Ab Initio Calculation of Exchange Interactions, Adiabatic Spin-Waves, and Curie Temperature of Itinerant Ferromagnets Patrick Bruno Max-Planck-Institut für Mikrostrukturphysik, Halle, Germany People involved: Josef Kudrnovský (Prague and Halle) Vaclav Drchal (Prague) Ilja Turek (Brno) Marek Pajda (Halle) Leonid Sandratskii (Halle)
Summary: motivations adiabatic spin-waves the ``magnetic force theorem results: spin-wave energies and Curie temperature of bulk Fe and Ni results: 2D systems results: dilute magnetic semiconductors (Ga 1-x Mn x As) outlook papers: M. Pajda et al., PL 85, 5424 (2000) M. Pajda et al.., PB 64, 174402 (2001) P.B., cond-mat/0207592 L. Sandratskii and P.B., PB (in press), cond-mat/0207050 (p)reprints: www.mpi-halle.de
Motivations? thermodynamics of itinerant ferromagnets? (temperature dependence of M s, Curie temperature,...) M S M bulk transition metals low dimensional systems (ultrathin films, surfaces,...) nanostructures (nanowires, clusters,...) T C T influence of a metallic substrate and of a capping layer Fe Cu substrate
Experimental background influence of a Cu coverage on the Curie temperature of an Fe film. Vollmer et al., PB61, 1303 (2000) Cu thickness (ML) Curie temperature >313K 300 290 280 270 260 250 240 230 220 210 200 190 180 170 160 150 140 130 <130K Fe wedge Cu wedge Cu (001) substrate Fe thickness (ML)
thermodynamics of magnetic systems partition function: Z ( k T) = exp ε / n 1 Z M = kbt Z B n B energy energy range of interest all degrees of freedom magnetic excitations simplified model
Important constraints Theory Nernst theorem: ( ) S = 0 at T = 0 B (for quantum systems) 2 2 M F F S = = = = T T B B T B T= 0, B= 0 T= 0, B= 0 T= 0, B= 0 T= 0, B= 0 0 Goldstone theorem: ( q) lim 0 q 0 ω = Mermin-Wagner theorem: no LO at T >0forshort-range, isotropic interactions, in 1D- or 2D-systems generalization for long-range oscillatory interactions (e.g., KKY): (P.B., PL 87, 137203 (2001)) Experiment low temperature limit: MT ( ) T 3/2 (Bloch law) Curie temperature, critical behavior (??) high-temperature limit: Curie-Weiss law (local moment)
elevant magnetic excitations energy Stoner excitations Stoner excitations magnons!!! magnons wave vector two-step approach: 1. mapping onto a (classical) effective Heisenberg model H = J u u eff ij i j i j 2. statistical physics calculation M s (T), T C exchange interactions: J ij essential parameter for understanding the Curie temperature metals complicated oscillatory KKY behavior need of performing ab initio calculations
Calculation of thermodynamic properties MFA 2 mean-field approximation: kt B C = J0 j 3 j 0 poor description of the low-temperature regime (no collective excitations) violation of the Mermin-Wagner theorem (2D) T C is overestimated Green s function method + random phase approximation (PA, Tyablikov): spin-wave regime correctly described (T<<T C ) reasonable extrapolation to T C M 1 6 1 1 = PA kt B C MS N q ω M T 3/2 ( q) Mermin-Wagner theorem well satisfied M S ( correct behavior for 2D systems) incorrect critical exponents ( = spherical model) T C T
Alternative approaches disordered local moments (DLM) (Gyorffy, Staunton, 1985, 1992) coupled spin-fluctuations (Kübler, 1996) DMFT (Lichtenstein, Katsnelson, Kotliar, 2001)
Adiabatic approximation neglect of the precession motion of the atomic magnetic moments calculation of the energy of static spin-waves adiabatic approximation valid if spin-wave precession time >> spin-precession time in exchange field interatomic hopping time i.e., if ω ( q) <<, W exchange splitting bandwidth
Constrained density functional theory constraining field B Hohenberg-Kohn energy functional: constrained energy functional: (Dederichs et al., 1984) [ ρ] = [ ρ] + [ ρ] + [ ρ] + [ ρ] E T E E E F HK 0 ext H xc [ ] + E o = HK ρ, B EHK ρ c ns ρ, B ( ) E cons, B B drm r ρ Ω Lagrange parameters
(Liechtenstein et al., 1987) Magnetic force theorem idea: try to use the variational properties of the HK functional to avoid doing self-consistent total energy calculations problem: we cannot calculate chosen apriori 0 [ ρ] (requires the knowledge of the input potential which gives ρ ( n, m) as output ) T for a charge- and spin-density ρ ( n, m) solution: use instead the Harris functional (Harris, 1985): [ ] [ ] [ ρ] w = V σ B σ eff eff 0 eff F Harris ρ, ρ, B = T B + Eext ρ + EH ρ + E xc + Econs ρ, B N el i = 1 ( ) ( w ) n( ) V ( ) ( ) ( ) T ρ, = εi eff ρ, d eff B B r r r m r Beff r [ ρ]( r) V ( r) σ B ( r) eff eff eff ( E [ ] [ ] [ ] ) ext + EH + E xc + Econs, B δρ ( r) δ ρ ρ ρ ρ w =
properties of the Harris functional: ( ) F Harris ρ, HK ρ, B = F B for the values ρ, B which minimize the HK functional the Harris functional is stationary (but necessarily minimal) with respect ( ) to ρ and B near the self-consistent solution ρ, B : F Harris ρ, HK ρ, B = F B + O2, ( δρ δb ) the Harris functional has been found to provide a better approximation than the HK functional for the ground state energy
ferromagnetic configuration 0 0 ( ) u = u constrained rotated configuration (infinitesimal rotation) 0 u = u + δu we look for a second order expansion: ( δu) A δu δu O ( δu) E = + 2 with A = J + δ J and A = A = 0 approximate the total energy energy change by using the Harris functional with ( ) trial input ρ 0, B within LDA, one obtains: N el { ( ) ( [ ])} i eff 0, 0 0, i eff 0, 0 0,0 O2 (,, ) ( ) ( ) E δu = ε w n m + δ ε w n m + δn δm δ u u B u B i = 1
choice of Liechtenstein et al.: input values of constraining field B = 0 with and ( δu) δu δu O2 ( δ, δ, B ) E = A + n m A = J + δ J ε F 1 J = dε Im Tr B G B G π ( xc xc ) estimation of the error: δn = O 2 δm = O B 2 = O ( δu) ( δu) ( δu) 1 O ( δn, δm, B ) = O ( δu) 2 2 therefore J J bare exchange parameters true exchange parameters
improved choice: input values of constraining field with and B ( = exact values) εf 1 ( xc xc ) = dε Im Tr π + εf 2 dε Im Tr( G G ) = π ( δu) δu δu ( δ ) B δu χ B B O ( δu) E = A + M K + K G B G G B G B χ calculation of the constraining fields: 1 ( ) M ( K ) δu = δu + χ B = X M K δu M K with = 4 final result: (using some matrix notations) T 1 ( ) ( ) 1 A = A + M K X M K 2 renormalized magnetic force theorem
equation of motion: spin-wave energies = eigenvalues of Physical interpretation 4 2 define: J = d Bxc ( ) m( ) exchange splitting on site M M r r r Ω 4J simple approximation: K 8J 1 and χ A = MIxc ( 1 χixc ) M 2 1 1 M 1 1 ( ) ( ) 1 1 A = A 1 4M A = J 4M J 4 1 1 1 1 1 1 = MIxc ( 1 χixc )( χixc ) M = M( χ Ixc ) M = Mχ M 2 2 2 M dδu E = 2 u0 = 4 A δu u0 dt d u u δ 1/ 2 1/ 2 Ω 4M AM approximation 1 ( ) 1 Ω = Ω 1 Ω for systems with one atom per cell ω( q) = ω( q) 1 ω( q)/
Technical details about calculations density functional theory (local density approximation) tight-binding LMTO method utilisation of the ``magnetic force theorem + + Ji, j Im dε f ( ε ) Tr V ( ε i0 ) ( ε + i0 i V i G ij V j V j G ji integration over energies complex in the complex plane integration over the Brillouin zone using up to 5 x10 6 k-points calculation of J i,j for 172 (fcc), 195 (bcc) or 101 (2D) shells of neighbors (~7to10xa) (a = lattice parameter) + )
4µ Spin-wave dispersion of bulk bcc Fe E( q) = B J ( 1 exp(iq )) m s j 0 0, j bcc Fe oscillatory KKY interactions Kohn anomalies C.K. Loong et al., JAP55, 1895 (1984) J.W. Lynn, PB 11, 2624 (1983)
Spin-wave dispersion of bulk fcc Ni fcc Ni H.A. Mook and D.McK. Paul, PL 54, 227 (1985)
Convergence with number of shells (bcc Fe)??
Calculation of the spin-wave stiffness constant E 4µ ( q) = B J ( 1 exp(iq )) m s j 0 0, j for small q with 2 D = µ B 3m ( ) 2 E q D q s J 2 Problem: the calculation of D by using D = lim max 2µ B 3m s 0< < max J 2 is non convergent due to the long-range of the oscillatory KKY interaction for a weak ferromagnet: J sin k + F k F 3 + φ
D = lim max 2µ B 3m s 0< < max J 2
regularization of D = lim max 2µ 3m B s 0< < max J 2 by substituing with the formally equivalent expression D = lim η 0 lim max 2µ 3m B s 0< < max J 2 exp ( η ) (numerically convergent) ~ Ewald summation technique
D = lim η 0 lim max 2µ 3m B s 0< < max J 2 exp ( η )
D = lim η 0 lim max 2µ 3m B s 0< < max J 2 exp ( η )
D = lim η 0 lim max 2µ 3m B s 0< < max J 2 exp ( η )
D = lim η 0 lim max 2µ 3m B s 0< < max J 2 exp ( η )
esults for bulk systems good agreement metal D th (mev.å 2 ) D exp (mev.å 2 ) T C MFA (K) T C PA (K) T C exp (K) Fe (bcc) 250 280, 330 1414 950 1045 Co (fcc) 663 580 1645 1131 1388-1398 Ni (fcc) 753 422, 555 397 350 624-631? origin of the strong disagreement for Ni? importance of Stoner excitations E(q) () 2 E D q q adiabatic approximation rather poor failure of the ``bare magnetic force theorem correlation effects? q
Effect of renormalization of exchange interaction parameters bcc Fe renormalized spectrum bare spectrum
fcc Ni renormalized spectrum bare spectrum
enormalized Curie temperature Curie temperature obtained from the renormalized parameters Jij Curie temperature obtained from the bare parameters J ij metal exp ( ) T C K PA C T C (K) PA T C (K) Fe (bcc) 1045 950 1057 Ni (fcc) 624-631 350 634 ω( ) ω( ) = q q 1 ω( q)/ 1 6 1 1 = PA kt B C MS N q ω ( q) k PA PA PA 6kBTC BTC = kt B C 1 M 1
Two-dimensional systems free standing Fe or Co monolayer (theorist's fantasy) Fe or Co monolayer Cu substrate Cu cap layer embedded Fe or Co monolayer Cu cap layer of variable thickness Cu substrate Fe or Co monolayer Cu substrate
exchange interactions for an atomic layer system m s (µ B ) J 0,1 (mev) T C MFA (K) vac/fe/vac 3.06 46.4 1265 vac/fe/cu 2.83 36.8 1068 Cu/Fe/Cu 2.59 35.9 1189 bulkfe(bcc) 2.26 19.6 1414 vac/co/vac 2.02 38.7 1300 vac/co/cu 1.79 31.9 1043 Cu/Co/Cu 1.58 27.4 797 bulk Co (fcc) 1.61 14.7 1645 exch. interactions much larger in 2D than in bulk mean-field approx. much too high T C in 2D need for better approx. (PA)
spin-wave dispersion for an Fe atomic layer free-standing layer layer deposited on a Cu (001) substrate layer deposited on a Cu (001) substrate and covered with a Cu cap
free-standing layer layer deposited on a Cu (001) substrate layer deposited on a Cu (001) substrate and covered with a Cu cap
Curie temperature (PA): spin-wave energy: E k B 1 6µ = PA T m C B s 1 N 1 E q ( q) 4µ () = + B ( ) q anis m s j 0 J 1 exp(i q ) 2 TC D ln( J J / anis ) Co monolayer T C PA 0 for anis 0 (Mermin-Wagner) typical order of magnitude of the anisotropy: anis 2π V M 2 PA much better than MFA
oscillation of the Curie temperature (PA) as a function of the cap layer thickness Fe or Co monolayer Cu cap layer Cu (001) substrate
Qualitative interpretation of the results Ji, j Im dε f ( ε ) Tr V i V i G ij V j V j G ji for bulk (3D): G ij exp ( i k ij ij ) j i for 2D: G ij ( exp i kij ( ) 1/2 ij )
Analogy to oscillatory interlayer exchange coupling
Dilute magnetic semiconductors (DMS) Mn S = 5 / 2 + 1 hole J p-d 1.2 ev (AF) Ga Mn As 1 x x H. Ohno, Science 281, 951 (1998)
H. Ohno et al., Nature 408, 944 (2000)
Calculation of exchange interactions and Curie temperature of DMS Ga Mn As 1 x x supercell approach (ordered compound) x = 0, 3.125%, 6.25%, 12.5%, 25%, 100% range accessible experimentally augmented spherical waves (ASW) method (calculations performed by Leonid Sandratskii)
magnetic moment (in µ B )
( ) ( ) ( ) ( ) ( ) ( ) Im e ex p p x i F F F F F F J G G k G G ε ε ε κ ε + + + +
(mean-field) Curie temperature for x = 6.25% excess of electrons per Mn atom
Conclusions and outlook ``renormalized magnetic force theorem corrects the systematic error introduced by the ``bare magnetic force theorem no signficant additional computational effort suitable for complex systems simple approximation yields simple renormalization of spin-wave energies and Curie temperature Curie temperature of Fe and Ni are in good agreement with experimental ones (?) application to: - disordered systems (e.g., permalloy, NiPd,...) - films and surfaces - nanostructures - new materials (e.g., GaMnAs,...) needed improvements: local moments at T > T C quantum fluctuations? combination with the DMFT approach?
Acknowledgements many thanks to: Helmut Eschrig Josef Kudrnovský Sasha Lichtenstein Leonid Sandratskii Valeri Stepanyuk for numerous helpful discussions and suggestions