Spin and Orbital Magnetism of Rare Earth Atoms Adsorbed on Graphene and Metal Substrates

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1 Spin and Orbital Magnetism of Rare Earth Atoms Adsorbed on Graphene and Metal Substrates Alexander Shick Institute of Physics ASCR, Prague, CZ ü Acknowledge collaboration with D. Shapiro, Inst. of Electronics, RAS; A. Lichtenstein, Uni.Hamburg

Outline: u f-electron challenge: understanding of the physical and

character of Sm@GR and Ho@Pt DFT and DFT+U is not good enough for

Diagonalization of Anderson Impurity model.

2 Outline: u f-electron challenge: understanding of the physical and chemical properties of rare-earth materials u Electronic/magnetic character of Sm@GR and Ho@Pt DFT and DFT+U is not good enough for the 4f-materials u Beyond DFT: combining DFT and Hubbard-I, Exact Diagonalization of Anderson Impurity model. DFT+HIA and DFT+ED can serve instead of phenomenological crystal-field theory 2 6 January 218

3 Localized nature of 4f-electrons

4 u Interactions Between 4f-adatom (RE) and surface u Atomic physics + band theory u DFT+ Exact Diagonalization (ED) of Anderson Impurity Model Sm@GR & Ho@Pt

5 Rare-earth on graphene u Liu et al., PRB (21) Nd, Gd, Eu and Yb adatoms on graphene. 4x4 supercell, f-states in PAW structural optimization with VASP. Energetically most favorable hollow adsorption site. u Li et al., Physica E 75, 169 (216) La, Ce, Pr, Nd, Pm, Sm, Eu, Gd embedded in graphene, 7x7 supercell, GGA+U with VASP. All studied atoms are magnetic: Sm Eu Gd M s M L

Gapless Semiconductor sp 2 hybridized σ states π-states DOS 1.8.6.4.

6 Gapless Semiconductor sp 2 hybridized σ states π-states DOS DOS Energy (ev) Conical points (K) at E F Symmetry protected (T and I) Massless Dirac Fermions

Toy model: RE adatom on graphene (RE@GR) Xiaojie Liu et al.

structural optimization with VASP + no spin-orbit coupling + PAW-PBE+U

7 Toy model: RE adatom on graphene Xiaojie Liu et al., PRB 82, (21) ü Hexagonal hollow position ü 4x4x1 supercell structural optimization with VASP + no spin-orbit coupling + PAW-PBE+U (Dudarev) Sm:U=6.87 ev J=.76 ev in the ballpark of commonly accepted values for the RE Z[RE-Gr] a.u. Sm 4.58

8 Different flavours of LDA+U

9 E ee = 1 2 P n 1 2 Rotationally invariant DFT+U V ee1 3; 2 4 V ee1 3; 4 2 n 3 4 n 1 2 n m1 1,m 2 2 includes all spin-diagonal and spin-off-diagonal elements

10 FLL-LSDA+U Magnetic moment n f M S M L M X M Y M Z Sm@GR: AMF-LSDA+U Magnetic moment n f M S M L M X M Y M Z like, magnetic with M J =2.9 µ B f 6 -like: non-magnetic What is the magnetic state of Sm@GR? Different flavours of DFT+U give different answers

self-consistency over charge density AIM

charge density Full-Potential Linearized

11 [n] imp = [n] loc Anderson Impurity Solver Exact Diagonalization DFT+U + self-consistency over charge density AIM Exact Diagonalization : v Spin-orbit coupling + Crystal Field v Full Coulomb vertex DFT + U: v Self-consistency over charge density Full-Potential Linearized Augmented Plane Wave (FLAPW) basis n f [n] = 1 Im E F dz Tr G(z)

Anderson Impurity Model: Exact Diagonalization H imp = X kmm + X + 1 2 mm + X kmm X mm m m [ k ] mm b km b km + X m bath l s + CF + ex s z f f m SOC Crystal

12 Anderson Impurity Model: Exact Diagonalization H imp = X kmm + X mm + X kmm X mm m m [ k ] mm b km b km + X m bath l s + CF + ex s z f f m SOC Crystal Field d-f Exchange [V k ] mm f m b km +h.c. Hybridization Coulomb Interaction U mmmmf f m -chemical potential mm f m f m m f m f m f m - Hybridization strength

13 Choice of the ED parameters DOS/Hybridization strength: ( ) = 1 N f = Tr[G 1 LDA ( + i) -1/πTr[ImG LDA ] LDA DOS -1/πTr[ImG -1 LDA ] V 5/2 (mev) V 7/2 (mev) ε 5/2 (mev) ε 7/2 (mev) 25-5 Hybridization x 1 chosen to reproduce LDA n 5/2 & n 7/2 occupations Very weak hybridization close to the atomic limit

14 Ground state: H imp Ψ(N) = E N Ψ(N); N=n f + n bath GS>=Ψ(N=14), SINGLET, S=2.92 L=2.92 J=.3 (f 6 -like) + 5 mev 1EX>=Ψ(N=14), TRIPLET non-magnetic character of Sm@GR f-shell No difference between FLL and AMF. 1 8 (B) 5/2 7/2 fdos DOS (1/eV) Energy (ev)

15 Probe for Valence and Multiplet structure: XAS&XMCD 4f Branching Ratio B 3d Dipole selection rule Sm@GR n f n 5/2 f n 7/2 f B DFT+U-FLL DFT+U-AMF DFT+ED-FLL DFT+ED-AMF atomic LS atomic jj LS-coupling should work for RE!

16 Difficulties of DFT+U u Different flavours of DFT+U suggest different magnetic character of Sm@GR: FLL-LDA+U yields magnetic Sm adatom AMF-LDA+U non-magnetic but in jj-coupling u DFT+ED calculations show that Sm@GR has a singlet f 6 -like LS-coupled non-magnetic ground state A. Kozub, A. B. Shick, F. Maca, J. Kolorenc, A.I. Lichtenstein, Phys. Rev. B 94, (216)

17 Science 352, 318 (April 216) Magnetic remanence in single atoms F. Donati, 1 S. Rusponi, 1 S. Stepanow, 2 C. Wäckerlin, 1 A. Singha, 1 L. Persichetti, 2 R. Baltic, 1 K. Diller, 1 F. Patthey, 1 E. Fernandes, 1 J. Dreiser, 1,3 Ž. Šljivančanin, 4,5 K. Kummer, 6 C. Nistor, 2 P. Gambardella, 2 * H. Brune 1 * Fig. 1. Ho atoms on MgO films. (A) Constantcurrent STM image of Ho atoms on 2 monolayer (ML) MgO/Ag(1) (tunnel voltage V t =1mV, tunnel current I t =2pA,T =4.7K,Hocoverage Q Ho =.5±.1ML).(B) Adsorption geometry of Ho atoms on top of O on 2-ML MgO/Ag(1) as simulated with DFT, together with a schematic of the XAS experiment. (C) Splitting of the lowest quantum levels of Ho atoms from multiplet calculations. Zero-field values of hj z i are reported. (D)XAS and XMCD at the M 4,5 edges for an ensemble of individual Ho adatoms. The arrow points to the maximum of the XMCD signal that is recorded as a function of magnetic field to obtain the magnetization curves shown in (E)(field sweep rate db/dt) = 8mT/s,photonfluxf =1 1 2 nm 2 s 1, T =6.5K, Q Ho =.1ML,MgOcoverageQ MgO =7.ML).

18 XAS and XMCD d, f Probe spin and orbital moments + multiplet structure Sum rules p, d Dipole selection rule - magnetic dipole moment

analysis of spontaneous transitions. Even below the first excitation energy, t dro 5 1 15 2 25 3 Time (s) voltage. This is due to the exponentially in 7.

This leads to a sho XAS XMCD Ho measurement in spite of the lower temp bias voltage, the lifetime is much longer, 3 1 nm 2.

19 analysis of spontaneous transitions. Even below the first excitation energy, t dro Time (s) voltage. This is due to the exponentially in 7. tunnelling electrons at higher bias voltage CoAP excite the atom to the first excited state. O to the Co island, the Ho atom experienc Pt CoP stray field of the island. This leads to a sho XAS XMCD Ho measurement in spite of the lower temp bias voltage, the lifetime is much longer, 3 1 nm 2.6 An interesting side effect of the forbi conduction electrons is that magnetic int 2 Experiment ( ) Experiment ( ) bi-stability for Ho/ Magnetic -1 mediated by the Rudermann Kittel Ka Donati et al., PRL (214) 18 Experiment (55 ) Experiment 1, do not cause the exchange of an action (55 ) 1-2 order, M such that the observed t is near (H) J = 8, J = ±6 XMCD z arrangement of the Ho atoms, as discu Information. 4-1 In= conclusion, general symmetry requi Calc. J z = ± 6 ( ) Calc. J ± 6 ( ) z Normal -4 ± stabilize 6 (55 ) the magnetic moment of a Calc. J z = ± 6 (55 ) Calc.way J z =to 1 Grazing.5-2 it from conduction electrons and nuclea 3 Experiment ( ) be used in data storage and qu J zpotentially =±6 Experiment (55 ) (mv) Voltage -3 = ±a8 magnetic field, the interactions cou J zing ure 3 Lifetimes of adsorbed Ho atoms as function of external allowing controlled quantum manipulati Ho@Pt(111) dl/dv (arbitrary units) Magnetic bi-stability for Ho/Pt(111) - XMCD Energy (mev) XAS (arb. u.) XMCD (arb. u.) o/pt(111) - XMCD -4 rameters. a, Spontaneous ground states for Calc. J = ± 8 ( ) Calc. J z transitions = ± 8 ( ) between the two -7 z -3 erent magnetic fields along the surface normal observed with spin-polarized 1 METHODS SUMMARY -.5 = ± 8 (55 ) = ± 8 (55 ) Calc. J Calc. J z z dv signal recordedcalc. at.7j K, 5 mv and 5 na. b, Spontaneous transitions -5 z = ± 6 ( ) -4-8 without applied The Pt(111) sample was cleaned by argon-ion sp ween the two ground states for two different temperatures 85 K. The sample surfaces were checked for Calc. J z = ± 6 (55 ) im gnetic field recorded at 5 mv and 5 na. c, Spin-polarized di/dv map of Ho deposition of Ho. Ho atoms were deposited onto ms and Co islands on Pt(111) (V 5 3 mv, I na). Two distinct Unpolarized parallel (P) and in the-2stm STM tips 6 were prepar nals are observed on135 Co islands magnetized antiparallel (AP) cleaned Field in situ(t) by flashing to above 2,5 K. Spi Energy (ev) he tip. Positions ofenergy Pt atomic(ev) steps are indicated by dashed lines. d, Lifetimes -6

DFT+ED: Ho@Pt(111) ü Ho in fcc and hcp positions 1st layer hcp fcc 2nd layer

20 DFT+ED: ü Ho in fcc and hcp positions 1st layer hcp fcc 2nd layer a.u. 3x3x1 supercell hcp 4.39 fcc 4.35 U=7.3 ev J=.83 ev S. Lebegue et al., PRB (26)

Anderson Impurity Model - modified l s + CF + ex s z Δ EX = J df m 5d ~ 1 mev (J df =.1 ev) removing the interacting DFT+U potential and SOC hm S i hm L i hm S i+hm D i hj z i DFT+U [Miyamachi213] 4.

21 Anderson Impurity Model - modified l s + CF + ex s z Δ EX = J df m 5d ~ 1 mev (J df =.1 ev) removing the interacting DFT+U potential and SOC hm S i hm L i hm S i+hm D i hj z i DFT+U [Miyamachi213] FT+U [Khajetoorians216] ex = 5 mev ex = 1 mev ex = 15 mev XMCD [Donati214] 2.28± ± ± ±.8 DFT+ED improves agreement with XMCD R LS =M L /[M S +M D ]=1.2 XMCD: R LS =

Energy, mev 9 8 7 6 5 4 3 2 1 ED CF fit 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 Total moment J Z Fit to Crystal Field Theory DFT+ED: J=8, J z =7> Energy (mev) hl z i hs z i hj z i. 5.32 1.67 7. 9.8 4.

22 Energy, mev ED CF fit Total moment J Z Fit to Crystal Field Theory DFT+ED: J=8, J z =7> Energy (mev) hl z i hs z i hj z i

Magnetic stability Ho@Pt(111) Rate Equations for the populations

23 Magnetic stability Rate Equations for the populations P i of the different states of H M Reduced Quantum Master Equation

6 5 Energy, mev 4 3 2 1 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 Total moment J Z Ground State: J=8, J z = u Δ EX = J df m 5d

24 6 5 Energy, mev Total moment J Z Ground State: J=8, J z = u Δ EX = J df m 5d -> : The magnetic state relaxes to non-magnetic state overruling the existence of long living magnetic moment for Ho@Pt.

25 u The <J z > = 6.8 spin-polarized state in an external magnetic field. u Reasonable agreement with experimental XMCD data. u The role of 5d-4f interorbital exchange polarization in modification of the 4f shell energy spectrum is ephasized. u Once the magnetic field is removed, the system goes to <J z > = non-magnetic state. No magneticaly stable Ho@Pt. A. B. Shick, D. S. Shapiro, J. Kolorenc, A. I. Lichtenstein, Scientific Reports 7, 2751 (217).

Tuning magnetic anisotropy, Kondo screening and Dzyaloshinskii-Moriya interaction in pairs of Fe adatoms

Tuning magnetic anisotropy, Kondo screening and Dzyaloshinskii-Moriya interaction in pairs of Fe adatoms Department of Physics, Hamburg University, Hamburg, Germany SPICE Workshop, Mainz Outline Tune magnetic