Spin-orbit coupling fields in Fe/GaAs heterostructures Outline motivation a simplified model of the Fe/GaAs heterostructure extracting spin-orbit coupling parameters spin-orbit coupling field conclusions Martin Gmitra Alex Matos-Abiague Claudia Ambrosch-Draxl Jaroslav Fabian University of Regensburg, Germany University of Leoben, Austria Spin Phenomena in Reduced Dimensions Workshop in Regensburg, September -6, 8 SFB 689
Motivation to our work Tunneling anisotropic magnetoresistance (TAMR) effect in F/SC heterostructures φ I F Outline motivation a simplified model of the Fe/GaAs heterostructure extracting spin-orbit coupling parameters spin-orbit coupling field conclusions TMR = R(φ) R() R() relative angle (Co/MgO/Co) TAMR [ref] = R(φ) R [ref] R [ref] absolute angle (GaMnAs/GaAs) N M. Bode et al., PRL 89, 37 () L. Brey et al., APL 8, 996 () C. Gould et al., PRL 93, 73 () F SC [] φ F []
Motivation to our work TAMR in Fe/GaAs junction bring effect to RT perfect interface (small mismatch) J. Moser et al., PRL 99, 66 (7) Outline motivation a simplified model of the Fe/GaAs heterostructure extracting spin-orbit coupling parameters spin-orbit coupling field conclusions C v Why the bias flips the symmetry axis??? Fe (cubic) + GaAs (cubic) =? two-fold
Bychkov-Rashba SIA Spin splittings induced by inversion asymmetry in d-like systems H = σ Ω(k) ɛ σ (k) Dresselhaus + = C v symmetry BIA SIA+BIA ɛ σ (k) ɛ σ ( k) ɛ σ (k) = ɛ σ ( k) Structure inversion asymmetry (SIA) surfaces, interfaces H BR = α(ẑ k ) σ = α(σ y σ x ) Bulk inversion asymmetry (BIA) zinc-blende structure - GaAs H D σ κ = γ(σ y σ x ) k I. Zutic, J. Fabian, S. Das Sarma, RMP 76, 33 ()
A simplified model of F/SC z [] - x [] H = H + H so + V z y [] φ n H = [ ] m n σ H so = H BR + H D Y S ɛ so Γ k θ X ɛ σ n(k) = k m + σ n n + σk [α n sin(φ θ) + γ n sin(φ + θ)]
Method of study vacuum Å.Å 6Å First-principles calculation scheme (WIENk): DFT (Density Functional Theory) GGA (Generalized Gradient Approximation) FP-LAPW (Full Potential Linearized Augmented Plane Wave) Spin-orbit coupling (second variational method)
Band structures of the x Fe/GaAs As-terminated interface model vacuum Å.Å 6Å E [ev].. -. -. S Γ n [] X.. -...7.6 -. DOS [/(ev spin atom)] E [ev]
Mapping to the simplified model.... E [ev].3 (a).. -. -. S 3 (b) M / Γ X / Γ band label (n) 3 α n [evå].7..6.3 -. γ n [evå].6.9.7..7 3 (c).3.. -. -. M / X no spin-orbit coupling case. -. -. -. E [ev] X / TAMR αγ [ cos(φ) ] Γ X / J. Fabian et al., Acta Phys. Slovaca 7, 6 (7)
d Fermi surface contour plots no SOC π/a π a π/a n x = [ ] π/a π/a π a π/a π/a π/a π a π/a π/a π/a π/a π/a C v symmetry of the Fe/GaAs interface π a π/a π/a π/a n y = []
Reversal of the C symmetry axis! l (ev Å ) 6 - - - - V bias (mv) A. Matos-Abiague, J. Fabian, arxiv: 7387 J. Fabian et al., Acta Phys. Slovaca 7, 6 (7) nelling across our metal/semiconductor heterojunction: Here H = H + H Z + H BR + H D. () [ ] H = m(z) + V z, () with m(z) the electron effective mass [in terms of the bare electron mass m 3 we assume m = m c =.67 m in the - central (GaAs) region and m = m l = m r m in the left (Fe) and right (Au) regions] and V (z) the conduction band profile defining the potential barrier along the nelling across our metal/semiconductor heterojunction: growth direction (z) of the heterostructure [see Fig. 3(a)]. H = H + H Z + H BR + H D. () The Zeeman spin splitting due to the exchange.field Here (in the Fe region) and the external magnetic field in the [ ] Fe and Au (the Zeeman energy in GaAs is much smaller H = m(z) than all the other energy scales characterizing the system + V z, () and we-. can therefore neglect its effect) is given by with m(z) the electron effective mass [in terms of the bare H Z = (z) n σ. (3) electron mass m we assume m = m c =.67 m in the - central (GaAs) region and m = m l = m r m in Here the (z) represents the Zeeman energy in the different left (Fe) and right (Au) regions] and V (z) the conduction band profile defining the potential barrier along regions, σ is a vector whose components are the Pauli matrices, the and n is a unit vector defining the spin quantization axis determined by the in-plane magnetization. growth direction (z) of the heterostructure [see Fig. 3(a)]. The Zeeman spin splitting due to the exchangedirection field in Fe. (in the Fe region) and the external magnetic field in the The Bychkov-Rashba SOI due to the structure inversion asymmetry at the interfaces can be written as [] Fe and Au (the Zeeman energy in GaAs is much smaller than all the other energy scales characterizing the system -. and we can therefore neglect its effect) is given by H BR = α i (σ x p y σ y p x )δ(z z i ), () H Z = (z) i=l,r n σ. (3) - where, α l (α r ) denotes k the SOI strength at the left (right) Here (z) represents the Zeeman energy in the different interface z l = (z r = x /k * w/k * d). We note that inside the GaAs regions, σ is a vector whose components are the Pauli barrier, away from the interfaces, there is also a Bychkov- FIG. : a) φ-scan of the tunneling resistance at. K and matrices, and n is a unit vector defining the spin quan- T and aaxis theoretical determined fit; b) by φ-scan the in-plane at +9 mv; magnetization Rashba SOI contribution induced by the applied bias. -9 mv bias in a saturation magnetic field B =. T and B = tization However, this contribution is negligible for our system ɛ σ n(k) c) and we neglect it. ɛ () φ-scans for direction different inbias Fe. voltages. Symbols correspond to The Dresselhaus SOI resulting from the bulk inversion experimental The results Bychkov-Rashba for -9 mv, -SOI mv, due mv, to the 9 mv structure and inversion solid asymmetry lines correspond at the interfaces to theoretical canresults be written with 3 mv bias; asymmetry in GaAs is incorporated in the model through as [] α l =.3 ev Å, α l =.8 H BR = ev Å, α l =.6 ev Å, α l = the term [3,, 6, 7, n + σw 8] n n 7. ev Å, α l =. ev Å respectively. α i (σ x p y σ y p x )δ(z z i ), () H D = i=l,r (σ xp x σ y p y ) ( γ(z) ), () z z J. Moser et al., PRL 99, 66 (7) /k * - - - - -.......... 3.....3.. -. -. -.3 -. -.! X/ E-E F [ev]
(α, α 3, γ, γ 3 ) SOC field captures higher order terms (,,,) (,,,) `SOCF for band n= within st order approximation (,,,). (,,,)!/a!/a... (,,,) (,,-,) (,,,) (,,-,) -!/a -!/a!/a -!/a!/a -!/a k=!/a k=!/7a k=!/a 6 6 ɛ so [ (8γ 3 a γ)k 3 sin(φ 3θ) + (8α 3 a α)k 3 sin(φ + 3θ) + 3k(a k 8)(α sin(φ θ) + γ sin(φ + θ)) ] / X.Cartoixa et al., PRB 73, 3 (6)
Conclusions determination of the effective Bychkov-Rashba & Dresselhaus spinorbit coupling parameters in Fe/GaAs ab-initio support of the phenomenological TAMR model calculation of the spin-orbit coupling fields - work in progress direct impact on k-resolved experiments: ARPES, STS, etc.