Anomalous Kondo Spin Splitting in Quantum Dots

0.36 MB·English

by  J. M. Aguiar Hualde

Anomalous Kondo Spin Splitting in Quantum Dots
J. M. Aguiar Hualde,1 G. Chiappe,2,1 and E.V. Anda3
1Departamento de F´ısica, Facultad de Ciencias Exactas, Universidad de Buenos Aires,
Ciudad Universitaria, Pabellon I, Buenos Aires, 1428, Argentina.
2Departamento de F´ısica Aplicada, Universidad de Alicante,
San Vicente del Raspeig, Alicante 03690, Spain.
3Departamento de F´ısica, Pontificia Universidade Cat´olica do R´ıo de Janeiro,
6
0 22452-970 R´ıo de Janeiro, Caixa Postal 38071, Brasil.
0 (Dated: February 6, 2008)
2
The Zeeman splitting of localized electrons in a quantum dot in the Kondo regime is studied

n using a new slave-boson formulation. Our results show that the Kondo peak splitting depends on
a the gate potential applied to the quantum dot and on the topology of the system. A common fact
J
of any geometry is that the differential susceptibility shows a strong non linear behavior. It was

0 shownthatthereexistacriticalfieldabovewhichtheKondoresonanceissplittedout. Thiscritical
1 fieldrapidlydiminisheswhenthegatepotentialislowered, asaconsequenceofthereductionofthe
Kondotemperatureandasubsequentstrongenhancementinthedifferentialsusceptibilityoccurring

] at low fields. The critical field is also strong depedent on the topology of the circuit. Above this
l
e criticalfieldtheZeemansplittingdependslinearlyuponthemagneticfieldanddoesnotextrapolate
- tozeroatzerofield. ThemagnitudeoftheY-interceptcoordinatedependsonthegatepotentialbut
r
t the slope of this function is not renormalized being independent of the value of the gate potential.
s
Ourresults are in agreement with very recent experiments.
.
t
a
m PACSnumbers: 73.63.Fg,71.15.Mb
-
d
The observation of Kondo effect in quantum dots has g,µ andB isthegyromagneticfactor,theBohrmagne-
n B
o already a large history [1, 2]. The effect appears when a ton and the magnetic field respectively. In others, [5, 6],
c net spin becomes localized at a quantum dot (QD) cou- bellowacriticalvalue,theeffectofthemagneticfieldwas
[ pled with metallic leads [3, 4]. Local spin fluctuations not observed and above it the splitting was larger than

1 (LSF) in the vicinity of the QD allows the spin of the ∆B and does not extrapolate to zero at zero field.
v electrons in the leads to screen the localized spin, as a Thesplittingwasinitiallytheoreticallypredicted[7]to
2 magnetic impurity is screened in a host metal. A sharp be ±gµ B, in line with [2]. However,morerecentcalcu-
9 B
resonance in the local density of states (LDOS) is devel-
lations [8] shows that the splitting would be observable
1
oped at the Fermi level of the system. When the QD
1 only above Bc, when the magnetic Zeeman energy ∆B
0 is connected as a bridge between two metallic leads, the becomes competitive with the Kondo temperature T .
K
6 embeddedgeometry(EG),theresonanceprovidesachan- Moore and Wen [9] established the relationship between
0 nel of conduction and the conductance reaches the value the field induced spin splitting in the spectral function
at/ G = 2G0, being G0 = eh2 the quantum of conductance. of the system in equilibrium and the splitting appear-
When the QD is side connected (SCG), two channels of
m ing in the differential conductance when the system is
conduction interfere destructively (one direct and other slightly out of equilibrium. They predict also that the
-
acrosstheQD,fromleadtolead)andtheconductanceis
d spin screening characteristic of the KR should reduce
n zero for the value of the gate potential, for which the in- the magnetic field splitting, i.e ∆<∆ . Unfortunately
B
o terference is complete. The existence and characteristics these theories were not capable of explaining the phe-
c of the Kondo resonance can be studied through conduc-
nomenology seen in the latest experiments [5, 6].
:
v tance G of the system.
Inthisletterwedevelopedatheoreticalstudythatcor-
i
X To better understand the Kondo effect it is important rectly explains these last experimental results. We use
r to study its behavior when an external magnetic field is differentapproachestostudytheproblem. Weproposea
a
applied. It is expected the LSF to be quenched and the new slave-boson formalism(SB) that considers two inde-
Kondo regime to disappear for large enough fields. For pendentbosonfields,eachoneassociatedtoupanddown
intermediate values,the Kondoresonanceis splitted and spin. We solve the problem using as well the embedded
its effect can be observed in the differential conductance clustermethod(ECM),whichisapowerfulltooltostudy
ofthe system[2]. Inrecentexperiments[5,6]itwaspos- strong correlated systems [11], and we compare the re-
sible toobtainveryprecisevalue of∆, the Zeemansplit- sults of these two approaches. We study the system for
ting, (ZS) for localized electrons in a QD in the Kondo thetwogeometriesmentionedabove(sideconnectedand
regime (KR). However,these experiments are controver- embedded). The value of the splitting results to be de-
sial. Inoneofthem,[2],aboveacriticalvalueofthemag- pendent on the geometry of the circuit and has a strong
netic field, it was found that ∆ ∼ ∆ = 2gµ B, where dependence on the gate potential applied to the QD and
B B
2
Hˆ = Va† c b +c.c. (3)
T i;σ 0σ σ
Xi;σ h i
InordertoavoiddoubleoccupancyattheQD,constraint
conditions have to be imposed. They are written,
2n +b† b −b†b +b†b b† b = 1. (4)
0;σ −σ −σ σ σ σ σ −σ −σ
When both spins are equivalent, the above equations
merge into the constraint of the standard slave-boson
formalism. The Hamiltonian is treated in the mean-field
approximationforthe bosonfields (MFSB), substituting
FIG.1: Sketchofthegeometryofthetwotopologiesconsider
in thiswork. Upperpanel, SCG. Lower panel, EG. in equations (3) and (4) the operator bσ by its mean-
value z =<b >. The mean-field versions of equations
σ σ
(4) are incorporated into the Hamiltonian using the La-
the coupling between the leads and the QD. We also ob- grange multipliers γσ. Minimizing the total energy with
tainthesolutionintheHubbardI(HI)aproximation[10] respecttothesemultipliersweobtaintwonewequations,
that adequately describes the situation for T >T .
k KV <a† c >+z2(γ +γ )+γ −γ =0 (5)
The system is described by the Anderson impurity 1;σ 0σ σ σ −σ σ −σ
Hamiltonianincluding the effect ofanexternalmagnetic
where K = 1 for the SCG and K = 2 for the EG. We
field. We identify three contribution to the total Hamil-
solveequations(5)self-consistentlyandobtaintheGreen
tonian: the leads (H ), the QD term (H ) and the con-
L C function at the QD and the transmission T(E) across it
nections among the leads and the dot(H ). H is the
T L ( G=T(E )).
f
Hamiltonian of two semi-infinite chains for the EG con-
We have chosen the parameters to obtain a similar
figuration or of a one dimensional infinite chain for the
Kondo resonance for the SCG and the EG configura-
SCG case. The Hamiltonian H is given by,
C tions. They are V = 0.35 for the EG and V = 0.65
for the SCG. Increasing V the system goes from Kondo
g
to mixed valence regime.
HˆC = ǫσn0;σ+Un0;↑n0;↓ (1) Fig.2 (upper panels) shows the transmission at zero
Xσ field obtained using the approximations MFSB and
ECM. The effect of the Kondo resonance is clear: G=2
wherethe subindex0denotesthe QDsite,ǫ±21 =Vg±B or G = 0 for the EG or the SCG configurations respec-
is the spin dependent QD energy level, V is the gate
g tively, and a resonance or anti-resonance width roughly
potential at the QD site and U the Coulomb repulsion
similar for both geometries and both methods of calcu-
that, for simplicity, is supposed to be infinite. We have
lations. When the magnetic field is increased the Kondo
adopted |gµ |=1. Finally,
B peak is splitted and eventually the system goes out of
resonance. In the medium panels of Fig. 2 we show the
transmission for B = 0.18. The value of the magnetic
Hˆ = Va† a +c.c. (2)
T i;σ 0σ fieldsplitting obtainedbythetwomethodsareverysim-
Xi;σ h i ilar,andfortheECMtheresultsarealmostindependent
of the cluster size exactly diagonalized.
where i is −1 and 1 for EG or i=1 for the SCG. In the
To study how this splitting develops we write down
first case −1 and 1 denotes the first site of the left and
the expressionforthe Greenfunctionatthe QD.Forthe
right semi-infinite one dimensional leads. In the second
MFSB solution we have,
case 1 is the site of an infinite one dimensional lead to
which the QD is connected and V is the hopping proba-
bility(see Fig. 1). All the energies are expressedin units 1
G (E)= (6)
of the hopping among the leads sites t, D denotes the 0;σ E−ǫ −2γ +K(Vz )2g (E)
σ σ σ L
bandwidth of the leads and the Fermi energy is E =0.
f
Here g (E) is the undressed Green function of the lead.
In the new slave-boson formulation we define a boson L
From equation (6) we can obtain E , the values of the
field associate to each electron spin. The fermion opera- σ
tors a† is written as the product, a† = c† b†, where renormalized energies of the QD level. They are,
0;σ 0;σ 0;σ σ
c and b denote the quasi-fermion and boson operators.
Following a similar formulationto the standardone [12], ǫ +2γ
σ σ
E = . (7)
HT is transformed into σ 1−((K−1)Vzσ/t)2
3
2 V
g
1.5 EG
T (E) 01..05215 SEBCSEMBC, 1M5 sites B = 0.18 SCG 0.2 VVVggg === ---000...358 -1 -BB0 .==8 00..002155-0.6 00..1146∆
T(E) 1 ECM. 11 sites ∆/2
0.5 ECM 14 sites
B = 0.18 ECM 10 sites
0 SB
2
B = 0.025
1.5 B = 0.01
T(E) 1 B = 0.0025 0 z
0.5 BB == 00..001025 0.8 ∆
B = 0.001 n
0
-0.8 -0.4 0 0.4 0.8 -0.8 -0.4 0 0.4 0.8 0.6
E E
0.4
2 ∆
Γ
FIG.2: LeftcolumnEGandrightcolumnSCG.Upperpanels: 0.2
transmission as a function of energy for B = 0. Continuous
0
lineSB,brokenlineECM(11sites). Mediumpanels: thesame z
-0.2
as above but for B = 0.18 and two different cluster sizes. 0 0.01 0.02
Lower panels: thesameas aboveforthreedifferentmagnetic
fields B1 < B2 < B3 obtained with MFSB. For more details B
see thetext.
FIG.3: ResultsforEG.Upperpanel: Thefunction∆(B)for
three different values of Vg. The semi-sum of the resonance
The term Γ = 2γσ can be interpreted as the widthsW,forVg=−0.5,isshownwiththickpointline. The
spindependσentr1e−n(oKrm−1a)l(iVzazσt)io2noftheresonancelevelby equation W(B)= ∆(B) defines Bc. The point line shows ∆
using the Hubbard I approximation. Lower panel: evolution
the interaction. Depending on the geometry (K = 1 or
of the parameters with B, for Vg = −0.8. The inset shows
2)thereisalsoamultiplicativecorrectioninthetermǫ .
σ ∆(Vg) for two magnetic fields.
The ZS is given by the difference ∆=E −E ,
↓ ↑
ures 3 and 4). This implies that, as known [6], ∆ in-
V +B+2γ V −B+2γ
g ↓ g ↑
∆= − . (8) creases logaritmically by lowering the Kondo tempera-
1−(K−1)(Vz )2 1−(K −1)(Vz )2
↓ ↑ ture since V ∼ (V2/D)log(D/T ). As ∆ > ∆ , the
g k B
slope α is larger than the value predicted in [8], and in
In Fig. 3 and 4 (lower panel) we show the parameters
agreement with recent experiments [5, 6]. Increasing V ,
of the problem as a function of the magnetic field. For g
we enter into the mixed valence regime and ∆ recovers
both configurations z → 1 ,z → 0 when B → ∞. ∆
↑ ↓
a nearly linear evolution that almost extrapolates to the
and ∆ = Γ −Γ show a rapid increase at low fields,
Γ ↓ ↑
origin. In this regime ∆ is not constant anymore but
more evident for the SCG case. ∆ results to be al- Γ
Γ
increasesmonotonicallywiththefield. Asaconsequence,
mostconstantforalargerangeofvaluesofthe magnetic
the slope of the linear relationship between ∆ and B is
field. As a consequence, ∆(B) tends asymptotically to a
renormalizedforbothgeometrieswithrespecttothenon-
linear function. The slope of this linear function is qual-
interactingcase. Inspiteofthis,itisimportanttorealize
itatively the same as for the non interacting regime (for
that ∆ is always lower than the value it acquires in the
the EG configuration there is a small correction propor-
tional to V2 with respect to the non-interacting case). Kondo regime.
To determine when the ZS can be experimentally de-
For low fields, ∆ rises due to the Zeeman effect but also
tected the relevant quantity to be study is W, the semi-
because the local level at the QD suffers a spin depen-
sum of the resonance width of each Kondo peak. It can
dent renormalization as the magnetic field progressively
be expressed as,
quenches the LSF, giving an additional contribution to
the splitting. The differential magnetic susceptibility (Vz )2 (Vz )2
↓ ↑
(DMS) is strongly enhanced for low magnetic fields, (see W =K + .(9)
(cid:20)1−(K−1)(Vz )2 1−(K−1)(Vz )2(cid:21)
↓ ↑
the curve n − n = ∆ in figures 3,4). For larger
0,↑ 0,↓ n
fields, when the spin fluctuations are quasi totally sup- The ZS is not detectable if ∆ < W. The value of the
pressed, ∆ recovers its linear function appearance, but critical magnetic field B satisfies the equation ∆(B )=
c c
shiftedupwithrespecttotheorigin,duetotheCoulomb W(B ). In figure 3 and 4 we plot W(B) for V = −0.5
c g
interactionactingoneachspinlevel. LoweringV ,theY- and V = −0.4, respectively. The values of W obtained
g g
intercept coordinate of this linear function is increased, from (9) at B = 0 are the same to those obtained with
although the slope does not change. For a fixed value theusualformulaT =Dexp(−xV /2KV2)[12]wherexis
k g
of the field B, ∆ results to depend linearly on V with πforEGor2πforSGC.TheB obtainedforeachconfig-
g c
a slope α almost independent of B (see the inset of fig- uration corresponds to a ∆(B ) << T in line with the
c k
4
Vg of the slope of the function ∆(B), the magnetization in
-0.8 -0.6 thishightemperatureregimeisverydifferent. TheDMS
0.4 0.18
V = -0.3 atzerofieldismuchlessathightemperaturethanatlow
g 0.16
0.3 V = -0.4 ∆ temperature.
g 0.14
/2 0.2 Vg = -0.5 BB == 00..002155 0.12 Summarizing, we have studied the magnetic response
∆ of a QD connected with two different geometries SCG
0.1 and EG. To do so we developed a slave-bosonformalism
that takes into account a different boson field for each
0 spin. In the Kondo regime there is a critical magnetic
z field B above which the Kondo resonance can be seen
0.8 c
splitted. B depends upon the topology of the system,
c
∆
n being larger for the EG than for the SCG configuration.
0.4 The Zeeman energy associated to this field is lower that
z
2 ∆ the Kondo temperature. For B < B , ∆ suffers a non-
Γ c
linear and rapid growth with the magnetic field that oc-
0 curs due to the spin dependent local energy Coulomb
0 0.01 0.02
renormalization and the quenching of the LSF induced
B
by the magnetic field. When the spin fluctuations are
completely quenched, B > B , the spin splitting is pro-
c
FIG. 4: Results for the SCG. Upper panel: The function
portional to the magnetic field with a slope independent
r∆e(soBn)anfocretwhirdetehsdWiffe,rfeonrtVvgal=ues−0o.f5V,igs.shTohwenswemitih-stuhmickotfrtahcee on temperature and gate potential. The Y-interception
coordinateisnonzeroandincreaseswhenthegatepoten-
line. The equation W(B)=∆(B) definesBc. The point line
shows ∆ using the Hubbard I approximation. Lower panel: tial is reduced. At a fixed magnetic field, ∆(Vg) results
evolutionoftheparameterswithB,forVg =−0.5. Theinset to be a linear function with a B independent slope. The
shows ∆(Vg) for two magnetic fields. value of ∆ is always greater than ∆B for T <Tk due to
the spins correlations implied in the Kondo effect. Our
results are in good agreement with recent experimental
experimental observation. Note that B is appreciably
c results.
smaller for the SCG than for the EG topology because
Financial support by the argentinian CONICET and
inthis case∆hasanegativecontributioncommingfrom
UBA (grant UBACYT x210) and the spanish pro-
V (8), which is absent for the SCG, and also ∆ has a
g Γ
gramn Ramon y Cajal of MCyT and Brazilian agen-
more rapid increasing with B for the SCG. In figure 2
cies, FAPERJ, CNPq, and CNPq(CIAM project) are
(lowerpannels)itisshownhowthesplittingdevelopsfor
gratefully acknowledged. Also funding from Generali-
each geometry.
tat Valenciana, (grant GV05/152), and by the spanish
We have also solved the problem in the Hubbard I
MCYT (grants MAT2005-07369-C03-01 and NAN2004-
approximation [10]. It describes the situation for T >
T , i.e when the system is outside the Kondo regime. 09183-C10-08)are acknowledged.
K
The local Green function results to be:
1
G (E)= (10)
0;σ E−ǫ +KV2(1−n )g (E)
σ −σ L [1] D. Goldhaber-Gordon et.al, Nature 391, 156 (1998).
[2] S.M. Cronenwett et.al, Science 281, 540 (1998).
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[3] L.I.Glazmanet.al,JETPLett.47,482(1988);L.I.Glaz-
function of the external field [10]. ∆ results to be:
man et.al, Europhysics Lett. 19, 623 (1992).
[4] T.K. Nget.al, Phys.Rev.Lett. 61, 1768 (1988).
V +B V −B
∆= g − g (.11) [5] A. Kogan et.al, Phys. Rev.Lett. 93, 166602 (2004).
1−(K−1)V2(1−n↑) 1−(K−1)V2(1−n↓) [6] S. Amasha et.al, cond. mat. 0411485 (2005).
[7] Y. Meir, N.S. Wingreen and P.A. Lee Phys. Rev. Lett.
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slope is equal to that of the low temperature solutions [9] J.E. Moore and X.-G. Wen, Phys. Rev. Lett. 85, 1722
obtained with MFSB, while for the EG there is a small (2000).
[10] W.JonesandN.MarchinTheoreticalSolidStatePhysics,
renormalization of it. This shows that the essential dif-
(Wiley-interscience, London, 1973).
ference between the low and high temperature regime as
[11] V. Ferrari et.al, Phys. Rev. Lett. 82, 5088 (1999);
far as the ZS is concerned is the existence of a non-zero G.ChiappeandJ.Verges,J.CondsMatter158805(2003);
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5
(2005). (Cambridge Univ.Press, 1993).
[12] A.C.Hewson,inTheKondoProblemtoHeavyFermions,
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