Kondo effect with non collinear polarized leads: a numerical renormalization group analysis

0.27 MB·English

by  P. Simon

Kondo e(cid:27)e
t with non
ollinear polarized leads:
a numeri
al renormalization group analysis
1 2 3 4
P.1Simon , P. S. Cornaglia , D. Feinberg , and C. A. Balseiro
Laboratoire de Physique et Modélisation des Milieux Condensés,
7 2 CNRS and Université Joseph Fourier, BP 166, 38042 Grenoble, Fran
e
0 Centre de3Physique Théorique, É
ole Polyte
hnique, CNRS, 91128 Palaiseau
edex, Fran
e
0 Laboratoire d'Etudes des Propriétés Ele
troniques des Solides, CNRS,
2 asso
iated to U4niversité Joseph Fourier, BP 166, 38042 Grenoble, Fran
e and

n Instituto Balseiro and Centro Atómi
o Barilo
he,
a Comisión Na
ional de Energía Atómi
a, 8400 Barilo
he, Argentina
J (Dated: February 6, 2008)

0
The Kondo e(cid:27)e
t in quantum dots atta
hed to ferromagneti
leads with general polarization
3
dire
tions is studied
ombining poor man s
aling and Wilson's numeri
al renormalization group

methods. Weshowthatpolarizedele
trodeswillleadingeneraltoasplittingoftheKondoresonan
e
]
l inthequantumdotdensityofstatesex
eptforasmallrangeofangles
losetotheantiparallel
ase.
l
a We also show that an external magneti
(cid:28)eld is able to
ompensate this splitting and restore the
h unitarylimit. Finally,westudytheele
troni
transportthroughthedevi
einvariouslimiting
ases.
-
s
PACSnumbers: 75.20.Hr,72.15.Qm,72.25.-b,73.23.Hk
e
m
t. I. INTRODUCTION Mostofthepreviousworkshavefo
usedquitesurpris-
a inglyonlyonparallelandantiparallelpolarizations. Ser-
m 10
gueev et al.
onsidered non
ollinear polarizations but
One of the most remarkable a
hievements of re
ent
- didnottakeintoa

ountthespinsplittingofthedoten-
d progress in nanoele
troni
s has been the observation of
ergy level whi
h naturally gives rise to spurious results.
n the Kondo e(cid:27)e
t in a single semi
ondu
tor quantum dot
19
o (QD).1,2 Sin
e these early experiments, the Kondo ef- In a re
ent preprint, Swirkowi
z et al. analyze the non
c 3,4,5
ollinear situation using the equation of motion method
fe
thasalsobeenobservedin
arbonnanotubeQDs,
[ 6 supplemented by an e(cid:27)e
tive ex
hange (cid:28)eld that has to
and mole
ular transistors
onne
ted to normal metalli
bedeterminedself-
onsistently. Su
hmethodgivesqual-
2 leads. The su

ess of these experiments has opened the T TK TK
v way to a systemati
analysis, both theoreti
ally and ex- itative results for temperature ∼ , where is the
4 Kondotemperature. Themainresultisthatthesplitting
perimentally,ofthee(cid:27)e
tsoftheleads
orrelationsonthe
9 of the zero bias Kondoanomalyde
reasesmonotoni
ally
Kondoe(cid:27)e
t. Ofparti
ularinterestaretheexperimental 2θ
7 when in
reasingthe angle between the two lead mag-
7 realizations of the Kondo e(cid:27)e
t
onne
ted to super
on-
7 8 netizations. In this paper, we go beyond the equation
0 du
tingleads ortoferromagneti
ele
trodes. Thelatter
of motion method by
ombining a poor man s
aling ap-
6 isparti
ularlyimportantsin
eitis
onne
tedto thenew
0 emerging(cid:28)eld
alled(cid:16)spintroni
s(cid:17).9 Onegoalofspintron- proa
h and the numeri
al renormalization group (NRG)
t/ i
s is to provide devi
es where the transport properties approa
h. The poormans
alingprovidesus withan an-
a are
ontrolled by magneti
degrees of freedom. In this alyti
alexpression forthe Kopndotemθperature asa fun
-
m tion of the lead polarization and [see Eq.(12)℄ while
sense, aKondoquantum dot
onne
tedto ferromagneti
- ele
trodes may be regarded as an interesting setup to NRG allows us to
ompute dot spe
tral densities and
d the near equilibrium transport properties like the
on-
study intera
tion e(cid:27)e
ts in spintroni
s devi
es.
n
du
tan
e. The planofthepaperisthefollowing: InSe
.
o From the theoreti
al point of view, it has been pre-
II we des
ribe our model Hamiltonian. In Se
. III, we
c di
ted, despite some early
ontroversialresults,10,11,12,13
: use poor man s
aling to estimate the Kondo tempera-
v that ferromagneti
ele
trodes with parallel polarization
ture. In Se
. IV, we present the NRG resultsfor the dot
14,15,16
i produ
easplittingoftheKondozerobiasanomaly
X density of states and for the
ondu
tan
e. Finally, Se
.
whereas polarized ele
trodes with antiparallel polariza-
V
ontains a brief summary of the results.
r 14,16,17
tion generally do not. Some of these predi
tions
a
60
have been
he
ked experimentally by
onta
ting C
8
mole
uleswithferromagneti
ni
kelele
trodes. The un-
derlying me
hanism responsible of the splitting of the
Kondo resonan
e is quite
lear: virtual quantum
harge II. MODEL HAMILTONIAN
(cid:29)u
tuations between the dot and the leads be
ome spin
dependent. As a result, an e(cid:27)e
tive ex
hange magneti
In order to des
ribe this system we use the following
(cid:28)eld appears in the dot whi
h generally lifts the spin
general Hamiltonian
degenera
y on it. For antiparallel
on(cid:28)gurations, the

ontributions of ea
h ele
trode to the e(cid:27)e
tive (cid:28)eld may
H =H +H +H ,
leads tun dot
eventually
an
el ea
h other for symmetri

ouplings. (1)
2
H
leads 14
where denotes the Hamiltonian of the
ondu
tion in the leads. The spin degenera
y of the Anderson en-
ele
trons in the leads and is de(cid:28)ned by ergy levels is therefore lifted with an e(cid:27)e
tive Zeeman
Hleads = εk±c†k,γ,±ck,γ,±. (2) splitting
δε =ε ε pΓcos(θ)ln(D/Λ).
k,γ,± d d↑ d↓
X − ∼ (6)
c†
Iεn this equaγtio=n Lk,,Rγ,±
reates an ele
tro~nn with en~nergy This splitting prevents from rea
hing of the strong
ouB-
k± in lead with spin along ± γ where γ is pling regime. Nonetheless, an external magneti
(cid:28)eld
a unitarγy ve
tor parallel to the~nmagnet~nization2θmoment
an be applied lo
ally to
ompensate tεh↑is inteεrn↓al (cid:28)eld
in lead . The angle between L and R is . Sin
e and restore the degenera
y between and . One
the leads are assumed to be polarized, there is a strong
an then perform a S
hrie(cid:27)er-Wol(cid:27) transformation. The
ρ (ω)
spin asymmetry in the density of states γ± . The Kondo Hamiltonian so obtained is similar to the usual
tunneling jun
tions between the leads and the dot may Kondo Hamiltonian:
1
be des
ribed by a standard tunneling Hamiltonian H = (Jz↑c† c Jz↓c† c )Sz
Htun = (tγc†k,γ,σdσ+H.c.), (3) K 2k,kX′,α,β kα↑ kβ↑− kα↓ kβ↓
k,γX,σ=↑,↓ + J⊥(c†kα↓ckβ↑S++c†kα↑ckβ↓S−). (7)
d σ =
σ
where destroys an ele
tron in the dot with spin ↑
, The bare values of the Kondo
ouplings are
↓. Herethepossiblevaluesforthedotspinarequantized
~e = (~n +~n )/~n +~n
calong the axis z L R | L R|. The ocperator J⊥ =Jz↑ =Jz↓ 2t2 U =J,
k,γ,σ is a linear
ombination of the operators k,γ,s=± ∼ (cid:18)(|εd|(εd+U)(cid:19) (8)
and reads:
(cid:18)ccαα,,kk,,↑↓ (cid:19)=(cid:18)csoins((θθ22αα)) −cosisn((θ2αθ2α))(cid:19)(cid:18)ccαα,,kk,,+− (cid:19), (4) pwehreproesεitdio∼n εodf↑ψ∼L+εda↓n.dNψoLte−.alOsonteh
aatnψtLh↑enispaerlifnoermartshue-
se
ond stage of the RG analysis dire
tly on the Kondo
θ = θ H
where α ± here. dot is the usual Anderson Hamil- Hamiltonian. The RG equations
an be obtained by in-
tonian and reads tegratingΛo0utthεedhighΛenergydegreesoffreedombetween
H = ε d†d +Ud†d d†d gµ BSz, a
ut-o(cid:27) ∼ and and read:
dot d,σ s s ↑ ↑ ↓ ↓− B (5) dλ 1+p
σX=↑,↓ z↑ = 2 cos2(θ/2)+ sin2(θ/2) λ2,
dlnl 1 p ⊥ (9)
where Sz = (d†↑d↑ − d†↓d↓). The last term of Eq. (5) dλz↓ = 2(cid:20)cos2(θ/2)+ 1−−psin2(θ/2)(cid:21)λ2,
des
ribes the Zeeman energy splitting of the dot. In dlnl 1+p ⊥ (10)
(cid:20) (cid:21)
what follows we mainly
onsider a symmetri
situation dλ 1 p
in wΓh=i
hΓLtγ+=ΓtR. We deΓnLot=e tπhte2(tρo+ta+l tρu−n)neling strength dln⊥l = (cid:20)cos2(θ/2)+ 1+−psin2(θ/2)(cid:21)λ⊥λz↑
by with . The e(cid:27)e
tive
1+p
magneti
(cid:28)eldsgeneratedbytheferromagneti
ele
trodes + cos2(θ/2)+ sin2(θ/2) λ λ ,
~e =(~n +~n )/~n +~n ⊥ z↓
z L R L R 1 p (11)
isthenalongtheaxis | |. Wenext (cid:20) − (cid:21)
perform a poor man s
aling analysis.
where wλ⊥e h=av√eρi+ntρr−oJdu⊥
eλdzt↑h/↓e=dimρ±enJszi,o↑/n↓less Klon=doΛ0
/oΛu-
plings , and .
III. POOR MAN SCALING ANALYSIS
The dimensionless Kondo
ouplings are all driven to
T
K
strong
oupling at the same energy s
ale whi
h
an
The analysis extends in a straightforward manner the bedeterminedexpli
itlybyintegratingouttheaboveRG
one developed in Ref. [14℄. In this part we negle
t the equations. The Kondo temperature
an be expressed as:
energyρdependρen
e in the density of states and also sup- arctanh(pcos(θ))
ppo=se(ρ+γ± ρ≈−)/±(ρ.+I+nρt−h)e ferromagneti
leads, the ratio TK(p,θ)=Λ0exp − 2λ0pcos(θ) , (12)
− isin generaldi(cid:27)erentfrom zero (cid:18) (cid:19)
and as shown bellow it is one of the relevant parame-
λ0 =J0(ρ++ρ−)
ters to des
ribe the e(cid:27)e
t of the spin polarization. For where . Thisexpressiongeneralizesthe
θ
an energy dependent density of states, the situation is one obtained in Ref. [14℄ to all values of . The depen-
p,θ
more
omplex as will be shown in Se
. IV. One per- den
e of the Kondo temperature with appears as a
pcos(θ)
forms a two stage renormΛalizationDgroup (RG) analy- fun
tion of the singleθparTameter and istherefTor0e
sis by redu
ing the
uto(cid:27) from the bandwidth to auniformfun
tionof . K takesitsmaximumvalue K
ε¯ =ε +ε p θ = π/2
d d↑ d↓
whi
h renormalizesthe parameters of the independent of at whi
h
orresponds to an-
ε ε T θ = 0
↑ ↓ K
Andersonmodel. Ingeneral, and renormalizedi(cid:27)er- tiparallel magnetizations and is minimum for
ently due to the spin dependent density of states (DOS) as expe
ted.
3
f(ω)
IV. NUMERICAL RENORMALIZATION where istheFermifun
tion. Notethatthedensities
GROUP ANALYSIS ofstatesintheleadsarenolonger
onstantwhi
hismore
20
realisti
to des
ribe ferromagneti
leads. For the sake
In order to
ompute dot spe
tral densities and trans- ofsimpli
ity, from hereonwetakethe
hemi
alpotential
portpropertieswepro
eedwithaNRGanalysis. Atthis equal to zepro. Note that for this parti
ular
hoi
e the
pointitis
onvenienttoperformtwounitarytransforma- parameter de(cid:28)ned in the previous se
tδioǫn is zero while
tions on the initial Hamiltonian de(cid:28)ned in Eq. (1). We the spin polarization is not. For small anPd tem4δpǫera-
(cid:28)rst make an even/odd transformation introdu
ing the tures the magnetization is simply given by ≃ πD. In
operators the present
ase, the most important e(cid:27)e
t of Bthe leads
int
c =(c c )/√2. magnetismistheo

urren
eofaninternal(cid:28)eld that
k,e/o,± k,L,± k,R,±
± (13)
reatesthe e(cid:27)e
tive Zeeman splitting alreadymentioned
above.
We next introdu
e a new basis de(cid:28)ned by
To
al
ulatethedotdensityofstatesweneedthelo
al
bk,e,↑/↓ =ck,e,± , bk,o,↑/↓ =ck,e,∓. Green fun
tion. In the standard form, we have
(14)
G−1(ω)=G−1(ω) Σ(ω),
σ 0σ − (19)
The tunneling Hamiltonian
an be written in the new
basis as: Σσ(ω) G0σ(ω)−1
where is the self-energyand isthe non-
H = (t b† d +H.c.)
tun η kη,σ σ (15) intera
ting Green fun
tion
k,ηX=e/o,σ 1
G0σ(ω)= .
te = t√2cos(θ/2) to = t√2sin(θ/2) ω−ǫd,σ−gµBσ−∆σ(ω) (20)
where , and . The
θ

ru
ial point is that the angle arising due to the non ∆σ(ω)
The hybridizationfun
tion isspin-dependentand

ollinear lead magnetizations is now hidden in the tun-
te to has in general a non-zero real part that is given by
neling amplitudes and .
[∆ (0)]=σ4t2δǫcos(θ)/D2,
Note that in this formulation the dot is
oupled with R σ (21)
a single e(cid:27)e
tive lead with an angle and spin dependent
ω 0
bulk spin DOS given by for our
hoi
e of the DOS and ∼ . This is equivalent
ρeff =ρ cos2(θ/2)+ρ sin2(θ/2), to an angle-dependent e(cid:27)e
tive magneti
(cid:28)eld
↑/↓ ± ∓ (16) µ B =8t2δǫcos(θ)/D2,
B int
(22)
whi
h is mixture of the initial bulk spin densities of
states. This allows us to treat this problem in a sim- that polarizes the dot's spin and destroys the Kondo ef-
µ B &T
B int K
pli(cid:28)ed way using the numeri
al renormalization group fe
t for .
21,22,23
method with the generalizationsto treat arbitrar- The (cid:28)rst thing we want to
he
k is whether the polar-
θ
24
ily shaped densities of states. ized leads with a given angle are able to split the dot
ρ (ω)= [G (ω)]/π
σ σ
We insist that su
h formulation has been made possi- DOS[ −I ℄. Ithasbeenshownthatpo-
bleonlybe
ausewe
onsidersymmetri
tunnelingampli- larized leads with antiparallel polarized dire
tions (
or-
t = t θ =π/2
L R
(tθud=esπ/2) we h.aNveotρee↑ftfha=t ρfoe↓rffanatnidpatrhaelleel(cid:27)pe
otliavreizmatoiodnesl orefssptoantdesin.1g3,1t4o Is there hae(cid:28)ren)itdeoinntoetrvspallitartohuenddotθd=enπsi/ty2
has spin symmetry. where this property holdρs? We have plotted in Fig. 1
↑
the dot spe
tral density in the spin up se
tor for dif-
θ
ferent values of . As expe
ted the low energy peak of
ρ ω =0 θ =0
↑
A. Study of the dot density of states is pinned at for but is shifted away from
ω = 0 θ
as soon as is de
reased. This is mu
h
learer in
From hereon we will
onsider the DOS along the po- the right inset of Fig. 1 where the peak is zoomed. ρ =
larizationdire
tion in ea
hlead to be given by the tight- ρ T+hρe splitting of the total dot density of states
↑ ↓
binding expressions is shownθin the left inset of Fig. 1 for four difθ-
ferent values of . The splitting gradually in
rease as
2 θ = π/2 θ = 0
2 ω+δǫ
ρ±(ω)= 1 ± , is de
reased from aθnd is maxπim/2um for .
πDs − D (17) For very small deviations of around , the e(cid:27)e
tive
(cid:18) (cid:19) µ B .T
B int K
magneti
(cid:28)eld andnosplittingisexpe
ted.
where δǫ+ = −δǫ− = δǫ is the spin-dependent shift of For
ollinearpolarizationdire
tions,ithasbeenshown
the bands with respe
t to the Fermi level. The quantity that this splitting
an be
ompensated by an external
δǫ P 15
magneti
(cid:28)eld restoring the Kondo e(cid:27)e
t. For general

hara
terizes the magnetization of the leads given
by dire
tionsoftheleadpolarizations,ane(cid:27)e
tivemagneti
∞ (cid:28)eld is also generated. This e(cid:27)e
tive magneti
(cid:28)eld
an
P =Z−∞dωf(ω)[ρ+(ω)−ρ−(ω)], (18) binet
hoemppleannseat(e~ndLb,y~nRa)ppwlyitihngaandireex
tteironnalopmpaogsniteeti
to(cid:28)e~elzd.
4
25 20
20
50 25
20ρ(ω) 234000 ρ(ω)↑121005 θ=θ0.=405.54θπ8=5ππ/2 15 ρ(ω)↑1105 µµµΒΒΒBBB === 148.. δ26ε6 δ tεδ2 εt 2t2
15 10 5 5
ω) θ=0 ω) θ=0
(↑ -00.01 -0.005 0 0.005 0.01 -00.01 -0.005 0 0.005 0.01 (↑10 0
ρ 10 ω/D ω/D ρ -0.005-0.0025ω0/D0.00250.005
θ=0
5
5
θ=π/2
0 0
-0.2 -0.1 0 0.1 0.2 -0.2 -0.1 0 0.1 0.2
ω/D ω/D
FIG.1: Dotdensityofstatesforthθe=spi0n,-0u.p45se5
πt,o0r.a4s85fuπn,
πti/o2n FIG.2: Dotdensityofstatesforthespin-upse
torasfun
tion
of energy for di(cid:27)erent values ofδǫ=0.5D . ofenergyfor di(cid:27)erentvaluesoθft=he0magneti
(cid:28)eldinthe
ase
ThesUhif=to0f.2theǫdb=an−dsUi/s2(cid:28)xedtot = 0.071 . Otherparameter with parallel magnetizations ( ). Otherparametersas in
are: , , and . Right inset: zoom Fig. 1. The spin polarization in the dot and the splitting of
ρ↑ θ
of for thρe↑same values of . Wθe
learly seeθth=esπh/if2t of the theKondopeak
anbe
ompensatedwith a magneti
(cid:28)eld.
maximum when de
reasing away from . Left
20
inset: Total densityof states as fun
tion of frequen
y for the 20
θ
tafoesumsropoaenfroaratesumtrhee.entei(cid:27)one
etdivveamluaegsnoefti
.(cid:28)TehldeidselnarsgiteyrothfastnattheseiKsosnpdliot 15 (ω) 1105 µµµµµµΒΒΒΒΒΒBBBBBB ==== == 0303 55....8484..995 5 δδ δδεεδδεε εεtt tt 22tt2222
↑
ρ
5
We therefore expe
t that the splitting of the Kondozero ω) θ=π/8
( 10
bias anomaly indu
ed by the polarized ele
trodes
an be ρ↑ -00.005-0.0025 0 0.00250.005
ω/D

orre
ted by an adequate magneti
(cid:28)eld for all values of
θ
. We have
he
kedthatthis propertyholdsusingNRG.
ρ
↑ 5
We have presented in Fig. 2 the DOS as fun
tion of
θ =0
energyfor (wherethesplittingismaximum)andfor
di(cid:27)erentvalueoftheexternalmagneti
(cid:28)eld. Foragiven
value of the external magneti
(cid:28)eld, the splitting of the 0
-0.2 -0.1 0 0.1 0.2
Kondo peak vanishes and the Kondo peak is restored at ω/D
ω =0 θ
. Thisisalsotrueforageneralangle (seeFig.3).
Thesepropertiesofthedot'sspe
traldensity will
learly θ=π/8
FIG. 3: Same as Fig. 2 with .
re(cid:29)e
t in the transport properties of the system as we
will see.
ρ (0) = ρ (0)
d↑ d↓
see. In our NRG formulation we have .
Sin
etheDOSintheleadsareenergydependent(seeEq.
B. Transport properties (17)), this equality does not prevent to have a non zero
p = 0
polarization following Eq. (18) though . For this
Γσσ′ = (Σσσ′)
Using the Keldysh formalism, it has been shown that
σa,sσe′,=the,hybridizationmatri
es −I (where
for the
ollinear magnetizations, as in the
ase of non- ↑ ↓) are diagonaland angle independent. This al-
T =0
25
magneti
leads, the (cid:28)nite temperature
ondu
tan
e
an lowsthe useof the Meir Wingreen formula.GT(0h)e2
beputintermsonthedot'sspe
traldensity.25Thisisnot
ondu
tan
eisthussimplyproportionalto| d | . The
T = 0
the
aseforarbitraryangles. Theproblemisthatfornon
ondu
tan
e has been
omputed using NRG and
L θ

ollinear magnetizations, the two spin
hannel of the is shown in Fig. 4 as a fun
tion of for di(cid:27)erent values
R δε
and leadsaremixedandthestandardapproa
husedto of the splitting . The
ondu
tan
e rea
hesthe unitary
θ = π/2
eliminate the lesser propagator
an not be applied. To limit for (antiparallel
ase) and is minimum
our knowledge, in these
ir
umstan
es, the only dire
t forparallelmagnetizations. Thoughthis resultmaylook
wayofevaluatingthe
ondu
tan
eat anytemperatureis unusual, this is expe
ted sin
e the splitting is zero for
θ = π/2
through the Kubo formalism. This requires evaluating restoring the Kondo e(cid:27)e
t and is maximum for
θ = 0 p(0) = 0
two parti
le propagators, a matter that requires sophis- . Remember that with the de(cid:28)nition of
ti
ated numeri
al
odes. There are however some limit- the DOS we used in NRG.
ing
ases that illustrate the general behavior as we will Thisresultshouldbe
ontrastedtothesituationwhere
5
1
δε=0.1D anomalous denominator
omes from the fa
tGthre imagi-
δε=0.25 D narypartofthe dotretardedGreenfun
tion dσ issim-
δε=0.5 D 1/Γσσ T =0 θ
pply − Tat T (wp,hθi)
h depends on the ang(lTe/Tan)d2
K K
. At (cid:28)nite ≪ , deviations of order
G
u
from are expe
ted using the Fermi liquid theory. On
2e/h) the other hand, at high temperature T ≫ TK(p,θ), the
20.5
(
ondu
tan
e is mu
h smaller and
an be
omputed by
G/
renormalized perturbation theory:
2e2 3π2/16
G= (1+p2cos(2θ)) .
h ln2(T/T ) (24)
K
00 π/4 π/2 3π/4 π Noti
e thattheexpressionforthe hightemperature
on-
θ du
tan
e has the same angular expression as the one for
a tunnel jun
tion between two ferromagneti
leads
on-
versely to the low temperature
ase.
FIG. 4: Condu
tan
e as a fuδnǫ
=tio0n.1Dof,t0h.2e5aDn,g0le.5fDor di(cid:27)er-
ent values of the bands shift . Other
parameters as in Fig. 1.
V. CONCLUSION
anexternalmagneti
(cid:28)eldabletorestoretheKondozero
In this paper, we have studied a quantum dot in the
bias anomaly is added. We
onsider
onstant bulk DOS
as in Se
. III. Ele
trons with spins ↑ have a di(cid:27)erent Kondo regime between two non
ollinear ferromagneti
DOS at the Fermi energy than ele
trons with spins ↓, ele
trodesby
ombiningpoormans
alingandNRGte
h-
p(0) = 0 T = 0
resulting in 6 (as in [13,14,15℄). At , the niques. We have shown using NRG that the Kondo zero
bias anomaly is in generalsplitted for all angles between
Kondo singlet is therefore formed and the Fermi liquid
G
the two lead magnetizations (ex
ept for the antiparallel
theory
an be applied. The
ondu
tan
e through the
T = 0 G
u
ase). This splitting
an be
ompensated by an external
devi
e rea
hes at its maximum
ondu
tan
e
magneti
(cid:28)eld. Based on the spe
tral densities, we have
whi
h reads:
2e2 1 p2 addressedtransportpropertiesforboth
ompensatedand
Gu = h 1 p2−cos2(θ) . (23) un
ompensated
ases.
− Note added Some of the results presented in this pa-
G θ = 0 per have been also independently obtained in a re
ent
Noti
e that here reap
hes the unitary limit foθr= π/2 preprint27.
whatever the value of and is minimum for
p G
at (cid:28)xed . In order to determine to determine u, we A
knowledgments. (cid:21) PS would like to thank interest-
have extended the s
attering approa
h developed by Ng ing
ommuni
ations with J. Martinek. Part of this re-
and Lee26 to polarized ele
trodes following [15℄. This sear
hwassupportedbythe
ontra
tPNANO`QuSpins'
of the Fren
h Agen
e Nationale de la Re
her
he.
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