Table Of ContentSpin-flip Effects in the Mesoscopic Spin-Interferometer.
A.Aldea∗,†, M.Tolea∗ and J.Zittartz†
∗ National Institute of Materials Physics, POBox MG7, Bucharest-Magurele, Romania
† Institut fu¨r Theoretische Physik, Universit¨at zu K¨oln, D-50923 K¨oln, Germany
5 We investigate the properties of the electron spin-transmission through an Aharonov-Bohm in-
0 terferometer with an embedded two- dimensional multilevel quantum dot containing magnetic im-
0 purities. A suitable formalism is developed. The amplitude and the phase of the flip- and nonflip-
2 transmittance are calculated numerically as function of the magnetic field and the gate potential
n appliedonthedot. Theeffectsinducedbytheexchangeinteractiontospin-dependentmagnetocon-
a ductancefluctuationsand transmittance phase are shown.
J
9 PACSnumbers: 85.75.-d,73.23.-b
1
] I. INTRODUCTION
l RING-DOT coupling
l
a
h The spin interferometry in mesoscopic systems is ex- ( 2 )
- pected to give new insights in the field of nano-physics.
s (in) (out)
e We approach this problem in a ring-dot geometry with
m theaimtoidentifynewpropertiesoftheAharonov-Bohm
. oscillations due to the presence of magnetic impurities ( 1 )
t
a withina multiple leveldot; weshowalsothe ensembleof
m parameters that determine the problem.
- The Aharonov-Bohm interferometer with embedded
FIG. 1: The sketch of the spin interferometer; the incident
d
quantum dot has been used to study the phase of the
n electron is spin-polarized; the dot contains a localized mag-
o electron transmission as a voltage applied on the dot is neticimpuritywhichcoupleswiththespinofthefreeelectron.
c varied [1, 2], however no attention was paid till now to Amagneticfieldpiercesperpendicularlytheringandthedot.
[ the spin transmittance or even to magnetoconductance
fluctuationsinsuchtunablesystems,wherethefinitesize
3
v of the dot plays an effective role. The set-up is sketched thatpiercesboththeringandthedotandonthegatepo-
3 inFig.1;thedotcontainsmagneticimpurities,whichim- tentialappliedonthedot;atthesametime,itisstrongly
0 pliesanexchangeinteractionbetweenthespinofthefree controlled by the strength of the ring-dot coupling (τ)
2 incident electron and the localized spins in the dot. In and the energy of the incident electron(EF).
2
the real systems, the dot has a finite size so that the As explained throughout the paper, the range of pa-
0
3 electroncrossesthedotalongdifferentpaths,withimpli- rameters is large and different regimes can be studied.
0 cationsfortheresultinginterferencepattern. Thisiswhy For instance, in the case of small τ, one may consider
/ theusualmodelwhichassimilatesthedotwithaone-site thatthering-dotcouplingonlyproducesbroadeningand
t
a impurity (with at most two-orbitals) [3] cannot capture slight shifts in the spectrum of the individual dot. In
m
alltheinterferenceeffectsinactualring-dotsystems. The the opposite case, this picture is no longer valid and the
- price to be paid for considering a realistic geometry for ring-dot system has to be regardedas a unique coherent
d
n the dot is to make a single electron approach; we men- system,meaningthatτ mustbe consideredinallpowers
o tion in this respect the recent paper by Nakanishi et al oftheperturbationseries. Thehybridizationdependson
c [4]whousesacontinuousmodelforthestudyoftheFano theelectricpotentialVg sincetheappliedpotentialshifts
v: effects. the dot levels. However, for a two-dimensional quantum
i The spin-dependent transmittance amplitude can be dot, the same can be achieved by the variation of the
X
written as a superposition of all possible paths: magnetic flux, as put into evidence by a recent experi-
r ment carried out in the same ring-dot geometry [5]. We
a
tσ,σ′ =t1δσ,σ′ +Xtkσ,σ′ ei(Φ+∆φkσ,σ′), (1) shallfocusourstudyonthespindependentmagnetocon-
ductance as function of flux at fixed gate potential.
k∈2
Since severalenergyscalescome intothe problem,one
where t1 is the amplitude of the transmittance through has to establish from the very beginning the assump-
thelowerarmwhichconservesthespin,Φisthemagnetic tions of the approach. We consider that the main pro-
flux that pierces the ring and ∆φ is the supplementary cesses which are to be taken into account are the orbital
contribution of the dot, where spin-flip processes may motion in magnetic field along the ring and inside the
occur. dot, and the exchange interaction J. We assume that
Theaspectoftheoutputdependsonthemagneticfield the Zeeman effect is less effective than the exchange, i.e.
2
gµ B/J <1 [6];thespin-orbitinteractionisneglected, rying the spin σ passes through the interferometer and
B
too. Weuseatight-binding(TB)descriptionthatproved interacts via the exchange J with the localized spin S.
to be efficient in describing interference effects for differ- So we have to project the Hamiltonain on the product
entlyshapedsystemsandinthe presenceofdisorderand Hilbert space describing a single free spin and a single
interactions [7, 8]. The magnetic field (in the Landau localizedspin: σ S . This space is spannedby
H2N ⊗ H2S+1
gauge) appears as a Peierls phase Φ in the hopping in- thebasis i,σ > χ . Forthisreason,inthepresent
ij S
{| }⊗{ }
tegral,andonlythenext-neighborhoppingisconsidered. sectionweshallusethebra/ketrepresentationinsteadof
Thedotisanislandof3 5sitesattachedontheexternal the creation/annihilationoperators.
×
sideofthering,withthemagneticimpurityplacedinthe For the calculation of the transmittance (in the
middle. Thedotarearepresents15%fromtheareaofthe Landauer-Bu¨ttiker formalism) additional terms are
wholedevice. The Hamiltonianofthering-dotsystemin needed in order to describe the external leads and their
perpendicular magnetic field reads: contacts to the ring-dot system. It is however sufficient
touseaneffectiveHamiltoniandependingonlyonthede-
H = X wij,σ ei2πΦijc†i,σcj,σ+ X Vgc†i,σci,σ− greesoffreedomofthering-dot,butcontainingthewhole
hi,ji,σ σ,i∈QD information about the leads and lead-ring coupling in a
J (c† c c† c )Sz J (c† c S++H.c.), non-Hermitian term [9]:
− n↑ n↑− n↓ n↓ n− n↓ n↑ n
n QD. (2)
∈ Heff =H +τ2 e−iq ασ ><ασ , (3)
0 X | |
where c† (c ) are creation (annihilation) operators in α,σ
i i
localized states indexed by i ; the index n is devoted to
where q is defined by E =2cos(q) and α denotes the
the site of the dot where the magnetic impurity -called F { }
sites where the leads are connected to the system. τ
alsothe’flipper’-isplaced. Thefirsttermdescribesboth 0
is the parameter describing the lead-ring coupling and
the 1D ring and the dot (i,j = 1..N), the second term
it will be taken τ = 1. Taking into account that the
allowsforthegatepotentialonthedotandthelastterms 0
lead-ring coupling conserves the spin, only the diagonal
represent the local spin-spin interaction. The hopping
elements are modified by the coupling τ , and one can
integral will be taken as energy unit, i.e. w = 1 for 0
ij,σ
define the non-flip effective Hamiltonian :
any pair (i,j) excepting at the contacts between the dot
and the ring, where w =τ [0,1].
∈ Heff = w ei2πΦij i ><j + V i ><i
The outline of this paper is as follows. In section II ↑↑ X ij,↑ | ↑ ↑| X g| ↑ ↑|
we adapt the formalism used previously for the spinless hi,ji i∈QD
problem [9] to the spin transportthrough meso systems. JSz n ><n +τ2 e−iq α ><α , (4)
This yields a fast way to obtain numerical results in the − n | ↑ ↑| 0 X | ↑ ↑|
α
presence of spin scattering based on the calculation of
the resolvent in a reduced Hilbert space and Landauer- while the spin-flip Hamiltonian is simply:
Bu¨ttiker formalism. The existence of different regimes
is shown. Section III introduces the singlet and triplet H↑e↓ff =H↑↓ =−JSn−|n↑><n↓|. (5)
operatorsandexpressesthe flipandnon-flipprocessesin
termsoftwo-timeGreen(singlet&triplet)functionswith In order to calculate the spin transmittance through the
their corresponding equations of motion. We also show complexsystemoneneedstoknowtheretardedresolvent
how the single electron scattering problem is recovered Green function:
from the general many-body scheme. An application is
done for an analytically soluble model in sec IV, with G+(E)=(E Heff +i0)−1.
−
the aimto produce hints for the ring+2Ddotinterferom-
eter. This complex system is studied numerically in sec Then, in order to express the different flip and non-flip
V, where spectral and transport properties are shown, tunnelingprocesseswedefinethe2x2matrixGreenfunc-
indicating the role of different parameters like the Fermi tion
energy and the coupling constant τ. The main conclu-
sions are presented in the last section. Gσ,σ′(z)=:<σ G(z)σ′ >
| |
for which Dyson equations can be written immediately
II. SPIN TUNNELING IN THE RESOLVENT andtheexpressionsforthenonflipG (z)andflipG (z)
↑↑ ↑↓
REPRESENTATION AND DISCUSSION OF read :
DIFFERENT REGIMES
−1
G (z)= z Heff H (z Heff)−1H
In what follows we shall reduce the many-body prob- ↑↑ (cid:16) − ↑↑ − ↑↓ − ↓↓ ↓↑(cid:17)
lem to the physical situation when a ”test” electron car- G (z)=G (z)Heff(z Heff)−1. (6)
↑↓ ↑↑ ↑↓ − ↓↓
3
(a)
Using (5) these equations can be further expressed as:
3
G (z)= z Heff J2S− n >
↑↑ (cid:16) − ↑↑ − n| ↑ 2.5
1 −1 2
G (z)=<JnG↓|z(−z)SH−↓e↓fnf|n>↓<>n<n↑|Sn+1(cid:17) ,. (7) 11.5Magnetic flux
↑↓ − ↑↑ n| ↑ ↓|z Heff 0.5
In order to proceed we shall consider S−=↓1↓/2. Since -2 -1.5 -1 -0.5Vgate0 0.5 1 1.5 2 0
n (b)
the exchange interaction conserves the total spin and its
projectionon the z-axis,there arethree different tunnel-
ing processes between the following in and out states: 3
2.5
a) α > α′ > (non flip process)
cb)) |||αα↑↑↑⇓⇓⇑>> −−−→→→ |||αα′′↓↑↑⇑⇓⇑>> ((fnloinp−−prfocliepssp)rocess) (8) 11.25Magnetic flux
Tmhitetafinrcset twwhoilperothceesstehsircdonptrroibcuestse ttoo tthheenfloinp-fltirpantsrmanist-- -2 -1.5 -1 -0.5Vgat0e 0.5 1 1.5 2 00.5
tance. We remind that the pair of sites (α,α′) represent
FIG. 2: The map of the nonflip-transmittance in the plane
the points where the ring is connected to the terminals (Vgate, Magnetic flux) for two values of the Fermi energy:
and that the incoming spin is supposed to be polarized (a) EF = 0 and (b) EF = 0.5. The transition from the
in the up( )-state. The notation >, > stands for resonant to anti-resonant behavior can be noticed; the other
the eigenve↑ctors of the impurity sp|in⇑ope|ra⇓tor Sz. parameters: τ =0.2,J =−1.
n
Finally,thespintransmittanceisobtainedbycalculat-
ianngdtohuetmstaattreisx,ealnedmeunsitnsgofthGemσ,σi′ninth(e7)gebneetrwaeleenxpthreessseioinn by the external leads). The multitude of parameters
may give rise to different regimes and as an illustration
of the Landauer-Bu¨ttiker formula:
we show in Fig.2 a map of the nonflip-conductance for
Tσ,σ′S,S′(E,Φ)=4τ04sin2q |<α,S|G+σσ′(E)|α′,S′ >|2 tawroesdoinffaernetntbevhaaluveiosrof(pEaFn.elOan)etnootaincesanthtie-rcehsoannagnetfroonme
(9)
(panelb). Atthesametime,wehavefoundthattheflip-
As an explicit example of how the matrix elements look
conductancekeepstheresonantcharacterforbothvalues
like, we give :
of the Fermi energy.
<α G (z)α′ >= The understanding of the 3D picture in Fig.2 is rel-
σσ
↑⇑| | ↑⇑
atively simple; one has to analyse separately the role
<α z w ei2πΦij i ><j
↑| (cid:16) −X ij,↑ | ↑ ↑|− played by the ring and by the dot in the transmittance
hi,ji process. Let us call T(I) the transmittance of the device
J −1 σσ′
τ 2 eik α ><α + n ><n α′ >. in the case τ = 0 (when the dot is completely detached
0 X| 1 ↑ 1 ↑| 2 | ↑ ↑(cid:17) | ↑ fromthering)andexpressformallythetransmittanceas
α1
the sum of two contributions :
(10)
The value of this Green function element can be cal- Tσσ′(E,Φ)=Tσ(Iσ)′ +τ2 Tσ(IσI′), (11)
culated by the direct numerical inversion of the matrix
where the second term is added when τ =0. Obviously,
which is now completely known. 6
T(I) doesnotdependonthe fluxbutmaydependonthe
The Aharonov-Bohm interference is impossible when σσ′
Fermi energy. Let us consider two situations :
the dot does not transmit, since the circulation of the
a) the energy of the incident electron is such that
vector potential is to be considered along a closed con-
tour. The transmittance of the dot can be controlled by T↑(↑I) ≈ 0; then the second term describes the resonances
changing the gate potential applied on the dot. There- of the ring-dot system, with a gate potential applied on
fore,thetransmittancespectrumintheplaneofthevari- the dot only.
ables V (gate potential) and Φ (magnetic flux through b) the energy is such that T(I) < 1; since the total
g ↑↑
the ring) has to be analyzed first. transmittance cannot be larger than∼1, the only possible
Thespaceofparametersis coveredbyτ (ring-dotcou- effectwhichcanbeproducedbyT(II)isanantiresonance.
↑↑
pling), J (exchange) and E (the Fermi level imposed We succeeded to catch these two extreme cases in
F
4
Fig.2a and b, respectively, by choosing two different val- by the following zero temperature two-time Green func-
ues of the Fermi energy. More complicated 3D patterns tions:
appear in intermediate situations. In what concerns the
flip transmittance, obviously T(I) = 0 so that only the Gaij(E)=≪d↑ci↑;c†j↑d†↑ ≫E
↑↓
situation in panel (a) may occur and the flip transmit- Gb (E)= d c ;c† d† (16)
ij ≪ ↓ i↑ j↑ ↓ ≫E
tance will show always a resonant character.
Gc (E)= d c ;c† d† ,
ij ≪ ↑ i↓ j↑ ↓ ≫E
where the ground state on which the Green functions
III. SINGLET-TRIPLET REPRESENTATION FOR are defined should be explained. From now on the state
THE FLIP AND NON-FLIP PROPAGATORS. > , on which the Green functions are defined, will be
0
|
chosen as the vacuum with respect to both c- and d-
In what follows we shall explain the Singlet-Triplet fermions. This is the so-called ’empty band’ approach
structure of the energy spectrum, the implication of the for the electronic problem. With this choice the prop-
spectrum for the transport properties and the physical agators (16) describe the propagation of a ”test” elec-
conditions (T = 0 and ’empty band’ ground state [12]) tron through the interferometer containing the localized
under which one can get rigorous results. spin. The same single electron approach is considered
ForS=1/2thenaturalformulationoftheproblemisin by Menezes et al [15] and Joshi et al [16] when using
terms of singlet-triplet operators (see for instance [13]). the waveguide approachfor the ring+flipper problemin
One can define the fermion operators d†,d†,d ,d for thecontinuumrepresentation(theydonotemphasisethe
thelocalizedspin,providedoneproject{so↑uta↓llt↑he↓st}ates natural singlet-triplet formulation of the problem). Us-
with occupancy different from 1, i.e. one has ing the same approachNolting obtained rigorous results
forthethe electronexcitationspectrumofferromagnetic
n =d†d +d†d =1. (12) semiconductors [12].
d ↑ ↑ ↓ ↓
Thegenericequationofmotionsatisfiedbythesefunc-
tions is (see the discussion in Appendix):
Then, Sz, S+and S− in eq (2) can be written as:
n n n
E A;B =<[A,B] > + [A,Heff] ;B .
1 ≪ ≫E + 0 ≪ − ≫E
Sz = (d†d d†d ) (17)
n 2 ↑ ↑− ↓ ↓ As an example, the equation of motion for Gc reads:
S+ =d†d , S− =d†d . (13)
n ↑ ↓ n ↓ ↑
(E V )Gc = w (Φ)Gc +τ2e−iq δ Gc
− g ij X ik kj 0 X iα ij
Simultaneously we introducethe singletoperatorΣi and k α
the triplet operators Tp (p=1,2,3) J J
i +δ Gc δ JGb δ d†d d c ;c† d†
in2 ij − in ij − in2 ≪ ↓ ↓ ↑ i↓ j↑ ↓ ≫
Σi = √12 (d↑ci↓−d↓ci↑) +J ≪c†n↓cn↑ci↓d↓;c†j↑d†↓ ≫+J ≪d†↑d↑d↓ci↑;c†j↑d†↓ ≫
J
1 d ( c† c c† c )c ;c† d† (18)
Ti1 = √2 (d↑ci↓+d↓ci↑) −2 ≪ ↑ n↑ n↑− n↓ n↓ i↓ j↑ ↓ ≫
T2 =d c With our choice for the ground stare, exact results are
i ↑ i↑ obtainedastheequationofmotiongetclosedwithoutany
T3 =d c . (14)
i ↓ i↓ approximation. Indeed,nowall”higher”Greenfunctions
The above operators may be used in order to write d ( c† c c† c )c ;c† d† ,
≪ ↑ n↑ n↑− n↓ n↓ i↓ j↑ ↓ ≫
the Hamiltonian(2)inthe followingway(where wehave
c† c c d ;c† d† , (19)
used also n =d†d +d†d =1): ≪ n↓ n↑ i↓ ↓ j↑ ↓ ≫
d ↑ ↑ ↓ ↓
etc, vanish.
H = (wΣ Σ†Σ +wT Tp†Tp) We remind that if the Fermi sea is considered as the
X ij i j ij X i j
ground state, the ”higher” Green functions do not van-
hi,ji p
ishanymoreandthe solutioncanbe foundonlyapprox-
3
wiΣj =wij ei2πΦij +δijVg+ 2Jδijδin imately, using specific decoupling procedures ( see, for
instance [13, 14]).
1
wiTj =wij ei2πΦij +δijVg− 2Jδijδin . (15) Eq.(18) becomes :
J
In the representation of {c,d}-fermions, the propaga- X(cid:2)(cid:0)E−Vg−τ02e−iqXδiα− 2δin(cid:1)δik−wik(cid:3)Gckj(E)
tion of the free electron corresponding to non-flip pro- k α
cessesa)andb)andtothespin-flipprocessc)isdescribed = δ J Gb (E) . (20)
− in ij
5
One can verify that <Σ H Tp† > = 0, meaning that
i j 0 2.5
the Hilbert space H2σN ⊗ H2S splits as follows: 2 T
1.5
S
H2σN ⊗ H2S =HNΣ ⊕H3TN. 1
gies 0.5
The singlet and triplet-type eigenenergies can be ob- ner 0 T
tained by the diagonalization of the matrices wiΣj and Eigene-0.5 S
wT, respectively. -1
ij T
Itisusefultodefinethesingletandtripletpropagators -1.5
S
-2
GΣ(E)= Σ ;Σ† , GT(p)(E)= T(p);T(p)† -2.5
ij ≪ i j ≫E ij ≪ i j ≫E 0 0.5 1 1.5Magne 2tic Flux 2.5 3 3.5 4
asthese functions satisfydecoupledequationsofmotion:
FIG.3: Theenergyspectrumofthesimplest(triangular)spin
N interferometer as function of the magnetic flux. The singlet
(Eδ wΣ τ2e−iq δ δ GΣ(E)=δ ,(21) (S) and the triplet (T) eigenvalues are indicated. The three
X(cid:2) ik− ik − 0 X iα ik(cid:3) kj ij horizontal lines represent possible positions of E ; exchange
k=1 α parameter J =−0.5. f
N
Eδ wT τ2e−iq δ δ GT(p)(E)=δ ,(22)
X(cid:2) ik− ik− 0 X iα ik(cid:3) kj ij
k=1 α arealsodegeneracypointswhichareindependentofJ as
wherewΣ andwT aredefinedineq(17). TheGreenfunc- we shall explain below.
Inordertostudythetransport,weconnectidealleads
tions for the three tripletstates are identical, so we shall
to the sites we shall refer to as ”1” and ”3” using the
skip the upper index of T. When obtaining equations
hopping constant τ . The flipper is located on the site
(21) and (22) the symmetry relations 0
”2”. The sites of the triangle are interconnected by the
d c ;c† d† = d c ;c† d† hoppingparametere±i2πΦ/3 (thetotalencircledfluxΦis
≪ ↓ i↑ j↓ ↑ ≫ ≪ ↑ i↓ j↑ ↓ ≫ measured in quantum flux units). We find the solution
d c ;c† d† = d c ;c† d†
≪ ↓ i↑ j↑ ↓ ≫ ≪ ↑ i↓ j↓ ↑ ≫ of eq.(21-22) as :
have been used. e−i2πΦ+(E αΣ,T)
It turns outfromtheir definitions that the three prop- GΣ,T(E)=ei2πΦ/3 − , (24)
13 ∆
agators describing the processes a), b) and c) can be ex-
pressedinterms ofsingletandtripletpropagatorsasfol-
lows ∆=(E τ2e−iq)2(E αΣ,T) 2(E τ2e−iq)
− 0 − − − 0
(E αΣ,T) 2cos(2πΦ), (25)
Ga =GT − − −
Gb =(GT +GΣ)/2 (23) with αΣ = J3 and αT = J1 . One can notice that,
2 − 2
Gc =(GT GΣ)/2 while the module of GΣ,T(E) is symmetrical under flux
− reversal and has the Φ =1 flux periodicity, its phase is
0
The above relations show that the non-flip process a) not symmetrical under flux reversal and has a 3Φ0 pe-
is purely triplet-type since only the matrix wT enters riodicity due to the factor ei2πΦ/3. On the other hand,
the dynamics of the propagator Ga, while the non-flip the phase difference between GT and GΣ i.e., ∆λTΣ =
(cid:0)
processb)andtheflipprocessc)involveboththe triplet atan(ImGT/ReGT) atan(ImGΣ/ReGΣ) ,whichisim-
− (cid:1)
andsingletstates. Wenotethatfrom(22)werecoverthe portant for the processes b) and c) (as expressed by
expression of GT given by (10) in the previous section. eq.23), is periodic with Φ but remains asymmetric un-
0
der flux reversal. This explains why the transmittances
assignedtotheprocessesb)andc)areasymmetricunder
IV. THE SIMPLEST SPIN INTERFEROMETER flux reversal,as also found by Joshi et al[16].
For τ = 0 the zeros of ∆ in Eq.27 give the energy
0
By the use of eq.(21-22) and (23) it is now very easy spectrum. For integer (semi-integer) values of the flux
toanalyseanalyticallythesimpleexampleofatriangular the energy E = 1(E = 1) is an eigenvalue indepen-
−
spin interferometer placed in magnetic field. In this way dentofαΣ,T meaningthatthe triplet andthe singletare
we get hints for understanding the complex system. degenerated as it can be seen in Fig.3.
We present first the energy spectrum (with Singlet- Looking at the spectrum, we can realize that GΣ and
Triplet structure) versus the enclosed flux (Fig.3). The GT (that is also the propagator for the a) process ) will
distance between a Singlet-Triplet pair of levels is, of show,eachofthem,oneortwopeaksperperiod,depend-
course,controlledby the exchangeparameter, but there ing on the E (see the horizontal lines in the spectrum
f
6
figure). As a result, Gc and Gb may show up to four
1 1
peaks per period in the case of the week coupling with
a
the leads. However, for strong coupling, neighbouring 0.5
singlet and triplet peaks will overlap with either con-
0
structive or destructive effect, depending on the phase
difference ∆λ (see Fig.4c). -0.5
TΣ
As the magnetic flux is changed, the T and S levels
-1
cross the Fermi energy giving rise to the addition or loss
of one electron at each cross. Taking into account that 1
the addition of an electron corresponds to a phase in- se b
a 0.5
crease of π (and vice versa) the evolution of ∆λTΣ in Ph
Fig.4ais easily understood,whenthe peaks arewellsep- e, 0
c
arated in the case of weak coupling. As we increase the an
coupling, the peaks are broadening and we can see in mitt -0.5
s
Fig.4c that the first pair of peaks evolves into a single an -1
bigpeakwhile thesecondpairvanishescorrespondingto Tr
1
a phase difference ∆λTΣ 0. c
≈
0.5
0
V. THE SPECTRUM AND SPIN
TRANSMITTANCE OF THE RING+2D DOT -0.5
INTERFEROMETER
-1
Inthefollowingweshallcontinuethenumericalanaly- 0 0.2 0.4 0.6 0.8 1
sisofthe realsystemconsistingofthe ring+2Dquantum Magnetic Flux
dot(withmagneticimpurity). Inordertoshowthecom-
FIG. 4: Flip transmittance (solid line) for the triangular
plexity of the problem, we present first a detail of the spin interferometer, plotted together with the phase differ-
energyspectrumofoursystem(Fig.5). InFig.5athedot ence ∆λTΣ (dashed line, in π units) for three couplings with
is decoupled from the ring; the levels of the 2D finite the leads : a)τ0 = 0.2, b)τ0 = 0.4, c)τ0 = 1.0; the other
dot (shown in solid line) depend on the magnetic field parameters are J =−0.5, Ef =0.
and can be indexed as Triplet and Singlet levels, while
the levels of the truncated ring (dotted line) are invari-
ant with Φ. Obviously, when we increase the ring-dot for studying the AB oscillations.
coupling τ the levels undergo hybridization. As long as Since the spin-up and the spin-down waves cannot in-
the ring-dot hybridization is weak (as in Fig.5b) and we terfere, being orthogonal and corresponding to two dif-
aresufficientlyfarawayfromtheself-avoidingpoints,the ferent channels, the flip-interference process can only
eigenvalues can still be identified as ring-like or dot-like. takeplacebetweentheflippedelectronwavetransmitted
The ring-like levels begin to oscillate with the (approx- through the dot and the flipped wave reflected by the
imate) Φ period and each one splits also into a Sin- dot; this introduces a difference in phase compared to
0
gletandaTriplet;obviouslythesplittingdependsonthe thenonflipprocessandexplainswhytheflip-interference
value of the exchange J, and on the hybridization de- may be destructive at values of the magnetic field where
termined by the ring-dot coupling τ and on the relative the direct (nonflip)process is constructive. Another ef-
distance between levels. fect is the accumulation of phase in the dot which will
For strong hybridization (see Fig.5c) the dot-like lev- affect the Fourier spectra of the spin-dependent magne-
els begin also to exhibit Φ oscillations superimposed toconductance fluctuations.
0
over their usual slow dependence on the flux observed The interference pattern as function of the magnetic
in Fig.5a. One can notice that the dot-like levels are in- flux through the ring depends specifically on the charac-
tercalatedbetweensinglet-tripletpairsofring-likelevels; teristics of the quantum dot, namely, by its coupling to
this occursbecause the exchangesplitting of the dot-like the ring and the properties of the energy spectrum de-
levels is much bigger than the splitting of the ring-like scribedabove. Implicitly,the positionoftheFermilevel
levels. In fact, for strong ring-dot coupling, the spec- becomes important.
trum becomes complicated and it is difficult to identify For weak ring-dot coupling, the AB oscillations come
any more the origin of different eigenvalues. from the participation in transport of the ring-like lev-
Now we continue the numerical analysis of the spin els that have a (approximate) Φ periodicity and show a
0
transmittance in the limit of perfect ring-leads coupling Singlet-Tripletsplitting (the caseshowninFig.5b). This
(τ =1), which meets the usual experimental conditions splitting is however very small since the localized mag-
0
7
0.4 (a) 1
T
0.3 0.9
non-flip
0.2 0.8
S
es 0.1 0.7
gi e
er 0 nc 0.6
n a
ene-0.1 T mitt 0.5
Eig-0.2 ans 0.4
-0.3 S Tr 0.3
-0.4 0.2
-0.5 0 5 10 15 20 25 0.1 flip
0.4 (b)
0
0.3 0 5 10 15 20 25
Magnetic flux
0.2
es 0.1
gi
er 0 FIG. 6: The two components -nonflip and flip- of the trans-
n
ne-0.1 mittanceasfunction of themagnetic fluxΦthroughthering
Eige-0.2 (parameters: τ = 0.4,J = −1,Vgate = 0,EF = −0.35). The
non-flip transmittance exhibits usual oscillations, while the
-0.3
flip has peaks when the dot levels cross theFermi energy.
-0.4
-0.5
0 5 10 15 20 25
0.4 (c)
0.3
0.2 1
es 0.1 0.9 (a)
gi
er 0 0.8
Eigenen--00..21 mittance 00..67
-0.3 ns 0.5
a
-0.4 p tr 0.4
-0.5 0 Magn e5tic flux 1(0in quan 1t5um flux 2 0units) 25 Nonfli 00..23
0.1
0
FIG. 5: Spectrum of the ring-dot (with magnetic impurity)
system for the cases: a) The dot is decoupled from the ring 0.07 (b)
(τ =0). Thelevelsofthe2Ddot(solidline)showfluxdepen-
dence and strong Singlet(S)-Triplet(T) splitting ; the levels 0.06
e
ofthetruncatedring(dottedline)areconstant. b)Moderate nc 0.05
a
ring-dotcoupling(τ =0.4). Bothdotandringlevelsundergo smitt 0.04
hybridization. Thelevelsoftheringhavea(approximate)Φ0 an
oscillationperiodandaverysmallS-Tsplittingthatbecomes p tr 0.03
morevisibleinthecase c)Strongring-dotcoupling(τ =1.0) Fli 0.02
with strong hybridization of levels.
0.01
0
0 5 10 15 20 25
Magnetic flux (in quantum flux units)
netic impurity is placed in the dot and acts like a small
perturbationforthe”ring”levels. Then,onecanassume FIG.7: Thetwocomponentsofthetransmittanceasfunction
that GT GΣ and the phase difference λ 0, ex- of the magnetic flux Φ through the ring: (a) nonflip T↑↑, (b)
cept |for t|h≈os|e in|tervals of flux where the doTt,Σle≈vels get flipT↑↓(parameters: τ =1,J =−1,Vgate=0,EF =0). Note
the doublepeak structure of the flip and the complementary
involved in transport. As a result the flip oscillations,
windows in therange [15-20Φ0].
being proportionalto GT GΣ , are much smaller than
| − |
the nonflip oscillations given by Ga or Gb.
The positionofthe Fermienergyis veryimportantfor nonfliptransmittancebehavessimilarly. Onthecontrary,
the general aspect of the oscillation pattern. When the the flip transmittance does notshow oscillationsbut has
Fermi energy crosses (or is close to) the oscillating ring- onlypeaksatvaluesofthefluxwherethedotlevelscross
like levels,both the nonflip andflip transmittancesshow E . In this way the energy levels of the dot can be
F
regularoscillations(muchreducedfortheflipcase),with identified by the flip transmittance. We illustrate this
anomalies in those domains of the flux where dot-like situation in Fig.6.
levels are also involved in transport . For the strong ring-dot coupling τ = 1, the spectrum
Evenif the Fermienergyis farfromthe ring levelsthe showninFig.5cplaysthegameandtheresultconsistsin
8
strongoscillationsforanyFermilevel. Fourpeaksperpe-
1
riod cannot be observedsince the splitting is small com-
0.9
pared to the broadening induced by the strong coupling (a)
0.8
of the system to the leads τ = 1. The general picture
0 0.7
is that one of single or double peak per period both for nce 0.6
the flip and non-flip transmittance, depending however mitta 0.5
on the EF. The double peak is usually asymmetric as in ans 0.4
Fig.4b of the triangle model. Tr 0.3
0.2
Apeculiarsituation,whenthespineffectisstrong,oc-
0.1
curs at E = 0 and is shown in Fig.7 . The difference
F 0
3 3.5 4 4.5 5 5.5 6
noticedbetween the panels (I) and(II) is due to the fact Magnetic flux
0.1
that the scattering processes (b=nonflip) and (c=flip) (b)
combine in different ways the singlet and triplet propa- 0.08
gators. The situation is presented at a better resolution m
u
inFig.8a,anditcanbe noticedthatT↑↓ is nearlyzeroin ectr0.06
p
zthereosaomf tehpelaflciepwtrhaenrsemTi↑t↑ta≈nc1e. oWccituhrsnwecheesnsityG,TthequGasΣi- ourier s0.04
| | ≈ | | F
and λT,Σ 0. Under the same circumstances, the non- 0.02
≈
flip process (b) described by GT +GΣ has a maximum
value. Thisexplainsthedestr|uctiveinte|rferenceofheflip 0
(c)
process at the same value of the flux where non-flip in-
0.008
terference process is constructive. It can also be noticed
m
that the transmittance varies strongly in some domains u
l(aatsiofonrinindsitcaantecsetbheattwtehene t1r5an−sm20isΦs0io)n.oAf tsheepadroatteiscmaluccuh- urier spectr00..000046
o
suppressed in these domains, while the amplitude of T F
↑↑
0.002
throughthe lowerarmis close to 1; these two facts yield
the picture shown in Fig.7a. On the other hand, when
0
0 1 2 3 4 5
the transmission through the dot is low, the probability
Frequency
of the flip process is reduced explaining the complemen-
FIG. 8: (a) The double-peak structure of T embraced by
↑↓
tary window of the T↑↓ in Fig.7b. the large peak of T↑↑ . The Fourier spectrum corresponding
We have checked that such windows are a combined tothecurvesshowninFIG.3: (b)thenonflipcomponent. (c)
effect of the presence of the dot and of the magnetic im- the flip component shows more noise and a reduced value of
purity;itturnsout,howeverthatforE =0thewindows the first maximum as the result of the specific interference
in the interference pattern comes fromFthe exchange. process (parameters: τ =1,J =−1,Vgate=0,EF =0).
One canobserve in the top panel of the Fig.8 that the
largepeakrepresentingthenonflip-interferenceembraces
the double peak representing the flip-process. The split- the Fourier spectrum of the AB oscillations, namely the
ting increases with the exchange J and hence the width positions of the maxima do not coincide any more with
of the large peak depends also on this parameter. theintegermultiplesofthe2π/Φ0,whicharecharacteris-
Animportantinstrumentforthe analysisofthetrans- tic to the individual ring. The shift can be noticed both
mittance oscillations is the Fourier transform [10]. The in Fig.8b and Fig.8c for the non-flip and flip transmit-
Fourier spectra of T and T are shown in Fig.8b and tance, respectively. In the mesoscopic physics, the Fano
↑↑ ↑↓
Fig.8c in the case of the perfect coupling between the effectistheconsequenceofthehybridizationbetweenthe
dot and the ring (τ =1). The fact that already the first continuum spectrum of the leads and the discrete spec-
interferenceprocessoccurswiththeparticipationofare- trum of the meso-system [11]. The asymmetric shape of
flectedwaveresultsinthereductionofthefirstmaximum thetransmittancepeaks,theoccurrenceofzeros,andthe
in the Fourier spectrum for the flip-channel. behavior of the transmittance phase on the resonances
The coupling τ is important for the aspect of the and between them are the aspects of interest.
Fourier spectrum because any constriction at the dot In the system under consideration, the dependence on
gives rise to a reflection of the spin-up electron, and the gate potential (at fixed magnetic field) of the trans-
changes the aspect of the Fourier spectrum of T , that mittance phases corresponding to different flip and non-
↑↑
might become similar to the spectrum of T . flip processes show the usual behavior and we shall not
↑↓
When the dot is attached to the ring the positions of insist on these aspects: the phase increases with π on
the interference peaks are shifted due to an accumula- each resonance and there are also discontinuous jumps
tion of phase in the dot; this effect is clearly visible in with π between the resonances with the same parity,
±
9
where the transmittance vanishes [8]. Since in our case 1 5 1
2
the magnetic field is non-zero,these jumps do not repre- a b
0.8 4
senttrueFanozerosbecausetherealandimaginarypart 0.5
onoffeoRtuheseTlytratanhndesmsIimgitntT.anoNcceecvuaermrsthpaeltiltecuslods,seewcvhaaenlnuneotstheocfhcathhnaegneggeastiimenubsliitagasn-, Transmittance 00..46 1 0 2 23Phase 0 -0 .05 Im T
0.2 1
thevariationofthephaseisstillabrupt. Then,thejump
1
of the phase can be upwards (with a gain of π) or down- 0 3.4 3.6 3.8M a4gne t4i.c2 flu x4.4 4.6 4.8 5 -1 -0.5 Re 0 T 0.5 1 -1
0.5 5
wards(withalossofπ);obviously,thedifferenceisgiven 0.4
by the sign of ImT at ReT =0 (which is positive in the 0.4 c 4 d
fibresOhtancvaitoshreeaanostdhfunenrecghtaiaotninvde,oifnthmtehaegpnhseeatcsioecnhdfiaeovldne.ea).Svoemrye caonmalpylseixs Transmittance 00..23 23Phase- 00 ..022 Im T
0.1 1
canbemadeforthesimpletrianglemodel,wherewecan
-0.4
studyseparatelythephaseofthedenominatorandofthe 0 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 -0.4 -0.2 0 0.2 0.4
Magnetic flux Re T
numeratorinEq.(24). The two contributionsareeasy to
FIG. 9: a) Transmittance and phase for the impurity-free
be identified for τ << 1 when the resonances are nar-
0 system (J=0); b) The corresponding evolution of the pha-
row. Then the phase of the denominator ∆ (eq(25)) has
sor (ImT vs ReT). The points 1,0,2 correspond to the same
a sudden evolution with π and π on consecutive reso- points in the panel (a). The arrows indicate the evolution
−
nances, but remains constant in between. On the other of the phasor with increasing flux in the range [3.3−5.0Φ0].
hand,the phaseofthe numerator,whichinfactdoesnot c) Transmittance and phase for the Triplet (solid line) and
depend on τ , has a very small evolution on the narrow theFlip (dashedline);d)Thecorresponding evolutionofthe
0
phasors. The phasor of the Flip (amplified for better visibil-
resonances and is responsible for the monotonic increase
(or decrease,depending onE ) ofthe phase ofGΣ,T be- ity) has four turning points corresponding to the four peaks
F 13 of the flip-transmittancein panel(c).
tween resonances. We note that the numerator of these
propagatorsvanishesonlyincidentallyforE αΣ,T = 1
− ∓
and integer (semi-integer) values of the flux, In the limit
interferometer consisting of a ring with embedded two-
τ 1 the width of the resonancesincreasescausing the
0
→ dimensional dot doped with magnetic impurities. For
overlapofneighbouringresonancesanda’bump’ instead
the time being, there are no experiments to be com-
of a π evolution of the phase at the variation of the
∓ paredwith,buttheyareexpectedinthenearfuture,and
flux.
our goal is to make predictions. The study is based on
The non-monotonicvariationofthe phaseonthe peak
the spectral analysis and a suitable transport formalism
indicatesthatthephasorofthetransmittanceinthecom-
which has been developed for the single-particle case.
plex plane (ReT,ImT)has a turning point (as shown in
Fig.9b and Fig.9d). TheaspectoftheAharonov-Bohmoscillationsdepends
When extending this analysis to the complex system strongly on the Fermi energy and the ring-dot and ring-
we realize that the numerator becomes a very compli- leadscouplingparameters. Generally,theoscillationsex-
cated trigonometric function and exact zeros, and even hibitsingleordoubleasymmetricpeaksperperiodasitis
quasi-zeros, may exist only incidentally. So, the general suggested by the analytically solvable triangular model.
aspect of the phase consists in a smooth dependence on Thefliposcillationsaremuchreducedinamplitudecom-
flux between two consecutive peaks and a bump on each paredtothenonflipones. Thefinitesizeofthedot,com-
peak. bined with the effect of the impurity, gives rise to win-
dows in the interference pattern where the flip and non-
The special situation when the phase indicates quasi
flip components of the transmittance behave in a com-
FanozeroshasbeenfoundnumericallyatE =0forthe
F
plementary way. In the case of weak ring-dot coupling
case of impurity free dot (J = 0); however, the usual
theABoscillationsoftheflipchannelmayceasetoexist,
behavior is recovered in the presence of the exchange
the flip magnetoconductance showing instead peaks cor-
(J = 0). The two situations are shown in Fig.9. In
6 respondingtothelevelsofthedot. Thegeneralaspectof
the same figurewe give alsothe phase ofthe process(c),
thetransmittancephaseoftheABoscillationsconsistsin
which is obtained by combining the phases and modules
of the GT and GΣ for the complex ring-dot system. bumpsontheoscillationpeaksandamonotonicbehavior
between them.
In order to explain these properties we haveexpressed
VI. CONCLUSIONS thedifferentphysicaltunnelingprocessesintermsofsin-
glet and triplet propagators,which is the naturalformu-
In conclusion, in this paper we report the first calcu- lation of the problem.
lation of the spin-dependent transmittance through an Acknowledgments The support of DFG /SFB 608
10
andthe RomanianCERESProgrammeisacknowledged. The solution of eqs (30) is
WethankM.NitaandV.Dinuforveryusefuldiscussions.
g =τ e−iq(n+1) d c ;c† d† . (31)
APPENDIX n 0 ≪ ↑ α↓ j↑ ↓ ≫
WeintroducetheHamiltonianofthesemi-infinitenon- so that, finally,
interacting leads HL and the lead-ring coupling HLR .
The different leads are indexed by α, which stands also
for the site where the lead is attached. Then, ≪d↑cα0↓;c†j↑d†↓ ≫=τ0e−iq ≪d↑cα↓;c†j↑d†↓ ≫, (32)
∞
HL =XXcαnσ†cαn+1σ +H.c. (26) which concludes the demonstration that in this case the
ασ n=0 use of Heff is equivalent to the use of Htotal.
HLR =τ cα†c +H.c., (27)
0X 0σ ασ
ασ
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n n−1 n+1
