannier interpolation also enables systematic and efficient Brillouin zone convergence of these predictions which were not possible previously. Similar to holes in Si, the Δs of Fe also exhibits a broad distribution extending from 0 to ℏ in the contribution to the total spin relaxation rate (Fig. 1d). Therefore, spin relaxation in transition metals are not purely spin–flip transitions, and we expect this effect to be even more pronounced in 4d and 5d metals with stronger SOC than the 3d magnetic metal considered here. Finally, Fig. 1a–c shows that τ_{s} is approximately proportional to momentum relaxation time τ_{m} for both Si and bcc Fe, which is expected for spin relaxation in systems with EY mechanisms^{1}.
Systems with inversion symmetry: graphene
Graphene is of significant interest for spinbased technologies, and significant recent work with model Hamiltonians seeks to identify the fundamental limits of spin coherence in graphene^{32}. Estimates vary widely from theoretical estimates on the order of microseconds to experiments ranging from picoseconds to nanoseconds^{33,34,35,36}, with the discrepancies hypothesized to arise from faster extrinsic relaxation in experiments. However, previous model Hamiltonian studies required parametrization of approximate matrix elements, and focus on specific phonon modes (e.g. flexural modes) for spin–phonon relaxation. Here we predict intrinsic electron–phonon spin relaxation time for freestanding graphene to firmly establish the intrinsic spin–phonon relaxation limit free of specific model choices or parameters.
Figure 2 shows the predicted spin–phonon relaxation times as a function of temperature and Fermi level position. At room temperature, our calculated lifetimes are of the same magnitude (in microseconds) as previous predictions^{33} indicating that faster relaxation is likely extrinsic in experiments. However, in addition to the flexural phonon mode^{33,35}, inplane acoustic (A) phonon modes have a strong and nonnegligible contribution, while optical modes (O) have an overall smaller effect (Fig. 2b). We also find that the ratio between inplane and outofplane spin relaxation times range from 0.5 to 0.7 (Fig. 2a, c), consistent with experimental measurements^{34}. As evident from Fig. 2c, longer spin relaxation time of up to microseconds is achievable at low temperatures in pristine and freestanding graphene. However, at low temperatures, competing effects from substrates and disorder can make overall measured spin relaxation faster than theoretical predictions^{35}.
Finally note that while the ratio between inplane and outofplane spin relaxation times is nearly 1/2, which is often considered to be a signature of the DP mechanism, freestanding graphene is inversion symmetric and does not exhibit the DP mechanism. Figure 2d shows that the spin relaxation time is mostly insensitive to the extrinsic scattering rates, instead of the linear relation (inverse relation with scattering time) expected for the DP mechanism, as discussed below in further detail. The spin relaxation of graphene may be switched to the DP regime by adding substrates or external electric fields to break inversion symmetry^{36,37}, which will be investigated in detail using this theoretical framework in future work.
Systems without inversion symmetry: outofplane τ _{s} of MoS_{2} and MoSe_{2}
The twodimensional transition metal dichalcogenides (TMDs) exhibit extremely longlived spin/valley polarization (over nanoseconds)^{38}, with long valleystate persistence attributed to spinvalley locking effects. A fundamental understanding of spin/valley relaxation mechanisms is now required to utilize this degree of freedom for valleytronic computing^{39}. Next we investigate spin relaxation τ_{s} of systems without inversion symmetry from firstprinciples, starting with two TMD systems—monolayer MoS_{2} and MoSe_{2} as prototypical examples. (Unless specified, τ_{s} represents outofplane spin relaxation time τ_{s}_{,zz} for TMDs.)
In both systems, valence and conduction band edges at K and K′ valleys exhibit relatively large SOC band splitting, with nearly perfect outofplane spin polarization. Timereversal symmetry further enforces opposite spin directions for the bandedge states at K and K′. Previous studies using model Hamiltonians consider the DP mechanism to dominate spin relaxation in these materials^{17}, but in our firstprinciples approach, we do not need to a priori restrict our calculations to EY or DP limits.
In Fig. 3, we show the outofplane spin (τ_{s}) and momentum (τ_{m}) relaxation time of conduction electrons in two monolayer TMDs as a function of temperature, along with their intervalley/intravalley contributions and experimental values. First, the overall agreement between our calculations and previous experiments by ultrafast pumpprobe spectroscopy is excellent^{38,40,41}. Note that ultrafast measurements of TMDs obtain coupled dynamics of spin and valley polarizations according to the selection rules with circularly polarized light, necessitating additional analysis to extract τ_{s}, e.g., a phenomenological model fit to experimental curves in ref. ^{38}. On the other hand, our firstprinciples method simulates τ_{s} directly without model or input parameters. This provides additional confidence in the experimental procedures of extracting τ_{s}, and lends further insights into different scattering contributions in the dynamical processes as we show below. Moreover, special care is necessary when comparing with certain low temperature measurements with lightly doped samples, which access spin relaxation of excitons rather than individual free carriers, as discussed in refs. ^{42,43}; we focus here on spin relaxation of free carriers.
Next, comparing the relative contributions of intervalley and intravalley scattering for spin relaxation time, we find that the intravalley process dominates spin relaxation of conduction electrons in both TMDs: the intravalley only spin relaxation time (black squares) in Fig. 3 is nearly identical with the net spin relaxation time (red circles), while the intervalley contribution alone (blue triangles) is consistently more than an order of magnitude higher in relaxation time (lower in rate). Furthermore, with decreasing temperature, the relative contribution of the intervalley process decreases because the minimum phonon energies for wave vectors connecting the two valleys exceed 20 meV, and the corresponding phonon occupations become negligible at temperatures far below 300 K.
Previous theoretical studies of MoS_{2} with model Hamiltonians^{17} obtained (outofplane) τ_{s} two orders of magnitude higher than our predictions, which agree with experimental data^{38}. Such significant deviations are possibly because of the approximate treatments of electronic structure and electron–phonon coupling in their theoretical model. In addition, our firstprinciple calculations treat all phonon modes on an equal footing. Table 1 shows that the relative contributions of each phonon mode to τ_{s} varies strongly with temperature. Full electron and phonon band structure is therefore vital to correctly describe spin–phonon relaxation with varying temperature, while model Hamiltonians that select specific phonon modes have limited range of validity^{17}.
Hole–spin relaxation in MoS_{2} and MoSe_{2} has not been previously investigated in detail theoretically. Figure 4 presents our predictions of hole τ_{s} and τ_{m} in the two TMDs, indicating that hole τ_{s} is much longer than that for electrons at all temperatures, exceeding 1 ns below 100 K. In contrast to the electron case, the intervalley process is relatively much more important and dominates spin relaxation at low temperature in MoS_{2} and at all temperatures in MoSe_{2}. This is because large SOC splitting at the valence band maximum makes the intravalley transition between two valence bands nearly impossible based on energy conservation in the electron–phonon scattering process. Experimental measurements also observe long spin relaxation times dominated by intervalley scattering in tungsten dichalcogenides^{44}, which may facilitate applications in spintronic and valleytronic devices.
External magnetic fields can serve as tools tuning material properties^{45} and are an inherent component of spin dynamics measurements^{38,44}. Systems with broken inversion symmetry in particular may strongly respond to magnetic fields. We therefore investigate the effects of an external field B on τ_{s} by introducing a Zeeman term (g_{s}μ_{B}/ℏ)B ⋅ S to the electronic Hamiltonian interpolated using Wannier functions (approximating g_{s} ≈ 2), just prior to computing τ_{s} with Eq. (3). Figure 5 shows that the outofplane τ_{s} of conduction electrons of MoS_{2} decreases with increasing inplane magnetic field B_{x}, in agreement with experimental work on MoS_{2}^{38} and in general consistency with previous theoretical studies of τ_{s} for systems with broken inversion symmetry^{17,46}.
This strong magnetic field response has a simple intuitive explanation: in TMDs, the spin splitting of bands can be considered as the result of the internal effective magnetic field \({B}_{{\mathrm{so}}}\hat{{\bf{z}}}\) due to broken inversion symmetry. Applying a finite B_{x} perpendicular to \({B}_{{\mathrm{so}}}\hat{{\bf{z}}}\) will cause additional spinmixing and increase the spin–flip transition probability, thereby reducing the spin relaxation time. The degree of reduction depends on the detailed electronic structure of MoS_{2} and MoSe_{2} as shown in Supplementary Figs. 2 and 3: MoSe_{2} exhibits a larger spin splitting of conduction bands and a higher internal magnetic field, and is therefore less affected by external B_{x}. Similarly, hole–spin relaxation in both MoS_{2} and MoSe_{2} (not shown) exhibit very weak dependence on B_{x} because of the large spin splitting and high internal effective magnetic field B_{so} for valence bandedge states compared to those near the conduction band minimum. This insensitivity of hole τ_{s} to magnetic fields is also consistent with experimental studies of hole τ_{s} in WS_{2}^{44} and WSe_{2}^{47}.
Finally, outofplane magnetic field B_{z} has a negligible effect on spin relaxation for TMDs (not shown), unlike the inplane magnetic field B_{x} or B_{y}. This is because electronic states around band edges are already polarized along the outofplane direction under a strong internal \({B}_{{\mathrm{so}}}\hat{{\bf{z}}}\). High experimental external magnetic fields ~1 Tesla are relatively weak in contrast and only slightly change the spin polarization of the states, rather than introducing a spinmixing that leads to spin relaxation.
Systems without inversion symmetry: inplane τ _{s} of MoS_{2}
In all cases, the spin–phonon relaxation time decreases with increasing temperature, approximately proportional to the momentum relaxation time τ_{m}. This is expected because both scattering rates, \({\tau }_{{\mathrm{s}}}^{1}\) and \({\tau }_{{\mathrm{m}}}^{1}\), are proportional to phonon occupation factors which increase with temperature. The intrinsic inplane spin relaxation time (τ_{s}_{,xx}) in MoS_{2} also shows the same trend with temperature (Fig. 6a), but exhibits a fundamental difference from the previous cases when considering additional extrinsic scattering.
Specifically, Fig. 6b shows the dependence of spin relaxation times in MoS_{2} conduction electrons as a function of extrinsic scattering rates, which enter Eq. (3) through an additional contribution to the smearing width γ of the energyconserving δfunctions (in addition to the intrinsic electron–phonon contributions computed from firstprinciples). This additional smearing physically corresponds to the reduced lifetime and increased broadening of the electronic states in the material due to scattering against defects, impurities etc^{37}. Importantly, the inplane spin relaxation time τ_{s}_{,xx} increases linearly with extrinsic scattering rate, or inversely with extrinsic scattering time, which is a hallmark of the DP mechanism of spin relaxation^{48}.
Note that this inverse relation competes with the phonon occupation factors in determining the overall temperature dependence of spin relaxation time. At higher temperature, increased phonon occupation factors lower the intrinsic relaxation times of both carrier and spin, as stated above. The lowered intrinsic carrier relaxation time increases the finite smearing width in Eq. (3), which contributes towards increasing the spin relaxation time within the DP mechanism (inverse relation). However, the direct contribution of phonon occupation factors in the spin relaxation rate in Eq. (3) overwhelms this secondary change and results in a net decrease of spin relaxation time, consistent with all calculations above and experiments^{49,50}.
In contrast with the inplane case, the outofplane spin relaxation τ_{s}_{,zz} is mostly insensitive to the extrinsic scattering rate (and broadening γ), as all previous spin relaxation results in Kramersdegenerate materials discussed above (e.g. for graphene in Fig. 2d). Note that τ_{s}_{,xx} is also overall much shorter than τ_{s}_{,zz}, because the strong internal magnetic field in TMDs stabilizes spins in the z direction as discussed above. Large anisotropy in spin lifetimes due to a similar spinvalley locking effect has been theoretically predicted^{51} and experimentally measured^{52} previously in graphene–TMD interfaces as well.
Systems without inversion symmetry: GaN
Finally, we show spin relaxation in GaN as an archetypal example of the DP mechanism. Fig. 7 shows that both inplane (τ_{s}_{,xx}) and outofplane (τ_{s}_{,zz}) spin lifetime of GaN are proportional to extrinsic scattering rates, or inversely proportional to extrinsic scattering time. Most importantly, the ratio between τ_{s}_{,zz} and τ_{s}_{,xx} is exactly 1/2 for this material, which is an additional feature of the conventional DP mechanism^{1}. Note that, in contrast, the 2D TMDs are more complex due to strong SOC splitting and anisotropy, did not exhibit this 1/2 ratio, and exhibited the extrinsic scattering dependence only for inplane spin relaxation. Overall, these results indicate that the general densitymatrix formalism presented here elegantly captures the characteristic DP and EY mechanism limits, as well as complex cases that do not fit these limits, all on the same footing in a unified framework.
Discussion
In summary, we have demonstrated an accurate and universal firstprinciples method for predicting spin relaxation time of arbitrary materials, regardless of electronic structure, strength of spinmixing and crystal symmetry (especially with/without inversion symmetry). Our work goes far beyond previous firstprinciples techniques based on a specialized Fermi’s golden rule with spin–flip transitions and provides a pathway to an intuitive understanding of spin relaxation with arbitrary spinmixing. In TMD monolayer materials, we clarify the roles of intravalley and intervalley processes, which are additionally resolved by phonon modes, in electron and hole–spin relaxation. We predict longlived spin polarization from resident carriers of MoS_{2} and MoSe_{2} and show their strong sensitivity of electron spin relaxation to inplane magnetic fields.
The predictive power of firstprinciples calculations is crucial for providing fundamental understanding of spin relaxation in new materials. The same technique can be applied to predict spin relaxation in realistic materials with or without defects useful for quantum technologies, wherever spin relaxation is dominated by electron–phonon scattering. We have already considered the general impact of disorder and electronimpurity scattering on spin–phonon relaxation through carrier broadening, but impurities can contribute an additional channel for spin relaxation, especially in the Kramersdegenerate case and at lower temperatures^{18}. The extension of this technique to directly predict electronimpurity scattering for specific defects is relatively straightforward using supercell calculations, but computationally more demanding, while predicting the impact of electron–hole interaction^{53,54,55} and electron–electron scattering^{56,57} is additionally challenging. Finally, a robust understanding of ultrafast experiments may require simulation of realtime dynamics to capture initial state effects, probe wavelength effects and beyondsingleexponential decay dynamics, which is a natural next step within the general Lindbladian densitymatrix formalism presented here.
Methods
Computational details
All simulations are performed by the opensource planewave code  JDFTx^{58} using pseudopotential method, except that the Born effective charges and dielectric constants are obtained from opensource code QuantumESPRESSO^{59}. We firstly carry out electron structure, phonon and electron–phonon matrix element calculations in DFT using Perdew–Burke–Ernzerhof exchangecorrelation functional^{60} with relatively coarse k and q meshes. The phonon calculations are done using the supercell method. We have used supercells of size 7 × 7 × 7, 4 × 4 × 4, 6 × 6 × 1, 6 × 6 × 1, 6 × 6 × 1, 4 × 4 × 4 for silicon, BCC iron, graphene, monolayer MoS_{2}, monolayer MoSe_{2} and GaN, respectively, which have shown reasonable convergence for each system (<20% error bar in the final spin relaxation estimates). SOC is included through the use of the fully relativistic pseudopotentials^{61}. For monolayer MoS_{2} and MoSe_{2}, the Coulomb truncation technique is employed to accelerate convergence with vacuum sizes^{62}.
We then transform all quantities from plane wave to maximally localized Wannier function basis^{63} and interpolate them^{24,25,64,65} to substantially finer k and q meshes (with >3 × 10^{5} total points) for lifetime calculations. Statistical errors of lifetime computed using different random samplings of kpoints are found to be negligible (<1%). This Wannier interpolation approach fully accounts for polar terms in the electron–phonon matrix elements and phonon dispersion relations using the approaches of Verdi and Giustino^{66} and Sohier et al.^{67} for the 3D and 2D systems.
Data availability
All relevant data are available from the authors upon request.
Code availability
Firstprinciples methodologies available through opensource software, JDFTx^{58}, and postprocessing codes available from authors upon request.
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Acknowledgements
We thank Mingwei Wu, Oscar Restrepo, and Wolfgang Windl for helpful discussions. This work is supported by National Science Foundation under Grant Nos. DMR1760260, and startup funding from the Department of Materials Science and Engineering at Rensselaer Polytechnic Institute. This research used resources of the Center for Functional Nanomaterials, which is a US DOE Office of Science Facility, and the Scientific Data and Computing center, a component of the Computational Science Initiative, at Brookhaven National Laboratory under Contract No. DESC0012704, the lux supercomputer at UC Santa Cruz, funded by NSF MRI Grant AST 1828315, the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DEAC0205CH11231, the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI1548562^{68}, and resources at the Center for Computational Innovations at Rensselaer Polytechnic Institute.
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J.X. and A.H. implemented the codes and performed the major part of ab initio calculations; S.K. and F.W. contributed to part of the calculations and data analysis; J.X., A.H., R.S. and Y.P. wrote the manuscript with contributions from all authors. Y.P. and R.S. designed and supervised all aspects of the project.
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Xu, J., Habib, A., Kumar, S. et al. Spinphonon relaxation from a universal ab initio densitymatrix approach. Nat Commun 11, 2780 (2020). https://doi.org/10.1038/s41467020160635
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