Nonequilibrium Kondo Effect in a Quantum Dot Coupled to Ferromagnetic Leads

0.42 MB·English

by  Yasuhiro Utsumi

Nonequilibrium Kondo Effect in a Quantum Dot Coupled to Ferromagnetic Leads
Yasuhiro Utsumi1,2, Jan Martinek1,3,4, Gerd Scho¨n1, Hiroshi Imamura5, and Sadamichi Maekawa3
1 Institut fu¨r Theoretische Festk¨operphysik, Universita¨t Karlsruhe, D-76128 Karlsruhe, Germany
2 Max-Planck-Institut fu¨r Mikrostrukturphysik, Weinberg 2, D-06120 Halle (Saale), Germany
3 Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
4 Institute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznan´, Poland
5
5 Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, Japan
0
(Dated: February 2, 2008)
0
2
We study the Kondo effect in the electron transport through a quantum dot coupled to ferro-

n magnetic leads, using a real-time diagrammatic technique which provides a systematic description
a ofthenonequilibriumdynamicsofasystemwithstronglocalelectroncorrelations. Weevaluatethe
J theoryinanextensionofthe‘resonanttunnelingapproximation’,introducedearlier,byintroducing

9 the self-energy of the off-diagonal component of the reduced propagator in spin space. In this way
we develop a charge and spin conserving approximation that accounts not only for Kondo correla-

] tions but also for the spin splitting and spin accumulation out of equilibrium. We show that the
l
l Kondoresonances, split bytheapplied biasvoltage, maybespinpolarized. Aleft-right asymmetry
a in the coupling strength and/or spin polarization of the electrodes significantly affects both the
h
spinaccumulation andtheweight of thesplit Kondoresonances out ofequilibrium. Theeffectsare
-
s observable in the nonlinear differential conductance. We also discuss the influence of decoherence
e on the Kondoresonance in theframe of thereal-time formulation.
m
PACSnumbers: 75.20.Hr,72.15.Qm,72.25.-b,73.23.Hk
.
t
a
m
I. INTRODUCTION creaseoftheTMRwithincreasingtemperature[17]. The
- modelwasrevivedrecently,andledtoaseriesofpublica-
d
tions [18, 19, 20, 21, 22, 23, 24]. In the early stage, due
n Thecontinuingexperimentalprogresswithmesoscopic
o electronicdeviceshasstimulatedanewtheinterestinfun- to different approximations used, the question remained
c damental quantum mechanical questions [1]. A good controversial whether the Kondo resonance splits in the
[ presence of spin-polarized leads [19, 20] or not [21, 22].
example is the Kondo effect [2] in electron transport
The splitting was confirmed eventually by numerical-
1 through quantum dots (QD). The effect, which nowa-
v days is well established experimentally [3], has remained renormalization-group(NRG) techniques [23, 24].
2 achallengingproblemfornonequilibriumquantumtrans- TheoriginofthesplittingoftheKondoresonanceslies
7 an exchange coupling induced by spin-dependent quan-
port theory [4, 5, 6, 7]. After early experiments, which
1 tum charge fluctuations. Let us consider a single-level,
concentrated on semiconductor QDs, the Kondo effect
1
originally spin-degenerate QD with energy ǫ <0 (mea-
0 wasobservedinsingle-atom[8]andsingle-moleculetran- 0
sured relative to the Fermi level of the lead electrons)
5 sistors [9], as well as in carbon nanotube QDs [10], all of
and strong Coulomb interaction, U → ∞. Then, dou-
0 which were coupled to metallic leads. More recently, the
/ Kondo effect was demonstratedin QDs coupled to ferro- ble occupancy is suppressed and the QD behaves as a
t
a magnetic leads [11, 12]. In this article, we will discuss magnetic impurity. But there still exist quantum charge
m fluctuations due to the tunneling between the QD and
the nonequilibrium Kondo effect in such systems as an
the leads. An electron with majority-spin in the QD
- example of the nontrivial collective many-body physics
d can tunnel between the QD and the leads easier than an
in magnetic nanostructures.
n
electronwith minority-spin,thus gaininga largerkinetic
Research on electronic properties in magnetic systems
o
energy. As a result, a ferromagnetic exchange interac-
c has evolved into an active field, “spintronics” [13]. In
: spintronicsdevicesmagneticpropertiescontroltransport tion between the QD spin and the lead magnetization
v is induced. It lifts the spin degeneracy of the QD level
properties via the electron spin degree of freedom. One
i
X example is the tunnel magnetoresistance (TMR) in fer- by ∆ǫ. Such a spin splitting quenches spin-flip scatter-
ingprocessesforlowenergies,below|∆ǫ|,andsuppresses
r romagnetic tunnel junctions [14], which relies on switch-
a the Kondo correlations. It alsoleads to a splitting ofthe
ing the magnetization of the electrodes between paral-
Kondo resonance. The qualitative two-stagescaling the-
lel (P) and antiparallel (AP) orientations by an applied
ory presented in Ref. [19] indicates that the high-energy
magnetic field. In magnetic multilayers,the intrinsic ex-
part of quantum charge fluctuations, between |ǫ | and a
change coupling between layers [15] influences the mag- 0
cutoff energy D (of the order of the itinerant electron
netic structure and consequently the transport proper-
band width) is responsible for a spin splitting given by
ties [16]. The Kondo effect in transport through a QD
coupled to ferromagnetic leads (Fig. 1) provides a prime Γ D
exampleofmany-bodyeffectsinspintronicsdevices. The ∆ǫ∼−P ln , Γ≡(Γ↑+Γ↓)/2. (1)
π |ǫ |
theoretical model, a local spin coupled to two ferromag- 0
netic leads, was originally introduced to explain the de- Here we assumed the coupling strength Γ between the
σ
2
QD level and the leads to be spin dependent. The split- (a) (b)
ting of the level results in a corresponding spin splitting L R L R
of the Kondo peak in the spectral density. Remarkably,
it is possible to compensate the spin splitting ∆ǫ by ap-
plyinganexternalmagneticfield,andthustorecoverthe
full Kondo effect [19].
In spite of the interest in the problem, a systematic
approximation which can account for the spin splitting FIG.1: Asinglelevelquantumdotcoupledtotwoferromag-
netic leads for (a) parallel and (b) antiparallel alignments of
oftheKondoeffectinnonequilibriumtransportproblems
lead magnetizations. We allow for a left-right asymmetry in
has not yet been presented.
tunnelbarriers and/or spin polarization factors.
- The NRG is a powerful tool for the investigation of
the Kondo problem, but is limited to the equilibrium
state. The validity of extensions to the nonequilibrium
The outline of this paper is as follows. In Sec. II, we
state [19, 20, 22] remains to be proven.
introduce the model Hamiltonian andshortly review the
- The slave-bosonmean-field approximationis correctin
real-timediagrammatictechniqueaswellastheresonant
the limit of large degeneracy and able to capture the
tunneling approximation. We introduce an extension of
physics of the Kondo singlet at zero temperature [25].
the RTA, which is needed to account for the spin split-
However, it does not account for the spin splitting since
ting. In Sec. III we present numerical results. We will
thedescriptionofquantumchargefluctuationsisbeyond
show the splitting of the zero-bias anomaly for various
mean-field theory.
temperatures (Sec. IIIA). We provide a comprehensive
- The non-crossing approximation (NCA), a systematic
discussion of the restoration of the Kondo resonance by
conserving approximation [26, 27], has been successfully
anappliedmagneticfield(Sec.IIIB)andthenonequilib-
adoptedtothenonequilibriumstate[5,6]. NCAaccounts
rium Kondo effect (Sec. IIIC). In Sec. IIID, we will dis-
forspin-dependentquantumchargefluctuationsandthus
cuss the effect of asymmetries in system parameters and
for the spin splitting due to ferromagnetic leads. But, it
their consequencesfor nonequilibriumstates. InSec.IV,
also generates an additional spurious peak at the Fermi
we will discuss decoherence effects on the Kondo reso-
leveleveninequilibrium[20]aswellasaspuriouspeakin
nance out of equilibrium within the framework of the
the presence of magnetic field [5]. Such a shortcoming is
real-time diagrammatic technique. Sec. V is devoted to
dangerous in the context of transport, which is sensitive
adiscussionoftherelationbetweenourresultsandrecent
to the electron states near the Fermi energy. The spuri-
experiments [10, 11]. We summarize in Sec. VI.
ous peak canbe made to disappear,if vertex corrections
are properly accounted for [28]. However, the general-
ization of NCA described in Ref. [28] does not appear II. FORMULATION
tractable in nonequilibrium situations.
- The equation-of-motion (EOM) approach, which was
A. Model Hamiltonian
adopted in Ref. [19], is able to capture Kondo correla-
tions as well as – in principle – the spin splitting. How-
Our model Hamiltonian consists of several parts de-
ever, within the standard decoupling scheme, the spin
scribingasingle-levelQD,H , the left(andright)reser-
splittingislost[18,21],anditrequiresintroducinganad- D
voir, H , and tunneling, H ,
ditional self-consistent equation to determine the renor- L(R) T
malized QD-level energy [19].
H =H + H +H . (2)
D r T
In this situation, a systematic approximation, free of
r=XL,R
the mentioned drawbacks in nonequilibrium, is needed
for further progress in the exploration of the Kondo ef- The Hamiltonians for the isolated QD and the leads are
fect in spintronics devices. In this paper, we formulate
the problem using a real-time diagrammatic technique, H = ǫ d†d +Ud†d d†d , (3)
D σ σ σ ↑ ↑ ↓ ↓
which is a systematic method suitable to describe the Xσ
nonequilibriumtimeevolutionofasystemwithstronglo- H = ε a† a , (4)
cal electron correlations [7, 29]. We evaluate the theory, r rkσ rkσ rkσ
Xrkσ
accounting for Kondo correlations and spin-dependent
quantumchargefluctuations,inaconservingapproxima- respectively (we use unit where ~ = k = 1). Here,
B
tionmakinguseofanextensionoftheresonanttunneling d is the annihilation operator of electrons with spin-σ
σ
approximation (RTA) [7]. The RTA was shown to pro- (σ=↑,↓) in the QD, and U denotes the onsite Coulomb
duce qualitatively reasonable results in the presence of interaction. The energy of the single QD level with spin
a magnetic field at not too low temperatures, sufficient ↑/↓ is ǫ = ǫ ± E /2+ a eV/2. The first term, ǫ ,
↑/↓ 0 Z c 0
to allow for a comparison with experiment [30]. This may be tuned by a gate voltage. The Zeeman energy is
property motivated us to apply the RTA for the present givenbyE =2gµ B,wheretheexternalmagneticfield,
Z B
problem. B, is assumed to act (only) on the spin in the QD. The
3
applied bias voltage between the left and right leads, V, The central quantity is the reduced density matrix,
mayshifttheQDlevelbya eV/2[7]. Herea isanasym- whichisderivedfromthe totaldensitymatrixbytracing
c c
metry factor due to different capacitive couplings of the out the lead electron degrees of freedom,
QDtotheleftandrightleads. Theoperatora annihi-
latesanelectronintheleadr=L,Rwithwaverknσumberk Pχχ′(t)=TrLTrRTrD ρL0ρR0ρD0 TCSCXχ′χ(t)I , (8)
and spin σ. We assumed that the ferromagnetism in the Xχ′χ(t)I ≡eiH0t|χ′i(cid:2)hχ|e−iH0t, (χ,χ′=0,↑,↓(cid:3)). (9)
leads can be treated within mean-field approximation,
i.e., it is accounted for by a spin-dependent dispersion The combination of a trace over dot states, Tr , and a
D
ε and, hence, spin-dependent density of states (DOS) projectionoperator,X , selects the (χ,χ′)-component
rkσ χ′χ
ν (ω)= δ(ω−ε ). of the reduced density matrix Pχ. The subscript I de-
rσ k rkσ χ′
The tuPnneling Hamiltonian, finally, is given by notes the interaction picture with respect to the unper-
turbed Hamiltonian H ≡H +H +H . The contour
HT = Vrkd†σarkσ +H.c. (5) ordering operator TC is0definLed onRthe cDlosed time-path
Xrkσ C starting from the initial time, t , to time t on the up-
i
WeassumedthatthetunnelingamplitudeVrk isindepen- perbranchoftheKeldyshcontourandreturningtoti on
dent of spin. Furthermore, for simplicity, we will ignore thelowerbranch. TheKeldyshS-matrixisdefinedalong
a k dependence of the tunneling amplitude V =V . this contour by
rk r
The important parameter of this model is the spin-
dependentcouplingstrengthtothe ferromagneticleadr, S ≡exp −i dtH (t) . (10)
C T I
(cid:18) Z (cid:19)
C
Γrσ(ω)≡2π|Vr|2νrσ(ω). (6) Initially, at time ti, the density matrix is assumed to be
factorizedinto a productof equilibrium density matrices
The spin dependence is the maingeneralizationrequired of the leads, ρr, and the density matrix of the QD, ρD.
0 0
for the present problem as compared to that of a QD The former are characterized by the temperature T and
coupled to normal metal leads [7]. The parameter Γσ, chemical potential µr (µL=−µR=eV/2) as,
which appeared in Eq. (1), is given by Γ ≡ Γ ,
σ r rσ
whereΓrσ≡Γrσ(0)isthe valueatthe FermienerPgy. The ρr ≡e−β(Hr− kσµra†rkσarkσ)/Zr , (11)
spin-dependentcouplingstrengthsarerelatedtothespin 0 P 0
polarization of the leads by whereβ≡T−1. ThenormalizationfactorZr ensuresthat
0
Tr ρr=1.
P =(Γ −Γ )/(Γ +Γ ). (7) r 0
r r↑ r↓ r↑ r↓ Equation (8) can be evaluated in a perturbative ex-
In this paper, we will analyze two configurations of the pansion in terms of HT. Under the conditions that the
lead magnetizations, the parallel (P) and antiparallel initial density matrix of the QD is diagonal, hχ′|ρD0|χi=
(AP) alignments. We always take a positive value for Pχ0δχ′,χ, and that the Hamiltonian conserves the spin,
PL. The P (AP) alignment then corresponds to positive the time evolution reduces in the limit of ti →−∞ to a
(negative) values of P . master equation:
R
PstΣ =0, (12)
χ χ,χ′
B. Real-time diagrammatic technique χ=X0,↑,↓
for the (stationary) probability Pst which is the diago-
ForinfinitelystrongCoulombinteraction,U=∞,dou- χ
nal (χ,χ)-component of the stationary density matrix.
ble occupancy of the QD level is suppressed, and the
Theself-energyofthemasterequationΣ isrelatedto
Hilbertspaceisrestrictedtotheempty state|0iandtwo χ,χ′
transition probabilities from a state |χi to a state |χ′i.
singly occupied states |σi. In the presence of strong in-
Themaintaskistodetermineanappropriateself-energy
teractioneffect,theapplicationofWick’stheoremispro-
of the master equationin the frame of the diagrammatic
hibited, andthestandardapproachforthecalculationof
expansion.
nonequilibrium currents through a tunnel junction [31]
Figure 2 shows examples of diagrams. Basic build-
is not suitable. We, therefore, proceed using the ‘real-
ing blocks are the free evolution (horizontal lines on the
time diagrammatic technique’ developed by Schoeller et
Keldyshcontour)andthetunnelingprocesses(remaining
al. [7, 29]. Apartfromprovidingatransparentclassifica-
directed lines) of electrons with spin σ into or out of the
tionofvariousprocessesitallowsconstructingaconserv-
QD to the leads r [7]. In Fourier space the latter carry
ing approximation out of equilibrium. In the following,
a factor obtained by Fermi’s golden rule,
we will briefly outline the technique. It is closely re-
lated the Keldysh formalism and the Feynmann-Vernon γ±(ω)=Γ (ω)f±(ω−µ )/(2π), (13)
rσ rσ r
approach (see for example Ref. [32]). The technique en-
ables one to perform a systematic diagrammatic expan- wheref±(ω)=1/(e±βω+1)istheelectron(hole)distribu-
sion of the reduceddensity matrix in terms of H in the tionfunction. Inodertoregularizetheexpressionsweim-
T
real-time domain. poseaLorentziancutoffreplacingthecouplingstrengths
4
(ii) (iii) σ σ σ σ σ σ
σ 0 σ' 0 σ'' 0 σ'''
(a) π ω = ω + π σ ω
⋯
(i) σ σ σ σ σ σ
σ σ 0 σ 0 σ' 0 σ
0 σ 0 σ 0 σ' 0 σ'' 0 σ''' (b) σω = σω + σ ω +Σ σ ω +Σ σ ω + ⋯
FIG. 2: Diagrams for various tunneling processes: (i) quan- σ' σ'
tumchargefluctuations,(ii)theco-tunnelingprocessand(iii) σ σ σ 0 σ 0 σ' 0
the resonant tunneling process. The horizontal lines on the
Keldysh contour represent the free propagators of the quan- FIG. 3: (a) Dyson equation for πσ,σ and (b) the diagram-
σ¯,σ¯
tum dot. Remaining directed lines are tunnelinglines. matic expansion of the irreducible self-energy σσ,σ. Directed
σ¯,σ¯
dotted lines carry theenergy ω.
of Eq. (6) by Γ ρ (ω) where ρ (ω)=D2/(ω2+D2). The
rσ c c
t (t>t ) is described by the reduced propagator
effectivebandwidthD wasalreadyintroducedinEq.(1). i
In Fig. 2, each sector of the diagram corresponds to πσ,σ(t,t )=−ihσ¯|Tr Tr ρLρRT S X (t) |σi.
a physical tunneling process. The ‘bubbles’ in sector (i) σ¯,σ¯ i L R 0 0 C C σ¯σ I
(cid:2) (cid:3) (14)
representquantumchargefluctuations,whereanelectron
Since the tunneling Hamiltonian Eq. (5) conserves the
with spin σ virtually fluctuates between the QD and the
spin, spin is conserved at each vertex. E.g., the spin in-
leads. A singleline connectingthe upper andlowerhori-
dicesofanincoming/outgoingtunnelinglineandanout-
zontallines (not shown)describes a sequentialtunneling
going/incoming free propagator for each vertex are the
process. Two overlapping parallel lines shown in sector
same. Therefore, after Fourier transformation, Eq. (14)
(ii) representa co-tunnelingprocess,whereinone coher-
can be expressed as a Dyson equation,
ent transitionanelectronwith spin σ leavesthe QD and
anelectronwith spinσ′ fills the empty level. A‘train’of πσ,σ(ω)=1/(ω+i0+−ǫ +ǫ −σσ,σ(ω)), (15)
overlapping lines [sector (iii)] represents a process with σ¯,σ¯ σ σ¯ σ¯,σ¯
electrons going back and forth many times between the where σσ,σ denotes an irreducible self-energy and 0+ is
QD and the leads. σ¯,σ¯
a positive infinitesimal number. The diagrammatic rep-
resentation of Eq. (15) is given in Fig. 3(a). Directed
dottedlines carryingthe energyω correspondto the fac-
torexp(iωt)oftheFouriertransformation. Thediagrams
C. Extension of resonant tunneling approximation for σσ,σ are shown in Fig. 3(b), in which any auxiliary
σ¯,σ¯
vertical line cuts at least one tunneling line.
In order to account for the logarithmic behavior of Now, we observe that within RTA in any diagram for
Kondocorrelations,wehavetosumupasuitableclassof the self-energy of the master equation Eq. (12) a sector,
diagramsoftheexpansionuptoinfiniteorder. Thestart- wherethereducedpropagatorπσ,σ appears,isalwaysac-
σ¯,σ¯
ing point of our approximation is the class of diagrams companied by two tunneling lines, a left-going tunneling
denotedas‘resonanttunnelingapproximation’(RTA)[7]. linewithspinσandaright-goingtunnelinglinewithspin
Inthediagrammaticlanguage,RTAincludesalldiagrams σ¯, which are disconnected from πσ,σ itself. It is a con-
σ¯,σ¯
where any auxiliary vertical line cuts at most two tun- sequence of the requirement that any auxiliary vertical
neling lines [An example is presented in Fig. 2(iii)]. At line must cut the same number of left- and right-going
not too low temperatures RTA was shown to provide tunneling lines or free propagators with the same spin
a qualitatively reasonable approximation for the Kondo indices. This in turn follows from the fact that the spin
correlations with analytical expressions for the nonlin- is conservedat eachvertex, andthat the initial QDden-
ear differential conductance for arbitrary level degener- sity matrix is assumed to be diagonal. As a result, the
acy. It predicts also qualitatively reasonable behavior in irreducible self-energy σσ,σ does not appear in RTA [7],
σ¯,σ¯
the presence of Zeeman splitting. However, the origi- because in its presence, any auxiliary vertical line laying
nal RTA developed in Ref. [7] does not describe the spin onasectorofσσ,σ wouldcutatleastthreetunnelinglines.
σ¯,σ¯
splitting since, as we will discuss below, it does not take RTA can account for the Zeeman splitting of the Kondo
intoaccountthe subclassofdiagramswhichaccountsfor resonance,whichisalreadyincludedinadenominatorof
spin-dependent quantum charge fluctuations. the bare reduced propagator πσ,σ. However, RTA omits
σ¯,σ¯
To find the origin of the shortcoming, let us examine further information on the spin dynamics. As we will
the QD spin dynamics. In general, in order to describe discuss in the following, the further information, such as
thesuperpositionofspinstates|↑iand|↓i,itisnecessary the spinsplitting ∆ǫ causedby spin-dependentquantum
to consider also off-diagonal components of the density charge fluctuations, is included in σσ,σ beyond the origi-
σ¯,σ¯
matrix in spin space, e.g., Pσ, where σ¯ is the opposite nal RTA.
σ¯
spin of σ. The time evolutionof Pσ between time t and In order to obtain an expression for σσ,σ we proceed
σ¯ i σ¯,σ¯
5
in a perturbative expansion in terms of the coupling Γ, Here πσ,σ(1) is defined by Eq. (15) with the self-energy
σ¯,σ¯
givenbyEq.(17). Theresultissimilartothe self-energy
σσσ¯,,σσ¯ =σσσ¯,,σσ¯(1)+σσσ¯,,σσ¯(2)+··· . (16) of the Green function obtained previously using EOM
(Eq. (8) in Ref. [19]). The first term of Eq. (21) is re-
In lowest order we find the first two diagrams in the
lated with spin σ [the first and the second diagrams in
right-hand side of Fig. 3(b) describing spin-dependent
the right-hand side of Fig. 4(a)]. The real part of this
quantum charge fluctuations. From them we obtain the
term causes a slight shift of the QD-level and the imag-
first-order term
inary part gives a level broadening. The second term
σσ,σ(1)(ω)= dω′ γσ−(ω′) [the third diagram in the right-hand side of Fig. 4(a)]
σ¯,σ¯ Z ω+i0+−ω′+ǫ is associated with the Fermi distribution function of the
σ¯
+ dω′ γσ¯−(ω′) otoppodsiωtef+sp(ωin)/σ¯(ωth+rio0u+gh∓γEσ¯+ a∓nd∆ǫis)rfoorugσh=ly↑/p↓roaptoerVtio=n0a.l
Z ω+i0++ω′−ǫ Z
σ CoRnsequently, the second term leads to the logarithmic
= γ˜σ−(ω+ǫσ¯)−γ˜σ¯−(ǫσ−ω)∗, (17) behavior of Kondo correlations. Our result for σσ,σ pro-
σ¯,σ¯
whereγ±= γ± andγ˜±= γ˜±. Hereweintroduced vides a microscopic justification to lowest-order expan-
γ˜r±σ, whσich iPs trherσHilbertσtranPsfrormrσof γr±σ, sion in terms of Γ for the spin splitting σ∆ǫ˜postulated
in the EOM scheme (Eq. (9) of Ref. [19]).
γ±(ω′) Theself-energyofthemasterequation(12)isgivenby
γ˜±(ω)= dω′ rσ
rσ Z ω+i0+−ω′
= Γ2rπσ ρ(ω−µr)(cid:26)±ψ(cid:18)21 + 2DπT(cid:19) Σχ,σ = −2iImZ dω sX=±Λsχχ,,σ0(ω)γσs(ω), (22)
∓ Reψ(cid:18)21 +iω2−πTµr(cid:19)+πω2−Dµr(cid:27)−iπγr±σ(ω), (18) atinodnsΣΛχ+,0a=n−dPΛ−σ=a↑r,e↓Σdeχt,eσrm(χin=ed0,b↑y,↓s)o.lvTinhgetvheertdeixagfruanmc--
matic equations depicted in Fig. 4(b). In the diagram
andψ(x)denotesthedigammafunction. Onecanexpect for Λ+/−, an open line carrying energy ω (dotted line)
that the lowest order is sufficient for the description of connects to the upper/lower horizontal line [35]. The
the spin splitting. Indeed, for EZ = T = eV = 0 and diagrammatic equations can generate diagrams for the
ferromagnetic leads (PL=PR=P>0), the spin splitting resonanttunneling processes in a recursivemanner. The
∆ǫobtainedbythescalingtheory[Eq.(1)]isreproduced, explicit forms are
sσe↓↑l,,f↓↑-(e1n)e(ωrg)y≈g∆ivǫenfobry|ωE/qǫ.0(|1≪7)1a.lsWoedefsucrrtihbeersnthoeteretdhuactttiohne Λ+χχ,,σ0(ω)=π0σ,,0σ(ω)(cid:26)δχ,0−Z dω′ Λsωχχ,,+σ0(ωi0′+)∗−γσsω(ω′′)
of the Kondo resonance splitting below the value of the sX=±
Zeeman energy E for normal (nonmagnetic) leads. For
Z − dω′ Λsχ,σ¯(ω′)∗γs(ω′)πσ,σ(1)(ω−ω′) ,
PL=PR=0 and T=eV =0, the self-energy then reads, Z χ,0 σ¯ σ¯,σ¯ (cid:27)
sX=±
σ↑,↑(1)(ω)≈−Γtanh−1 EZ , Γ= Γrσ, (19) (23)
↓,↓ π (cid:18)−2ǫ0(cid:19) Xσ,r 2 Λ−χχ,,σ0(ω)=π0σ,,0σ(ω)
for ǫσ <0 and |ω/ǫσ| ≪ 1, which renormalizes and re- × −δ − dω′ Λsχχ,,σ0(ω′)∗γσs(ω′) . (24)
duces EZ. Such a reduction was derived before for the (cid:26) χ,σ Z sX=± ω+i0+−ω′ (cid:27)
Anderson [33] and Kondo [34] models.
Next,wediscussthenatureanddetailsoftheextended Againinthe right-handside ofEq. (23), we insertedthe
RTA.Theextensionamountstoinsertingintoallpossible self-energy σσ,σ(1) via πσ,σ(1).
σ¯,σ¯ σ¯,σ¯
positions in the diagrams of the original RTA expansion From Pst, which is obtained by solving the master
σ
the lowest-order irreducible self-energy σσ,σ(1) given by equationEq.(12),wefindtheaverageoccupationN and
σ¯,σ¯
Eq. (17). The reduced propagator describing the time the local magnetization M of the QD,
evolutionofPσ isgivenbyEq.(14)withthereplacement
0 N =Pst+Pst , M =(1/2)(Pst−Pst). (25)
of σ¯ by 0: ↑ ↓ ↑ ↓
The spin-dependent nonequilibrium DOS of the QD is
πσ,σ(ω)=1/(ω+i0+−ǫ −σσ,σ(ω)). (20)
0,0 σ 0,0 defined as
The resulting self-energy of the reduced propagator is 1
ρ (ω)=− {G>(ω)−G<(ω)}, (26)
represented by the diagrams from Fig. 4(a). It yields σ 2iπ σ σ
γs(ω′) where the lesser/greater Green functions are expressed,
σσ,σ(ω)= dω′ σ
0,0 Z ω+i0+−ω′ using Eqs. (23) and (24), as
sX=±
+ dω′γ+(ω′)πσ,σ(1)(ω−ω′). (21) G≷σ(ω)=2iIm PχstΛ±χχ,,σ0(ω). (27)
Z σ¯ σ¯,σ¯ χ=X0,↑,↓
6
σ σ 0 σ σ III. RESULTS AND DISCUSSIONS
σ
(a) = σ + ω + π(1) ω
ω ω σ σ In this section, we will present results obtained by
0 0 0 σ σ solving the integral equations numerically. In all results
χ σ σ χ 0 σ we tested the numerical accuracy by checking the spin
and charge conservation ( the relative error is smaller
Σ
(b) Λ+ σ,ω = π σ,ωδχ0+ Λs* π σ,ω than 5×10−6). We also checked the spectral sum rule,
s=± σ dωρ (ω)=Pst+Pst, which follows from the fact that
χ χ σ 0 σ
0 0 χ 0 σσ 0σ ΛR±χχ,,σ0(ω) is analytic in the upper half plane and decays
Σ at infinity as 1/ω. The sum rule is always satisfied with
+ s=± Λs* σ π(1) π σ,ω accuracy better than 5×10−4.
χ In Sec. IIIA, we will discuss the splitting of the zero-
σ 0
bias anomaly. Subsections IIIB and IIIC provide a dis-
χ Λ− σ,ωσ = πσ σ,ωδχσ+Σ χ Λs* σ0 πσ σ,ω caussosifopnroofptehrteiersesotfortahteionnoonfeqthueiliKbroinudmorKeosonndaonecffeeacst.weIlnl
s=± theses subsections, we will assume a left-right symmetry
χ χ
0 0 σ 0 inthecouplingstrengthsΓL=ΓR[Γr≡(Γr↑+Γr↓)/2]and
spinpolarizationfactorsP =|P |,anda =0. Inequilib-
L R c
FIG. 4: (a) The irreducible self-energy of the reduced prop- riumthesesymmetriesarenotimportantbecause,asone
agatorπσ,σ. (b)Diagrammaticequationsforthevertexfunc- canseefromaunitarytransformation[23],anasymmetry
0,0
tions Λ+χ,σ and Λ−χ,σ. An open dotted line connects to the givesjustaconstantprefactortothelinearconductance.
χ,0 χ,0
upper/lower horizontal line for Λ±. Shaded boxes represent However, this is not the case for a nonequilibrium state,
the off-diagonal component of the reduced propagator in the as will be shown in Sec. IIID.
spin space πσ,σ(1).
σ¯,σ¯
A. Splitting of the zero bias anomaly
Also the transport current flowing into the QD through
thejunctionr,I ,canbeexpressedintermsoftheGreen
r
functions. The spin-resolved components read Figure 5(a) shows the nonlinear differential conduc-
tance G (V)≡dI(V)/dV for P alignment and symmet-
P
Irσ =−ieZ dω{γr+σ(ω)G>σ(ω)+γr−σ(ω)G<σ(ω)}. (28) ric coupling, ΓL=ΓR, and spin polarization, PL=PR.
The splitting of the zero-bias anomaly is well observable
One can prove that the approximation satisfies spin for low temperatures. The enhancement of the peak at
and charge conservation [36]. Equations (22), (27) and low temperatures is a clear sign of Kondo correlations.
(28)combineto I =ie PstΣ ,whichbyvirtue The value of the splitting is independent of temperature
r rσ χ χ χ,σ
ofthemasterequPation,Eq.(P12)leadstothespin-resolved and almost coincides with the prediction of the scaling
current conservation, I =0. The conservation laws theory Eq. (1), 2∆ǫ≈0.204Γ. At high temperature, the
r rσ
for charge and spin, P two peaks are smeared out. However, the magnitude of
the splitting is not affected by the temperature.
IL=−IR=I, (Ir↑−Ir↓)=0, (29) Figure 5(b) shows the corresponding plot for the local
Xr magnetization M. For small bias voltage |eV|.|∆ǫ|, it
follows the spin-resolved current conservation. Equation increaseswithdecreasingtemperature. Atthesametime,
(29) is valid only for collinear magnetizations,where the the differential conductance decreases [Fig. 5(a)]. This
spin is conserved at each vertex in the diagrams of the indicates that the suppression of the zero-bias anomaly
self-energyforthemasterequation,regardlessofthesign can be attributed to the quenching of the spin-flip scat-
ofP . Thisfactensuresthattheself-energyofmaster tering at low temperatures. When the applied bias volt-
L(R)
equation does not mix diagonal and off-diagonal states ageexceedsthespin-splittingenergy,|eV|&|∆ǫ|,aspin-
and allows one to formulate a closed equation for the flip channel opens. As a result, the local magnetization
diagonal components of the reduced density matrix. starts to decrease and peaks appear in the nonlinear dif-
The linear conductance, Glin ≡ lim dI (V)/dV ferential conductance.
σ V→0 σ
[I =I ], probes the equilibrium spin-dependent DOS,
σ rσ
ρeq(ω), around the Fermi energy:
σ
B. Restoration of Kondo resonance
Glin Γ Γ ∂f+(ω)
σ =2π Lσ Rσ dω − ρeq(ω), (30)
G Γ +Γ Z (cid:26) ∂ω (cid:27) σ
K Lσ Rσ A remarkable feature of the spin splitting induced by
where G ≡ e2/(2π~) is the conductance quantum. We ferromagnetism is that it can be compensated by an ap-
K
symmetrized the expression Eq. (30) using the spin and pliedmagneticfield[19,23]. Atthepointoffullcompen-
charge conservation, Eq. (29). sation, the strong coupling limit can be retrieved. The
7
(a) (a) Glin+Glin
↑ ↓
0.4 Glin
↑
0.3 K G↓lin
K G
G /GP linG /0.2
0.2 0
(b)
1 (b) 0.5
T/Γ=0.005 )
0.01 ℏ
/
ℏ) 0.025 (2 0
2/0.6 0.05 M
(
M
0.5
0.2 0 0.1 0.2 0.3 0.4
E /Γ
Z
-0.4 -0.2 0 0.2 0.4
eV/Γ
(c) 0.2 ρeq+ρeq
↑ ↓
0.3 ρeq
↑
FIG. 5: (a) The nonlinear differential conductance and (b) 0.1 ρ↓eq
the local magnetization M for the P alignment (PL= PR= 0.2
q
0.1) andforsymmetriccouplingΓL=ΓR. Solid,dashed,dot- ρe 0
ted and dot-dashedlines are results for various temperatures -0.4 -0.2 0 0.2 0.4
0.1
T/Γ = 0.005,0.01,0.025, and 0.05, respectively. Parameters
are: ǫ /Γ=−2,D/Γ=50.
0
0
(d) 0.2
Kondo temperature in the leading logarithmic approxi- 0.3
mation is given by [19]
0.1
q 0.2
e
ρ
−πǫ˜arctan(P)
TK(P)≈ǫ˜exp(cid:26) Γ P (cid:27). (31) 0.1 -0.2 -0.1 0 0.1 0.2
Here we considered the case P = P = P. The ratio 0
L R
Γ/(πǫ˜)correspondsthevalueJ0(ν↑+ν↓)ofRef.[19]. The -8 -6 -4 -2 0
ω/Γ
cutoff energy ǫ˜follows from ǫ˜=−ǫ +Γ/(2π)ln(ǫ˜/D).
0
Figure 6(a) and (b) show the linear conductance Glin
and the local magnetization M as a function of Zeeman FIG. 6: (a) The linear conductance Glin and the local mag-
energyEZ in equilibriumforthe parallelalignment. The netizationM asafunctionoftheZeemanenergyEZfortheP
averageoccupation N depends slightly on EZ and varies alignment(PL=PR=0.2). Inpanel(a),thespin-resolvedcon-
between N≈0.88 and N≈0.91. At E =0, the direction ductanceisplottedbydashed(dotted)lineforup(down)spin.
Z
TheequilibriumDOSfor(c)E /Γ=0and(d)E /Γ=0.205.
ofthelocalmagnetizationM andleadmagnetizationsare Z Z
Solid and dashed (dotted) lines show the total DOS and the
aligned. WithincreasingZeemanenergythelocalmagne-
spin resolved DOS for up (down) spin. Insets in these pan-
tizationM changesthedirection. Atapeakinthelinear
els are blow-ups of the Kondo resonance. Parameters are:
conductance Glin, where spin-flip processes are expected
T/Γ=0.005, ǫ /Γ=−2,D/Γ=50.
0
tobeenhanced[solidlineinFig.6(a)],weobservethelo-
cal magnetization M=0 in agreement with results from
Ref. [23]. The spin state of the QD is responsible for the
slight asymmetry in the conductance around the M=0 (dotted) lines show the total DOS ρ(ω)≡ ρ (ω) and
σ σ
point. In the left (right) side of the peak, because the the up-spin (down-spin) component of thPe DOS ρσ(ω),
QDlevelisoccupiedbyanup-spin(down-spin)electron, respectively. Figure 6(c) and its inset demonstrate that
the co-tunneling currentfor up-spin (down-spin)compo- the Kondo resonance splits at E = 0. The two res-
Z
nent is dominant [dashed (dotted) line in Fig. 6(a)]. As onances, which appear rather well resolved, should be
the up-spins are majority spins, a larger conductance is smeared by decoherence effects. These effects are be-
obtained on the left side of the peak. yond the present approximation, as we will discuss in
WeanalyzedtheDOSatE =0[Fig.6(c)]andatE = Sec. IV. The enhancement/suppression of the up/down
Z Z
0.205Γ, where M≈0 [Fig. 6(d)]. The solid and dashed spin (dashed/dotted line) component of the QD-level
8
tDivOeSlo(cbarlomadagpneeatkizaartoiounn,dMω/>Γ0=. −2) indicates the posi- PA, P , EE ZZ == 00.205Γ (a) AP, PE, ZE = Z =0 .0205Γ (c) 0.4
K0.3
At EZ =0.205Γ, the Kondo resonance is restored at G 0.3
the Fermi energy [solid line in Fig. 6(d)]. We can ob- G/
servethattheresonancefortheminority-spincomponent 0.2 0.2
(dotted line) is higherthan thatof the the majority-spin
cspomeapkoinnge,ntit(idsasbheecdaulsienet)he[inmseintoirnityFi(gm.a6j(odr)i]t.y)Lspooinseilny )0.4 (b) AP, P , EE ZZ == 00.205Γ 0.2
ℏ
the QD is well (poorly) screened by majority (minority) 2/ 0 P, E Z = 0 0
spin in leads. This tendency is consistent with the zero- M( AP, E Z = 0.205Γ
-0.4 (d)
temperatureresultofNRG[23]andtheFriedelsumrule,
-0.2
which predicts ρ (0)∼1/Γ [23].
AsfortheQD-σlevelDOS,σup-spinanddown-spincom- -0.4 -0.2 eV0/Γ 0.2 0.4-0.4 -0.2 eV0/Γ 0.2 0.4
ponents are almost the same (at EZ = 0.205Γ), which AP (e) P (f)
indicates M=0 [dotted and dashed lines in Figs. 6(d)]. 0.2 ρ↑+ρρ↓
However, though the QD level can be spin polarized de- ρ↑
↓
pending on E , the total QD-level DOS is not affected ρ0.1
Z
by EZ, i.e. the solid lines in Figs. 6(c) and (d) can al- E Z = 0.205Γ E Z = 0.205Γ
most overlap around the broad peak. Such a feature is 0
also obtained from the NRG work [24]. (g) (h)
0.2
eV=EZ eV=−EZ
ρ0.1
C. Kondo resonance out of equilibrium
0
In this subsection, we will discuss the nonequilibrium -0.4 0 0.4 -0.4 0 0.4
ω/Γ ω/Γ
Kondoeffect. Fig.7showsthenonlineardifferentialcon-
ductance, G [(a) and (c)], and the local magnetization,
FIG.7: ThenonlineardifferentialconductanceG,(a)and(c),
M [(b) and (d)], for symmetric coupling. As discussed
and thelocal magnetization M,(b) and (d),for a symmetric
in Sec. IIIA, we can observe the splitting of the zero-
coupling ΓL=ΓR. Solid lines and dashed lines in (a) and (c)
bias anomaly for parallel alignment [solid line in (a)].
are plots for E =0 and E =0.205Γ, respectively. The solid
Z Z
However,forantiparallelalignment[solidline in(c)], the
lines in (a) and (c) are for the P alignment (PL=PR=0.2)
splitting vanishes. Moreover,for positive (negative) bias and the AP alignment (PL=−PR=0.2)], respectively. The
voltage spin accumulates in the QD with positive (nega- dashedlinesin(a)and(c)arefortheAPalignmentandtheP
tive) local magnetization [solid line in (d)]. alignment, respectively. The average occupation is N≈0.89
As discussedinSec.IIIB,the zero-biasanomalyis ex- for all parameter regimes. The panels (e) and (f) show the
pected to be restoredwhen the spin splitting is compen- corresponding DOS ρ(ω) of the solid and the dashed lines
in panel (c) at eV/Γ=0.205. The dashed and dot-dashed
sated by an applied magnetic field. The dashed line in
(c)showsthecaseforthePalignmentwithE =0.205Γ. linesareforρ↑(ω)andρ↓(ω),respectively. Panels(g)and(h)
Z showplotsoftheDOSρ(ω)ofthedashedlineinpanel(a)at
Figures7(e)and(f)showthenonequilibriumDOSρ (ω)
σ eV =E (g),andateV =−E (h). Theotherparametersare
Z Z
for the AP alignment and that for the P alignment with
thesame as for Fig. 6.
the Zeeman energy E = 0.205Γ. The total DOS has
Z
two peaks (solid lines) at two chemical potentials, simi-
larinshapetothatofanonequilibriumKondoDOSsplit
by an applied voltage. However, now the two peaks are the other hand, at negative bias voltages eV ≤−EZ, the
spinpolarized,i.e. thereiscleardifferenceintheweights spin-flip tunneling processes are between minority spin
for the spin-up component ρ (ω) (dashed lines) and the states. Hence the peak in the differential conductance is
↑
spin-down component ρ (ω) (dot-dashed lines). weak.
↓
For the AP alignment with the Zeeman energy E = AmorepreciseexplanationtakingintoaccountKondo
Z
0.205Γ, the splitting persists [dashed line in (a)], but in correlations can be given based on the spin-polarized
thiscasethesplittingiscausedbytheZeemanenergy[5], nonequilibrium Kondo resonance. Figure 7(g) shows
and a strong asymmetry between two peaks is obtained. ρ(ω) for eV = E = 0.205Γ. The two spin polarized
Z
Intuitively,theresultcanbeunderstoodbythefollowing Kondo resonances out of equilibrium split further into
argument: In the regime |eV|.E , a spin-down occu- four peaks (solid line) by the energy eV [5]. The lower
Z
piestheQDlevel. AboveapositivebiasvoltageeV ≥E , two peaks at −E +eV/2 and −E −eV/2 are for spin-
Z Z Z
spin-flip tunneling events from a left lead spin-up state down(dot-dashedline), relatedto the leftandrightlead
to a right lead spin-down state contribute to the cur- chemical potential at ±eV/2, respectively. The peak at
rent. In this case, because both initial and final states −E +eV/2 is larger than the peak at −E −eV/2, be-
Z Z
are majority-spin states, a large peak is expected. On cause the spin-down electron in the QD can be screened
9
80
0.4 (a) ΓR /ΓL =1 P (c) AP
0.5
40 K0.3 0.25
G
G/
R 0.2
0
M
0.1
T-40 0.4 (b) a c =1 P (d) AP 00..64
-80-0.4 -0.2 0 0.2 0.4 G/GK00..23 0.5 0.02M(2/)ℏ
eV/Γ 0.1 -0.2
-0.4 -0.2 0 0.2 0.4-0.4 -0.2 0 0.2 0.4
eV/Γ eV/Γ
FthIeGb.i8a:s vTolhteagteunVn.elSpminagpnoeltaorriezsaitsitoanncfeacTtoMrsRaraesPfuLn=ct|iPoRn|o=f (e) ΓR /ΓL =0.5 (f) ΓR /ΓL =0.25 eeVV <> 00
0.2. Thefurther parameters are thesame as for Fig. 6. 0.2
ρ
0.1
bymajorityspinsoftheleftleadwithspin-up. Thepeaks
-0.2 0 0.2 -0.2 0 0.2
atEZ±eV/2areforspin-upelectrons(dashedline), and ω/Γ ω/Γ
a larger amplitude of the peak at E −eV/2 can be ex-
Z
plained in the same way as above. Under the condition
FIG. 9: (a) The differential conductance for the P align-
eV =EZ,thepositionsofthetwolargepeaksatEZ−eV/2 ment GP(V) (PL= PR=0.2). Solid lines, dashed lines and
and −EZ+eV/2 coincide with the two chemical poten- dotted lines are results for ΓR/ΓL = 1,0.5, and 0.25. (b)
tials. Thus, tunneling electrons with energy between the ThedifferentialconductanceGP(V)forΓR/ΓL=0.5withthe
left and right chemical potentials [shaded region in (g)] asymmetry factor ac =0 (dashed line) and 0.5 (dot-dashed
can profit from the larger DOS inside the QD. Conse- line). (c) The differential conductance with the AP align-
quently, the differential conductance G (V) at positive ment GAP(V) (PL= −PR=0.2) and (d) the nonequilibrium
AP
local magnetization M. (e), (f) The total DOS ρ(ω) in the
biasvoltageeV =E isenhancedmorestrongly. Finally,
Fig. 7(h) shows theZDOS ρ(ω) for eV =−E =−0.205Γ. APalignment: (e) The solid (dashed) line is for ΓR/ΓL=0.5
Z at eV/Γ =−0.08 (0.07). (f) The solid (dashed) line is for
By repeating the same arguments as presented above,
ΓR/ΓL=0.25 at eV/Γ=−0.13 (0.11). The other parameters
we can explain the fact that at negative bias voltage
are thesame as for Fig. 6.
eV =−E the DOS ρ(ω) of the QD at energies between
Z
the left and right Fermi levels [shaded region in (h)] is
lower, and consequently the peak of the differential con-
conductance. In the following we discuss several exam-
ductance G (V) is smaller as well.
AP ples.
The anomalousbehaviorof the nonlinear conductance
Figure 9(a) shows the nonlinear differential conduc-
determines the shape of the TMR. Figure 8 presents the
tance G (V) for the P alignment (P = P = 0.2) for
P L R
TMR defined as
varying values of the ratio of coupling strengths Γ /Γ ,
R L
whilekeepingthetotalcouplingΓ=Γ +Γ unchanged.
L R
TMR≡(G (V)−G (V))/G (V). (32)
P AP AP The values of the splittings are almost the same for all
cases. The reason is the following: The spin splitting
We find a large negative TMR at zero bias. This is in
Eq.(1) is expressedas the sum of the left and right con-
contrast to the TMR of ferromagnetic tunnel junctions,
tributions ∆ǫ≡∆ǫ +∆ǫ (one can obtain the relation
L R
where one finds usually a maximum of the TMR at zero
for ∆ǫ using Eq. (1) and adding a lead index r to the
r
bias and then a decrease with increasing bias voltage
Γ and P). For P =P =P, the value of the splitting
L R
V [13, 37].
∆ǫ is proportionalto P(Γ +Γ ), which is independent
L R
of the ratio of coupling strength. The asymmetric ca-
pacitive coupling does not alter the conductance G(V)
D. Left-right asymmetry in system parameters by much. Figure 9(b) shows curves for the asymmetry
factor a =0 (dashed line) and 0.5 (dot-dashed line) for
c
In the previous subsection we showed that the spin- ΓR/ΓL=0.5 [the dashed line is the same as that in (a)].
polarization of the Kondo resonance and the spin- The asymmetry factor just tilts the curve slightly.
accumulation for a nonequilibrium situation depend on Figure 9(c) presents the nonlinear conductance
the orientation of the lead magnetizations. We can ex- G (V) for the AP alignment for different values of
AP
pect that a left-right asymmetry in coupling strengths, Γ /Γ , and a = 0. For an asymmetric coupling, we
R L c
Γ 6=Γ , and spin polarization, P 6=P , or an asymme- find again a splitting of the zero bias anomaly. Such
R L R L
try in the applied voltage could affect these properties a splitting was observed experimentally [11] (details are
as well and leave footprints on the nonlinear differential given in Sec. V). It can be understood because the spin
10
splitting due to the exchange interaction with the left (a) (c) P = 0.4, P = -0.2
L R
∆ǫL and right lead ∆ǫR cannot compensate each other eV < 0
∆ǫ 6= −∆ǫ for Γ 6= Γ . Actually, with decreasing K0.3 eV > 0 0.2
L R L R G ρ
Γ /Γ , ∆ǫ increases and thus the splitting is enhanced. G/
R L
0.2
Weobservetheasymmetryinthetwopeaks’amplitude
0.1
similarto thecaseofthe APalignmentwiththe Zeeman
splitting [dashed line in Fig. 7(a)]. Actually, this asym- 0.6 (b) (d) PL = 0.6, PR = -0.4
metrycanbeexplainedalmostinthesameway. Theonly 0.4
ℏ) 0.2
differenceisthatthelargepeakisatnegativebiasvoltage 2/0.2 ρ
[dashed and dotted lines in Fig. 9(c)] as opposed to the M( 0 PL = 00..41, PR = -00..12
dashedline in Fig. 7(a), because, ∆ǫ is negative whereas -0.2 0.6 -0.4 0.1
the Zeeman energy is positive. The solid (dashed) lines
-0.4 -0.2 0 0.2 0.4 -0.2 0 0.2
inpanels(e)and(f)showthetotalDOSρ(ω)fortheAP eV/Γ ω/Γ
alignment at negative (positive) bias voltage. We can
checkthatthe largerDOSatnegativebias voltage(solid FIG. 10: (a) The nonlinear conductance G(V) and (b) the
lines) is responsible for higher conductance peak. localmagnetizationM forvariousspinpolarizations(PL,PR)
Next, we will look at consequences of an asymmetry and for symmetric coupling ΓR=ΓL. The average value of
thepolarization factorsintheleftandrightleadsarefixedas
in left and right spin polarization factors P 6=P . The
L R (PL+PR)/2=0.1. The solid, dashed, and dotted lines are
dashed and dotted lines in Fig. 10 show the differential for PL=0.1, 0.4, and 0.6, respectively. The total DOS ρ(ω)
conductance G(V) (a) and local magnetization M (b) at eV/Γ=−0.1 (solid line) and eV/Γ=0.1 (dashed line) for
for various spin polarization factors but for symmetric PL=0.4 (c) and PL=0.6 (d). The other parameters are the
coupling strength Γ =Γ . For each curve, the average same as for Fig. 6.
R L
value of polarization factors in the left and right leads
was kept constant (P +P )/2 = 0.1. The solid lines
L R
The first diagram of the second-order expansion, de-
show the result for the equal polarizations (P =P =
L R
picted in Fig. 3(b), is evaluated as
0.1). As the spin polarization of the left lead increases
(tPheLlo=ca0l.4mafogrnedtaizsahteidonliinnecsr,eaPsLes=(d0ec.6refaosresd)oftotredpolsiinteivse) σσσ¯,,σσ¯(2-1)(ǫσ−ǫσ¯)= Z dω′dω′′ǫ −ω1′+i0+
(negative) bias voltage [Fig. 10(b)] and accordingly, the Xσ′ σ
difference betweenthe twopeaksgrows[Fig.10(a)]. The γ+(ω′′)γ−(ω′)
× σ′ σ . (34)
solid (dashed) lines in panels (c) and (d) are the total (ǫ −ǫ −ω′+ω′′+i0+)(ǫ −ω′+i0+)
σ σ′ σ
DOS at negative (positive) bias voltage [(c) for P =0.4
L
For simplicity, we assume here a flat DOS with cutoff
and (d) for 0.6]. It confirms that with increasing P ,
L
energy D, i.e., we replace the Lorentzian cutoff function
thedifferenceintheintensityincreases,causingthelarge
by ρ (ω)→θ(ω+D)θ(D−ω) with |ǫ |≪D. Then, the
difference in peak heights. c rσ
imaginary part of Eq. (34) reads,
− Imσσ,σ(2-1)(ǫ −ǫ )
σ¯,σ¯ σ σ¯
IV. DECOHERENCE EFFECT
ΓrσΓr′σ′ ǫrσ−ǫr′σ′
≈ π θ(ǫ −ǫ ). (35)
In this section, we will turn our attention to the ef- σX′rr′ (2π)2 ǫrσǫr′σ′ rσ r′σ′
fectsofdecoherence. Sincethetransversespinrelaxation
Theseconddiagramofthesecond-orderexpansionisob-
time, T2, is defined as the decay time of the off-diagonal tained from Eq. (34) as σσ,σ(2-2)(ω)=−σσ¯,σ¯(2-1)(−ω)∗.
componentsofthereduceddensitymatrixPσ,onemight σ¯,σ¯ σ,σ
σ¯ In addition to the above two diagrams there are four
identify it within the Markov approximation with the
other diagrams (Fig. 11); however their imaginary part
imaginarypartof the correspondingoff-diagonalcompo-
vanishes for ǫ <0:
nent of the self-energy in Fourier space, rσ
Imσσ,σ(2-3)(ǫ −ǫ )=
1/T =−Imσσ,σ(ǫ −ǫ ). (33) σ¯,σ¯ σ σ¯
2 σ¯,σ¯ σ σ¯ γ−(ω′)γ−(ω′′)
Im dω′dω′′ σ¯ σ
The relevantfrequency in the argumentǫσ−ǫσ¯ is the en- Z ω′−ω′′+ǫσ−ǫσ¯ +i0+
ergy difference between the states with spin σ and the 1 1 2
opposite spin σ¯. From Eq. (15), one can conclude, that × +
(cid:18)ω′−ǫ +i0+ ǫ −ω′′+i0+(cid:19)
σ¯ σ
1/T is closely related to the broadening of Kondo res-
2
Γ Γ θ(ǫ ) θ(ǫ )
onance. When evaluating the self-energy we note that ≈ rσ r′σ¯ r′σ¯ + rσ =0. (36)
the lowest-order contribution to σσσ¯,,σσ¯ does not contain Xrr′ 4π (cid:18) ǫrσ ǫr′σ¯ (cid:19)
an imaginary part. In the following, we therefore, per-
form the second order perturbation expansion in terms By adding the contributions from σσ,σ(2-1) and σσ,σ(2-2),
σ¯,σ¯ σ¯,σ¯
of Γ at zero temperature T=0 and ǫ ≡ǫ −µ <0. one obtains the transverse spin relaxation rate in 2nd
rσ σ r