Asymmetry-induced effects in Kondo quantum dots coupled to ferromagnetic leads

0.52 MB·English

by  K. P. Wojcik

Asymmetry-induced effects in Kondo quantum dots
3
1 coupled to ferromagnetic leads
0
2
n K. P. W´ojcik1, I. Weymann1, J. Barna´s1,2
a 1 FacultyofPhysics,AdamMickiewiczUniversity,61-614Poznan´,Poland
J 2 Institute ofMolecularPhysics,PolishAcademyofSciences,60-179Poznan´,
8 Poland
1
E-mail: [email protected]

]
l
l Abstract. We study the spin-resolvedtransport through single-level quantum
a
dots strongly coupled to ferromagnetic leads in the Kondo regime, with a focus
h
on contact and material asymmetry-related effects. By using the numerical
-
renormalization group method, we analyze the dependence of relevant spectral
s
e functions, linear conductance and tunnel magnetoresistance on the system
m asymmetry parameters. In the parallel magnetic configuration of the device
the Kondo effect is generally suppressed due to the presence of exchange field,
t. irrespectiveofsystem’sasymmetry. Intheantiparallelconfiguration,ontheother
a hand,theKondoeffectcandevelopifthesystemissymmetric. Weshowthateven
m relativelyweakasymmetrymayleadtothesuppressionoftheKondoresonancein
theantiparallelconfigurationandthusgiverisetonontrivialbehaviorofthetunnel
-
d magnetoresistance. Inaddition,byusingthesecond-orderperturbationtheorywe
n derivegeneral formulasfortheexchange fieldinboth magneticconfigurations of
o thesystem.
c
[

1 PACSnumbers: 72.25.Mk,73.63.Kv,85.75.-d,73.23.Hk,72.15.Qm
v
2
1
5
4
.
1
0
3
1
:
v
i
X
r
a
Asymmetry-induced effects in Kondo quantum dots with FM leads 2
1. Introduction
Transportpropertiesofnanoscopicobjects,suchasquantumdotsormolecules,tunnel-
coupled to external leads have attracted a lot of attention from both theoretical
as well as experimental sides [1, 2, 3, 4, 5, 6]. This is mainly due to possible
applications in nanoelectronics and spintronics, and due to a unique possibility to
study various many-body correlation effects between single charges and spins. When
the coupling between the quantum dot and external leads is relatively strong, the
electronic correlations may lead to the Kondo effect if the dot’s occupation number
is odd [7, 8, 9, 10]. For quantum dots coupled to ferromagnetic leads, on the
other hand, it was shown that the Kondo resonance can be suppressed due to the
presence of an effective exchange field, ∆εexch, that leads to the spin splitting of the
quantum dot level [11, 12, 13, 14, 15, 16, 17, 18, 19]. This suppression occurs if
the magnetic moments of external leads form a parallel (P) magnetic configuration
and when |∆εP | & T , where T is the Kondo temperature and ∆εP denotes
exch K K exch
the level splitting due to exchange field in the parallel configuration [15]. For an
antiparallel magnetic configuration of the device, the exchange field was found to
AP
vanish, ∆ε → 0, since the effective coupling to external leads becomes then spin-
exch
independent [12, 13, 15, 20]. This is however true only for fully symmetric systems,
while, as we show in this paper, for systems exhibiting some asymmetry, either the
left-rightcontactasymmetryorthematerial’sasymmetry,theexchangefieldmayalso
develop in the antiparallel configuration. Since experimentally it is very difficult to
buildatrulysymmetricdevice,itseemsdesirabletoanalyzetheeffectsofcontactand
material’s asymmetry on spin-resolvedtransport properties of quantum dots.
In the present paper we thus thoroughly study transport through quantum dots
coupledtoferromagneticleadsintheKondoregime,focusingespeciallyonasymmetry-
induced effects. To obtain the correct picture, we employ the Wilson’s numerical
renomalization group (NRG) method [21, 22, 23] with the idea of a full density-
matrix (fDM) [24]. By using NRG, we calculate the dependence of relevant spin-
resolved spectral functions, the linear conductance in the parallel and antiparallel
configurations, and the tunnel magnetoresistance (TMR) on the asymmetry between
the couplings to the left and right leads and for different spin polarizations of the
electrodes. We show that even relatively small asymmetry may fully suppress the
Kondoresonanceintheantiparallelconfiguration,leadingtonontrivialdependenceof
theTMReffectontheasymmetryparameters. Wealsoshowthatalthoughasymmetry
generally destroys the Kondo effect in the antiparallel configuration, there is a range
of asymmetry parameters, when the Kondo resonance can be restored. In addition,
by using the second-order perturbation theory, we derive general formulas for the
exchange field in both magnetic configurations depending on asymmetry parameters.
2. Theoretical description
The system consists of two ferromagnetic leads coupled to a single-level quantum
dot, see Fig. 1. The magnetizations of the leads are assumed to be collinear and
they can form two magnetic configurations: the parallel (P) and antiparallel (AP)
ones. Switching between different magnetic configurations of the device can be
obtained by sweeping the hysteresis loop, provided the left and right ferromagnets
have different coercive fields. The system considered can be described by the single-
Asymmetry-induced effects in Kondo quantum dots with FM leads 3
Figure 1. (Color online) The schematic of a single-level quantum dot coupled
to the left and right ferromagnetic electrodes. The magnetizations of electrodes
areassumed to be collinear and they can formeither parallel(P) or antiparallel
(AP)magneticconfiguration,asindicatedinthefigure. Thedotiscoupledtothe
leftand rightleadwiththe coupling strength Γrσ (r=L,R). Thedot level has
energyεd andU denotes Coulombcorrelationonthedot.
impurity Anderson Hamiltonian
H = εrkσa†rkσarkσ+εd nσ+Un↑n↓
r=XL,RXkσ Xσ
+ Vrkσa†rkσdσ+h.c. , (1)
r=XL,RXkσ (cid:16) (cid:17)
where a†rkσ creates an electron with spin σ momentum k and energy εrkσ in the
left (r = L) or right (r = R) lead. The energy of the dot level is denoted by ε , U
d
describestheCoulombinteractionoftwoelectronsresidingonthedot,andn =d†d
σ σ σ
is the particle number operator, with d† creating a spin-σ electron on the dot. The
σ
last term of the Hamiltonian (1) describes tunneling processes between the dot and
the ferromagnetic electrodes, with Vrkσ being the relevant hopping matrix elements.
Due to the coupling to external leads, the dot level acquires certain width, which is
described by the spin-dependent hybridization function, Γ =πρV2 , for lead r and
rσ rσ
spinσ. Here, ρ isthe densityofstates atthe FermilevelandweassumedthatVrkσ is
independent ofmomentumk[12,13]. Theconductionbands ofthe leadsareassumed
to be energyandspin independent, ρ=1/(2W),where W ≡1 is used as energyunit.
TheseassumptionsarejustifiablesinceweareinterestedintheKondoregimewhichis
mainly relatedwiththe conductionelectronstatesaroundthe Fermienergy. Next,by
introducingthespinpolarizationofleadr, p =(Γ −Γ )/(Γ +Γ ),thecouplings
r r↑ r↓ r↑ r↓
can be expressed as, Γ =(1+σp )Γ , with Γ =(Γ +Γ )/2. Here, Γ denotes
rσ r r r r↑ r↓ rσ
the coupling to the spin-majority (σ =↑) or spin-minority (σ =↓) conduction band of
the ferromagnetic lead r. For convenience, we have incorporated the effect of leads’
ferromagnetism into the spin-dependent tunnel matrix elements. This assumption is
commonly used in NRG calculations [12, 20].
The asymmetry induced effects to be studied in the present paper will include
the left-rightcontactasymmetry,Γ 6=Γ ,andthematerialasymmetry,i.e. different
L R
spin polarizations of the leads, p 6= p . In particular, we will study how the linear
L R
response transport properties depend on the parameters of the system, especially on
the couplings’ ratio Γ /Γ and spin polarizations’ ratio p /p . The linear-response
R L R L
spin-dependent conductance of the system can be found from the Meir-Wingreen
Asymmetry-induced effects in Kondo quantum dots with FM leads 4
formula [25],
e2 4Γ Γ ∂f(ω)
Lσ Rσ
G = dω − πA (ω), (2)
σ σ
h Γ +Γ Z (cid:18) ∂ω (cid:19)
Lσ Rσ
wheref(ω)denotestheFermifunctionandA (ω)isthespectralfunctionofdotlevel.
σ
The spectralfunctionis givenby A (ω)=−1ℑmGR(ω), where GR(ω)is the Fourier
σ π σ σ
transform of the retarded d-level Green’s function, GR(t)=−iθ(t)h{d (t),d†(0)}i.
σ σ σ
Another quantity of interest is the tunnel magnetoresistance (TMR), which
describes the change in spin-dependent transport properties when the magnetic
configuration of the device is varied. The TMR is defined as [26],
GP−GAP
TMR= , (3)
GAP
where GP/AP is the total linear conductance in the P/AP magnetic configuration. In
addition, we will also study the spin filtering properties of the system, which can be
related to the spin polarization of the linear conductance,
GP/AP−GP/AP
PP/AP = ↑ ↓ , (4)
GP/AP+GP/AP
↑ ↓
where PP/AP denotes the spin polarization in the parallel or antiparallel magnetic
configuration of the device.
For the single-impurity Anderson model, it is very convenient to perform an
orthogonaltransformationfromtheleft-rightbasisintoaneven-oddbasis,whereonly
the even linear combination of the lead operators couples to the dot, while the odd
combination is completely decoupled. The new effective spin-resolved couplings for
the parallel and antiparallel configuration can be then expressed as
ΓP/AP =Γ 1+σpβP/AP , (5)
σ
(cid:16) (cid:17)
where p denotes the average spin polarization p = (p +p )/2 and Γ = Γ +Γ ,
L R L R
while the parameter βP/AP is given by
(Γ −Γ )(p −p )
P L R L R
β =1+ , (6)
(Γ +Γ )(p +p )
L R L R
(Γ −Γ ) (p −p )
AP L R L R
β = + . (7)
(Γ +Γ ) (p +p )
L R L R
From the above formula follows that the couplings are generally different for each
spin direction, ΓP/AP 6= ΓP/AP, which results in different level renormalization for
↑ ↓
spin-up and spin-down. This difference leads to an effective spin-splitting of the dot
level, known as contact-induced exchange field, ∆εP/AP [12, 13, 14]. In the parallel
exch
configuration, ΓP 6= ΓP, and consequently, ∆εP 6= 0, once the spin polarization
↑ ↓ exch
is finite, p > 0, for any value of left-right contact asymmetry. In the antiparallel
r
configuration, on the other hand, the exchange field can occur, ∆εAP 6= 0, provided
exch
there is an asymmetry in the system, so that ΓAP 6= ΓAP. The asymmetry can be
↑ ↓
either related to the contacts, Γ 6= Γ , or to the material, p 6= p . Moreover, it
L R L R
turns out that even in the presence of asymmetry, in the antiparallel configuration
there is a parameter range where the exchange field can still vanish, which happens
for βAP =0. This occurs precisely when the following condition is met
Γ −Γ p −p
L R L R
=− . (8)
Γ +Γ p +p
L R L R
Asymmetry-induced effects in Kondo quantum dots with FM leads 5
The spin-dependence of the effective couplings gives rise to the exchange field
∆εexch, which can be found analytically by calculating second-order corrections δεdσ
to the energy of quantum dot levels, ∆εexch ≡ δεd↑−δεd↓. At low temperature, one
then gets for the parallel and antiparallel magnetic configurations
∆εP/AP = 2βP/APpΓln εd . (9)
exch π (cid:12)ε +U(cid:12)
(cid:12) d (cid:12)
(cid:12) (cid:12)
There are actually two factors that deter(cid:12)mine the(cid:12) strength of the exchange field: the
first one, ∼ ln|ε /(ε +U)|, is related to the gate voltage, which can be used to
d d
changepositionofthe dotlevel. Thisfactorleadsto the cancelationof∆εP/AP atthe
exch
particle-hole symmetry point of the model, i.e. ε = −U/2, irrespective of magnetic
d
configuration and asymmetry. The second factor, ∼ βP/AP, on the other hand, is
associatedwith asymmetryinthe systemanddepends onmagnetic configuration,see
Eqs.(6)-(7). It may either increase or suppress the exchange field, depending on the
magnetic configuration and parameters of the model.
AscanbeseenfromEq.(2),themainquantitytobecalculatedisthespin-resolved
spectral function of the dot-level. This is performed with the aid of the numerical
renormalization group method with full density matrix (fDM-NRG). This method,
knownasthemostpowerfulandversatiletostudyvariousquantumimpurityproblems,
allows us to determine the dependence of the spectral function on parameters of
the system in most accurate and reliable way. The starting point for the NRG is
logarithmic discretization of the conduction band of the leads and mapping of the
initial Hamiltonian to the Hamiltonian of a tight-binding chain with exponentially
decayinghoppings,theso-callWilsonchain[21]. ThechainHamiltonianisthensolved
in aniterative manner andits discardedeigenstatesare usedfor the constructionofa
full density matrix [24], which enables the calculation of relevant static and dynamic
quantitiesatarbitrarytemperature. Inourcalculationswehaveinparticularemployed
the flexible density-matrix numericalrenormalizationgroupcode [27]. In calculations
we kept 1024 states at each iteration and used the Abelian symmetries for the total
spin zth component and the total charge.
3. Results and discussion
Inthefollowingwepresentanddiscussthenumericalresultsonthelinearconductance
and spin polarization of the current in both magnetic configurations as well as the
resultingTMReffect. First,weanalyzetheeffectsrelatedwiththecontactasymmetry,
Γ 6= Γ , and then the effects due to different spin polarizations of the left and
L R
right lead, p 6= p , are discussed. Finally, we present the general case when both
L R
asymmetries are present.
3.1. Effects of left-right contact asymmetry, Γ 6=Γ
L R
By changing the ratio Γ /Γ one changes both the magnitude of exchange field
R L
as well as the Kondo temperature. However, the dependence of both quantities
on the strength of coupling Γ = ΓL + ΓR is different: while |∆εexch| ∼ Γ,
the Kondo temperature T depends on Γ in an exponential way [28, 29], T =
K K
UΓ/2 exp[πε (ε +U)/(2UΓ)] (for p = p = 0). Tuning the contact asymmetry
d d L R
rpatio will thus change the ratio |∆εexch|/TK, which conditions the occurrence of the
Kondo effect, as discussed and presented in the following.
Asymmetry-induced effects in Kondo quantum dots with FM leads 6
GP[e2/h]
2 2
a)
1.8
1.6
1.5
1.4
1.2
ΓR 1 1
ΓL 0.8
0.6
0.5
0.4
0.2
0 0
-1.5 -1 -0.5 0 0.5
GAP[e2/h]
2 1.8
b)
1.6
1.5 1.4
1.2
ΓR 1 1
ΓL 0.8
0.6
0.5 0.4
0.2
0 0
-1.5 -1 -0.5 0 0.5
TMR
2 1
c)
0.8
1.5 0.6
0.4
ΓR 1 0.2
ΓL 0
-0.2
0.5 -0.4
-0.6
0 -0.8
-1.5 -1 -0.5 0 0.5
εd/U
Figure2. (Coloronline)Thezero-temperaturelinearconductanceintheparallel
(a) and antiparallel (b) magnetic configuration and the resulting TMR (c) as a
functionofthelevelpositionεd/U andtheleft-rightcouplingsratioΓR/ΓL. The
parameters are: U =0.12W,ΓL =0.005W andpL=pR =0.4,withW ≡1the
bandhalfwidth.
Figure 2 shows the linear conductance in the parallel and antiparallel
configuration,as wellas the resultingTMR effect asa function ofthe couplings’ratio
Γ /Γ and the level position ε /U. Experimentally, the position of the dot level can
R L d
be changed by tuning the gate voltage. When Γ /Γ =0, the dot is coupled only to
R L
asingle(left) leadandtheconductancethroughthesystemisobviouslyequaltozero.
With increasing the coupling to the right lead, the linear conductance becomes finite
and exhibits a strong dependence on the level position. In the elastic cotunneling
regime, i.e. for ε /Γ ≫ 0 or (ε + U)/Γ ≪ 0, the conductance in the parallel
d d
configuration is GP ∼ (1+p p )Γ Γ , while for the antiparallel configuration one
L R L R
gets, GAP ∼(1−p p )Γ Γ , yielding TMR=2p p /(1−p p ) [20, 30], which for
L R L R L R L R
Asymmetry-induced effects in Kondo quantum dots with FM leads 7
GP[e2/h] GAP[e2/h]
2 ↑ 1 2 ↑ 1
a) d)
0.9 0.9
0.8 0.8
1.5 1.5
0.7 0.7
0.6 0.6
Γ
R 1 0.5 1 0.5
ΓL 0.4 0.4
0.3 0.3
0.5 0.5
0.2 0.2
0.1 0.1
0 0 0 0
-1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5
GP[e2/h] GAP[e2/h]
2 ↓ 1 2 ↓ 1
b) e)
0.9 0.9
0.8 0.8
1.5 1.5
0.7 0.7
0.6 0.6
Γ
R 1 0.5 1 0.5
ΓL 0.4 0.4
0.3 0.3
0.5 0.5
0.2 0.2
0.1 0.1
0 0 0 0
-1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5
2 PP 1 2 PAP0.6
c) 0.9 f)
0.4
0.8
1.5 0.7 1.5 0.2
Γ 0.6 0
R 1 0.5 1
ΓL 0.4 -0.2
0.5 0.3 0.5 -0.4
0.2
-0.6
0.1
0 0 0 -0.8
-1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5
ε /U ε /U
d d
Figure 3. (Color online) The spin-resolved linear conductance in the parallel
(a,b)andantiparallel(d,e)magneticconfigurationforthespin-up(a,d)andspin-
down (b,e) channels, and spinpolarization of the current in the parallel (c) and
antiparallel(f)configuration asafunctionofεd/U andΓR/ΓL. Theparameters
arethesameasinFig.2.
parametersassumedinFig.2gives,TMR≈0.38. Note thatthis value isindependent
of the couplings asymmetry, provided the current is mediated by elastic cotunneling
events.
IntheCoulombblockaderegime,ontheotherhand,thesituationismorecomplex,
since the dotis singly occupiedandthe electroniccorrelationsmayleadtothe Kondo
effect. Moreover, the occurrence of the Kondo effect is conditioned by the ratio of
the Kondo temperature and the exchange field induced splitting. In the parallel
configuration, the exchange field is always present, irrespective of Γ /Γ , and the
R L
Kondo resonance is suppressed, except for the particle-hole symmetry point, ε =
d
P
−U/2. By moving away from this point, the conductance drops once |∆ε | & T .
exch K
Since T depends on the coupling strength Γ, increasing Γ /Γ raises the Kondo
K R L
temperature, which leads to larger width of the Kondo peak in the middle of the
Coulombblockaderegime,seeFig.2(a). Intheantiparallelconfiguration,theexchange
fieldvanishesif the systemis symmetric,andthere isabroadKondoresonanceinthe
Asymmetry-induced effects in Kondo quantum dots with FM leads 8
whole localmomentregime,seeFig.2(b) forΓ /Γ ≈1. Nevertheless,ifΓ /Γ 6=1,
R L R L
the Kondo effect becomes suppressed when |∆εAP | & T . This happens faster for
exch K
Γ /Γ <1 than for Γ /Γ >1, since for smaller coupling Γ the Kondo temperature
R L R L
islowerandtheaboveconditioncanbefulfilledforrelativelysmallasymmetriesofthe
couplings. As already mentioned, for the particle-hole symmetry point the exchange
field vanishes in both magnetic configurations, see Eq. (9), the zero-temperature
conductance is then just given by, GP/AP = e2/h 4Γ Γ /(Γ +Γ )2, with
σ Lσ Rσ Lσ Rσ
the couplings correspondingly dependent on magnPetic configuration of the device.
For symmetric couplings, Γ /Γ = 1, both GP and GAP reach the maximum, with
R L
GP = 2e2/h and GAP = (1−p p )2e2/h. The respective behavior of conductance
L R
in both magnetic configurations leads to the corresponding dependence of the TMR,
which is shown in Fig. 2(c). Generally, the TMR is negative in the whole blockade
regime, exceptfor the particle-holesymmetry point, which is associatedwith the fact
that|∆εP |>|∆εAP |,andconsequentlyGP <GAP. Onlyforε =−U/2,whenthe
exch exch d
exchange field is suppressed, one finds a typical spin-valve effect with positive tunnel
magnetoresistance. Thus, tuning the position of the dot level and the asymmetry
factors, one can obtain a device with desired magnetoresistive properties.
Anotherquantitydescribingthespin-resolvedtransportpropertiesofthesystem,
interesting from an application point of view, is the spin polarization PP/AP of the
current flowing through the device, which is shown in Figs. 3(c) and (f) for both
magnetic configurations. The behavior of spin polarization can be understood from
the spin-resolved conductance. In the parallel magnetic configuration the coupling
of the spin-up level is much stronger than the coupling of the spin-down level, since
spin-up electrons belong to the spin-majority band. As a consequence, the spin-up
channel gives the main contribution to the conductance, see Figs. 3(a) and (b). The
difference between GP and GP is most visible around the resonances, ε ≈ 0 and
↑ ↓ d
ε ≈ −U, where the spin polarization PP takes large positive values, approaching
d
unity for Γ /Γ ≪ 1. This is contrary to the region around the particle-hole
R L
symmetry point where GP ≈GP and the spin polarization is suppressed, PP →0. In
↑ ↓
the antiparallel configuration the situation is slightly more complex, since the spin-
resolved couplings depend greatly on the system’s asymmetry, see Eq. (5). For equal
spin polarizations of the leads, p =p , as assumed in Fig. 3, one finds the effective
L R
couplings, ΓAP =Γ+σp(Γ −Γ ). In consequence, the spin-resolvedcouplings fulfill
σ L R
the following relations, ΓAP > ΓAP for Γ /Γ < 1, and ΓAP < ΓAP for Γ /Γ > 1,
↑ ↓ R L ↑ ↓ R L
withΓAP =ΓAPforthesymmetriccase,Γ =Γ . Thisgivesrisetothecorresponding
↑ ↓ L R
behavior of the spin polarization: PAP ≶ 0 for Γ /Γ ≶ 1, and PAP ≈ 0 for
R L
Γ /Γ ≈ 1. In the region around ε = −U/2, where the effective field vanishes,
R L d
the behavior of PAP is however different. The spin-dependent conductances are then
given by, GAP =4e2/h(1−p2)Γ Γ /[Γ+σp(Γ −Γ )]2, which yields negative spin
σ L R L R
polarization PAP < 0 for Γ /Γ < 1, and positive spin polarization PAP > 0 for
R L
Γ /Γ > 1. Note that this is just opposite to the case when the exchange field is
R L
present, i.e. ε 6= −U/2. The above analysis clearly demonstrates that by properly
d
engineeringthecouplingsbetweenthedotendelectrodes,andbytuningtheoccupancy
ofthedotwiththegatevoltage,onecanobtaindesiredspinpolarizationsoftheflowing
current, spanning almost the whole range from −1 to 1.
From an experimental point of view, it may be important to know how large
the asymmetry should be to suppress the Kondo resonance in the antiparallel
configuration. To address this question, in Fig. 4 we plot the linear conductance
Asymmetry-induced effects in Kondo quantum dots with FM leads 9
2
1.5
h]
2[e/ 1
P
A
G
0.5
0
0 0.5 1 1.5 2
ΓR/ΓL
Figure4. (Coloronline)Thelinearconductanceintheantiparallelconfiguration
as a function of left-right contact asymmetry ΓR/ΓL for different values of the
couplingtotheleftleadΓL forpL=pR=0.4. Eachlinecorrespondstodifferent
valueofΓL,increasinginthedirectionindicatedbythearrowfromΓL=0.0025W
to ΓL = 0.0075W in step of 0.0005W. The thick black line corresponds to
ΓL = 0.005W, value used in previous figures. The parameters are the same
asinFig.2withεd=−U/3.
as a function of Γ /Γ for different values of the coupling Γ . Since T depends
R L L K
exponentially on Γ, changing Γ corresponds to a huge change in the Kondo
L
temperature (note that T also depends on Γ /Γ ). On the other hand, the
K R L
dependence of the exchange field on Γ and the ratio Γ /Γ is only algebraic. It can
L R L
be seen that with lowering Γ , the suppressionof the Kondo effect occurs for smaller
L
asymmetries, e.g. for very weak coupling even relatively small asymmetry between
the left-right contacts can fully suppress the linear conductance in the antiparallel
configuration. Proper and very careful implementation of a quantum dot/molecular
deviceisthereforenecesseryinordertoobservedesiredeffects,suchase.g. restoration
of the Kondo effect when switching the magnetic configuration from parallel into
antiparallel one [15].
3.2. Effects of different spin polarizations of the leads, p 6=p
L R
Up to now we have focused on the asymmetry related to the left-right contacts,
however, the asymmetry can be also present if the electrodes have different spin
polarizations, p 6= p . The corresponding transport characteristics are shown in
L R
Fig. 5 for equal couplings Γ = Γ and different spin polarization ratio p /p ,
L R R L
with p = 0.4. The maximum value, p /p = 2.5, corresponds then to fully spin-
L R L
polarized right lead, while p /p = 0 corresponds to nonmagnetic right electrode.
R L
Now, by changing p /p , one can tune the magnitude of the exchange field, while
R L
the relevant Kondo temperature is constant. Generally, with increasing the ratio
p /p , the average spin polarization increases, and so does the exchange field. This
R L
intuitive behavior is however only valid for the parallel magnetic configuration and is
nicely visible in Fig. 5(a). It can be seen that GP displays a Kondo resonance at the
particle-hole symmetry point, whose width decreases with increasing strength of the
Asymmetry-induced effects in Kondo quantum dots with FM leads 10
GP[e2/h]
2.5 2
a)
1.8
2 1.6
1.4
1.5 1.2
pR 1
pL 1 0.8
0.6
0.5 0.4
0.2
0 0
-1.5 -1 -0.5 0 0.5
GAP[e2/h]
2.5 2
b)
1.8
2 1.6
1.4
1.5 1.2
pR 1
pL 1 0.8
0.6
0.5 0.4
0.2
0 0
-1.5 -1 -0.5 0 0.5
TMR
2.5 1.5
c)
2 1
1.5 0.5
pR
pL 1 0
0.5 -0.5
0 -1
-1.5 -1 -0.5 0 0.5
εd/U
Figure 5. (Color online) The conductance in the parallel (a) and antiparallel
(b)configurationandtheresultingTMR(c)asafunctionofεd/U andtheleads’
spin polarization ratio pR/pL. The parameters are the same as in Fig. 2 with
ΓL=ΓR=0.005W andpL=0.4.
exchange field, i.e. with increasing the ratio p /p . In the antiparallel configuration,
R L
on the other hand, the exchange field is a nonmonotonic function of p /p : it is
R L
maximum for p = 0, vanishes for p = p , and again reaches local maximum for
R R L
p =1. Consequently, GAP displaysthe Kondoeffectin the whole Coulombblockade
R
regime when p ≈ p , which becomes then suppressed with changing the materials’
L R
asymmetry ratio from the point p /p = 1, see Fig. 5(b). Moreover, it can be seen
R L
that the Kondo resonance around ε = −U/2 is now broader than in the case of
d
P AP
parallel configuration since generally, |∆ε | > |∆ε |. The different dependence
exch exch
of GP and GAP on the spin polarization ratio p /p reflects itself in a nontrivial
R L
behavior of the TMR effect, see Fig. 5(c). The TMR is positive in the whole elastic
cotunnelingregimeandgivenbytheJullierevalue,whileittakeslargenegativevalues