Kondo effect in quantum dots coupled to ferromagnetic leads

0.24 MB·English

by  J. Martinek

Kondo effect in quantum dots coupled to ferromagnetic leads
J. Martinek,1,2,3 Y. Utsumi,4 H. Imamura,4 J. Barna´s,3,5 S. Maekawa,2 J. K¨onig,1 and G. Scho¨n1
1Institut fu¨r Theoretische Festk¨orperphysik, Universit¨at Karlsruhe, 76128 Karlsruhe, Germany
2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
3Institute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznan´, Poland
4 4Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
0 5Department of Physics, Adam Mickiewicz University, 61-614 Poznan´, Poland
0
(Dated: February 1, 2008)
2
WestudytheKondoeffectinaquantumdotwhichiscoupledtoferromagneticleadsandanalyse
n
its properties as a function of the spin polarization of the leads. Based on a scaling approach we
a
J predictthatforparallelalignmentofthemagnetizationsintheleadsthestrong-couplinglimitofthe
Kondoeffectisreachedatafinitevalueofthemagneticfield. Usinganequation-of-motiontechnique
9
westudynonlineartransportthroughthedot. Forparallelalignmentthezero-biasanomalymaybe
2
splitevenintheabsenceofanexternalmagneticfield. Forantiparallelspinalignmentandsymmetric

coupling, the peak is split only in the presence of a magnetic field, but shows a characteristic
]
l asymmetry in amplitudeand position.
l
a
h PACSnumbers: PACSnumbers: 75.20.Hr,72.15.Qm,72.25.-b,73.23.Hk
-
s
e
m The Kondo effect [1] in electron transport through a and in the QD, Vrk is the tunneling amplitude, Sz =
quantum dot (QD) with an odd number of electrons is (d†d d†d )/2, and the last term is the Zeemanenergy
. ↑ ↑− ↓ ↓
t experimentally well established [2, 3]. Screening of the ofthedot. (Strayfieldsfromtheleadsareneglected.) We
a
m dot spin due to the exchange coupling with lead elec- assume identical leads and symmetric coupling, VLk =
trons yields, at low temperatures, a Kondo resonance. V . The ferromagnetism of the leads is accounted for
- Rk
d The main goal of the present work is to investigate how by different densities of states (DOS) ν (ω) and ν (ω)
r↑ r↓
n ferromagneticleadsinfluencetheKondoeffect. Intheex- for up and down-spin electrons.
o
treme case of half-metallic leads, minority-spin electrons
c In the following we study the two cases of parallel (P)
[ are completely absent, i.e., the screening of the dot spin
is not possible, and no Kondo-correlatedstate can form. and antiparallel (AP) alignment of the leads’ magnetic
2 moments. For the AP configuration and zero magnetic
What happens, however,for the generic case of partially
v
fieldandbiasvoltage,the modelisequivalent(bycanon-
spin polarized leads? How does the spin-asymmetry af-
6
ical transformation[8]) to a QD coupled to a single lead
0 fect the Kondo effect? Is there still a strong coupling
0 limit, and how are transport properties modified? with DOS νL↑+νR↑ =νL↓+νR↓. Inthis case, the usual
0 Based on a poor man’s scaling analysis we first show Kondo resonance forms, which is the same as for non-
1 magnetic electrodes [1].
thatthe strong-couplinglimitcanstillbe reachedinthis
2
caseifanexternalmagneticfieldisapplied. Thisisfamil-
0 This changes for the P configuration. In this case,
/ iar from the Kondo effect in QDs with an even number
t thereisanoverallasymmetryforupanddownspins,say
a of electrons [4, 5, 6, 7], which occurs at finite magnetic
ν +ν >ν +ν . Tounderstandhowthisasymme-
m fields, although the physical mechanism is different in L↑ R↑ L↓ R↓
try affects the Kondo physics we apply the poor man’s
- the present case. In the second part of the paper we an-
d scaling technique [9], performed in two stages [10]. In
alyzewithinanequation-of-motion(EOM)approachthe
n the first stage, charge fluctuations dominate and lead to
nonlinear transport through the QD. We find that for
o arenormalizationoftheQD’slevels. Sincetherenormal-
c parallel alignment of the lead magnetizations the zero- ization for the spin-down level is stronger than for spin-
: bias anomaly is split. This splitting can be removed by
v up, a level splitting between the two spin orientations is
i appropriately tuning the strength of an external mag- generated. Thisisoneofthekeymechanismforalltheef-
X
neticfieldB. Intheantiparallelconfigurationofthelead
fects discussedbelow. Toreachthe strong-couplinglimit
r magnetizations no splitting occurs at zero field.
a it is, therefore, essential to apply an external magnetic
TheAndersonHamiltonianforaQDwithasinglelevel
field to compensate for the generated spin splitting. In
at energy ǫ coupled to ferromagnetic leads is
0 the second stage, the resulting model is mapped onto a
H = ε c† c +ǫ d†d +Ud†d d†d KondoHamiltonianandthedegreesoffreedominvolving
rkσ rkσ rkσ 0 σ σ ↑ ↑ ↓ ↓
Xrkσ Xσ spinfluctuationsareintegratedout. Forsimplicityweas-
+ (V d†c +V∗c† d )+gµ BS , (1) sume for the scaling analysis flat DOSs νrσ and neglect
rk σ rkσ rk kσ σ B z the k-dependence of the tunnel amplitudes V =V.
Xrkσ rk
where c and d are the Fermi operators for electrons First we reduce the cutoff D from D , which is the
rkσ σ 0
with wavevector k and spin σ in the leads, r = L,R, smaller value of the bandwidth and the onsite repulsion
2
U [10]. Chargefluctuations leadto the scalingequations Finally, we point out an interesting consequence of
the spin polarization in the leads. With nonmag-
dǫ Γ
σ = σ¯ (2) netic leads, the Kondo Hamiltonian couples the spin
dln(D /D) 2π
0 of the QD to the spin of the leads only, but not
where we defined Γ = 2π V 2 ν , and σ¯ is oppo- to its charge. To analyze the analogous situation
σ | | r rσ in our case, we introduce the (pseudo) spin ~σ =
site to σ. This yields the soluPtion ∆ǫ = ǫ↑ ǫ↓ =
(1/π)PΓln(D0/D)+∆ǫ0,whereP =(νr↑ νr↓)−/(νr↑+ (1/2) kk′rr′σσ′c†krσσσσ′ck′r′σ′/(2D√νσνσ′), where the
−ν measures the spin polarization inethe−leeads,eΓ = spin-dPependent normalization factor is crucial to ensure
r↓) e
(Γ +Γ )/2 and ∆ǫ = gµ B is the Zeeman splitting. theproperspincommutationrelations,andthe(pseudo)
↑ ↓ 0 B
Theempty-dotstate|0ihybridizeswithstateswherethe chargeen=e kk′rr′σc†krσck′r′σ/(2Dνσ). The lastterm
dot is singly occupied 1σ with either spin up or down, inEq.(3)can,Pthen,bewrittenas2D(ν J +ν J )σ S
| i e ↑ z↑ ↓ z↓ z z
while the singly-occupiedstate 1σ only hybridizes with plus D(ν J ν J )nS . The first term is analogous
the empty-dot state 0 (for U| iǫ - asymmetric An- to the Ko↑ndz↑o−mo↓dezl↓withznonmagneetic leads, while the
derson model). Beca|usie of the≫spi|n-|dependent DOS in secondeterm couples spin to charge. The latter does not
the leads the hybridization is spin-dependent, which is scale up and the associated additional renormalization
the physical origin of the generated ∆ǫ. ofthe Zeemansplitting, (1/π)PΓ,is negligible as com-
To describe Kondo physics (for ǫ < 0) we terminate pared to ∆ǫ in the limit−D ǫ.
e 0 ≫| |
[10] the scaling of Eq. (2) at D ǫ, and perform a The unitary limit for the P configuration can be
Schrieffer-Wolff transformation. U∼sein−g the renormalized achieved byetuning the magnetic field appropriately, as
parametersD andǫ we getthe eeffectiveeKondoHamilto- discussedabove. Inthiscase,themaximumconductance
nian throughtheQDisGP =e2/hperspin,i.e.,thesame
e e max,σ
HKondo = {J+S+c†rk↓cr′k′↑+J−S−c†rk↑cr′k′↓+ atusdfeorofntohnemKaognndeoticresleoandasn.ceTfohrisupyiaelnddsdtohwatntshpeinaamtpthlie-
rXr′kk′ Fermi level are different, since GP Γ (E )ρ (E ),
Sz(Jz↑c†rk↑cr′k′↑−Jz↓c†rk↓cr′k′↓)}, (3) andtherefore,ρ↑(EF)/ρ↓(EF)=(m1ax,σP∼)/(1σ+PF). FσortFhe
−
AP configuration, the maximal conductance is reduced,
plustheterm 2S D˜(ν J ν J )andapotentialscat-
teringterm. T−heiznitial↑vazl↑u−es↓forz↓thecouplingconstants GAmPax,σ =(1−P2)e2/h, and vanishes for P →1.
In the remainder of this article we analyze the DOS
are J = J = J = J = V 2/ǫ J . To reach
+ − z↑ z↓ | | | | ≡ 0 of the QD and address nonequilibrium transport. For
the strong-coupling limit we tune the external magnetic
a qualitative discussion we should employ the sim-
fieldB suchthatthetotaleffectiveZeeemansplittingvan-
plest technique which accounts for both the formation
ishes, ∆ǫ = 0 (the field B will also slightly modify the
of Kondo resonances and the influence of the spin-
DOS in the leads [6]). During the second stage of scal-
dependent renormalization of the dot level on spin fluc-
ing spinefluctuations will renormalize the three coupling
tuations. Theequation-of-motion(EOM)techniquewith
constants J = J J , J , and J differently. The
+ − ≡ ± z↑ z↓ the usual decoupling procedure [12, 13] for higher order
scaling equations are
Green functions (GF) satisfies the first requirement but
d(ν J ) not the second. We, therefore, extend this scheme by
± ±
= ν J (ν J +ν J ) (4)
± ± ↑ z↑ ↓ z↓ calculating the level splitting ∆ǫ self-consistently. For a
dln(D/D)
morequantitativeanalysisonecouldincludehigher-order
d(ν J )
σezσ = 2(ν J )2 (5) (than usually) Green’s functiones in the EOM approach
± ±
dln(D/D) or higher-order diagrams in the resonant-tunneling-
approximation [14], or use more advanced schemes such
ewqiuthateiνo±ns=we√oνb↑sνe↓r,veνσtha≡t (Pν±rJν±rσ)2[11(]ν.↑JTz↑o)(sνo↓lvJez↓t)h=es0e as real-time [17] or numerical RG [18] methods. These
− techniquesare,however,muchmorecomplex[15]andare
and ν J ν J = J P(ν +ν ) is constant as well.
↑ z↑ ↓ z↓ 0 ↑ ↓
− not pursued here.
I.e., there is only one independent scaling equation. All
Within the Keldysh formalism, the transport current
couplingconstantsreachthe stablestrong-couplingfixed
I = I through a QD for Γ (ω)=λ Γ (ω) is
point J± = Jz↑ = Jz↓ = at the Kondo energy scale, σ σ Rσ σ Lσ
∞
D k T . For the P configuration the Kondo temper- P
atu∼re iBn lKeading order, I = e dω ΓLσ(ω)ΓRσ(ω) [f (ω) f (ω)]ρ (ω) (7)
σ L R σ
¯hZ Γ (ω)+Γ (ω) −
Lσ Rσ
1 arctanh(P)
TK(P)≈Dexp(cid:26)−(ν↑+ν↓)J0 P (cid:27) , (6) whereρσ(ω)=−(1/π)ImGrσet(ω). Forstronginteraction
e (U ) the retarded Green’s function is
depends on the polarization P in the leads. It is a max- →∞
imum for nonmagnetic leads, P = 0, and vanishes for 1 n
Gret(ω)= −h σ¯i (8)
P 1. σ ω ǫ Σ (ω) Σ (ω)+i0+
→ − σ− 0σ − 1σ
3
whereΣ (ω)= V 2/(ω ε )istheself-energy For magnetic leads and parallel alignment, we find a
0σ k∈L,R| k| − kσ
for a noninteractPing QD, while splitting of the peak in the absence of a magnetic field
[Fig. 2(c)], which can be tuned away by an appropriate
V 2f (ε )
Σ (ω,∆ǫ)= | k| L/R kσ¯ (9) magnetic field [Fig. 2(d)]. In the AP configuration, the
1σ
k∈XL,Rω−σ∆ǫ−εkσ¯ +i¯h/2τσ¯ opposite happens, no splitting at B = 0 [Fig. 2(e)] but
e finite splitting at B > 0 [Fig. 2(f)] with an additional
appears for interacting QDes only. The average occupa- asymmetryinthepeakamplitudesasfunctionofthebias
tion of the QD with spin σ is obtained from nσ = voltage. This asymmetry is related to the asymmetry in
h i
−(i/2π) dωG<σ(ω). Extending the standard derivation theamplitudeofDOS[Fig.1(b)]. Allthesefindingsarein
[12],werReplacedonther.h.s.ofEq.(9)∆ǫ ∆ǫ,where goodagreementwithourscalinganalysis. InFig.2(g)we
→
ǫσ is found self-consistently from the relation showthe tunnel magnetoresistance(TMR) whichcanbe
e
muchlargerthanforconventionalTMRsystems. Finally,
eǫσ =ǫσ+Re[Σ0σ(ǫσ)+Σ1σ(ǫσ,∆ǫ)], (10) we find that the positions of the peaks in the AP con-
ewhichdescribestheerenormaleizeddeot-levelenergy,where figurationin the presence of a magnetic field are slightly
the real part of the denominator of Eq. (8) vanishes [1]. shifted as a function of the polarization P [Fig. 2(h)].
We emphasize that without this self-consistency relation This can be explained in the similar way as in Ref. [7]
theKondoresonanceswill,ingeneral,appearatdifferent byanadditionallevelsplittingδ∆ǫ=(1/4π)PΓ/ǫeV at
positions, which disagree with the conclusions from the finite bias voltages due to spin accumulation in the QD.
e e
scaling analysis [19]. The procedure simulates higher- We finally comment on the observability of our pro-
order contributions and the influence of the renormal- posal. Ferromagnetic systems usually have strong spin-
ization of the dot level on spin fluctuations. Follow- orbit coupling which could lead to spin-flip relaxation
ing Ref. [12] we introduce, in heuristic way, a lifetime and, thus, could destroy the Kondo effect. On the other
τ (µ ,µ ,ǫ ,ǫ ) which describes decoherence due to a hand, recent observation of Aharonov-Bohm oscillations
σ L R ↑ ↓
finite bias voltage V or level splitting ∆ǫ. It is obtained in a ferromagnetic ring [22] proved coherent transport,
in second-oerdeer perturbation theory and depends on the i.e., the formation of the Kondo cloud in ferromagnetic
electrochemical potentials in the leads,eµ , and the leads might be possible as well. How can one attach
L(R)
level positions. Again, we replace the bare levels by the ferromagnetic leads to a QD? A conceivable realization
renormalized ones. In the numerical results presented might be to put carbon nanotubes in contact to ferro-
below we use Lorentzian bands of width D =100Γ. magneticleads[23]. TheKondoeffecthasbeenobserved
Fornonmagneticleads,P =0andzeromagneticfield, alreadyfor nonmagnetic electrodes [5, 24]. Alternatively
B = 0, the proposed approximation is identical to the one might use magnetic tunnel junctions with magnetic
standard EOM scheme [12]. For finite magnetic field, impurities in the barrier,orspin-polarizedSTM [25, 26].
B = 0, the self-consistency condition yields a splitting In conclusion, we presented a qualitative study of the
6
of the Kondo resonances which is slightly smaller than Kondo effect in QDs coupled to ferromagnetic leads. In
2gµ B,inagreementwithbothexperimental[2]andthe- particular we found a splitting of the Kondo resonance
B
oretical findings [20, 21]. For B = 0 and P > 0 in the forparallelalignmentoftheleadsmagnetizations,evenin
parallel configuration, we obtain a value of the splitting the absenceofamagneticfield. Ourresultsarebasedon
∆ǫ comparable to the result from scaling, Eq. (2). a poor man’s scaling approach and an EOM technique.
In Fig. 1 we plot the DOS of the QD for spins in AP Further investigationsona morequantitative levelusing
aned P configurations with spin polarization P = 0.2 in more advanced techniques would be desirable.
theleads. IntheAPconfigurationthereisoneKondores- We thank L. Borda, J. von Delft, L. Glazman, B.
onance[Fig.1(a)]andtheDOSisthesameasforthecase Jones, Yu.V. Nazarov, A. Rosch, H. Schoeller, A. Tagli-
of nonmagnetic leads. For the P configuration, however, acozzo,M.VojtaandA.Zawadowskifordiscussion. Sup-
theKondoresonancesplits[Fig.1(c)],whichcanbecom- portbytheResearchProjectKBN5P03B09120andthe
pensated by an external magnetic field B [Fig. 1(d)]. In DFGthroughthe CFNandthe Emmy-Noetherprogram
thelattercase,theamplitudeoftheKondoresonancefor is acknowledged.
spin down significantly exceeds that for spin up (as dis-
cussed above). A finite bias voltage, µ µ =eV >0,
R L
−
againleadstoasplitting forboththe APandthe Pcon-
figurations, [Fig. 1(b) and (e)]. In the AP configuration,
the amplitude of the upper and the lower Kondo peak [1] A. C. Hewson, The Kondo Problem to Heavy Fermions,
Cambridge Univ.Press (1993).
appear asymmetric [Fig. 1(b)].
[2] D. Goldhaber-Gordon et al., Nature (London) 391, 156
In Fig. 2 we show the differential conductance as a
(1998).
functionofthetransportvoltage. Fornonmagneticleads, [3] S.M.Cronenwettetal.,Science281,540(1998);F.Sim-
there is a pronounced zero-bias maximum [Fig. 2(a)], mel et al.,Phys.Rev.Lett. 83, 804 (1999); J. Schmidet
whichsplitsinthepresenceofamagneticfield[Fig.2(b)]. al.,Phys.Rev.Lett.84,5824(2000);W.G.vanderWiel
4
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FIG. 2: Total differential conductance (solid lines) as well as
FIG. 1: Spin dependent DOS for spin up (solid line) and the contributions for spin up (dashed) and the spin down
spin down (dashed), calculated for P and AP alignment (as (dotted-dashed) vs. the applied bias voltage V at zero
indicated), for a spin polarization of the leads P =0.2. The magnetic field B = 0 (a,c,e) and at finite magnetic field
parts (d,e) include the effect of an applied magnetic field B (b,d,f,h) for normal (a,b) and ferromagnetic leads with par-
and(b,e)ofanappliedbiasvoltageV. Theotherparameters allel (c,d) and antiparallel (e,f,h) alignment of the lead mag-
are: T/Γ=0.005 and ǫ/Γ=−2. netizations. Part (g) shows the tunnel magnetoresistance,
TMR=(GP−GAP)/GAP,for thecases (c) and(e). (h) The
conductanceGAP/(1−P2)forseveralvaluesofP asindicated.
Otherparameters are as in Fig. 1
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