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Electron spin dynamics in strongly correlated metals Balázs Dóra1,2,∗ and Ferenc Simon1,† 1Budapest University of Technology and Economics, Institute of Physics and Condensed Matter Research Group of the Hungarian Academy of Sciences, H-1521 Budapest, Hungary 2Max-Planck-Institut fu¨r Physik Komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany The temperature dependenceof theelectron spin life-time, T1 and theg-factorare anomalous in alkalifullerides(K,Rb)3C60,whichcannotbeexplainedbythecanonicalElliott-Yafettheory. These materials are archetypes of strongly correlated and narrow band metals. We introducetheconcept of”complexelectronspinresonancefrequencyshift”totreatthesemeasurablesinaunifiedmanner 9 within theKuboformalism. Thetheory isapplicable for metals with nearly degenerate conduction 0 bandsandlargemomentumscatteringevenwithananomaloustemperaturedependenceandsizeable 0 residual value. 2 PACSnumbers: 74.70.Wz, 74.25.Nf,76.30.Pk,74.25.Ha n a J 8 Spintronics, i.e. the use of the spin degree of freedom Γspin = α∆g2Γ, where the constant α ≈ 1..10. Typi- ofelectronsforinformationprocessing[1],isarapidlyde- cally ∆g2 10−4..10−7 thus T is orders of magnitude 2 1 ≈ veloping field. Its research is motivated by the orders of longer than τ. The experimentally accessible measur- ] magnitudelongerconservationoftheelectronspinalign- ables are the CESR line-width: ∆B = Γ /~γ (where spin l e ment in metals as compared to their momentum conser- γ/2π=28.0GHz/T is the electrongyromagneticfactor) - vation time. The survival of spin orientation, character- and the resistivity, ρ Γ. Thus the Elliott-relation es- r ∝ t ized by T , determines the time window for spin manip- tablishesthattheCESRline-widthandtheresistivityare s 1 . ulation. The g-factor determines the magnetic energy of proportional,whichenabledanexperimentalverification t a the electrons and characterizes the conditions for mag- for most elemental metals by Monod and Beuneu [5, 6]. m netic resonance manipulations of spins. Much as the Elliott-Yafet theory has been confirmed, - The seminal theory of Elliott and Yafet (EY) was de- it is violated in MgB at high temperatures as therein d 2 n veloped in the 1950’s to explain the electron spin relax- ∆B and ρ are not proportional [7], which was explained o ation and g-factor in metals [2, 3]. These parameters [8] by extending the EY theory for the case of rapid mo- c are measured with conduction electron spin resonance mentum scattering: ∆ Γ. The EY theory neglects the [ ≈ (CESR) [4]. In CESR, the metal is placed in a magnetic roleof Γ with respectto ∆ as inusualmetals Γ 1 10 ≈ − 1 field and is irradiated with microwaves. Resonant mi- meV and ∆ 1 10 eV. The need for a generalized v crowave absorption occurs when the irradiation energy EY theory is≈even−more demanding in alkali fullerides 3 equals the Zeeman splitting of the electron spins. whose conduction band is composed of a triply degen- 5 The EY theory is based on the presence of spin-orbit erate molecular orbital making a small effective ∆ and 4 4 coupling as it mixes a near lying band with the conduc- wherelargeelectron-phononcouplingandstructuraldis- . tion band states. Thus conduction band states are ad- ordergivesrise to a largeΓ. The situation is sketchedin 1 0 mixtures of spin up and down causing i) a change in the Fig. 1. 9 magnetic energy of conduction electrons (i.e. a g-factor The ∆B and the g-factor are shown in Fig. 2. for 0 shift, ∆g) and ii) allowed transitions between the two K C and Rb C and show that the g-factor is tem- 3 60 3 60 : spin states (i.e. spin relaxation) when the electron is v perature dependent and the CESR line-width does not i scattered on defects or phonons with momentum relax- followthe resistivityinviolationofthe EYtheory. Here, X ation time τ. Elliott and Yafet showed with first order weintroducetheconceptofthe”complexESRfrequency r time-dependent perturbation theory that: a shift” which allows a simultaneous treatment of T1 and theg-factorwithinthe Kuboformalism. We alsoinclude 2 a significant residual momentum scattering rate (dirty L Γspin = α1 Γ, (1) limit)andwefindthatthetheoryexplainsquantitatively (cid:18)∆(cid:19) the experimental observables, which enables to establish L ∆g =g g = α , (2) the ”generalizedElliott-relation”. 0 2 − ∆ We prepared Rb C powder samples by a conven- 3 60 where α are band structure dependent constants of tionalsolidstatereactionmethod[9]usingstoichiometric 1,2 the order of unity, g = 2.0023 is the g-factor of the amountsofsublimationpurifiedC andelementalRbto 0 60 free electron,L is the matrix element ofthe SO coupling study its ESR properties up to high temperatures which between the near lying and the conduction band state, has not been performed yet. ESR data is available for separated by an energy gap ∆, Γ =~/T , Γ=~/τ. K C for the 4-800 K temperature range [10]. A sharp spin 1 3 60 Eqs. 1 and 2 are summarized in the Elliott-relation: superconducting transition and high Meissner shielding 2 y g er TABLE I: Residual, ρ0, and high temperature, ρ(Th) resis- n tivity and the corresponding momentum scattering rates, Γ, E Γ for K3C60 (Th =790 K) and Rb3C60 (Th =700 K) from Ref. small [15]. Plasma frequencies are from Ref. [17]. The coefficient Γ large A,defined in the text,is also given. ∆ ωpl ρ0 Γ0 ρ(Th) Γ(Th) A (eV) (mΩcm) (meV) (mΩcm) (meV) (meV/K2) 0 K3C60 1.2 0.2 39 1.5 285 3.94·10−4 Rb3C60 1.1 0.5 81 4.0 650 13.1·10−4 the Fermienergyatdifferentpoints andseparatedby ∆, FIG. 1: The schematic view of the single particle spectrum which yields: at small Γ(≪ ∆) addressed by the EY theory and at large Γ(∼∆),whichcallsforthegeneralizedEYtheory. Duetothe large overlap (green area), the effectiveband-gap is reduced. L2 Γ (=~γ∆B)= Γ , (5) spin (cid:28)∆2+Γ2 (cid:29) FS ina1mTmagneticfield(measuredwithmicrowavecon- ductivity)togetherwiththeobservationofthecharacter- 1 1 Γ+i∆ istic CESR signal of Rb3C60 [11] attest the high quality ∆g = πk T (cid:28)2LzImΨ′(cid:18)2 + 2πk T(cid:19)(cid:29) (6) of the material. A sample of 10 mg sealed under helium B B FS in a quartz tube was measured in a commercial X-band where the ... means Fermi surface (FS) averaging, FS h i (9GHz)ESRspectrometerinthe100-700Ktemperature Ψ′(x) is the first derivativeof Euler’sdigamma function, range. and all parameters L, ∆, and Γ are taken on the FS. The description of T1 and g-factor is based on a two- Lz =L↑,↑−L↓,↓ and L2 =L2z+2|L↓,↑|2. Eq. 6. can be band model Hamiltonian, H =H0+HSO, where: simplified if 2πkBT .Γ to give H = [ǫ (k)+~γBs]c+ c +H , 2Lz∆ 0 ν k,ν,s k,ν,s scatt ∆g = (7) kX,ν,s (cid:28)∆2+Γ2(cid:29)FS (3) HSO = Ls,s′(k)c+k,ν,sck,ν′,s′ Eqs. 5 and 6. return the corresponding EY results k,ν6=Xν′,s,s′ (Eqs. 1-2) when Γ ∆ and is regarded as a generaliza- ≪ tion of the EY theory. If Eqs. 5. and 7. can be handled Here ν,ν′ = 1 or 2 are the band, s,s′ are spin indices, with isotropic band-band separation and SO coupling, Ls,s′ istheSOcoupling,andBisthemagneticfieldalong the generalized Elliott-relation is: the z direction. H is responsible for the finite τ. We scatt use the Mori-Kawasakiformula[12, 13] to determine the T1 and ∆g = ~∆ωL/µBB which allows to introduce the Γ =α∆g2Γ 1+ Γ2 (8) ”complex ESR frequency shift”: spin (cid:18) ∆2(cid:19) which returns the conventional formula when Γ ∆. ≪ ∆ω :=∆ω i = −h[P,S−]i+GRPP+(ωL), (4) We proceed to analyze the line-width and g-factor in ESR L− T 2 S alkali doped fullerides. Knowledge of the temperature 1 z h i dependent Γ is required, which we determine from resis- e where S is the expectation value of the spin along the tivity data on single crystals by Hou et al. [15] (solid z h i magnetic field, ω = γB is the Larmor frequency, and blue curves in Fig. 2.) using theoretical plasma frequen- L GR (ω) is the retarded Green’s function of the P and cies, ω through ρ = 1/ǫ ω2τ (where ǫ is the electric PP+ pl 0 pl 0 P+ pair with ~P = [H ,S+]. Eq. 4 is evaluated with constant). These compounds are unique in two aspects: SO the unperturbed Hamiltonian, H , to yield the lowest i) the resistivity is high and it follows a quadratic tem- 0 non-vanishingcorrectionduetoSOcouplingasitismuch perature dependence up to the highest availabletemper- smaller than the temperature or the band-gap. We note atures, and ii) the residual resistivity is also high and it thatEq. 4isanalogoustothecomplexconductivitywith is not related to a residual impurity concentration but para- and diamagnetic terms [14]. is intrinsic. The high value and quadratic temperature To enable an analytic calculation [14], we assume two dependence of ρ was explained by the coupling of elec- linearbandswithdifferentFermivelocities,bothcrossing tronstothehighenergyintramolecularphonons,whereas 3 K C Rb C 3 60 3 60 2.0 4 r 60 1.5 ( m 1.5 W 3 T) cm 40 BD m 1.0 ) ( ( 1.0 2 D m B D m) 2 T) D 0.5 1 0.5 c 1 20 D Wm D 2 ( 1 r 0.0 0.0 0 0 0.0 0 D 1 D -0.5 2 30 -1.0 D -20 D 1 D g g 1 2 1 0 D -1.5 3 -2.0 -40 0 200 400 600 800 0 200 400 600 800 T (K) T (K) FIG. 2: (Color online) The temperature dependent resistivity (solid blue curve from Ref. [15]), CESR line-width ((cid:3)), and g-factor shift ((cid:13)) (Refs. ([10, 11, 16]) in the fullerides. ESR data on Rb3C60 above 300 K is from the present work. Solid curvesarecalculated withthemodelexplainedinthetext. Dashedcurvesarethecontributionsfrom thedifferent∆1 and∆2. Note thedifferent scales for thetwo compounds. the large residual value was associated with an inherent in Eqs. 5. and 6. with a sum of two components. In disorder of the C ball orientation (the so-called mero- Fig. 2. we show the calculated ∆B and ∆g with this 60 hedral disorder) [17, 18]. The latter is related to the twocomponentsum. ThefitparametersaregiveninTa- frustrated nature of the C icosahedra with respect to ble II. Calculations with a single ∆ fail to account for 60 the cubic molecular crystal lattice. The parameters of the data in both compounds. We judge that the fits are [23] Γ(T)=Γ +A T2, are given in Table I. in reasonable agreement with the experiment, given the 0 · simplifications to the general theory in terms of a two The g-factor is independent of the temperature and component sum. We note that Adrian [19] suggested ∆B increases monotonously in the EY theory. In con- thatarelationsimilar toEq. 5explains ∆B forRb C , trast, both measurables deviate from the expected be- 3 60 however apart from a qualitative hint, no attempt for a havior in K C and Rb C as shown in Fig. 2: ∆g 3 60 3 60 | | quantitative analysis was made. decreases with increasing temperature and ∆B does not followtheresistivity. Mostsurprisingly,∆B decreases on Theobtained∆valuesarecompatiblewiththe known increasing temperature in Rb C . The generalized EY small, < 1 eV band-width of alkali fullerides [17]. The 3 60 theory shows that ∆ Γ explains the saturating and conduction band is derived from the triply degenerate ≈ decreasing ∆B and decreasing ∆g (see Fig. 1). How- t molecular orbital of C , whose degeneracy is lifted 1u 60 | | ever,asmall∆alonecannotexplainthedataandFermi inthefulleridecrystal. However,themerohedraldisorder surface parts, where ∆>Γ, are also present. preventstheknowledgeofthebandstructureandweinfer band structure properties from our analysis. It shows To handle the complicated Fermi surface of the ful- that on some parts of the FS, the band crossing it has lerides the simplest possible way, we assume that the FS a neighboring band as close as 50 meV. On other parts, consists of two parts: one with large and another with the nearest neighboring band is as close as 0.35-1 eV. small∆(∆ and∆ ,respectively)withdifferentrelative 1 2 densityofstates(DOS),N andN . Weassumeuniform The two compounds only differ in the relative amount 1 2 L and Γ. This allows to approximate the FS averages ofsuchFSparts: forK C partswithsmall∆dominate 3 60 4 TABLEII:Bestfittingparametersusedtosimulatetheexperimentalline-widthandg-factordatainK3C60 andRb3C60. Note thedifferent relative DOSin thetwo materials. |L| (meV) Lz (meV) ∆ (eV) N (%) 1 2 1 2 K3C60 0.67(1) -0.63(1) 0.94(3) 0.047(2) 3.0(2) 97(2) Rb3C60 1.10(2) -3.7(1) 0.35(3) 0.050(2) 31(2) 69(2) whereas for Rb C the relative DOS for the two types lerides do not possess any of these properties, still our 3 60 of Fermi surfaces are almost equal. theoryaccountsforthemeasuredCESRparameters. The Only the L/∆ ratio is available in the EY theory but currentdescriptionisapplicabletoabroadrangeofmet- the correlated metals allow measurement of ∆ and L in- als thus we expect that it will lead to smart design of dependently. We note that both L and L contain band materials for future spintronics devices. z structure dependent constantsofthe orderofunity. The FSacknowledgestheBolyaiprogrammeoftheHungar- negative sign of L reflects the electron-like (as opposed ian Academy of Sciences for support. Work supported z to hole-like) character of the conduction states, which is by the Hungarian State Grants (OTKA) No. F61733, a common situation in e.g. alkali metals. L and L are K72613,and NK60984. z | | unequalalreadyintheEYframework[5,6],whichisalso the situation herein. For both compounds the SO cou- plings are about three orders of magnitude smaller than the corresponding values for elemental K (0.26 eV) and ∗ Electronic address: [email protected] Rb (0.9 eV) (Ref. [3]). This is due to the weak charac- † Electronic address: [email protected] ter of the alkali orbitals in the conduction band of C60 [1] I. Zˇutić, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. [20, 21]. On average,the corresponding SO coupling pa- 76, 323 (2004). rametersare 3.7timeslargerinRb C thaninK C [2] R. J. Elliott, Phys.Rev. 96, 266 (1954). 3 60 3 60 which is in g∼ood agreement with the 3.5 ratio found [3] Y. 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