Abstract
Spintronics is an emerging paradigm with the aim to replace conventional electronics by using electron spins as information carriers. Its utility relies on the magnitude of the spinrelaxation, which is dominated by spinorbit coupling (SOC). Yet, SOC induced spinrelaxation in metals and semiconductors is discussed for the seemingly orthogonal cases when inversion symmetry is retained or broken by the socalled ElliottYafet and D'yakonovPerel' spinrelaxation mechanisms, respectively. We unify the two theories on general grounds for a generic twoband system containing intra and interband SOC. While the previously known limiting cases are recovered, we also identify parameter domains when a crossover occurs between them, i.e. when an inversion symmetry broken state evolves from a D'yakonovPerel' to an ElliottYafet type of spinrelaxation and conversely for a state with inversional symmetry. This provides an ultimate link between the two mechanisms of spinrelaxation.
Similar content being viewed by others
Introduction
A future spintronics device would perform calculations and store information using the spindegree of freedom of electrons with a vision to eventually replace conventional electronics^{1,2,3}. A spinpolarized ensemble of electrons whose spinstate is manipulated in a transistorlike configuration and is read out with a spindetector (or spinvalve) would constitute an elemental building block of a spintransistor. Clearly, the utility of spintronics relies on whether the spinpolarization of the electron ensemble can be maintained sufficiently long. The basic idea behind spintronics is that coherence of a spinensemble persists longer than the coherence of electron momentum due to the relatively weaker coupling of the spin to the environment. The coupling is relativistic and has thus a relatively weak effect known as spinorbit coupling (SOC).
The time characterizing the decay of spinpolarization is the socalled spinrelaxation time (often also referred to as spinlattice relaxation time), τ_{s}. It can be measured either using electron spinresonance spectroscopy (ESR)^{4} or in spintransport experiments^{5,6}. Much as the theory and experiments of spinrelaxation measurements are developed, it remains an intensively studied field for novel materials; e.g. the value of τ_{s} is the matter of intensive theoretical studies^{7,8,9,10,11,12,13} and spintransport experiments^{14,15,16,17} in graphene at present.
The two most important spinrelaxation mechanisms in metals and semiconductors are the socalled ElliottYafet (EY) and the D'yakonovPerel' (DP) mechanisms. These are conventionally discussed along disjoint avenues, due to reasons described below. Although the interplay between these mechanisms has been studied in semiconductors^{3,18,19,20}, no attempts have been made to unify their descriptions. We note that a number of other spinrelaxation mechanisms, e.g. that involving nuclearhyperfine interaction, are known^{2,3}.
The EY theory^{21,22} describes spinrelaxation in metals and semiconductors with inversion symmetry. Therein, the SOC does not split the spinup/down states (↑〉, ↓〉) in the conduction band, however the presence of a near lying band weakly mixes these states while maintaining the energy degeneracy. The nominally up state reads: (here a_{k}, b_{k} are band structure dependent) and b_{k}/a_{k} = L/Δ, where L is the SOC matrix element between the adjacent bands and Δ is their separation. E.g. in alkali metals L/Δ ≈ 10^{−2}..10^{−3} (Ref. 22). Elliott showed using first order timedependent perturbation theory that an electron can flip its spin with probability (L/Δ)^{2} at a momentum scattering event. As a result, the spin scattering rate () reads:
where is the quasiparticle scattering rate with τ being the corresponding momentum scattering (or relaxation) time. This mechanism is schematically depicted in Fig. 1a.
For semiconductors with zincblende crystal structure, such as e.g. GaAs, the lack of inversion symmetry results in an efficient relaxation mechanism, the D'yakonovPerel' spinrelaxation^{23}. Therein, the spinup/down energy levels in the conduction bands are split. The splitting acts on the electrons as if an internal, kdependent magnetic field would be present, around which the electron spins precess with a Larmor frequency of . Here is the energy scale for the inversion symmetry breaking induced SOC. Were no momentum scattering present, the electron energies would acquire a distribution according to . In the presence of momentum scattering which satisfies , the distribution is “motionallynarrowed” and the resulting spinrelaxation rate reads:
This situation is depicted in Fig. 1b. Clearly, the EY and DP mechanisms result in different dependence on Γ which is often used for the empirical assignment of the relaxation mechanism^{24}.
The observation of an anomalous temperature dependence of the spinrelaxation time in MgB_{2}^{25} and the alkali fullerides^{26} and the development of a generalization of the EY theory highlighted that the spinrelaxation theory is not yet complete. In particular, the first order perturbation theory of Elliott breaks down when the quasiparticle scattering rate is not negligible compared to the other energy scales. One expects similar surprises for the DP theory when the magnitude of e.g. the Zeeman energy is considered in comparison to the other relevant energy scales.
Herein, we develop a general and robust theory of spinrelaxation in metals and semiconductors including SOC between different bands and the same bands, provided the crystal symmetry allows for the latter. We employ the MoriKawasaki theory which considers the propagation of the electrons under the perturbation of the SOC. We obtain a general result which contains both the EY and the DP mechanisms as limits when the quasiparticle scattering and the magnetic field are small. Interesting links are recognized between the two mechanisms when these conditions are violated: the EY mechanism appears to the DPlike when Γ is large compared to Δ and the DP mechanism appears to be EYlike when the Zeeman energy is larger than Γ. Qualitative explanations are provided for these analytically observed behaviors.
Results
The minimal model of spinrelaxation is a fourstate (two bands with spin) model Hamiltonian for a twodimensional electron gas (2DEG) in a magnetic field, which reads:
where α = 1 (nearby), 2 (conduction) is the band index with s = (↑), (↓) spin, is the singleparticle dispersion with effective mass and Δ band gap, Δ_{Z} = gμ_{B}B_{z} is the Zeeman energy. is responsible for the finite quasiparticle lifetime due to impurity and electronphonon scattering and L_{α}_{,α′,s,s′}(k) is the SOC.
The corresponding band structure is depicted in Fig. 2. The eigenenergies and eigenstates without SOC are
The most general expression of the SOC for the above levels reads:
where , L_{ss′}(k) are the wavevector dependent intra and interband terms, respectively, which are phenomenological, i.e. not related to a microscopic model. The terms mixing the same spin direction can be ignored as they commute with the S_{z} operator and do not cause spinrelaxation. The SOC terms contributing to spinrelaxation are
Table I. summarizes the role of the inversion symmetry on the SOC parameters. For a material with inversion symmetry, the Kramers theorem dictates (without magnetic field) that and thus , which term would otherwise split the spin degeneracy in the same band. When the inversion symmetry is broken, is finite and the previous degeneracy is reduced to a weaker condition: dictated by time reversal symmetry.
We consider the SOC as the smallest energy scale in our model (, L(k_{F})), while we allow for a competition of the other energy scales, namely Δ_{Z}, Γ and Δ, which can be of the same order of magnitude, as opposed to the conventional EY or DP case. We are mainly interested in the regime of a weak SOC, moderate magnetic fields, high occupation and a large band gap. We treat the quasiparticle scattering rate to infinite order thus large values of Γ are possible.
The energy spectrum of the spins (or the ESR linewidth) can be calculated from the MoriKawasaki formula^{27,28}, which relies on the assumption that the lineshape is Lorentzian. This was originally proposed for localized spins (e.g. Heisenbergtype models) but it can be extended to itinerant electrons. The standard (Faraday) ESR configuration measures the absorption of the electromagnetic wave polarized perpendicular to the static magnetic field. The ESR signal intensity is
where B_{⊥} is the magnetic induction of the electromagnetic radiation, is the imaginary part of the spinsusceptibility, μ_{0} is the permeability of vacuum and V is the sample volume. The spinsusceptibility is related to the retarded Green's function as
with S^{±} = S_{x} ± iS_{y}, from which the ESR spectrum can be obtained.
The equation of motion of the S^{+} operator reads as
where is the consequence of the SOC. The Green's function of S^{+}S^{−} is obtained from the Green's function of as
The second term is zero without SOC thus a completely sharp resonance occurs at the Zeeman energy. The lineshape is Lorentzian for a weak SOC:
where the selfenergy is
which is assumed to be a smooth function of ω near .
The spinrelaxation rate is equal to the imaginary part of Σ(ω) as
The correlator is obtained from the Matsubara Green's function of , given by
where ν_{m} is the bosonic Matsubara frequency. The effect of is taken into account in the Green's function by a finite, constant momentumscattering rate.
The most compact form of the spinrelaxation is obtained when the Fermi energy is not close to the bottom of the conduction band () and a calculation (detailed in the Methods section) using Eq. (13) leads to our main result:
Results in more general cases are discussed in the Supplementary Material.
Discussion
According to Eq. (15), the contributions from intra () and interband (L(k_{F})) processes are additive to lowest order in the SOC and have a surprisingly similar form. A competition is observed between lifetime induced broadening (due to Γ) and the energy separation between states (Δ(k_{F}) or Δ_{Z}). The situation, together with schematics of the corresponding bandstructures, is shown in Fig. 3. When the broadening is much smaller than the energy separation, the relaxation is EYlike, , even when the intraband SOC dominates, i.e. for a material with inversion symmetry breaking. This situation was also studied in Ref. 29, 30 and it may be realized in IIIV semiconductors in high magnetic fields. For metals with inversion symmetry, this is the canonical EY regime.
When the states are broadened beyond distinguishability (i.e. or Δ_{Z}), spinrelaxation is caused by two quasidegenerate states and the relaxation is of DPtype, , even for a metal with inversion symmetry, . For usual metals, the , criterion implies a breakdown of the quasiparticle picture as therein Δ(k_{F}) is comparable to the bandwidth, thus this criterion means stronglocalization. In contrast, metals with nearly degenerate bands remain metallic as e.g. MgB_{2} (Ref. 25) and the alkali fullerides (K_{3}C_{60} and Rb_{3}C_{60}) (Ref. 26), which are strongly correlated metals with large Γ. When the intraband SOC dominates, i.e. for a strong inversion symmetry breaking, this is the canonical DP regime. These observations provide the ultimate link between these two spinrelaxation mechanisms, which are conventionally thought as being mutually exclusive.
Similar behavior can be observed in other models (details are given in the Supplementary material), and remain valid in the two different limits but the intermediate behavior is not universal. A particularly compelling situation is the case of graphene where a fourfold degeneracy is present at the Diracpoint and both inter and intraband SOC are present thus changing the chemical potential would allow to map the crossovers predicted herein.
Methods
We consider Eq. (14) as a starting point. The Matsubara Green's function of can be written as
where
is the Matsubara Green's function of fermionic field operators in band (α) and spin (s). The effect of is taken into account by the finite momentumscattering rate, Γ.
Using the relationship between the Green's function and spectral density, the Matsubara summation in Eq. (16) yields
where
is the spectral density. By taking the imaginary part after analytical continuation, the energy integrals can be calculated at zero temperature. Then, by replacing momentum summation with integration, we obtain
where
and , A is the area of the 2DEG. The matrix elements of the operator are
We determine the expectation value of the zcomponent of electron spin following similar steps as
where
The matrix elements of the S_{z} operator are
The spinrelaxation rate can be obtained as
We note this is the sum of intra and interband terms which are described separately.
References
Wolf, S. A. et al. Spintronics: A spinbased electronics vision for the future. Science 294, 1488–1495 (2001).
Žutić, I., Fabian, J. & Sarma, S. D. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323–410 (2004).
Wu, M. W., Jiang, J. H. & Weng, M. Q. Spin dynamics in semiconductors. Phys. Rep. 493, 61–236 (2010).
Feher, G. & Kip, A. F. Electron Spin Resonance Absorption in Metals. I. Experimental. Physical Review 98, 337–348 (1955).
Johnson, M. & Silsbee, R. H. Coupling of electronic charge and spin at a ferromagneticparamagnetic metal interface. Phys. Rev. B 37, 5312–5325 (1988).
Jedema, F., Heersche, H., Filip, A., Baselmans, J. & van Wees, B. Electrical detection of spin precession in a metallic mesoscopic spin valve. Nature 416, 713–716 (2002).
HuertasHernando, D., Guinea, F. & Brataas, A. Spinorbit coupling in curved graphene, fullerenes, nanotubes and nanotube caps. Phys. Rev. B 74, 155426 (2006).
Ertler, C., Konschuh, S., Gmitra, M. & Fabian, J. Electron spin relaxation in graphene: the role of the substrate. Phys. Rev. B 80, 041405 (2009).
Gmitra, M., Konschuh, S., Ertler, C., AmbroschDraxl, C. & Fabian, J. Bandstructure topologies of graphene: spinorbit coupling effects from first principles. Phys. Rev. B 80, 235431 (2009).
Castro Neto, A. H. & Guinea, F. Impurityinduced spinorbit coupling in graphene. Phys. Rev. Lett. 103, 026804 (2009).
Dóra, B., Murányi, F. & Simon, F. Electron spin dynamics and electron spin resonance in graphene. Eur. Phys. Lett. 92, 17002 (2010).
Zhang, P. & Wu, M. W. Electron spin relaxation in graphene with random Rashba field: comparison of the D'yakonovPerel' and ElliottYafetlike mechanisms. New J. Phys. 14, 033015 (2012).
Ochoa, H., Castro Neto, A. H. & Guinea, F. Elliotyafet mechanism in graphene. Phys. Rev. Lett. 108, 206808 (2012).
Tombros, N., Józsa, C., Popinciuc, M., Jonkman, H. T. & van Wees, B. J. Electronic spin transport and spin precession in single graphene layers at room temperature. Nature 448, 571–574 (2007).
Han, W. et al. Tunneling Spin Injection into Single Layer Graphene. Phys. Rev. Lett. 105, 167202 (2010).
Han, W. & Kawakami, R. K. Spin relaxation in singlelayer and bilayer graphene. Phys. Rev. Lett. 107, 047207 (2011).
Yang, T.Y. et al. Observation of long spinrelaxation times in bilayer graphene at room temperature. Phys. Rev. Lett. 107, 047206 (2011).
Pikus, G. E. & Titkov, A. N. In Meier F., & Zakharchenya B. (eds.) Optical Orientation (NorthHolland, Amsterdam, 1984).
Averkiev, N., Golub, L. & Willander, M. Spin relaxation anisotropy in twodimensional semiconductor systems. J. Phys. Cond. Mat. 14, R271–R283 (2002).
Glazov, M. M., Sherman, E. Y. & Dugaev, V. K. Twodimensional electron gas with spinorbit coupling disorder. Phys. E 42, 2157–2177 (2010).
Elliott, R. J. Theory of the Effect of SpinOrbit Coupling on Magnetic Resonance in Some Semiconductors. Phys. Rev. 96, 266–279 (1954).
Yafet, Y. Conduction electron spin relaxation in the superconducting state. Physics Letters A 98, 287–290 (1983).
Dyakonov, M. & Perel, V. Spin relaxation of conduction electrons in noncentrosymmetric semiconductors. Soviet Physics Solid State, USSR 13, 3023–3026 (1972).
Tombros, N. et al. Anisotropic spin relaxation in graphene. Phys. Rev. Lett. 101, 0466011–4 (2008).
Simon, F. et al. Generalized ElliottYafet Theory of Electron Spin Relaxation in Metals: Origin of the Anomalous Electron Spin Lifetime in MgB2 . Phys. Rev. Lett. 101, 1770031–4 (2008).
Dóra, B. & Simon, F. Electronspin dynamics in strongly correlated metals. Phys. Rev. Lett. 102, 137001 (2009).
Mori, H. & Kawasaki, K. Antiferromagnetic resonance absorption. Progress of Theoretical Physics 28, 971–987 (1962).
Oshikawa, M. & Affleck, I. Electron spin resonance in antiferromagnetic chains. Phys. Rev. B 65, 134410 (2002).
Ivchenko, E. L. Spin relaxation of free carriers in a noncentrosymmetric semiconductor in a longitudinal magnetic field. Sov. Phys. Solid State 15, 1048 (1973).
Burkov, A. A. & Balents, L. Spin relaxation in a twodimensional electron gas in a perpendicular magnetic field. Phys. Rev. B 69, 245312 (2004).
Acknowledgements
We thank A. Pályi for enlightening discussions. Work supported by the ERC Grant Nr. ERC259374Sylo, the Hungarian Scientific Research Funds Nos. K101244, K105149, PD100373 and by the Marie Curie Grants PIRGGA2010276834. BD acknowledges the Bolyai Program of the Hungarian Academy of Sciences.
Author information
Authors and Affiliations
Contributions
P.B. carried out all calculations under the guidance of B.D. A.K. contributed to the discussion and F.S. initiated the development of the unified theory. All authors contributed to the writing of the manuscript.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Electronic supplementary material
Supplementary Information
Supplementary file
Rights and permissions
This work is licensed under a Creative Commons AttributionNonCommercialNoDerivs 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/byncnd/3.0/
About this article
Cite this article
Boross, P., Dóra, B., Kiss, A. et al. A unified theory of spinrelaxation due to spinorbit coupling in metals and semiconductors. Sci Rep 3, 3233 (2013). https://doi.org/10.1038/srep03233
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep03233
This article is cited by

Ultrafast optical investigation of carrier and spin dynamics in lowdimensional perovskites
Science China Technological Sciences (2024)

Detection of electronphonon coupling in twodimensional materials by light scattering
Nano Research (2021)

Spinphonon relaxation from a universal ab initio densitymatrix approach
Nature Communications (2020)

Window of opportunity
Nature Physics (2017)

Spinrelaxation time in materials with broken inversion symmetry and large spinorbit coupling
Scientific Reports (2017)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.