Spin-polarized transport through a single-level quantum dot in the Kondo regime

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by  R. Swirkowicz

Spin-polarized transport through a single-level quantum dot in the Kondo regime
R. S´wirkowicz1, M. Wilczyn´ski1, and J. Barna´s2∗
1Faculty of Physics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland
2Department of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Poznan´, Poland;
and Institute of Molecular Physics, Polish Academy of Sciences,
5 ul. Smoluchowskiego 17, 60-179 Poznan´, Poland
0 (Dated: February 2, 2008)
0
2 Nonequilibriumelectronictransportthroughaquantumdotcoupledtoferromagneticleads(elec-
trodes) is studied theoretically by the nonequilibrium Green function technique. The system is
n
described by the Anderson model with arbitrary correlation parameter U. Exchange interaction
a
between thedot and ferromagnetic electrodes is taken into account via an effectivemolecular field.
J
The following situations are analyzed numerically: (i) the dot is symmetrically coupled to two fer-
5
romagnetic leads, (ii) one of the two ferromagnetic leads is half-metallic with almost total spin
2
polarization of electron states at theFermi level, and (iii) oneof thetwo electrodes is nonmagnetic

whereas the other one is ferromagnetic. Generally, the Kondo peak in the density of states (DOS)
]
l becomesspin-split whenthetotalexchangefieldacting onthedot isnonzero. Thespin-splittingof
l
a the Kondo peak in DOS leads to splitting and suppression of the corresponding zero bias anomaly
h in the differential conductance.
-
s PACSnumbers: 75.20.Hr,72.15.Qm,72.25.-b,73.23.Hk
e
m
. I. INTRODUCTION of the Kondo anomaly in electrical conductance – simi-
t
a larly as an external magnetic field suppresses the effect
m
innonmagneticsystems. Suchasuppressionwasstudied
- The Kondo phenomenon in electronic transport recently by the Green function technique in the limit of
d throughartificialquantumdots(QDs)orsinglemolecules infinite correlation parameter U [19], and was also con-
n attached to nonmagnetic leads was predicted theoret- firmed by numerical renormalization group calculations
o
ically more than a decade ago [1]. Owing to recent [23, 24]. However, only QDs symmetrically coupled to
c
[ progress in nanotechnology, the phenomenon has been two magnetic leads were studied up to now. The Kondo
also observed experimentally [2, 3]. Several theoretical anomaly survives then in the antiparallel magnetic con-
1 techniques have been developed to describe this effect figuration and is significantly suppressed in the parallel
v
5 [4, 5, 6, 7, 8, 9, 10, 11]. The description is usually one. Recent experimental observations on C60 molecules
0 simpler in the linear response regime, where equilibrium attached to ferromagnetic (Ni) electrodes support these
6 methods can be applied, but it becomes more complex general theoretical predictions [25].
1 when the system is driven out of equilibrium by an ex- Some features of the non-equilibrium Kondo phe-
0 ternal bias voltage [7, 11, 12, 13, 14, 15]. One of the
nomenoninQDscoupledtoferromagneticleadshavenot
5 methodsusedtodescribenon-equilibriumKondoeffectis
been addressedyet. Therefore,in this paper we consider
0
thenon-equilibriumGreenfunctiontechnique[5,16,17].
amoregeneralsituation. Firstofall,weconsiderthecase
/
at To calculate density of states (DOS) and electric cur- when the two ferromagnetic electrodes are generally dif-
rentonethenneedstheretarded/advancedaswellasthe
m ferent. In other words, the dot is (spin-)asymmetrically
lesser (correlation) Green functions. These can be de-
coupled to the ferromagnetic leads. This leads to qual-
-
rived within some approximation schemes.
d itatively new results. Second, we consider the case of
n It has been shown recently that the Kondo effect can arbitraryU insteadofthe limiting situationofinfiniteU
o
also occur when replacing nonmagnetic leads by ferro- studiedin[19]. Third,weintroduceaneffectiveexchange
c
: magnetic ones [18, 19, 20, 21, 22], but ferromagnetism field to describe the dot level renormalization.
v of the electrodes generally suppresses the effect – either We analyze in detail three different situations. In the
i
X partially or totally [19, 22]. However, in some peculiar first case the dot is coupled to two ferromagnetic leads,
situations the effect remains almost unchanged. Sup- and the coupling is fully symmetric in the parallel mag-
r
a pression of the Kondo anomaly is a consequence of an neticconfiguration. WeshowthattheequilibriumKondo
effective exchange field due to coupling between the dot peak in DOS is then spin-split in the parallel configura-
and ferromagnetic electrodes. The exchange field gives tion,whereasnosplittingappearsintheantiparallelone.
rise to spin-splitting of the equilibrium Kondo peak in The splitting, however, is significantly reduced for small
DOS,andthe two componentsofthe splitted peak move values of the correlation parameter U. The correspond-
away from the Fermi level, which leads to suppression ing zero-bias anomaly in conductance becomes split in
the parallel configuration as well [19]. The second situ-
ation studied in this paper is the one with asymmetric
couplingtotwoferromagneticleads. Asaparticularcase
∗Electronicaddress: [email protected] we consider the situation when one of the ferromagnetic
2
electrodes is half metallic, with almost total spin polar- whereǫ denotesenergyofthe dotlevel(spin-dependent
σ
ization of electron states at the Fermi level. Such struc- ina generalcase),U denotes the electroncorrelationpa-
tures have been shown recently to have transport char- rameter, whereas d+ and d are the creation and anni-
σ σ
acteristics with typical diode-like behavior [26, 27]. Fi- hilation operatorsfor electrons on the dot. The levelen-
nally,wealsoanalyzethecasewhenoneoftheelectrodes ergy ǫ includes the electrostatic energy due to applied
σ
is nonmagnetic whereas the secondone is ferromagnetic, voltage, ǫ = ǫ +eUd, where Ud is the electrostatic
σ 0σ e e
andshowthatoneferromagneticelectrodeissufficientto potential of the dot, and ǫ is the level energy at zero
0σ
generate spin-splitting of the Kondo anomaly. bias.
The paper is organized as follows. The model and The electrostatic potential Ud of the dot will be de-
e
method are briefly described inSections 2 and 3,respec- terminedfully self-consistentlyfromthe followingcapac-
tively. Numericalresultsforthethreedifferentsituations itance model [28, 29]:
mentioned above are presented and discussed in Section
4. SummaryandgeneralconclusionsaregiveninSection
e n − n =C (Ud−UL)+C (Ud−UR),
5. σ 0σ L e e R e e
!
σ σ
X X
(3)
wheren andn arethedotoccupationnumbershd+d i
II. MODEL σ 0σ σ σ
calculatedforagivenbiasandforzerobias,respectively,
whereas C and C denote the capacitances of the left
L R
Weconsiderasingle-levelQDcoupledtoferromagnetic
and right tunnel junctions. Self-consistent determina-
metallic leads (electrodes) by tunnelling barriers. We
tion of the dot electrostatic potential makes the descrip-
restrict our considerations to collinear (parallel and an-
tion gauge invariant. This is particularly important for
tiparallel) magnetic configurations and assume that the
strongly asymmetric systems.
axis z (spin quantization axis) points in the direction of
The last term, H , in Eq.(1) describes tunnelling pro-
the net spin of the left electrode (opposite to the cor- T
cesses between the dot and electrodes and is of the form
responding magnetic moment). Antiparallel alignment
is obtained by reversing magnetic moment of the right
electrode. ThewholesystemisthendescribedbyHamil- H = V∗ c+ d +h.c., (4)
tonian of the general form T kβσ kβσ σ
kβσ
X
H =H +H +H +H . (1)
L R D T where V are the components of the tunnelling ma-
kβσ
trix, and h.c. stands for the Hermitian conjugate term.
The terms H describe here the left (β = L) and right
β
Hamiltonian(4)includesonlyspin-conservingtunnelling
(β = R) electrodes in the non-interacting quasi-particle
approximation, H = ε c+ c , where ε is processes.
β kσ kβσ kβσ kβσ kβσ
thesingle-electronenergyintheelectrodeβ forthewave-
numberkandspinσ(σP=↑,↓),whereasc+ andc are
kβσ kβσ III. THEORETICAL FORMULATION
the corresponding creation and annihilation operators.
The single-particleenergyε includes the electrostatic
kβσ
energy due to applied voltage, ε = ε0 + eUβ = Electriccurrentflowingfromtheβ-thleadtothequan-
kβσ kβσ e
ε0 +µ , whereε0 is the correspondingenergyinthe tum dot in a nonequilibrium situation is determined by
uknβbσiasedβsystem, Ukββσis the electrostatic potential of the theretarded(advanced)Grσ(a)andcorrelation(lesser)G<σ
e Green functions of the dot (calculated in the presence of
β-th electrode, e stands for the electron charge (e < 0),
coupling to the electrodes), and is given by the formula
and µ is the chemical potential of the β-th electrode
β
[30]
(the energyismeasuredfromthe Fermilevelofunbiased
system). Electron spin projection on the global quan-
ie dE
tization axis is denoted as ↑ for s = 1/2 and ↓ and Iβ = Γβ(E){G<(E)+f (E)[Gr(E)−Ga(E)]},
z σ ~ 2π σ σ β σ σ
for sz = −1/2. On the other hand, spin projection on Z (5)
the localquantizationaxis(localspinpolarizationinthe
where f (E) is the Fermi distribution function for the
ferromagnetic material) will be denoted as + for spin- β
β-th electrode. The retarded (advanced) Green func-
majority and − for spin-minority electrons, respectively.
tions can be calculated from the corresponding equation
When local and global quantization axes coincide, then
ofmotion. Thekeydifficultyiswithcalculatingthelesser
↑isequivalentto+and↓isequivalentto−. (Note,that
Green function G<(E).
thelocalquantizationaxisintheferromagnetisopposite σ
In a recent paper [31] we applied the equation of mo-
to the local magnetization.)
tion method to derive both Gr(E) and G<(E) Green
ThetermH inEq.(1)describesthequantumdotand σ σ
D functions within the same approximation scheme [32].
takes the form
However, the approximations for the lesser Green func-
HD = ǫσd+σdσ+Ud+↑d↑d+↓d↓, (2) tion G<σ(E) do not conserve charge current in asymmet-
rical systems. Therefore, in the following we assume
σ
X
3
constant (independent of energy) coupling parameters,
Γβ(E) = 2π V V∗ δ(E−ε ) = Γβ. As pointed
σ k kβσ kβσ kβσ σ dε Γβ Γβ
out in Refs [33, 34], it is then sufficient to determine = σ −i σ (12)
(dE/2π)G<P(E), while knowledge of the exact form of 2πE−ε 2
σ β=LRZ
G<(E) is not necessary. Current conservation condition X
σ
aRllows then to express the above integral by an integral and
including retarded and advanced Green functions only,
dε 1
which in turn allows to rewrite the current formula in Σr (E)= A Γβ −
iσ 2π i −σ ε−E−ǫ +ǫ −i~/τ
the commonly used form, β=LRZ (cid:20) −σ σ −σ
X
ie dE ΓLΓR
I = σ σ [Gr(E)−Ga(E)][f (E)−f (E)].
σ ~ 2π ΓL+ΓR σ σ L R 1
Z σ σ (6) +ε+E−ǫ−σ−ǫσ−U −i~/τ−σ(cid:21), (13)
Similarly, the occupation numbers, n = hd+d i, are
σ σ σ
(for i=1,3), where A = f(ε), A = 1, and τ is the
then given by the formula 1 3 −σ
relaxationtimeoftheintermediatestates[5]. Thisrelax-
dE ationtime is spin dependent and in the low-temperature
n =−i G<(E)
σ 2π σ limit is given by the formula [5]
Z
=i dEΓLσfL(E)+ΓRσfR(E)[Gr(E)−Ga(E)]. (7) τ1σ = 2π1~ βX,β′ Xσ′ ΓβσΓβσ′′Θ(µβ′ −µβ +ǫσ−ǫσ′)
2π ΓL+ΓR σ σ
Z σ σ
In the following the parametersΓβ will be used to pa- ×µβ′ −µβ +ǫσ−ǫσ′ , (14)
σ
rameterize strength of the coupling between the dot and (µβ −ǫσ)(µβ′ −ǫσ′)
leads. It is convenient to introduce the spin polariza-
where Θ(x)=0 for x<0 and Θ(x)=1 otherwise.
tion factors p defined as p = (Γβ −Γβ)/(Γβ +Γβ),
β β + − + − InthelimitofinfiniteU theGreenfunction(8)reduces
where Γβ+ and Γβ− are the coupling parameters for spin- to the well known form,
majority and spin-minority electrons in the lead β, re-
spectively. Accordingly, one may write Γβ+ =(1+pβ)Γβ Gr(E)= 1−n−σ , (15)
and Γβ− =(1−pβ)Γβ, with Γβ =(Γβ++Γβ−)/2. σ E−ǫσ−Σr0σ(E)−Σr1σ(E)
The retarded (advanced) Green function Gσr(a) of the where Σr (E) is given by Eq.(12) and Σr (E) by
0σ 1σ
dotcanbecalculatedonlyapproximately,forinstanceby
the equation of motion method. In the approximations Σr (E)= Σβr(E)
1σ 1σ
introduced by Meir et al [5] one finds
β=L,R
X
E−ǫ −U(1−n )
Gr(E)= σ −σ ,
σ [E−ǫσ−Σrσ(E)](E−ǫσ−U)−Un−σΣrσ(E) ≡ dε fβ(ε)Γβ−σ , (16)
(8) 2π−ε+E+ǫ −ǫ +i~/τ
where Σrσ is the corresponding self energy, β=XL,RZ −σ σ −σ
(E−ǫ )n Σr (E)−L Σr (E) and τσ defined by Eq.(14).
Σr(E)=Σr (E)+U σ −σ 03σ 0σ 01σ , The above derived Green functions are sufficient to
σ 0σ L (E)[L (E)−Σr (E)]
0σ 0σ 03σ describe qualitatively basic features of the Kondo phe-
(9)
nomenon in QDs attached to nonmagnetic leads [5].
with L = E − ǫ − U(1 − n ), and Σr (E) and
0σ σ −σ 01σ However,they are not sufficient to describe properly the
Σr (E) defined as
03σ Kondo phenomenon when the quantum dot is attached
Σr (E)=n Σr (E)+Σr (E), (10) to ferromagnetic leads. The key feature of the system,
01σ −σ 0σ 1σ which is not sufficiently taken into account is the split-
ting of the dot level due to spin dependent tunneling
processes [19]. The most natural way would rely on an
Σr (E)=Σr (E)+Σr (E). (11) extensionoftheGreenfunctioncalculationsbygoingbe-
03σ 0σ 3σ
yond the approximations used to derive Eq.(8). This,
The self energies Σr (E), Σr (E), and Σr (E) are de- however,leadstocumbersomeexpressions. Toavoidthis,
0σ 1σ 3σ
fined as onemaytreattheproblemapproximatelybyintroducing
’by hand’ the level splitting to the formalism described
1
Σr (E)= Σβr(E)= |V |2 above. One way to achieve such an objective was pro-
0σ 0σ kβσ E−ε +i0+
kβσ posed in Ref.[19] for the limit of infinite U, where the
β=L,R β=L,R k
X X X
4

bare dot level ǫ in Σr (E) and Σr (E) was replaced by 0.7
σ 1σ 0σ (a) p=0
therenormalizedenergyǫσ calculatedinaself-consistent 0.6 P p=0.1
way. Such a renormalization works well in the limit of h]0.5 p=0.2
itnhfiengiteeneUra.l Ictasies ohfowarebveitreranroytUcl.eaTrhheorwefotroe,exwteenddeciitdetdo 2G [e/diff0.4 p=0.4
0.3
to include the level splitting via an effective exchange
field. The exchange-induced spin-splitting of the level is 0.7
(b)
particularly important for the self-energies Σr (E) and 0.6
1σ AP
Σr (E), so we replace the bare energy levels in the self- h]0.5
men3aeσlrigzeiedsΣler1vσe(lsEǫ)σan=dǫΣσr3σ±(Egµ)BbyBetxh/e2c,owrrietshpothnedienxgcrheannogre- 2G [e/diff00..34
field calculated from the formula 0.2
0.0
e 1 dε
-0.1
B = Re f (ε)
ex β
gµB 2π -0.2 (c)
β=XR,L Z MR-0.3 p=0.1
T-0.4 p=0.2
p=0.4
-0.5
1 1
× Γβ↑ ε−ǫ −i~/τ − ε−ǫ −U −i~/τ -0.10 -0.05 0.00 0.05 0.10
(cid:20) (cid:18) ↑ ↑ ↑ ↑(cid:19) eV / D
FIG. 1: Bias dependence of the differential conductance in
1 1
−Γβ − . (17) the parallel (a) and antiparallel (b) configurations, and the
↓(cid:18)ε−ǫ↓−i~/τ↓ ε−ǫ↓−U −i~/τ↓(cid:19)(cid:21) corresponding TMR (c) for indicated four values of the lead
polarization pL=pR =p(p=0 corresponds tononmagnetic
When the bare dot level and spin relaxation time are leadssothecorrespondingTMRvanishesandisnotshownin
independent of the spin orientation, ǫ = ǫ = ǫ and part(c)). Theparametersassumedfornumericalcalculations
↑ ↓
τ↑ =τ↓ =τ, the formula (14) acquires the form [35] are: kT/D =0.001, ΓL/D =ΓR/D =0.1, ǫ0↑/D =ǫ0↓/D =
ǫ0/D =−0.35, U/D =500, and (e2/CL)/D =(e2/CR)/D =
1 dε 0.33.
B = Re f (ε)
ex β
gµ 2π
B
β=R,L Z
X
is included only via the spin asymmetry of the coupling
1 1 parametersΓL andΓR. Apartfromthis,inallnumerical
× Γβ −Γβ − . σ σ
↑ ↓ ε−ǫ−i~/τ ε−ǫ−U −i~/τ calculations we assumed kT/D = 0.001 and ǫ0↑/D =
(cid:16) (cid:17)(cid:18) (cid:19)(18) ǫ0↓/D=ǫ0/D =−0.35.
To some extent such an approach is similar to that Forpositive(negative)biasweassumetheelectrostatic
usedinRef.[19],andbothapproachesgivesimilarresults potential of the left (right) electrode equal to zero. In
in the limit of large U. This follows from the fact that other words, electrochemical potential of the drain elec-
the expression(17) for exchange field is basically the ex- trode is assumedto be zero while of the source electrode
pressionfortheselfenergyΣr (seeEq.(13)fori=1and isshiftedupby|eV|. Inthefollowingthebiasisdescribed
1σ
E = ǫ ). However, the approach based on the exchange by the corresponding electrostatic energy eV. Note that
σ
fieldallowstohandleeasilyalsothegeneralcaseoffinite positive eV correspondsto negative bias due to negative
U. electron charge (e<0).
IV. NUMERICAL RESULTS A. QD coupled to two similar ferromagnetic leads
Now, we apply the above described formalism to the Consider first the situation when both electrodes are
Kondo problem in a QD coupled to ferromagnetic leads. made of the same ferromagnetic metal, and the coupling
Inthe numericalcalculationsdescribedbelow the energy of the dot to both leads is symmetrical (in the parallel
is measured in the units of D, where D = D¯/50 and D¯ configuration). For numerical calculations we assumed
is the electron band width. For simplicity, the electron ΓL/D = ΓR/D = 0.12 for spin-majority electrons and
+ +
band in the leads is assumed to be independent of the ΓL/D = ΓR/D = 0.08 for spin-minority ones, which
− −
spinorientationandextendsfrom−25DbelowtheFermi corresponds to the spin polarization factor p = p =
L R
level (bottom band edge) up to 25D above the Fermi p=0.2, and ΓL/D =ΓR/D=0.1.
level (top band edge). The energy integrals will be cut It was shown in Ref.[19] that ferromagnetism of the
off at E = ±25D, i.e., will be limited to the electron electrodes leads to spin splitting of the Kondo peak in
band. Thus, the influence of ferromagnetic electrodes the density ofstates (DOS) in the parallelconfiguration,
5

whereas no splitting occurs for the antiparallel orienta-
0.4 (a) P
tion. Such a behavior of DOS has a significant influence
onthe transportproperties. Firstofall,the Kondopeak h]0.3
idnuDctOanSceleaGddsiftfoz=er∂oIb/i∂aVsa.nTomhiaslyaninomthaelydiiffsepraenrttiicaullcaornly- 2G [e/diff0.2
interesting in the parallel configuration, where the spin 0.1

splitting of the Kondo peak in DOS leads to splitting of
0.6 (b) AP
thedifferentialconductance,asshowninFig.1(a)forfour
0.5
cdpeiffanneterdreenfdotrvalatalruzgeeersoUobf.itaFhsoe.reWpleh=cetrn0odtthheeesrppeionilsaproionzllaaytriiaoznastifinaogcnlteofarpcebtaoekr- 2G [e/h]diff000...234
comes nonzero, the peak becomes split into two compo- 0.1

nents locatedsymmetricallyonbothsides ofthe original
peak,withthecorrespondingintensitiessignificantlysup- 0.0
pressed. The splitting of the Kondo anomaly increases
-0.2 (c)
with increasing p. Moreover, height of the two compo- R U/D=1
M U/D=2
nents ofthe Kondoanomalydecreaseswith increasingp. T-0.4 U/D=5
Ontheotherhand,intheantiparallelconfigurationthere -0.6 UU//DD==15500
is no splitting of the Kondopeak in the density of states -0.08 -0.04 0.00 0.04 0.08
andconsequentlyalsonosplittingoftheKondoanomaly eV / D
inthe differentialconductance(seeFig.1(b)). Forallpo-
larization values, the anomaly is similar to that in the FIG. 2: Bias dependence of differential conductance in the
case of QDs coupled to nonmagnetic leads. However, parallel(a)andantiparallel(b)configurations,andthecorre-
spondingTMR (c) calculated for indicated values of thecor-
intensity of the Kondo anomaly decreases with increas-
relation parameter U and for p=0.2. The other parameters
ing polarization. Difference between conductance in the
are as in Fig.1.
antiparallel and parallel configurations gives rise to the
TMR effect which may be described quantitatively by
the ratio (IP−IAP)/IAP, where IP and IAP denote the
current flowing through the system in the parallel and
B. QD coupled to one ferromagnetic and one
antiparallelconfigurationsatthe samebias,respectively.
half-metallic leads
The associated TMR effect is displayed in Fig.1(c). One
finds negativevaluesofthe TMRratio,whichis aconse-
quenceofthespin-splitting ofthe the Kondopeakinthe Assume now that one of the electrodes (say the left
parallel configuration and absence of such a splitting for one) is made of a 3-d ferromagnetic metal, the second
antiparallel alignment. It is worth to note, that in the (right) one is half-metallic, and the total coupling to the
absence of the Kondo anomalythe TMR effect wouldbe latter electrode is much smaller than to the former one.
positive. This is reflected in the spin asymmetry of the bare cou-
pling constants, for which we assume ΓL/D = 0.28 and
+
ΓL/D = 0.12 for the left electrode, and ΓR/D = 0.04
− +
and ΓR/D = 0.0002 for the right one. These parame-
−
ters correspond to p = 0.4, p = 0.99, ΓL/D = 0.2,
L R
andΓR/D≈0.02. Thus,the spinasymmetryofthecou-
plingtotherightelectrodeismuchlargerthantotheleft
TheKondoanomalyintransportcharacteristicsshown one. In Fig.3 we show DOS in the parallel (left column)
in Fig.1 was calculated for the limit of large U. In Fig.2 and antiparallel (right column) magnetic configurations,
weshowsimilarcharacteristicsasinFig.1,butfordiffer- calculated for vanishing as well as for positive and nega-
entvaluesofthe correlationparameterU andaconstant tivebiasvoltages. Considernowthemainfeaturesofthe
value of the polarization factor p. The splitting of the spectrainmoredetails,andletusbeginwiththeparallel
Kondo anomaly in the parallel configuration and inten- configuration (left column in Fig.3).
sity of the peaks (Fig.2(a)) decrease with decreasing U. AtV =0theKondopeakinDOSisspin-split,andthe
In the antiparallel configuration there is no splitting of intensity of spin-down peak is relatively large, whereas
the Kondoanomaly,but intensityofthe Kondopeak de- thatofthespin-uppeakismuchsmaller. Theasymmetry
creaseswith decreasingU. The associatedTMReffect is in peak intensities is a consequence of the spin asymme-
showninFig.2(c). Theeffectisnegativeinacertainbias tryinthecouplingofthedottometallicelectrodes–this
rangearoundthezerobiaslimit,butabsolutemagnitude coupling is larger for spin-up electron, which determines
oftheeffectbecomessmallerforsmallervaluesofU. For hightoftheKondopeakforspin-downelectrons. Whena
large bias there is a transition from negative to positive biasvoltageisapplied,eachofthetwoKondopeaksgen-
TMR with decreasing U. erally becomes additionally split into two components.
6

1.6
1.4 P eV/D=0.0spin down AP eV/D=0.0 spin down 0.02 (a) spin up
1.2 spin down
1.0 spin up spin up h] 0.01 total P
OS 0.8 D/ 0.00
D 00001.....02466 Current [e--00..0021 Occupation0000....3456
1.4 P eV/D=0.05 AP eV/D=0.05 -0.03 0.-20.2 -0.1eV0 ./0 D 0.1 0.2
1.2
1.0
DOS00..68 0.02 (b) ssppiinn udpown
0.4 h] 0.01 total AP
0.2 D/ 0.00
11110.....02460 P eV/D=-0 .05 AP eV/D=-0. 05 Current [e--00..0021 Occupation000000......344556505050
OS0.8 -0.03 0.3-00.2 -0.1eV0 ./0 D 0.1 0.2
D0.6 -0.2 -0.1 0.0 0.1 0.2
0.4
eV / D
0.2
0.0
-0.2 0.0 0.2 -0.2 0.0 0.2 FIG.4: Current-voltagecharacteristicsintheparallel(a)and
Energy / D Energy / D antiparallel (b) configurations, calculated for the parameters
as in Fig.3. Current in the spin-down (spin-up) channel in
FIG. 3: DOS for spin-up (thick lines) and spin-down (thin theparallel(antiparallel)configurationalmostvanishessothe
lines) electron states on the dot in the parallel (left column) total current flows in the spin-up (spin-down) channel (the
and antiparallel (right column) magnetic configurations, cal- curvespresenting the total and spin-up (spin-down) currents
culated for three indicated voltages and for ΓL+/D = 0.28, are not resolved. The insets show the corresponding occupa-
ΓL−/D = 0.12, ΓR+/D = 0.04, ΓR−/D = 0.0002, U/D = 500, tion numbers. The parameters as in Fig.3
(e2/CL)/D=(e2/CR)/D=10. The other parameters are as
in Fig.1.
with the right electrode). As before, the component as-
sociated with the coupling to the right electrode in the
Oneofthem(theoneassociatedwiththecouplingtothe
spin-down channel is not resolved.
sourceelectrode)movesupinenergy,whereaspositionof
the second one (the one associated with the drain elec- Consider now the antiparallelconfiguration (right col-
trode) remains unchanged. This is because we assumed umn in Fig.3), when magnetic moment of the right elec-
thattheelectrochemicalpotentialofthesourceelectrode trode is reversed. There is a nonzero spin splitting of
shifts up by |eV|, while of the drain electrode is inde- the Kondo peak at equilibrium, contrary to the case of
pendent of the voltage. For eV > 0 (negative bias), the symmetric coupling to the magnetic electrodes, where
splittingofthelarge-intensity(spin-down)peakisclearly the spin splitting in the antiparallel configuration van-
visible, although one component of the splitted peak is ishes [19]. Apart from this, the situation is qualitatively
relatively small. This is just the component associated similar to the one for parallel configuration. The main
with the coupling of the dot to the rightelectrode in the difference is that now the bias-induced splitting of the
spin-up channel. Since this coupling is relatively small, large-intensity peak is not resolved, whereas splitting of
the corresponding intensity is also small. The second the low-intensity peak is resolved.
component, in turn, is much larger because it is associ- Asinthecaseofsymmetricalcouplingdescribedabove,
atedwiththecouplingtotheleftelectrodeinthespin-up the Kondo peaks in DOS give rise to anomalous behav-
channel, which is the dominant coupling in the system iorofthecorrespondingtransportcharacteristics. Dueto
considered. Splitting of the low-intensity (spin-up) peak thesplittingoftheequilibriumKondopeak,theanomaly
is not resolved. Intensity of the component associated in DOS does not contribute to transport in the small
withthe couplingto the rightelectrodeinthe spin-down bias regime. The Kondo peaks enter the ’tunnelling
channelpracticallyvanishesbecausethiscouplingisneg- window’ at a certain bias, which leads to an enhanced
ligibleinthecaseconsidered. ForeV <0(positivebias), conductance. Such an enhancement is clearly visible
the situation is changed. Now the electrochemical po- in the current-voltage characteristics shown in Fig.4 for
tential of the left electrode is independent of the bias. bothparallel(a)andantiparallel(b)configurations(solid
Consequently, intensity of the components whose posi- lines), where for negative values of eV the enhancement
tion is independent of energy is significantly larger than is quite significant, but it is less pronounced for eV >0.
intensity of the other components (the ones associated This asymmetry is due to the difference in intensities of
7

0.20
(a) P 1.2 (a) spin up
0.16
spin down
h] 0.8
2G [e/diff00..0182 DOS 0.4 eV/D=0.0
0.04 0.0

0.3 (b) AP 1.2 (b)
eV/D=0.14
2e/h]0.2 OS 0.8
G [diff0.1 D 0.4
0.0
0.0

0.4
0.2 (c) 1.2 (c) eV/D=-0.14
0.8
0.0
R S
TM-0.2 DO 0.4
-0.4 0.0
-0.6
-0.2 -0.1 0.0 0.1 0.2 -0.4 -0.2 0.0 0.2 0.4
eV / D Energy / D
FIG. 5: Bias dependence of the differential conductance in FIG. 6: DOS for spin-up (solid lines) and spin-down (dot-
the parallel (a) and antiparallel (b) configurations and the ted lines) electron states of the dot, calculated for three
corresponding TMR (c), calculated for the same parameters different voltages, and for ΓL/D = 0.12, ΓL/D = 0.08,
+ −
as in Fig.3. ΓR+/D = ΓR−/D = 0.1, U/D = 500, and (e2/CL)/D =
(e2/CR)/D=0.33. Theother parameters are as in Fig.1.
thecorrespondingKondopeaksthatenterthe’tunnelling
window’. conductance and also TMR may be interesting from the
point of view of applications in mesoscopic diodes.
The differential conductance in the Kondo regime is
shown in Fig.5 for parallel (a) and antiparallel (b) mag-
netic configurations. In the parallel configuration the
C. QD coupled to one ferromagnetic and one
Kondo anomaly occurs in the spin-up channel and for
nonmagnetic leads
eV > 0 only. This may be easily understood by con-
sidering the relevant DOS (see Fig.3, left column). For
eV > 0 only the Kondo peak in spin-up DOS can enter A specific example of asymmetric systems is the case
the tunnelling window created by the bias. For eV < 0, where one electrode is ferromagnetic (typical ferromag-
on the other hand, the Kondo peak in spin-down DOS netic 3d metal) whereas the second one is nonmagnetic.
canenterthetunnelingwindow. However,thespindown For numerical calculations we assumed ΓL+/D = 0.12,
channel is almost non-conducting, so the corresponding ΓL/D = 0.08 for the left (magnetic) electrode, and
−
peak in the differential conductance is suppressed. In ΓR/D = ΓR/D = 0.1 for the right (nonmagnetic) one,
+ −
the antiparallel configuration the Kondo peak in differ- which corresponds to p = 0.2, p = 0, and ΓL/D =
L R
ential conductance occurs for eV < 0 only. This can ΓR/D = 0.1. As in the other asymmetrical situations
be accounted for by taking into account behavior of the studied in this paper, the equilibrium Kondo peak in
KondopeaksinDOSshowninFig.3(rightcolumn), and DOSbecomesspin-split. When abiasvoltageis applied,
the factthatnowthe spin-upchannelisnon-conducting. each component becomes additionally split, as shown in
ForeV >0onlytheKondopeakinthespin-upDOScan Fig.6. Variation of the spectra with bias voltage can be
enter the tunneling window, whereas for eV < 0 this is accounted for in a similar way as in the case of the dot
the Kondo peak in spin-down DOS (of large intensity). coupled asymmetrically to two ferromagnetic electrodes.
The correspondingTMR is shownin Fig.5(c). It is in- The only difference is that now all components of the
teresting to note that TMR is highly asymmetrical with peaks are clearly resolved. This is because all coupling
respect to the bias reversal. It becomes positive for eV constants are now of comparable magnitude.
exceeding a certain positive value, and negative below The corresponding differential conductance is shown
this voltage. This is a consequence of the fact that for in Fig.7. Due to the spin splitting of the Kondo peak in
positiveeV theKondopeakindifferentialconductanceis DOS, the Kondo anomaly in the conductance becomes
clearlyvisibleintheparallelconfiguration(seeFig.5(a)), split as well, as clearly seen in Fig.7. However, the
whereasforeV <0theKondopeakoccursintheantipar- splitting is asymmetric with respect to the bias reversal.
allel configuration(see Fig.5(b)). Such a behavior of the Thus, there is no need to have two ferromagnetic elec-
8

0.48 Acknowledgments
0.46
The work was supported by the State Committee
0.44 for Scientific Research through the Research Project
PBZ/KBN/044/P03/2001and 4 T11F 014 24.
0.42
0.40
h]
2e / 0.38
G [diff0.36
0.34
0.32
0.30
0.28
-0.10 -0.05 0.00 0.05 0.10
eV / D
FIG. 7: Bias dependenceof thedifferentialconductance, cal-
culated for the parameters as in Fig.6.
trodes to observe splitting of the Kondo anomaly, but it
is sufficient when only one lead is ferromagnetic.
V. SUMMARY AND CONCLUSIONS
In this paper we considered the Kondo problem in
quantumdotscoupledsymmetricallyandasymmetrically
to ferromagnetic leads. As an specific example of asym-
metrical systems, we considered the case when one elec-
trode is ferromagnetic, whereas the second one is non-
magnetic.
We showedthat ferromagnetismof the leads gives rise
to a splitting of the equilibrium Kondo peak in DOS for
all asymmetrical situations. This generally takes place
forbothmagneticconfigurationswhenthetwoelectrodes
aredifferent. Thesplittinginbothconfigurationsalsooc-
curswhenbothmagneticelectrodesareofthesamemate-
rial,but the correspondingcoupling strengths to the dot
aredifferent. Indeed, suchasplittinginparallelandalso
antiparallel configurations was recently observed experi-
mentally[25]. Whensimilarelectrodesaresymmetrically
coupled to the dot, the splitting occurs only in the par-
allel configuration. An interesting conclusion from the
experimental point of view is that the splitting also oc-
curs in the case when one electrode is nonmagnetic.
The spin-splitting of DOS can lead to characteristic
splitting of the zero bias anomaly in electrical conduc-
tance. This in turn can lead to negative (inverse) tun-
nelmagnetoresistanceeffect. Inhighlyasymmetricalsys-
temsTMRcanchangesignwhenbiasvoltageisreversed.
9
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