Fano kondo spin_filter_new_version

Fano-Kondo Spin Filter

A. C. Seridonio1,2 , F. M. Souza∗1,3 , J. Del Nero4,5 and I. A. Shelykh1,6
1 International Center for Condensed Matter Physics, Universidade de Bras´ ılia, 70904-970, Bras´
ılia, DF, Brazil
2 Instituto de F´ısica, Universidade Federal Fluminense, 24310-346, Niterí, RJ, Brazil
o
3 Instituto de F´ ısica, Universidade Federal de Uberlˆndia, 38400-902, Uberlˆndia, MG, Brazil
a a
4 Departamento de F´ ısica, Universidade Federal do Par´, 66075-110, Bel´m, PA, Brazil
a e
5 Instituto de F´ ısica, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil
6 Science Department, University of Iceland, Dunhaga 3, IS-107, Reykjavik, Iceland

Abstract
The spin-dependent scanning tunneling spectrum generated by a ferromagnetic tip on a metallic host with
a Kondo adatom is discussed. The combined effects of the Kondo screening, quantum interference, and tip
magnetism are studied. For short tip-adatom separations, the askew lineshape characteristic of quantum
interference in Kondo systems is split in two spin components. For the majority-spin component, the Fano
antiresonance is strongly enhanced. With the tip atop the adatom, a window of bias voltages is found
in which the tunneling current is nearly 100% spin polarized. In the vicinity of a Kondo adatom, the
ferromagnetic scanning tunneling microscope constitutes a powerful spin polarizer.

1. Introduction electrode magnetizations, and on the external mag-
netic field [14, 15, 16, 17, 18]. Current technology
The Kondo effect is one of the most fascinat- allows the construction of single QD devices and
ing manifestations of strong correlations in elec- routine observation of all such phenomena [19, 20].
tronic systems [1]. First observed as an enhance-
In an alternative nanoscale setup, a magnetic
ment of the resistance of dilute magnetic alloys
atom is adsorbed on a metallic host and probed by
below a characteristic temperature TK , the effect
a scanning tunneling microscope (STM) tip, either
found a more controlled environment in the early
normal or ferromagnetic. The quantum interference
90s, following the development of nanoscale minia-
between currents flowing from the tip to the host
turization techniques. Kondo effects in quantum
directly or by way of the adatom gives rise to a rich
dots (QDs) and QD arrays coupled to metallic or
variety of transport regimes. In particular, at tem-
semiconductor electrodes were reported [2]. A va-
peratures below TK , in analogy with the distortion
riety of mesoscopic phenomena associated with the
of a Lorentzian into a antiresonance in conventional
narrow resonance in the density of states of a QD
Fano systems [21], the interference distorts the con-
coupled to conduction-electron reservoirs were re-
tribution of the Kondo resonance to the adatom
ported [3, 4, 5]. In particular, the Kondo resonance
spectral function. Antiresonances with such charac-
was shown to an enhance the conductance at low
teristics have been widely reported in experiments
temperatures (T < TK ).
with nonmagnetic tips [22, 23, 24, 25, 26] and dis-
The mismatch between the electrode Fermi lev-
cussed theoretically [27, 28, 29, 30].
els resulting from the application of an external
bias splits the Kondo peak into two resonances The development of FM tips [31] makes adatom
[6, 7, 8, 9, 10, 11, 12, 13]. If the electrodes are systems potentially important actors in the rapidly
ferromagnetic, the large local exchange field splits developing field of spintronics [32]. Recently, fol-
further the resonances. The separation between the lowing a report by Patton et al of a theoretical
peaks now depends on the relative directions of the study of the conductance for a ferromagnetic tip in
the Kondo regime [33], the present authors inves-
tigated the dependence of the conductance on the
∗ Corresponding Author: fmsouza@infis.ufu.br tip-adatom separation in a full range of the Fano pa-
Preprint submitted to Physica E: Low Dimensional Systems and Nanostructures September 14, 2009

rameters, defined as the ratio between tip-adatom
and tip-host couplings [34].
Here, we analyze a spin-resolved conductance for
FM tip-adatom-host metal system in the weak-
coupling regime, in which tunneling amplitudes
can be treated as perturbations. To account for
the Kondo effect, we refer to the Doniach-Sunjic-
like expression for the adatom spectral function
[35, 36, 37, 38]. We find that very weakly mag-
netized tips can generate strongly spin-polarized
currents [39, 40]. For certain configurations, in an
appropriate bias interval, the current can be fully
spin-polarized, an effect due to the suppression of
one spin component by the spin splitting of the Figure 1: Scanning Tunneling Microscope (STM) device
Fano-Kondo resonance. The influence of the tip- with a ferromagnetic (FM) tip with an adsorbed magnetic
adatom separation on this spin-polarizing device is atom (Kondo adatom) on the surface of a normal metal (Host
surface). In the Host surface, the adatom site (c0 ) is hy-
also investigated. bridized with the metal conduction band (f0 ) through the
hopping v. The interference between the hopping elements
t0R and t leads to the Fano-Kondo profile in the conduc-
2. Theoretical Model tance.

We consider the following model Hamiltonian
We focus the Kondo regime, which favors the
H = HSIAM + Htip + HT , (1) formation of a localized magnetic moment at the
adatom, i. e., the conditions ε0 < εF , ε0 + U > εF
where (Htip ) represents the ferromagnetic tip, the
and Γ |ε0 |, ε0 + U . At temperatures T TK ,
HT , the host-tip coupling, and (HSIAM ) is the spin
the antiferromagnetic correlations introduced by
degenerate Anderson Hamiltonian representing the
the coupling to the host electrons screen the impu-
host and the magnetic impurity coupled to it [41]:
rity moment and add a narrow resonance, of half-
width kB TK , centered at the Fermi level, to the
HSIAM = εp a† apσ + ε0 c† c0σ adatom spectral density. The Kondo temperature
pσ 0σ
pσ σ
is approximately given by the expression [42]
+ U n0↑ n0↓ ΓU
† kB TK = exp[πε0 (ε0 + U )/2ΓU ]. (4)
+ v f0σ c0σ + H.C. , (2) 2
σ
We neglect the electron- electron interactions in
where apσ is a fermionic operator for a conduc- the FM tip and write
tion electron with momentum p and spin σ; c0σ
Htip = (εk + eV )b† bkσ , (5)
is a fermionic operator for an electron localized on kσ
adatom state; and ε0 and U are the energy of the kσ

adatom level and Coulomb repulsion between two where the bkσ ’s are he tip conduction states and V
electrons with opposite spins at the adatom site, is the tip bias voltage. For a ferromagnetic tip, the
respectively. spin-dependent density of states is
For simplicity, we consider a structureless con-
duction band with density of states ρ0 . The half ρσ = ρ0 (1 + σp) ,
tip (6)
bandwidth is denoted D. To describe the hybridiza-
where p is the magnetization. The tip-host-adatom
tion between the band and the impurity, we intro-
tunneling Hamiltonian reads
duced the shorthand  
1
f0σ = √ apσ , (3) HT = b† t ϕp (R)apσ + t0R c0σ 
N p kσ
kσ p

where N is the number of conduction states. + H.C., (7)
2

where t and t0R = t0 e−kF R are the tip-host and results from a competition between the two con-
tip-adatom hopping amplitudes, respectively. Fol- duction paths. As explained in detail in Ref. [34],
lowing [29], we let t0R decay exponentially with the Eq.(10) can be rewritten as
tip-adatom lateral distance R. The amplitude t de- 2
pends on the conduction wave function ϕp (R) at Fk F R 2
ρσ (eV ) = ρ0
0R 1− − (˜R )
q
the tip position R. ρ0
× πΓρσ (eV )
ad
3. The Conductance Formula FkF R
+ 2Γ qR
˜ GRet (eV )
adσ ,
ρ0
To derive an expression for the conductance, we
(14)
treat the tunneling Hamiltonian (7) to linear or-
der in perturbation theory [27, 28, 29, 30] under where
typical experimental conditions eV D. The low-
temperature conductance is then FkF R
qR = qR +
˜ q (15)
ρ0
ρσ
tip ρσ
0R
G(eV, T TK , R) = G0 , (8) is an eﬀective Fano parameter,
σ
ρ0 ρ0
1
where k eV −εk 1 eV + D
q= = ln (16)
πρ0 π eV − D
2 e2
G0 = (2πtρ0 ) (9) is the Fano parameter for the Anderson Hamilto-
h
nian (2) decoupled from the tip [27, 28, 29, 30] and
is the background conductance and
1
Fk F R
2
ρσ (eV )
˜ ρσ (eV )
ad = − GRet (eV )
adσ
ρσ (eV ) = ρ0 1− + 0R (10) π
0R 1
ρ0 ρ0 1 iΓK 2
=
is the density of states of the host-adatom-tip sys- πΓ (eV + σδ) + iΓK
tem. The function 1
= sin2 δeV
σ
(17)
πΓ
Fk F R = ϕk (R)δ(εk −εF ) = ρ0 J0 (kF R)(11)
is the adatom spectral density, here expressed in
k
three alternative ways: as the imaginary part of the
where J0 is the Bessel function of order zero, ac- interacting adatom Green’s function GRet (eV ),; by
adσ
counts for the dependence of the conductance on an analytical Doniach-Sunjic-like expression [35, 36,
the tip-adatom separation R. The last term within σ
37, 38]; or by the conduction-state phase shift δeV ,
curly brackets on the right-hand side of Eq. (10) is respectively.
The ferromagnetic exchange interaction between
FkF R
ρσ (eV ) =
˜0R | m| f0σ the tip and the magnetic adatom [14, 17] lifts the
m
ρ0 spin degeneracy of the adatom level ε0 and hence
+ πρ0 vqR c0σ |Ω |2 splits the Kondo resonance into two peaks sepa-
rated by an energy δ. A “poor man’s” scaling anal-
× δ(Em − EΩ − eV ), (12)
ysis [14, 33] yields an estimate of δ:
where |Ω (|m ) is the ground-state (an eigenstate)
δ = ε0↓ − ε0↑
of the Anderson Hamiltonian (2). To highlight the
↑ ↓
interference between the tunneling paths t0R and t γtip + γtip
built in Eq. (10), we have deﬁned the Fano param- = p ln(D/U )
2π
eter [29] γtip
= ln(D/U ) p exp (−2kF R) , (18)
π
−1 t0R
qR = (πρ0 v) = qR=0 e−kF R . (13) where
t
σ 2
With qR 1 (qR 1) the tip-adatom (tip-host) γtip = πρσ |t0R |
tip
current is dominant. With qR ≈ 1, the current = γtip (1 + σp) exp (−2kF R) . (19)
3

Since eV D, the right-hand side of Eq.(16) 1.0
p=1
is very small, which shows that it is safe to neglect 0.8
p=0.8
the last term on the right-hand side of Eq.(15). The 0.6
p=0.6

Spin polarization
spin-resolved conductance is therefore 0.4
p=0.4
0.2
σ
ρσ
tip ρσ (eV ) 0.0
p=0.2
G /Gmax = , (20) p=0
ρ0 ρ0 -0.2
-0.4
where
-0.6
2
Gmax = 1 + qR G0 (21) -0.8
-1.0
and -40 -30 -20 -10 0 10 20 30 40
Bias voltage
ρσ (eV ) = ρσ (eV )/ 1 + qR
0R
2
(22)
Figure 2: Spin polarization (℘) at R = 0 as a function of
To simplify Eq.(20), we now let the equivalence
the bias voltage eV in units of the Kondo half-width ΓK
tan δqR ≡ qR define a Fano-parameter phase shift for qR=0 = 1 and tip magnetizations ranging from p = 0
δqR and note that to p = 1. For nonzero p’s, ℘ reaches a unitary plateau in
a magnetic polarization dependent range centered at eV =
σ GRet (eV )
adσ
0. At negative voltages, the spin polarization dips to ℘ =
tan δeV = − . (23) −1. Following Refs. [28] and [33], we have set ε0 = −0.9eV,
GRet (eV )
adσ γtip = Γ = 0.2eV, U = 2.9eV, D = 5.5eV, TK = 50K, and
kF = 0.189˚−1 .
A
For R = 0 we then find that

Gσ /Gmax = (1 + σp) cos2 (δeV − δqR=0 ) , (24)
σ
The tip voltage V controls the spin polarization
which gives access to the spin polarization of the of the current, which can be tuned from −1 to
tunneling current: +1. The literature contains a number of proposed
spin filters [43, 44, 45, 46] and spin diodes [47], a
G↑ − G↓ few of which have been constructed in the labora-
℘= . (25)
G↑ + G↓ tory [48]; dynamic control of the current polariza-
tion by time-dependent bias voltages has also been
4. Results proposed [49, 50]. To the best of our knowledge,
however, ours is the first demonstration that the
To illustrate our discussion we have chosen the interplay between the Kondo effect and quantum
following set of model parameters: qR=0 = 1, ε0 = interference can control and amplify the spin polar-
−0.9eV, γtip = Γ = 0.2eV, U = 2.9eV, D = 5.5eV, ization of the current.
TK = 50K and kF = 0.189˚−1 [28, 33]. The Kondo
A Figure (3) shows the spin-resolved conductances
temperature was estimated using Eq.(4). G↑ and G↓ . For p = 0 we find G↑ = G↓ . The
Figure (2) shows the spin polarization ℘ as a interference between the two tunneling paths ac-
function of the bias voltage for differing tip mag- counts for the asymmetric Kondo resonance, a line
netization parameter p. As expected, ℘ vanishes shape frequently found with nonmagnetic metallic
for p = 0. For p = 0, two features of ℘ stand out: tips [22, 23, 24, 25, 26]. As p grows, the conduc-
(i) a plateau ℘ ≈ +1, with p-dependent width, cen- tances become increasingly distinct: G↑ (G↓ ) shifts
tered at eV = 0; and (ii) a narrow dip, which sinks to the left (right) and opens a window of completely
to -1, centered at a negative, p-dependent bias. As spin polarized conductance with G↓ ≈ 0, which cor-
p is increased, the plateau broadens, while the dip responds to the ℘ = 1 plateau in Fig. (2).
is shifted toward negative biases. For p = 1 (fully The narrow ℘ = −1 dip in Fig. (2) stems from
polarized tip), again as expected, the spin polar- the vanishing of G↑ at a negative voltage in each
ization curve saturates at ℘ = 1. Remarkably, the panel of Fig. (3). As the ferromagnetic polarization
100% polarization plateau persists even at weak tip p increases, while the amplitude of G↑ increases, G↓
magnetizations, e. g., p = 0.2. For sufficiently high is suppressed. This contrast is due to the density
voltages, the spin polarization becomes constant, of states at the tip: for σ = −1 the prefactor on
℘ = p. the right-hand side of Eq.(24) vanishes for p → 1,
4

G /G max
1,0 2

(b) up
(a) down

0,8 R=0
1 R=0
R=1A
0,6
R=2A

Spin Polarization
0,4 0 -40 -20 0 20 40

2

(c)
0,2

R=1A
0,0 1

V V V

-0,2

1A 2A
0
-40 -20 0 20 40

2
-0,4 (d)

-0,6 R=2A

1

-0,8

-1,0 0
-40 -20 0 20 40

-40 -20 0 20 40
Bias Voltage

Figure 3: Spin-resolved conductances (24) for tip atop the Bias voltage

adatom (R = 0), as functions of the bias voltage eV in units
of the Kondo half-width ΓK for qR=0 = 1. For a nonmag- Figure 4: (a) Spin polarization (℘) as a function of bias
netic tip (p = 0) the identical spin resolved conductances G↑ voltage eV in units of the Kondo half-width ΓK for fixed tip
and G↓ display the Fano-Kondo line shape. A ferromagnetic magnetization p = 0.4, qR=0 = 1 and differing tip-adatom
tip (p = 0) displaces the spin-resolved antiresonances to dif- lateral displacements R. Large separations R wash out the
ferent energies. A window of energies in which G↓ ≈ 0 while marked structure of the spin polarization visible at R = 0,
G↑ is maximum results. In this region, the spin polarization and force the polarization into coincidence with the ferro-
℘ is unitary. Model parameters as in Fig. (2). magnetic polarization p. Panels (b)-(d) show the evolution
of the spin resolved conductances, G↑ and G↓ , with the dis-
tance. As R increases, the energy splitting between the G↑
and G↓ antiresonances and the prominence of each antireso-
nance is reduced. A sketch of the experimental setup appears
in the inset. Model parameters as in Fig. (2).
which forces G↓ → 0.
5. Conclusions
Finally, Fig. (4) shows the spin polarization ℘
as a function of the tip-adatom separation R. For We studied theoretically the spin-resolved con-
large R , the minimum and the maximum ℘ are ductance for a ferromagnetic STM tip coupled to an
reduced, the current polarization ℘ is governed by adatom on a nonmagnetic metallic host. For each
the tip polarization p, and the polarization curve spin component, the spin-resolved conductance dis-
becomes flat. To explain the behavior at the po- plays a Fano antiresonance. The ferromagnetic po-
larization at smaller separations, the (b)-(d) on the larization nonetheless tends to shift the two antires-
right-side of the figure show the spin-resolved con- onances to different energies. A window of bias volt-
ductances as functions of the bias voltage at three ages arises in which G↓ ≈ 0, while G↑ is maximum.
different separations R behavior. The two strongly In that window, the spin polarization ℘ remains
asymmetric Fano-Kondo resonances in panel (b) close to unity. The application of an appropriately
(R = 0) become less pronounced as R increases; tuned bias voltage can therefore produce fully spin
moreover, the splitting between G↑ and G↓ is re- polarized currents. A ferromagnetic STM tip there-
duced. As R increases, therefore, the resulting po- fore constitutes a tunable spin polarizer.
larization loses its structure and approaches a con-
stant. We note that Ref. [22], which studied an
6. Acknowledgments
unpolarized tip, reported a similar R dependence
of the Fano-Kondo line shape: a Fano antireso- We thank L. N. Oliveira for revising the text of
nance for R ≈ 0, a less pronounced antiresonance the final version and M. Yoshida for fruitful discus-
for intermediate R and an approximately flat pro- sions. This work was supported by the Brazilian
file (background) for large R, a trend analogous to agencies IBEM, CAPES, FAPESPA and FAPERJ.
the evolution of the solid curves in panels (b)-(d)
and also to the evolution of the dashed curves. Due
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