Dimensionality effects in the LDOS of ferromagnetic hosts probed via STM: spin-polarized quantum beats and spin filtering

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by  A. C. Seridonio

Dimensionality effects in the LDOS of ferromagnetic hosts probed via STM:
spin-polarized quantum beats and spin filtering
A. C. Seridonio1,2, S. C. Leandro1, L. H. Guessi1, E. C. Siqueira1,
F. M. Souza3, E. Vernek3, M. S. Figueira4, and J. C. Egues5
1Departamento de Física e Química, Universidade Estadual Paulista, 15385-000, Ilha Solteira, SP, Brazil
2Instituto de Geociências e Ciências Exatas - IGCE, Universidade Estadual Paulista,
Departamento de Física, 13506-970, Rio Claro, SP, Brazil
3Instituto de Física, Universidade Federal de Uberlândia, 38400-902, Uberlândia, MG, Brazil.
4Instituto de Física, Universidade Federal Fluminense, 24210-340 Niterói, RJ, Brazil
5Instituto de Física de Sâo Carlos, Universidade de São Paulo, 13560-970, São Carlos, SP, Brazil.
3
1 We theoretically investigate the local density of states (LDOS) probed by a STM tip of ferro-
0 magnetic metals hosting a single adatom and a subsurface impurity. We model the system via
2 thetwo-impurity AndersonHamiltonian. By using theequation of motion with therelevant Green
functions,wederiveanalyticalexpressionsfortheLDOSoftwohosttypes: asurfaceandaquantum
n
wire. The LDOS reveals Friedel-like oscillations and Fano interference as a function of the STM
a
J tip position. These oscillations strongly depend on the host dimension. Interestingly, we find that
the spin-dependent Fermi wave numbers of the hosts give rise to spin-polarized quantum beats in
3
the LDOS. While the LDOS for the metallic surface shows a damped beating pattern, it exhibits

] an opposite behavior in thequantum wire. Dueto this absence of damping, the wire operates as a
l spatially resolved spin filterwith a high efficiency.
e
-
r PACSnumbers: 07.79.Fc,74.55.+v,85.75.-d,72.25.-b
t
s
.
t
a I. INTRODUCTION has been devoted to spin-polarized systems away from
m the Kondo regime and with two impurities.
- The local density of states (LDOS) of electronic sys- Thus we present in this work a theoretical description
d
tems with impurities can exhibit Fano line shapes, due of the systems sketched in Fig. 1. We show that in-
n
o to the quantum interference between different electron terestingphenomenasuchasthespin-polarizedquantum
c paths. Such interference arises from the itinerant elec- beats in the LDOS and the spin-filtering effect arise. To
[ trons that travel through the host and tunnel into the this end, we consider two distinct geometries consistent

impurity sites.1,2 For a single magnetic adatom in the with recent experiments: a metallic surface and a quan-
4
v Kondo regime3 probed by a scanning tunneling micro- tum wire. The 2Dcase emulates the Fe islandin Ref.[5].
6 scope (STM) tip, interesting features manifest when one The quantum wire on the other hand, mimics the “elec-
0 hasaspin-polarizedelectronbathpresent. Herewemen- tron focusing” effect investigated in Ref.[25]. Interest-
4 tion the splitting of the Kondo peak in the differen- ingly, we note that the pioneering quantum wire treat-
3 tial conductance due to the itinerant magnetism of the ment for “electron focusing” in a side-coupled geometry
.
1 host.4Such hallmark has already been found experimen- can be found in Ref.[29]. In this treatment the nonin-
1 tallyinanFeislandwithaCoadatom.5Additionally,the teracting single impurity Anderson model51 was solved
2 STMsystemcanalsooperateasaFano-Kondospin-filter in one dimension by considering the impurity above the
1
due to a spin-polarizedtip and a nonmagnetic host.6,7In wire. Weshouldalsopointoutthatthefullab-initiocal-
:
v the absence of a ferromagnetic host, the Fano-Kondo culationthat yields to “electronfocusing” in Ref.[25] can
i profile becomes doubly degenerate.8–23 Away from the be qualitatively recovered by the simple quantum wire
X
Kondo regime, a spin diode emerges.24 model adopted in Ref.[29].
r
a In the condensed matter literature on scanning mi- Here we extend this one-dimensional treatment of the
croscopy, there is a profusion of work discussing spin- Anderson Hamiltonian by including a spin-dependent
dependent phenomena employing ferromagnetic leads DOS for the wire, a second lateral impurity right be-
coupled to quantum dots or adatoms in the Kondo neathitandCoulombinteractioninbothimpurities. We
regime.4,6,7,31–50 Here we mention those with metallic perform our study in the framework of the two-impurity
samples and buried impurities, in which the anisotropy Anderson model by employing the equation of motion
of the Fermi surface plays an important role in electron approach to calculate the LDOS of the system. The
tunneling.25–30 Accordingto the experiment ofPrüser et Hubbard I approximation52 is used by assuming for the
al.25,suchanisotropyallowsatomsofFeandCobeneath sakeof simplicity infinite Coulombenergies atthe impu-
theCu(100)surfacetoscatterelectronsinpreferentialdi- rities. We show that the LDOS can be written in terms
rections of the material due to an effect called “electron of the Fano factor, the Friedel-like function for charge
focusing”. In this scenario,the STM becomes a new tool oscillationsand the spin-dependent Fermi wavenumbers
for the detection of the Fermi surface signatures in the of the host. Such quantities lead to spin-polarizedquan-
real lattice of a metal. In contrast, much less attention tum beats in the LDOS. We also show that this effect
2
is strongly correlated to the host dimensionality. Thus
the quantum beats in the LDOS of the metallic sur-
ftaoctehepruensednatmapeldonogn-erafnougenddainmtpheedqubaehntauvmiorwiinrecsoynsttreamst. H=X~kσ ε~kσc~†kσc~kσ+Xjσ εjdσd†jσdjσ +Xj Ujd†j↑dj↑d†j↓dj↓
SadsuespcuhemnddeidsotnbinytchttehfeedaiFmtauenrneossifoaoncrtiagoliirntyaantoedfftFrhoreimehdoetshltef.usTnpcheteciorifienfc,orwfeohrtimhches +Xj~kσ"√VjN~kσσφ~kσ(cid:16)R~j(cid:17)c~†kσdjσ + √VjN∗~kσσφ~∗kσ(cid:16)R~j(cid:17)d†jσc~kσ#,
metallic surface and the quantum wire become spatially (1)
resolved spin filters, where the latter displays a higher
efficiency due to the undamped LDOS. The spin-polarized electron gas forming the hosts is de-
scribed by the operator c~†kσ (c~kσ) for the creation (an-
nihilation) of an electron in a quantum state labeled by
the wave vector ~k, spin σ and energy ε . For the im-
~kσ
purities, d†jσ (djσ) creates (annihilates) an electron with
spin σ in the state ε , with j =1,2. The third term of
jdσ
Eq. (1) accounts for the on-site Coulomb interaction U
j
at the jth impurity placed at position R~ . In our calcu-
j
lations, we assume U = U (single occupancy of
1 2
→ ∞
the impurities). Finally, the last two terms mix the host
continuum of states and the levels ε . This hybridiza-
jdσ
tionoccursattheimpuritysitesR~ viathehost-impurity
j
couplings Vj~kσ and the plane waves φ~kσ R~j = ei~k.R~j.
σ is the number of conduction states fo(cid:16)r a(cid:17)given spin
N
σ. Theferromagnetichostsareconsideredspin-polarized
electron baths, characterizedby the polarization
Figure 1. (Color online) Side-coupled geometry with twoim- ρ ρ
FM FM
puritiesinpresenceofaSTMtip. Γtipisthetip-hostcoupling. P = ρ ↑+−ρ ↓, (2)
FM FM
a) left panel: 2D evanescent waves appear in the LDOS of a ↑ ↓
metallic surface. The system is treated as a two-dimensional
in which
electron gas showed in the right panel. b) left panel: the
confinement of 1D waves in specific directions (perpendicu- 1
ρ = =ρ (1+σP) (3)
lar wave fronts) is due to the “electron focusing” effect (see FMσ 2D 0
σ
Ref.[25]). Each direction is modeled by a quantum wire as
illustrated in theright panel. is the density of states of the hosts in a Stoner-like
framework53,54 expressedin terms of the spin-dependent
This paperis organizedasfollows. InSec. II,weshow half-width Dσ and the density ρ0 for the case P =0.
thetheoreticalmodeloftheferromagnetichostswiththe
impurities in the side-coupled geometry as sketched in
B. LDOS for the spin-polarized systems
Fig. 1andderivetheLDOSformulaforbothsystems,the
metallic surface and the quantum wire. The decoupling
scheme HubbardI52 for the Greenfunctions is presented To obtain the host LDOS we introduce the retarded
in Sec. III. In Sec. IV, we discuss the results for the Green function in the time coordinate,
quantum beats in the LDOS and the spin-filtering. The
i
conclusions appear in Sec. V. Gσ t,R~ =−~θ(t)ZF−M1 e−βEn
(cid:16) (cid:17) Xn
II. THEORETICAL MODEL ×hn| Ψ˜σ R~,t ,Ψ˜†σ R~,0 +|ni, (4)
h (cid:16) (cid:17) (cid:16) (cid:17)i
where
A. Hamiltonian
1
Ψ˜ R~ = φ R~ c (5)
InordertoprobetheLDOSoftheferromagnetichosts, σ √ σ ~kσ ~kσ
(cid:16) (cid:17) N X~k (cid:16) (cid:17)
we represent a STM tip weakly connected to hosts hy-
bridized to a pair of side-coupled impurities as outlined isthefermionicoperatordescribingthequantumstateof
in Fig. 1. The systems we investigate are described ac- the host site placed below the STM tip, ~ is the Planck
cordingtothetwo-impurityAndersonmodelgivenbythe constant divided by 2π, θ(t) the step function at the in-
Hamiltonian3 stant t, β = 1/k T with k as the Boltzmann constant
B B
3
andT thesystemtemperature, and n arethepar- Notice that we needto findthe mixed Greenfunction,
FM
titionfunctionandamany-bodyZeigenstat|eoifthesystem ˜σ (ε+). To this end, we define the advanced Green
Gdjcq~
Hamiltonian[Eq. (1)],respectively,and[ , ]+ isthe function
··· ···
anticommutator for Eq. (5) evaluated at distinct times.
FromEq.(4)thespin-dependentLDOSatasiteR~ ofthe
i
host [see Fig. 1] can be obtained as Fdσjcq~(t)= ~θ(−t)ZF−M1 e−βEn
n
1 X
ρσLDOS(ε,R)=−πIm G˜σ ε+,R~ , (6) ×hn| d†jσ(0),cq~σ(t) +|ni, (12)
n (cid:16) (cid:17)o h i
where ˜ (ε+,R~)isthetimeFouriertransformof (t,R~). which results in
σ σ
Here, εG+ = ε+iη and η 0+. In what followsGwe first
→ ∂ i
darebvietlroapryapgoesniteiroanlsfRo~rjmaanldismR~l;folartiemrpounrwiteiestalkoecathliezeldimaitt ∂tFdσjcq~(t)=−~δ(t)ZF−M1 n e−βEn
X
Ro~fjt=hisR~wlo=rk~0.,inordertotreatthe side-coupledgeometry ×hn| d†jσ(0),cq~σ(t) +|ni− ~iεq~σFdσjcq~(t)
To obtain an analytical expression for the LDOS we hi 1 i
apply the equation-of-motionapproachto Eq. (4). Thus + −~ √ Vlq~σφq∗~σ R~l Fdσjdl(t),
wesubstituteEq. (5)inEq. (4)andbegintheprocedure (cid:18) (cid:19) Nσ Xl (cid:16) (cid:17)
(13)
with
t,R~ = 1 φ R~ φ R~ σ (t) (7) whereweusedonceagainEq. (10),interchanging~k ~q.
Gσ σ ~kσ q∗~σ Gc~kcq~ Thus the Fourier transform of Eq. (13) becomes ↔
(cid:16) (cid:17) N X~kq~ (cid:16) (cid:17) (cid:16) (cid:17)
1
expressed in terms of ε−F˜dσjcq~ ε− =εq~σF˜dσjcq~ ε− + √ Vlq~σφq∗~σ R~l
σ
i (cid:0) (cid:1) (cid:0) (cid:1) N Xl (cid:16) (cid:17)
Gcσ~kc~q(t)=−~θ(t)ZF−M1 e−βEn ×F˜dσjdl ε− , (14)
n
X (cid:0) (cid:1)
with ε =ε iη. Applying the property
×hn| c~kσ(t),cq†~σ(0) +|ni. (8) − −
Performing ∂ on Eq. (8h) we find i G˜dσjcq~ ε+ = F˜dσjcq~ ε− † (15)
∂t
(cid:0) (cid:1) n (cid:0) (cid:1)o
∂ i to Eq. (14), we show that
∂tGcσ~kc~q(t)=−~δ(t)ZF−M1 e−βEn
×hn| c~kσ(t),cq†~Xσn(0) +|ni ε+G˜dσjcq~(cid:0)ε+(cid:1)=εq~σG˜dσjcq~(cid:0)ε+(cid:1)+ √N1 σ Xl Vl∗q~σφq~σ(cid:16)R~l(cid:17)
+ hi ε σ (t)i+ i ×G˜dσjdl ε+ (16)
(cid:18)−~(cid:19) ~kσGc~kc~q (cid:18)−~(cid:19) and (cid:0) (cid:1)
1
V φ R~ σ (t), (9)
× √ j~kσ ~∗kσ j Gdjcq~ V φ R~
Nσ Xj (cid:16) (cid:17) ˜σ ε+ = 1 l∗q~σ q~σ l ˜σ ε+ . (17)
Gdjcq~ √ ε+ ε(cid:16) (cid:17)Gdjdl
where we have used (cid:0) (cid:1) Nσ Xl − q~σ (cid:0) (cid:1)
∂ Now we substitute Eq. (17) into Eq. (11) and the
i~ c (t)= c , =ε c (t)
∂t ~kσ ~kσ H ~kσ ~kσ latter into Eq. (7) in the energy coordinate to obtain
+ (cid:2) 1 (cid:3) V φ R~ d (t). (10) 2
√ j~kσ ~∗kσ j jσ φ R~
Nσ Xj (cid:16) (cid:17) G˜σ ε+,R = 1 (cid:12)ε~k+σ(cid:16)ε(cid:17)(cid:12)
In the energy coordinate, we solve Eq. (9) for (cid:0) (cid:1) Nσ X~k (cid:12)(cid:12) − ~kσ(cid:12)(cid:12)
G˜cσ~kc~q(ε+) and obtain +(πρ0)2 (qjσ −iAjσ)(qj −iAjσ)
j
X
G˜cσ~kc~q(cid:0)ε+(cid:1)×=G˜εdσ+jcδ−~q~kq~εε~k+σ +. √N1 σ Xj Vj~kεσ+φ−~∗kσε(cid:16)~kRσ~j(cid:17)(11) ××+(GG˜˜πddσσρjjdd0jl)(2(εεXj))6=.l(qjσ −iAjσ)(qlσ −iAlσ)(18)
(cid:0) (cid:1)
4
Itisworthmentioningthattheimaginarypartofthefirst of Eq. (21), it reduces to the case of two decoupled sys-
term of Eq. (18) gives the background DOS of the host tems with one impurity each. The terms in Eq. (22),
[Eq. (3)] andthe others describeimpurity contributions, indeed, hybridize such single-impurity problems via the
with mixed Green functions ˜σ (ε). Thus the LDOS for-
Gdjdl
mula encodes the single Fano factor q , the Friedel-like
jσ
function A and the new interfering term ∆̺ .
V φ R~ φ R~ jσ jlσ
q = 1 j~kσ ~∗kσ j ~kσ (19) We close this section by recalling that the phase αjσ
jσ πρ ε(cid:16) ε(cid:17) (cid:16) (cid:17) is nonzero for the quantum wire system, as we shall see
0Nσ X~k − ~kσ later. The quantities q and A in the quantum wire
jσ jσ
beingtheFanoparameterduetothesinglecouplingV deviceareindeedfunctions thatexhibitundampedoscil-
j~kσ
lations as the tip moves away from the impurities. Con-
between the host and a given impurity. This factor en-
versely,dampedoscillationsarepredictedinthe metallic
codes the quantum interference originated by electrons
surface setup. In this case α = 0 and A becomes a
traveling through the ferromagnetic conduction band jσ jσ
real function. Thus the quantity A should be read
thattunneltotheimpuritystateandreturntotheband, jσ
| |
justasA inEqs. (21)and(22). Moreover,aferromag-
andthosethatdonotperformsuchtrajectory. Addition- jσ
neticenvironmentischaracterizedbytwospin-dependent
ally, we recognize
Fermiwavenumbers,namely,k andk ,whichatlow
F F
polarization P introduces a slig↑htly diffe↓rence between
1
A = V φ R~ φ R~ δ ε ε them. As a result, this feature leads to a full LDOS
jσ ρ0Nσ X~k j~kσ ~∗kσ(cid:16) j(cid:17) ~kσ(cid:16) (cid:17) (cid:0) − ~kσ(cid:1) cρa↑LnDObSe+daρm↓LDpOedS worituhnsdpainm-ppeodla,rdizeepdeqnudainngtuumpobneatthse,tshyast-
= A eiαjσ (20)
| jσ| tem dimensionality. We shall look more closely at these
features later.
as an expression that we call the Friedel function, be-
causeitleadstoFriedel-likeoscillationsintheLDOSwith In STM experiments, in particular within the linear
α as a spin-dependent phase. At the end, the Green responseregimeandneglecting tip-adatomcoupling,the
jσ
function ˜ (ε+,R) indeed depends on ˜σ (ε) and the differential conductance G = G↑ +G↓ is the observable
Gσ Gdjdj measured by the tip, whose spin component is given by4
mixed Green function ˜σ (ε). Finally, from Eqs. (6)
Gdjdl
and(18), the LDOSofthe ferromagneticsystemscanbe
recast as the expression
Gσ = e2πΓ +∞ρσ (ε) ∂f (ε φ) dε, (23)
h tipˆ LDOS −∂ε −
(cid:20) (cid:21)
ρσ (ε,R)=ρ +πρ2 A 2 q2 −∞
LDOS FMσ 0 | jσ| − jσ
Xj h(cid:16) (cid:17) whereeistheelectroncharge(e>0),Γtip isthetip-host
Im ˜σ (ε) +2q A coupling, f is the Fermi-Dirac distribution and φ is the
× Gdjdj jσ| jσ| appliedbias. Forφ<0thehostisthesourceofelectrons
sinnα + πo Re ˜σ (ε) andthetipisthedrain. Forφ>0,wehavetheopposite.
× jσ 2 Gdjdj It is useful to define the dimensionless LDOS
(cid:16) (cid:17) n oi
+πρ2 ∆̺ , (21)
0 jlσ
j=l
X6
where LDOS = ρ↑LDOS +ρ↓LDOS (24)
ρ +ρ
FM FM
∆̺ = q q Im ˜σ (ε) +[A q ↑ ↓
jlσ − jσ lσ Gdjdl | jσ| lσ
nπ o and the transport polarization,
cos α + + A A cos(α α )
jσ jσ lσ jσ lσ
× 2 | || | −
− qjσ(cid:16)|Alσ|cos (cid:17)αlσ + π2 Im G˜dσjdl(ε) PT = GG↑−+GG↓, (25)
(cid:16) π(cid:17)i n o ↑ ↓
+[q A sin α + +q A
jσ lσ lσ lσ jσ
| | 2 | | in order to investigate the spin-polarized quantum beats
π(cid:16) (cid:17)
sin α + + A A sin(α α ) as we shall see in Sec IV. Recall that, in the absence
jσ j l jσ lσ
× 2 | || | −
of the impurities, the transport polarization of Eq. (25)
(cid:16) (cid:17) i
Re ˜σ (ε) . (22) becomes P = P, as established by Eq. (2). In Sec.
× Gdjdl T
IV, we shall verify that Eq. (25) oscillates around P,
n o
The set of Eqs. (21) and (22) is the main analyti- exhibitingtwodistinctbehaviorsasaresultofthesystem
cal finding of this work. It describes the spin-dependent dimensionality – damped spin-polarized quantum beats
LDOS in ferromagnetic hosts with two impurities local- in the metallic surface setup and an undamped pattern
ized at distinct sites R~ . In the absence of the last term in the quantum wire device.
j
5
C. Fano and Friedel-like functions for the metallic
surface system
1
A2D = Im ¯ , (32)
jσ πρ Gjσ
In this section, we calculate the expressions for the 0
(cid:8) (cid:9)
Fano parameter [Eq. (19)] and the Friedel-like func- which allow us to close the calculation. Equations (31)
tion [Eq. (20)] in the metallic surface case, where no and (32) imply the relation
manifestation of “electron focusing” occurs. This calcu-
lation was previously performed in the single-impurity
1
problem4 and now we present an extension applied to q2D = ¯ iA2D. (33)
jσ πρ Gjσ − jσ
the double impurity system. We begin by solving Eq. 0
(20). Tothisend,weassumeφ~kσ(R~)=eikRcosθkσ forthe AstheFriedelfunctionisalreadyknownfromEq. (28),
electronic 2D wave function and use thequantity 1 ¯ (ε,R)providesarelationshipforthe
πρ0Gjσ
Fano parameter. To this end, we can write
J0(ξ)= 21π ˆ 2πeiξcosθkσdθkσ, (26) 1 ¯ = 1ρFMσV 2 ¯ , (34)
0 πρ Gjσ 2 ρ Gjlσ
0 0
l=1
X
the angular representation for the zeroth-order Bessel
tfuhnecfltiaotn-b.anTdhuDsO, iSn the wide-band limit |ε| ≪ Dσ with twaittihonG¯:jlσ(ε,R) obeying the following integral represen-
¯ = 1 +∞dε ∆2 1
ρ = S k dεkσ −1 , (27) Gjlσ πˆ−∞ kσ∆2+ε2kσ ε−εkσ−iη
FMσ Nσ2π ( (cid:18) dk (cid:19) )k=kFσ × H0(l) kσR˜ . (35)
(cid:16) (cid:17)
In the equation above we have used, for the sake of
expressed in terms of the spin-dependent Fermi wave
simplicity, the linear dispersion relation
number k and an element of area in the host, we
Fσ
S
find εkσ =DσkF−σ1(k−kFσ), (36)
ρ the Hankel functions H(1)(ξ) = J (ξ) + iY (ξ) and
A = FMσVJ k R˜ A2D, (28) 0 0 0
jσ ρ 0 Fσ ≡ jσ H(2)(ξ) = J (ξ) iY (ξ). We remark that Eq. (36) in
0 (cid:16) (cid:17) co0mbination0with−Eq.0(3) for the ferromagnetic host al-
fortheFriedel-likefunctionwithR˜ = R~ R~ astherel- lows us to find
j
−
ativecoordinatewithrespecttothejt(cid:12)himpur(cid:12)ity. Notice
(cid:12) (cid:12)
that, according to Eq. (20), phase α(cid:12)jσ is ze(cid:12)ro and Eq. 1 P
(28)isarealquantity. InthecaseoftheFanoparameter, kF = − kF . (37)
we start defining the advanced Green function ↑ r1+P ↓
In particular, for a small polarization P, Eq. (37) re-
sults in slightly different Fermi wave numbers and, con-
G¯jσ = 1 Vj~kσφε~∗kσ(cid:16)εR~j(cid:17)φi~kησ(cid:16)R~(cid:17) (29) sLeDquOeSntalsy,wienshspailnl-speeo.laLriozoekdinqguaatntEuqm. (b3e5a)t,swiencathlceulfautlel
Nσ X~k − ~kσ− the integral ¯j1σ bychoosingacounterclockwisecontour
G
over a semicircle in the upper-half of the complex plane,
by assuming the Lorentzian shape
whichincludesthe simplepoleε =+i∆. Applying the
kσ
residue theorem, we obtain
∆2 ∆
Vj~kσ =V ∆2+ε2kσ, (30) G¯j1σ =H0(1) k∆R˜ ε i∆, (38)
(cid:16) (cid:17) −
for an energy-dependent coupling4 in order to obtain an with k∆ = kFσ(1+iD∆σ). For the evaluation of G¯j2σ we
analyticalsolutionfor q . Notice that, inthe limit ∆ used a clockwise contour over a semicircle in the lower-
jσ
≫
|εkσ|, Eq. (30) recovers the case Vj~kσ =V. Thus we can half plane, including the poles εkσ = ε−iη and εkσ =
write the identities i∆, which yields
−
qjσ = π1ρ Re G¯jσ ≡qj2σD (31) G¯j2σ =2iH0(2) kεR˜ ∆2∆+2ε2
0
(cid:8) (cid:9) + ∆ (cid:16)H(2)(cid:17)k R˜ , (39)
and ε+i∆ 0 ∆∗
(cid:16) (cid:17)
6
with k =k 1+ ε . Taking into account the prop- with m as the effective electron mass and as a given
ε Fσ Dσ length in the wire. Additionally, in the widLe-band limit
erty H0(2)(ξ) =(cid:16) H0(1)(ξ(cid:17)∗) ∗ for the second term in Eq. ε Dσ,wefindthefollowingcomplexFriedel-likefunc-
| |≪
(39), Eq. (34) behcomes i tion:
ρ
π1ρ G¯jσ = ρFρMσV iH0(2) kεR˜ ∆2∆+2ε2 Ajσ =V FρM0σeikFσR˜ ≡A1jσD, (46)
0 0 (cid:20) (cid:16) (cid:17)
+ Re H(1) k R˜ ∆ . (40) characterized by a spin-dependent phase αjσ = kFσR˜.
0 ∆ ε i∆ Thisphaseresultsinnondamped,oscillatorybehavioras
(cid:26) (cid:16) (cid:17) − (cid:27)(cid:21) afunctionofR˜,whichalsoappearsinq1D. Thuswetake
jσ
Explicit calculation of the terms in the brackets of
into account Eqs. (44) and (45), rewriting Eq. (19) as
Eq. (40) leads to
iH0(2) kεR˜ ∆2∆+2ε2 =iJ0 kFσR˜ +Y0 kFσR˜ (41) qjσ = πρV εeikRε˜ =2V ρ L π~m2I ≡qj1σD, (47)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) 0Nσ X~k − k 0Nσ
and
with
∆
Re H(1) k R˜ = Y k R˜ , (42)
(cid:26) 0 (cid:16) ∆ (cid:17)ε−i∆(cid:27) − 0(cid:16) Fσ (cid:17) 1 +∞ eikR˜ sin kFσR˜
= ℘ dk = (48)
Iwnheorredewrethoavenesausrseumtheedl|iεm|i≪t VDσ,∆=≪V DinσEaqn.d∆(30≫),|wε|e. I −2π ˆ−∞ k2−kε2σ (cid:16)2kFσ (cid:17)
j~kσ
make the substitution of Eqs. (28), (40), (41) and (42) in the limit ε Dσ and with ℘ as the principal value.
| |≪
in Eq. (33), showing that Finally, we obtain the Fano factor
ρ
qj2σD =0 (43) qj1σD =2V FρMσ sin kFσR˜ , (49)
0
(cid:16) (cid:17)
for any value of kFσR˜. which also presents spin-dependent Fermi wave numbers
In summary, the zero value of the Fano parameter k andk asinEq.(46). Heretheyarestillconnected
F F
given by Eq. (43) and the zeroth-order Bessel function via↑Eq. (37↓), thus leading to undamped spin-polarized
J k R˜ foundinEq. (28)leadto long-rangedamped quantum beats in the full LDOS.
0 Fσ
spi(cid:16)n-polar(cid:17)izedquantumbeatsinthefullLDOS.Thisfea-
ture will be discussed in Sec. IV.
III. CALCULATION OF THE IMPURITY
GREEN FUNCTION
D. Fano and Friedel-like functions for the quantum
wire system Inthepresentsectionwecalculate ˜σ (ε)(j,l =1,2)
Gdjdl
that appear in Eqs. (21) and (22) for the LDOS. To
HerewedeterminetheFanoparameterinEq. (19)and handletheinteractingtermoftheHamiltonian,weadopt
theFriedel-likefunctioninEq. (20)forthequantumwire the Hubbard I approximation,52 which provides reliable
case. FollowingA.Weismann,29 weuseφ (R~)=eikR as results at temperatures abovethe Kondotemperature.52
~kσ
the electron wave function, in which the direction intro- We begin by repeating the equation-of-motion approach
ducedbyR~ definestheSTMtip-impuritydistancewhere for these Green functions, which results in
“electronfocusing” manifests. We alsouse the dispersion
relation
(ε ε˜ +iΓ ) ˜σ =1+U ˜
− jdσ jjσ Gdjdj jGdjσndjσ¯,djσ
ε = ~2k2 D (44) + ΣRljσ −iΓljσ ×G˜dσldj (50)
kσ 2m − σ Xl6=j (cid:0) (cid:1)
and the flat DOS and
(ε ε˜ +iΓ ) ˜σ =U ˜
dεkσ −1 m − ldσ llσ Gdldj lGdlσndlσ¯,djσ
ρFMσ = NσL2π (cid:18) dk (cid:19)k=kFσ = NσL2π~2kFσ, (45) + ΣRjjσ −iΓjjσ ×G˜dσjdj, (51)
(cid:0) (cid:1)
7
where ε˜ = ε + ΣR for l = j. In the equation and apply the equation-of-motion approach to
above, G˜jddlσσndlσ¯,djjdσσis a hjjiσgher-ord6er Green function ob- G˜c~kσd†lσ¯dlσ¯,djσ. In order to cancel the second term
tained from the time Fourier transform of with the last one in Eq. (55), we combine the approxi-
mations in Eqs. (57) and (58) simultaneously with the
i
Gdlσndlσ¯,djσ(t)=−~θ(t)ZF−M1 e−βEn property
n
X
×hn|hdlσ(t)ndlσ¯(t),d†jσ(0)i+|ni, (52) √1 σ¯Vl~kσφ~∗kσ¯ R~l = √1 σ¯Vl~∗kσφ~kσ¯ R~l , (59)
with ndlσ¯ =d†lσ¯dlσ¯ being the number operator of the lth X~k N (cid:16) (cid:17) X~k N (cid:16) (cid:17)
impurity with spin σ¯ (opposite to σ). Here whichisonly fulfilled ona metallic surfacesystem,while
for the quantum wire, it holds in the side-coupledgeom-
ΣR = 1 Vj∗~kσVl~kσφ~kσ R~j φ~∗kσ R~l (53) etry R~l = R~j =~0. Bearing this in mind, we can rewrite
ljσ ε (cid:16)ε (cid:17) (cid:16) (cid:17) Eq. (55) as follows
Nσ X~k − ~kσ
represents the real part of the noninteracting self-energy
(ε ε U +iη) ˜ =δ n
Σljσ and − ldσ− l Gdlσndlσ¯,djσ ljh dlσ¯i
1 1
ΣIljσ =−Γljσ =− σπ Vj∗~kσVl~kσφ~kσ R~j φ~∗kσ R~l + √ σVl~∗kσφ~kσ R~l ×G˜c~kσd†lσ¯dlσ¯,djσ.(60)
N X~k (cid:16) (cid:17) (cid:16) (cid:17) X~k N (cid:16) (cid:17)
δ ε ε (54)  
× − ~kσ Once again, employing the equation-of-motion approach
describes(cid:0)the corr(cid:1)esponding imaginary part, which plays for G˜c~kσd†lσ¯dlσ¯,djσ, we find
the role of a generalized Anderson parameter Γ . In
ljσ
orderto closethe systemofGreenfunctions inEqs. (50)
ε+ ε ˜ =
and (51), we first take the time derivative of Eq. (52) − ~kσ Gc~kσd†lσ¯dlσ¯,djσ
andthenperformthetime Fouriertransform. Withthat (cid:0) (cid:1) V 1 φ R~ ˜
we obtain l~kσ√ × ~∗kσ l Gdlndlσ¯,djσ
σ
N (cid:16) (cid:17)
1
+ V φ R~ ˜
ε+−εldσ−Ul G˜dlσndlσ¯,djσ =δljhndlσ¯i Xq~ l∗q~σ√Nσ¯ × q~σ¯(cid:16) l(cid:17)Gc~kσd†lσ¯cq~σ¯,djσ
(cid:0) 1 (cid:1) V 1 φ R~ ˜
+−X~k √Nσ¯Vl~kσφ~∗kσ¯(cid:16)R~l(cid:17)×G˜c~†kσ¯dlσ¯dlσ,djσ −Xq~ lq~σ√N1σ¯ × q∗~σ¯(cid:16) l(cid:17)Gcq†~σ¯dlσ¯c~kσ,djσ
  + V φ R~ ˜ . (61)
1 ˜j~kσ√ × ~∗kσ ˜j Gd˜jσndlσ¯,djσ
+ √ σVl~∗kσφ~kσ R~l ×G˜c~kσd†lσ¯dlσ¯,djσ X˜j6=l Nσ (cid:16) (cid:17)
X~k N (cid:16) (cid:17) HerewecontinuewiththeHubbardIscheme,proceed-
 
ing as in Eqs. (57) and (58) by making the following
1
+ √ σ¯Vl~∗kσφ~kσ¯ R~l ×G˜d†lσ¯c~kσ¯dlσ,djσ, approximations:
X~k N (cid:16) (cid:17)
  (55)
which also depends on new Green functions of the same G˜c~kσd†lσ¯cq~σ¯,djσ ≃ d†lσ¯cq~σ¯ G˜c~kσdjσ, (62)
D E
orderofG˜dlσndlσ¯,djσ andonthe averageoccupationnum- G˜cq†~σ¯dlσ¯c~kσ,djσ ≃ d†lσ¯cq~σ¯ G˜c~kσdjσ, (63)
ber hndlσ¯i, that is, calculated as G˜d˜jσndlσ¯,djσ ≃hDndlσ¯iG˜Edσ˜jdj (64)
hndlσ¯i=ˆ +∞dε −π1Im G˜dσ¯ldl f(ε). (56) and replacing Eq. (59) in Eq. (61) to show that
−∞ (cid:26) (cid:16) (cid:17)(cid:27)
Within the Hubbard I approximation we truncate the
GinrgeetontfhuencdteioconuspG˜licn~†kσg¯dslσ¯cdhleσm,djeσ and G˜d†lσ¯c~kσ¯dlσ,djσ accord- G˜c~kσd†lσ¯dlσ¯,djσ = Vl~kσ√ε+N1σ−φε~∗k~kσσ(cid:16)R~l(cid:17)G˜dlndlσ¯,djσ
(cid:0) V 1(cid:1)φ R~
G˜c~†kσ¯dlσ¯dlσ,djσ ≃Dc~†kσ¯dlσ¯EG˜dσldj (57) + P˜j6=l ˜jε~k+σ√−Nεσ~kσ~∗kσ(cid:16) ˜j(cid:17)
G˜d†lσ¯c~kσ¯dlσ,djσ ≃ c~†kσ¯dlσ¯ G˜dσldj, (58) ×hndlσ¯iG˜(cid:0)dσ˜jdj. (cid:1) (65)
D E
8
To close the original setup of Green functions in Eqs. IV. NUMERICAL RESULTS
(50) and (51), we substitute Eq. (65) in Eq. (60) and
obtain
A. Numerical Parameters
ε ε U +iΓ0 ˜ =δ n
− ldσ− l llσ Gdlσndlσ¯,djσ ljh dlσ¯i
Here we present the results obtained via the formu-
(cid:0) + n ΣR (cid:1)(ε) iΓ ˜σ , (66)
h dlσ¯i ˜jlσ − ˜jlσ ×Gd˜jdj lation developed in the previous section. The energy
X˜j6=l(cid:16) (cid:17) scale adopted is the Anderson parameter Γ. We em-
ploy the following set of model parameters: Γ =0.2 eV,
where
ε = ε = 10Γ and ε = ε = 4.5Γ.15,21 Such
1dσ 1d 2dσ 2d
− −
Γ0 =π 1 V 2δ ε ε , (67) values correspond to a Kondo temperature TK ≈ 50K
llσ l~kσ − ~kσ found in the system Co/Cu(111) with Coulomb interac-
σ
N X~k (cid:12) (cid:12) (cid:0) (cid:1) tion U = 2.9 eV.13,15,19 Thus the Hubbard I approxi-
(cid:12) (cid:12) mation is employed with T = Γ/10k = 231.1K just to
B
which allows us to determine all the necessary Green
avoid Kondo physics. Finally, in order to generate spin-
functions for the LDOS. Thus, to solve the system com-
polarizedquantumbeatsintheLDOS,wesubstituteEq.
posed by Eqs. (50), (51) and (66), we now assume the
(2) with P =0.1 in Eq. (37).
side-coupled geometry R~ = R~ = ~0 (Fig. 1). A ge-
l j
ometry with R~ = R~ = ~0 will be published elsewhere.
l j
6 6
By choosing the side-coupled configuration and assum-

ing constant symmetric couplings V = V = V, we
j~kσ l~kσ 1.1
verifyfromEqs. (54)and(67)thatΓ =Γ =Γ0 =
jjσ ljσ llσ metallic surface
Γ =ΓρFMσ depends on the standard Anderson param- 1.0
σ ρ0
eter Γ = πV2ρ0. Additionally, in the wide-band limit, 0.9 P = 0.1
Eq. (53) ensures ΣR = ΣR = 0. For the sake of sim-
plicity weconsidertjhjσe infinlijtσe Coulombcorrelationlimit OS 0.8 (a) kFR
(U1 =U2 →∞). Thus the direct Green function for the D 0.7 01
impurity j =1 reduces to the form L
Short-range limit 1.5
0.6 2
1 n 0.5
˜σ (ε)= −h d1σ¯i , (68)
Gd1d1 ε ǫ12dσ+i¯12σ 0.4
− △ -20 -15 -10 -5 0 5
where
/
ǫ =ε +Σ¯ (ε) (69)
12dσ 1dσ 12σ
1.1
representsarenormalizedenergyleveldressedbythereal (b)
1.0
part of the nondiagonalself-energy
0.9
Σ¯12σ(ε)=−(1(ε−hnεd1σ¯)i)(1−hnd2σ¯i) OS 0.8 Srlahimnogirtte- Intermediate-range limit Lonligm-riatnge
× (ε−ε−2dσ)22dσ+Γ2σΓ2σ (70) LD 0.7
0.6 / = -10
and
0.5
¯ =Γ (1 n )(1 n )Γ
△12σ σ− −h d1σ¯i −h d2σ¯i σ 0.4
Γ2 -2 0 2 4 6 8 10 12 14 16 18 20
σ (71)
× (ε−ε2dσ)2+Γ2σ kFR
Figure 2. (Color online) In both panels we use kBT = 0.1Γ.
is an effective hybridization function. The mixed Green (a) LDOS [Eq. (24)] of a metallic surface with P = 0.1 as a
function G˜dσ2d1(ε) becomes function ofε/Γ for differentvaluesof kFRin theshort-range
limit (see panel (b)). The Fanoprofile presents two antireso-
˜σ (ε)= iΓ 1−hnd2σ¯i ˜σ (ε). (72) nancesplacedatε=ε1d =−10Γandε=ε2d =−4.5Γ,which
Gd2d1 − σ (ε ε +iΓ ) Gd1d1 display an evanescent behavior for increasing distances. (b)
(cid:26) − 2dσ σ (cid:27) Keeping the energy at ε = −10Γ, Friedel oscillations appear
Notice that the other Green functions ˜σ and ˜σ in the LDOS.
Gd2d2 Gd2d1
canbe derivedbyswapping1 2inEqs. (68)and(72).
↔
9

2.5 quantum wire
metallic surface
1.000 P = 0.1 kFR
0
P = 0.1 2.0 (a) 0.5
S S 1
O 0.996 (a) kFR O 1.5

D 15 D 1.5
L Long-range limit 20 L
0.992 25
30 1.0
0.988
0.5
-20 -15 -10 -5 0 5 -20 -15 -10 -5 0 5
/ /

1.000 (b) 2.5 (b) / = -10
2.0
S 0.996 S
O O
1.5
D D
Long-range limit
L L
0.992 1.0
/ = -10
0.5
0.988
15 30 45 60 75 90 105 120 135 150 0 15 30 45 60 75 90 105 120 135 150
kFR kFR
Figure 3. (Color online) In both panels we use kBT = 0.1Γ. Figure 4. (Color online) (a) LDOS [Eq. (24)] at kBT =0.1Γ
(a) LDOS [Eq. (24)] of a metallic surface with P = 0.1 as a of a quantum wire with P =0.1 as a function of ε/Γ for dif-
function of ε/Γ for different values of kFR in the long-range ferent values of positions kFR. Fano-profiles appear around
limit (see panel (b)). The Fanoprofile presents two antireso- ε=ε1d =−10Γ and ε=ε2d =−4.5Γ. This pair of antireso-
nancesplacedatε=ε1d =−10Γandε=ε2d =−4.5Γ,which nances(blackline)canbetunedtoresonances(greenline)by
display an oscillatory behavior for increasing distances. (b) movinglaterallytheSTMtip. (b)Inoppositiontothemetal-
Damped spin-polarized quantum beats emerge in the LDOS licsurfacedevice,undampedspin-polarizedquantumbeatsin
as function of kFR with ε=ε1d =−10Γ. theLDOS at ε=ε1d =−10Γ manifest.
B. Metallic Surface tion. SuchadipintheLDOSisaresultofchargescreen-
ingaroundtheimpuritiesbyconductionelectrons,which
Definingk =k ,webegintheanalysisinthemetal- suppresses the LDOS of the host. Beyond the adatom
F F
lic surface app↓aratus by dividing our study into regions position, the LDOS is indeed dictated by Friedel oscilla-
we call short, intermediate and long-range limits. The tions,whichalsoleadtoastrongdecayinthelong-range
short-range limit presented in Fig. 2(a) reveals that the limit [see the Fig. 2(b)]. The evanescent feature of the
LDOS given by Eq. (24) as function of energy exhibits LDOS is a result of the interplay between the Friedel-
two Fano antiresonances. Each one corresponds to the like expression A2D and the Fano parameter q2D. These
jσ jσ
discrete levels of the adatom (j = 1) and the subsurface quantities are governed by Eqs. (28) and (43), where
impurity (j = 2). The main feature in this situation is theformerevolvesspatiallyaccordingtothezeroth-order
that the Fano profile conserves its line shape when the Besselfunction J0. SuchdampingintheLDOShasbeen
dimensionless parameter k R is changed. Additionally, already observed experimentally in a system composed
F
thisprofileissuppressedforincreasingdistances,tending by an Fe host and a Co adatom.5
to the DOS background of the host. In Fig. 3(a) we plot the Fano line shape in the long-
In Fig. 2(b) we look at how the LDOS evolves with range regime (k R > 15). The same dips at ε = 10Γ
F
−
k R exactly at the Fano antiresonance ε=ε = 10Γ. and ε = 4.5Γ are observed as in the short-range limit
F d1 − −
At the host site (k R=0), the LDOS presents a deple- [Fig. 2(a)]. However, a contrasting feature is found be-
F
10
tweenthesetwolimits. Whileintheshort-rangecasethe

dips become suppressed as k R increases, in the long-
F
range limit the dip oscillates with k R. This is a result
F
of the oscillatory profile observed in the LDOS for in- 0.104 (a) metallic surface
creasing k R [see Fig. 2(b)]. The oscillations of the dip
F P = 0.1 Long-range limit
can be more clearly visualized in Fig. 3(b), where we
0.102
show the LDOS at ε = ε = 10Γ. A peculiar beat-
d1 −
ing is observed in the LDOS due to the slightly different
Fermi wave numbers [see Eq. (37)]. PT0.100
0.098
C. Quantum Wire / = -10
0.096
Figure 4(a) shows the LDOS plotted against energy 20 40 60 80 100 120 140
fordifferentk Rvalues inthe short-rangelimit. Forthe
STMtipatkFR=0,theLDOSshowsthetwo-dipstruc- kFR
F

ture already observed in Fig. 2(a). In contrast, as k R
F
increases,the antiresonanceschangeto resonances,pass- 0.8
ing through intermediate profiles (asymmetric Fano line (b) quantum wire
0.6
shapes). We emphasize that this behavior in the LDOS
was recently observed in the experiment performed by 0.4
Prüser et al. with atoms of Fe and Co beneath the
0.2
Cu(100) surface.25
PT
We observe in Fig. 4(b) the evolution of the LDOS 0.0
withk R. Non-evanescentoscillationsoccur,modulated
F
-0.2
by an amplitude beating. This undamped behavior is
encoded by Eqs. (46) and (49), for Friedel-like oscil- -0.4
lations (A1D) and Fano interference (q1D), respectively.
jσ jσ
Thesequantitiesaresimpletrigonometricfunctionswith- -0.6
0 20 40 60 80 100 120 140
out damping. This feature is due to the absence of an
extradimensionforthescatteringofthe electronicwave. kFR
Onthe otherhand, in 2Dthis propagationis spreadin a Figure 5. (Color online) Transport polarization of the ferro-
plane leading to a spatial decay in the LDOS. Thus the magnetic hosts [Eq. (25)] at kBT = 0.1Γ and applied bias
amplitude ofthe undamped beats is muchlargerthan in φ=ε1d =−10Γ. (a) Damped spin-polarized quantum beats
themetallicsurfacedevice. Thismeansthatinsuchcase, appear in the polarization of the metallic surface device. (b)
the LDOS signalcan be more easily resolvedexperimen- Undamped type occur in the polarization of the quantum
tally. wire. In both situations, we have a spatially resolved spin-
filter, with a polarization that oscillates around P = 0.1. In
thequantumwire case, thisoscillation is more pronounced.
D. Transport Polarization
Another quantity we investigate is the spin- undamped and pronounced oscillations. The polariza-
polarization given by Eq. (25). As the differential tion in the latter case does not exceed P +0.62 or
T
≈
conductance of Eq. (23) is proportional to the LDOS fall below P 0.5. Therefore the polarized current
T
≈ −
[Eq. (24)], the ferromagnetic hosts filter electrons that through the junction formed by the STM tip and the
tunnels into (or out of) the STM tip. This filtering surface alternates from spins up (+0.62) to down ( 0.5)
−
is dominated by the majority spin component. Thus depending on the tip position. Additionally, along
devices without impurities behave as spin filters with a this probing direction, the polarization not only can
spatially uniform polarization that coincides with the invert the orientation of the majority spin component,
value given by Eq. (2). Here we adopt P = 0.1. Due but also becomes zero at some sites, where locally the
to the impurities in the side-coupled geometry and the unbalance of spins is totally suppressed. As a result, we
host dimensionality, this polarization is perturbed in have a tunneling current without polarization in specific
two different forms. In both the metallic surface andthe positions on the sample surface. On the other hand, as
quantum wire as we can see in Figs. 5(a) and 5(b) with wecanseeinFig. 5(a),theamplitudeofthebeatsinthe
applied bias φ = ε = 10Γ, the polarization oscillates metallicsurfacepolarizationisextremelysuppressedand
1d
−
around P = P = 0.1. Unlike the metallic surface does not change its signal (P >0). Thus the quantum
T T
system, where small deviations with damping occur, the wire operates as a spatially resolved spin filter, with a
“electron focusing” effect in the quantum wire leads to higher efficiency.