Spin-orbit coupling and Kondo resonance in Co adatom on Cu100 surface DFTED study A. B. Shick and M. Tchaplianka

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Spin-orbit coupling and Kondo resonance in Co adatom on Cu(100) surface:

DFT+ED study

A. B. Shick and M. Tchaplianka

Institute of Physics, Czech Academy of Science, Na Slovance 2, CZ-18221 Prague, Czech Republic∗

A. I. Lichtenstein

Institute of Theoretical Physics, University of Hamburg, 20355 Hamburg, Germany and

European X-Ray Free-Electron Laser Facility, Holzkoppel 4, 22869 Schenefeld, Germany

(Dated: October 4, 2022)

We report density functional theory plus exact diagonalization of the multi-orbital Anderson

impurity model calculations for the Co adatom on the top of Cu(001) surface. For the Co atom

d-shell occupation nd≈8, a singlet many-body ground state and Kondo resonance are found, when

the spin-orbit coupling is included in the calculations. The diﬀerential conductance is evaluated

in a good agreement with the scanning tuneling microscopy measurements. The results illustrate

the essential role which the spin-orbit coupling is playing in a formation of Kondo singlet for the

multi-orbital impurity in low dimensions.

I. INTRODUCTION

The electronic nanometer scaled devices require the

atomistic control of their behaviour governed by the elec-

tron correlation eﬀects. One of the most famous corre-

lation phenomena is the Kondo eﬀect originating from

screening of the local magnetic moment by the Fermi sea

of conduction electrons, and resulting in a formation of a

singlet ground state [1]. Historically the Kondo screening

was detected as a resistance increase below a characteris-

tic Kondo temperature TKin dilute magnetic alloys [2].

Recent advances in scanning tuneling microscopy (STM)

allowed observation of the Kondo phenomenon on the

atomic scale, for atoms and molecules at surfaces [3, 4].

In these experiments, an enhanced conductance near the

Fermi level (EF) is found due to the formation of a sharp

Abrikosov-Suhl-Kondo [5–7] resonance in the electronic

density of states (DOS).

One case of the Kondo eﬀect the most studied exper-

imentally and theoretically is that of a Co adatom on

the metallic Cu substrate [3, 8–10]. The experimental

STM spectra display sharp peaks at zero bias, or so called

”zero-bias” anomalies, similar to the Fano-resonance [11]

found in the atomic physics, which are associated with

the Kondo resonance. The theoretical description of the

Kondo screening in multiorbital dmanifold is diﬃcult

since the whole dshell is likely to play a role. Very re-

cently, theoretical electronic structure of the Co atom

on the top of Cu(100) was considered [10] using nu-

merically exact continuous-time quantum Monte-Carlo

(CTQMC) method [12] to solve the multiorbital sin-

gle impurity Anderson model [13] (SIAM) together with

the density-functional theory [14] as implemented in the

W2DYNAMICS package [15, 16]. However, the spin-orbit

coupling (SOC) was neglected. The peak in the DOS at

∗Electronic address: shick@fzu.cz

EFwas obtained in these calculations, and was inter-

preted as a signature of the Kondo resonance.

Alternative interpretation was proposed [17] which is

based on the spin-polarized time-dependent DFT in con-

junction with many-body perturbation theory. These au-

thors claim that the ”zero-bias” anomalies are not neces-

sarily related to the Kondo resonance, and are connected

to interplay between the inelastic spin excitations and

the magnetic anisotropy. Thus the controversy exists con-

cerning the details of the physical processes underlying

the Kondo screening in Co@Cu(100). In this work, we

revisit Co@Cu(100) case making use of the combination

of DFT with the exact diagonalization of multiorbital

SIAM (DFT+ED) including SOC. We demonstrate that

SOC plays crucial role in formation of the singlet ground

state (GS) and the Kondo resonance.

II. METHODOLOGY: DFT + EXACT

DIAGONALIZATION

The exact diagonalization (ED) method is based on

a numerical solution of the multi-orbital Anderson im-

purity model (AIM) [13]. The continuum of the bath

states is discretized. The ﬁve d-orbitals AIM with the

full spherically symmetric Coulomb interaction, a crystal

ﬁeld (CF), and SOC is written as,

H=X

kmσ

kmb†

kmσ bkmσ +X

mσ

dd†

mσdmσ

mm0σσ0ξl·s+∆CF σ σ0

mm0d†

mσdm0σ0

kmσVkmd†

mσbkmσ +h.c.(1)

mm0m00 m000 σσ0

Umm0m00 m000 d†

mσd†

m0σ0dm000 σ0dm00 σ.

The impurity-level position dwhich yield the desired

hndi, and the bath energies km are measured from the

arXiv:2210.00600v1 [cond-mat.str-el] 2 Oct 2022

FIG. 1: The ball model (top view) for Co@4Cu8] supercell. The speciﬁc choice of the Cartesian reference frame is show. With

this choice, the local Green’s function without SOC becomes diagonal in the basis of cubic harmonics m={xz, yz, xy, x2−

y2,3z2−r2}(a); orbitally resolved DOS (b); orbitally resolved hybridization Im ∆ (c) for the Co adatom on Cu(001),

chemical potential µ, that was set to zero. The SOC ξpa-

rameter speciﬁes the strength of the spin-orbit coupling,

whereas ∆CF matrix describes CF acting on the impu-

rity. The hybridization Vmk parameters describe the cou-

pling of substrate to the impurity orbitals. These param-

eters are determined from DFT calculations, and their

particular choice will be described below.

The last term in Eq.(1) represents the Coulomb in-

teraction. The Slater integrals F0= 4.00 eV, F2=7.75

eV, and F4= 4.85 eV are used for the Coulomb inter-

action [9, 10]. They correspond to the values for the

Coulomb U= 4 eV and exchange J= 0.9 eV for Co

which are in the ballpark of commonly accepted Uand

Jfor transitional 3d-metals.

The DFT calculation were performed on a supercell of

four Cu(100) layers, and the Co adatom followed by four

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摘要：

Spin-orbitcouplingandKondoresonanceinCoadatomonCu(100)surface:DFT+EDstudyA.B.ShickandM.TchapliankaInstituteofPhysics,CzechAcademyofScience,NaSlovance2,CZ-18221Prague,CzechRepublicA.I.LichtensteinInstituteofTheoreticalPhysics,UniversityofHamburg,20355Hamburg,GermanyandEuropeanX-RayFree-ElectronLaser...

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Spin-orbit coupling and Kondo resonance in Co adatom on Cu100 surface DFTED study A. B. Shick and M. Tchaplianka

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