Inelastic electron scattering off a quantum dot in the cotunneling regime: the signature of mesoscopic Stoner instability

4.5 MB·

by  E. V. Repin

Inelastic electron scattering off a quantum dot in the cotunneling regime: the
signature of mesoscopic Stoner instability
E. V. Repin1 and I. S. Burmistrov2,1
1Moscow Institute of Physics and Technology, 141700 Moscow, Russia
2L.D. Landau Institute for Theoretical Physics RAS, Kosygina street 2, 119334 Moscow, Russia
(Dated: January 26, 2016)
We explore the inelastic electron scattering cross section off a quantum dot close to the Stoner
instability. WefocusontheregimeofstrongCoulombblockadeinwhichthescatteringcrosssection
is dominated by the cotunneling processes. For large enough exchange interaction the quantum
dot acquires a finite total spin in the ground state. In this, so-called mesoscopic Stoner instability,
regime we find that at low enough temperatures the inelastic scattering cross section (including
the contribution due to an elastic electron spin-flip) for an electron with a low energy with respect
to the chemical potential is different from the case of a magnetic impurity with the same spin.
This difference stems from (i) presence of a low-lying many-body states of a quantum dot and
6
(ii) the correlations of the tunneling amplitudes. Our results provide a possible explanation for
1
absence of the dephasing rate saturation at low temperatures in recent experiment [N. Teneh, A.
0
Yu. Kuntsevich, V. M. Pudalov, and M. Reznikov, Phys. Rev. Lett. 109, 226403 (2012)] in which
2
existence of local spin droplets in disordered electron liquid has been unraveled.

n
a PACSnumbers: 75.75.-c,73.23.Hk,73.63.Kv
J

4
2 I. INTRODUCTION cupyingalocalizedlevel[9]. Themagneticimpuritywith
spinS >1/2canbemimickedbyatrapwithmanyelec-

] trons localized therein. Recently, such electron droplets
l Theelectronscatteringoffamagneticimpurityaffects
l with spin S ≈ 2 (per a droplet) have been detected in
a crucially properties of electron systems at low tempera-
h tures. The simplest model of a magnetic impurity is a two-dimensional(2D)electronsysteminSi-MOSFETby
- random vector of fixed length equal to S. Albeit this thermodynamicmeasurementsofasamplemagnetization
s
[10]. Inthepresenceofstrongexchangeinteractionin2D
e model ignores the quantum nature of a spin it is enough
m to produce interesting nontrivial effects, e.g. suppres- disorderedelectronsystematlowtemperatures, thespin
ofanelectrondropletcanbefiniteduetophenomenonof
. sion of the superconducting transition temperature due
t the mesoscopic Stoner instability [11,12]. The finite spin
a toelasticelectronspin-flip[1]. Typicallythisclassicalap-
m proximation is not adequate for the description of mag- of an electron droplet yields the Curie-type behavior of
the spin susceptibility. The temperature dependence of
- netic atoms in real systems since their spin is not large,
d S ∼ 1. Importantly, the quantum effects in dynamics of measured magnetization is consistent with the Curie law
n for the spin susceptibility of a single droplet provided
a spin makes electron scattering off a magnetic impurity
o their concentration is inversely proportional to tempera-
tobeinelastic. Forexample,theZeemansplittingmakes
c
ture [10].
[ the spin-flip scattering to be energy dependent and sup-
presses it due to polarization of the spin along magnetic Motivatedbytheseexperiments[10]weconsidertheef-
1 field [2]. The other well-known quantum effect is Kondo fectofsuchmany-electronpuddleswiththefinitespinon
v
renormalization of the interaction coupling between an transportpropertiesof2Delectronsystem. Inparticular,
1
electron spin and spin of an impurity that leads to non- weestimatecontributiontothedephasingtimeduetoin-
8
3 monotonic temperature dependence of resistivity (for a elasticelectronscatteringoffsuchdropletsatlowtemper-
6 review see [3]). atures(T). Forasakeofsimplicity,wemodelanelectron
0 The outcome of interaction between electrons and a puddle by a quantum dot described by the so-called uni-
1. magnetic impurity can be conveniently formulated in versal Hamiltonian [11] with large charging energy (Ec)
0 terms of the scattering cross section. For example, the and ferromagnetic exchange interaction (J >0). We as-
6 peculiarity of the Kondo problem can be seen in a non- sume that the quantum dot is weakly tunnel coupled to
1 monotonic behavior of the inelastic scattering cross sec- electrons participating in transport.
:
v tionwithenergyatzerotemperature[4]. Thisnonmono- Asaquantumdotisconcernedwefocusontheregime
i tonicity is translated into a nonmonotonic temperature of strong Coulomb blockade, E (cid:29) T, with an integer
X c
dependence of the electron dephasing rate due to rare number of electrons on the quantum dot. In this regime
r magnetic impurities. The contribution to the dephasing theleadingcontributiontotheelectronscatteringoffthe
a
rate due to inelastic scattering off magnetic impurities quantumdotcorrespondstotheforthorderinthetunnel-
affectsdependenceoftheweaklocalizationcorrectionon ingamplitudes. Thisissimilartothecotunnelingregime
temperature and magnetic field [2, 5–8]. in a standard analysis of transport through the strongly
InrealmaterialswithCoulombinteractionamagnetic Coulomb-blockadedquantumdot. Wecomparetwocases
impurity with spin 1/2 can be formed by an electron oc- of exchange interaction in the quantum dot: Heisenberg
2
interaction and Ising interaction. In the former case the II. FORMALISM
total spin of the quantum dot in the ground state can
be estimated as S ≈ J/[2(δ−J)] where δ denotes the We start with the following Hamiltonian
mean level spacing for single particle levels of the quan-
tum dot [11]. Near the macroscopic Stoner instability, H =H +H +H . (1)
QD R T
δ−J (cid:28) δ,J, the total spin of the quantum dot is large
S (cid:29) 1. For the Ising exchange the total spin in the Here the first term H describes electrons in a quan-
QD
groundstateiszeroforJ <δ,i.e. themesoscopicStoner tum dot. We consider a metallic quantum dot, i.e. with
instability is absent [11]. the large dimensionless conductance, g = E /δ (cid:29) 1,
Th Th
In general, the inelastic cross section consists of three where ETh denotes the Thouless energy. In this case,
terms: elastic spin-flip, inelastic spin-flip, and inelastic the quantum dot is accurately described by the so-called
non-spin-flipcontributions. Inthispaper,weconcentrate universal Hamiltonian [11,14]:
on the case of strong exchange interaction: the quantum
(cid:88)
dotisclosetothemacroscopicStonerinstability,δ−J (cid:28) H = (cid:15) d† d +E (nˆ−N )2−JS2. (2)
QD ασ ασ ασ c 0
δ,andlowtemperaturesT (cid:46)δ−J. Wefindthatforsmall α,σ
energy of incoming electron, ε(cid:28)δ:
Hered andd† aretheannihilationandcreationoper-
ασ ασ
(i) the elastic spin-flip contribution is the same as ators for electrons with an energy (cid:15)ασ =(cid:15)α+µBgLBσ/2
for a magnetic impurity with the spin S ≈ on the quantum dot, where σ =±1 denotes the spin in-
J/[2(δ−J)](cid:29)1; dex, gL and µB stand for the electron g-factor and the
Bohr magneton, respectively. The second term in the
(ii) at energies ε (cid:38) δ − J the inelastic spin-flip and right hand side of Eq. (2) accounts for the Coulomb
blockade. It involves the particle number operator,
non-spin-flip channels become active; they add the
contribution which is 1/S2 ∼ (1 − J/δ)2 smaller (cid:88) (cid:88) (cid:88)
nˆ = nˆ = nˆ = d† d , (3)
than one due to elastic spin-flip. σ α ασ ασ
σ α α,σ
The presence of Zeeman splitting which is large in com-
and the external charge N . The last term in the right
0
parison with temperature suppresses the elastic spin-flip
hand side of Eq. (2) describes the ferromagnetic Heisen-
contributionduetodestructionofthemesoscopicStoner
berg exchange interaction (J > 0). It is expressed via
phase [13]. Then we find that the inelastic cross sec-
the operator of the total spin on the quantum dot,
tion vanishes for energies |ε| (cid:46) δ −J. At higher ener-
gies δ−J (cid:46) |ε| (cid:28) δ, the inelastic cross section reaches 1(cid:88) 1 (cid:88)
S = s = d† σ d . (4)
the value which is of the order of elastic spin flip contri- 2 α 2 ασ σσ(cid:48) ασ
bution (without magnetic field) for a magnetic impurity α α,σ,σ(cid:48)
with spin 1/2. In the case of Ising exchange interac-
We do not consider here interaction in the Cooper
tion we find that the inelastic cross section at energies
channel which are responsible for superconducting cor-
|ε| (cid:46) δ −J is sensitive to the parity of the number of
relations in quantum dots [15–17].
electrons on the quantum dot: for odd number of elec-
Next the term H describes electrons in a reservoir.
trons there is the elastic spin-flip contribution similar to R
Forasakeofsimplicity,weneglectinteractionofelectrons
a magnetic impurity with spin 1/2. Surprisingly, we find
in the reservoir and write the Hamiltonian as
thatatenergiesδ−J (cid:46)|ε|(cid:28)δ theinelasticcrosssection
becomes almost insensitive to the parity of the number H =(cid:88)ε a† a . (5)
of electrons. R kσ kσ kσ
k,σ
The paper is organized as follows. In Sec. II we re-
view the formalism and present the general expression Herea† anda arethecreationandannihilationoper-
ασ ασ
for the inelastic cross section at nonzero temperature. atorsforelectronswithanenergyε =ε(k)+µ g˜ Bσ/2
kσ B L
Next (Sec. III) we apply the general formula and derive in the reservoir, where g˜ denotes the g-factor in the
L
the expression for the inelastic scattering cross section reservoir. Wenotethatallenergiesarecountedfromthe
for the cotunneling regime. As the simplest example we chemical potential.
consider the case of a single-level quantum dot and com- Finally,thetermH accountsforthecouplingbetween
T
pare our results to ones obtained before (see Sec. IIIA). the quantum dot and the reservoir. We choose it in a
Nextweconsidertheinelasticscatteringcrosssectionfor standard form of the tunneling Hamiltonian:
amany-levelquantumdotneartheStonerinstabilityfor
(cid:88)
Heisenberg (Sec. IIIB) and Ising (Sec. IIIC) exchange H = t d† a +h.c. (6)
T αk ασ kσ
interactions. We conclude the paper with discussion of
α,σ,k
relation of our results to the experimentally available se-
tups and with the summary of the main results. Some Weemphasizethatthereisnospin-flipofelectronduring
technical details are given in the Appendices. thetunnelingeventfromthequantumdottothereservoir
3
or vice versa. In what follows we neglect the effect of N andtheeffectivedimensionless(inunitse2/h)chan-
ch
electrons in the reservoir on dynamics of the total spin nel conductance g can be written as
ch
of the quantum dot (see Refs. [18,19]).
(trgˆ)2 trgˆ2
Following Ref. [6], the T-matrix for scattering of elec- N = , g = . (14)
ch trgˆ2 ch trgˆ
trons from the state |kσ(cid:105) with energy ε = ε to the
k,σ
state |k(cid:48)σ(cid:48)(cid:105) can be written in terms of the Green’s func- We assume that the total conductance of the tunneling
tions: junction is small, gT =gchNch =trgˆ(cid:28)1.
We stress that the T-matrix obtained in accordance
(cid:104) (cid:105)−1 (cid:104) (cid:105)−1
(cid:104)k(cid:48)σ(cid:48)|T|kσ(cid:105)=− G(0) (ε) GA (ε) G(0)(ε) . with Eq. (12) is averaged over the equilibrium den-
k(cid:48)σ(cid:48) k(cid:48)σ(cid:48);kσ kσ
sity matrix of the quantum dot and reservoir. In par-
(7)
ticular, this averaging involves summation over initial
whereG(0)andG arethefreeandfullmany-bodyGreen’s
statesofthequantumdotwiththeGibbsweight. Hence,
functions for electrons in the reservoir, respectively. Us-
a standard expression for the elastic scattering σ ∝
el
ingtheDysonequationfortheadvancedGreen’sfunction
|(cid:104)k(cid:48)σ|T|kσ(cid:105)|2, where ε = ε , is inapplicable for our
GA, Eq. (7) can be rewritten as follows kσ k(cid:48)σ
definition of the T-matrix. In what follows, we shall ex-
tract the inelastic part of the cross section directly from
(cid:104) (cid:105)−1
(cid:104)k(cid:48)σ(cid:48)|T|kσ(cid:105)=−δk(cid:48),kδσ(cid:48),σ G(k0σ)(ε) the final expression for the total cross section (see Sec.
IIIB).
(cid:88)
− t¯ GA (ε)t . (8)
k(cid:48)β βσ(cid:48);ασ αk
αβ
III. THE SCATTERING CROSS SECTION IN
Here GA (ε) is the exact advanced Green’s function THE COTUNNELING REGIME
βσ(cid:48);ασ
for electrons in the quantum dot. The corresponding
Matsubara Green’s function GβAσ(cid:48);ασ(iε) can be found in To the lowest order in Qσ (ε) the scattering cross sec-
αβ
the imaginary time as follows (see e.g., Ref. [20]): tionisdeterminedbytheGreen’sfunctionofelectronson
an isolated quantum dot, i.e. the Green’s function cor-
1 (cid:104) (cid:105)
G (τ)=− Tr e−τHd† e−(β−τ)Hd , (9) responding to the Hamiltonian H . Then, if quantities
ασ;βσ(cid:48) Z βσ(cid:48) ασ Qσ (ε)arereal,thescatteringcroQssDsectionisdetermined
αβ
where τ > 0, β = 1/T and Z = Tre−βH stands for the by the tunneling density of states for the isolated quan-
tum dot. In the case of Coulomb valley, this implies ex-
grand canonical partition function. The total scattering
ponentially small scattering cross section at low energies
crosssectionforanelectroninastate|kσ(cid:105)isrelatedwith
|ε|<E .
the T-matrix as [6] c
Tocalculatethescatteringcrosssectiontotheforthor-
2 derinthetunnelingamplitudesletusintroducethebasis
σσ = Im(cid:104)kσ|T|kσ(cid:105). (10)
tot v of the exact many-body eigenstates |i(cid:105) for the Hamilto-
F
nian (2) of the isolated quantum dot: H |i(cid:105) = E |i(cid:105).
QD i
HerevF isthevelocityofelectronsinthereservoiratthe ThencomputingtheGreen’sfunctionofelectronsonthe
Fermi level. In our problem of electron scattering off the quantum dot to the second order in tunneling (see Ap-
quantumdotitismoreconvenienttostudythefollowing pendix A) we find the following result for the total scat-
quantity tering cross section:
Aσtot(ε)=(cid:88)δ(ε−εkσ)Im(cid:104)kσ|T|kσ(cid:105), (11) Aσ (ε)=π[1+e−βε] (cid:88) (cid:88) p (cid:90) dε(cid:48) Qσβα(ε)Qσγη(cid:48)(ε(cid:48))
k tot i 1+e−βε(cid:48)
αβγηi,f,σ(cid:48)
which is the scattering cross section averages with the 1 1
×(cid:104)i|d† d +d d† |f(cid:105)
single-particle density of states in the reservoir. Using γσ(cid:48)ε(cid:48)−E +H ασ ασε+E −H γσ(cid:48)
i QD i QD
Eq. (8), we can express the quantity Aσ (ε) as
tot ×(cid:104)f|d† 1 d +d 1 d† |i(cid:105)
(cid:88) βσε(cid:48)−E +H ησ(cid:48) ησ(cid:48)ε+E −H βσ
Aσ (ε)= Im Qσ (ε)GA (ε). (12) i QD i QD
tot βα ασ;βσ ×δ(ε+E −E −ε(cid:48)). (15)
αβ i f
(cid:80)
Here p = exp(−βE )/Z, where Z = exp(−βE ), is
Here we introduce the matrix i i i i
theGibbsprobabilityfortheinitialstatesofthequantum
Qσ (ε)=(cid:88)δ(ε−ε )t t¯ . (13) dot. Wementionthattheresult(15)canalsobeobtained
αβ kσ αk kβ withinthegeneralizedFermigoldenruleapproachforthe
k
T-matrix (see Appendix B). As discussed above, we will
This matrix characterizes the tunnel junction in the beinterestedintheinelasticscatteringonly,whichmeans
following way. Let us define the matrix gˆ = that we will always be considering different initial and
αβ
(4π2/δ)(cid:80) Qσ (ε). Then for an electron with the en- final states of the quantum dot, i (cid:54)= f. In what follows
σ αβ
ergy ε the effective number of open tunneling channels we neglect possible dependence of Qσ on spin index σ.
βα
4
A. Single-level quantum dot reservoiras(2n /ν)Aσ (ε),wereproducetheresultof
s inel,sf
Ref. [22].
Toillustratethegeneralexpression(15)forthescatter-
ingcrosssectionweconsiderasimpleexampleofasingle
level quantum dot. In this case there are four many- B. Many-level quantum dot near Stoner instability
body states: the state without electrons, |0(cid:105), two states
with single electron, | ↑(cid:105) and | ↓(cid:105), and the state with
Now we consider the many-level quantum dot de-
two-electrons with opposite spins, | ↑↓(cid:105). We note that
scribed by the universal Hamiltonian (2). We remind
although the universal Hamiltonian (2) is not justified
that the charging energy E is large, E (cid:29)T,ε,δ,J, and
c c
for a single level quantum dot, the general expression
the external charge N has an integer value. Then, the
0
(15) written in terms of exact many-body eigenstates is
energyofintermediatestatesintherighthandsideofEq.
correct. Then, we find from Eq. (15)
(15) is equal to the charging energy, H −E = E .
QD i c
(cid:20) (cid:21) Dropping the elastic contribution, i.e. the term with
p +p p +p
Aσ (ε)=πQ2(ε) 0 σ + σ¯ ↑↓ |i(cid:105) = |f(cid:105), from Eq. (15), and using the commutation
tot (ε+E0−Eσ)2 (ε+Eσ¯ −E↑↓)2 relation [d†ασ,dβσ(cid:48)]=δαβδσσ(cid:48) −2dβσ(cid:48)d†ασ, we rewrite the
1+e−βε inelastic contribution to the scattering cross section as
+πQ(ε)Q(ε+E −E )
σ σ¯ 1+e−β(ε+Eσ−Eσ¯)
4π (cid:88) (cid:88)
×p (cid:20) 1 + 1 (cid:21)2. (16) Aσinel(ε)= E2 Qγα(ε)Qαγ(ε+Ei−Ef)
σ E −E −ε ε+E −E c α,γ f(cid:54)=i,σ(cid:48)
σ¯ 0 σ ↑↓
p [1+e−βε]
Here σ =↑,↓ and σ¯ =↓,↑, respectively. The first term in × 1+ie−β(ε+Ei−Ef)(cid:104)i|d†ασ(cid:48)dασ|f(cid:105)(cid:104)f|d†γσdγσ(cid:48)|i(cid:105)
the right hand side of Eq. (16) describes elastic spin-flip
4π (cid:88) (cid:88)
of electron with energy ε and spin projection σ after the + Q (ε)Q (ε+E −E )
E2 αα γγ i f
scattering off the single level quantum dot. The second c α(cid:54)=γf(cid:54)=i,σ(cid:48)
term corresponds to the scattering with spin-flip. In the p [1+e−βε]
absence of magnetic field the two states with single elec- × 1+ie−β(ε+Ei−Ef)(cid:104)i|d†γσ(cid:48)dασ|f(cid:105)(cid:104)f|d†ασdγσ(cid:48)|i(cid:105)
tron have the same energy, E =E , and the result (16)
↑ ↓ (18)
coincidewiththeresultofRef. [21]forthefulltransmis-
sionprobability. Inthepresenceofmagneticfieldspinup
Herewetakeintoaccountthattheinitialandfinalstates
and spin down states are not equivalent, E (cid:54)= E , and
↑ ↓ of the quantum dot has the same number of electrons.
the spin-flip scattering becomes inelastic. In the absence
Equation(18)constitutesthemainresultofourpaper.
ofinteractiontheenergyofthestatewithtwoelectronsis
We note that it can be applied to computation of the in-
expressedviatheenergiesofthestateswithoneandzero
elastic cross section for an arbitrary Hamiltonian which
electrons, E =E +E −E . Then, the spin-flip term
↑↓ ↑ ↓ 0 describes a quantum dot provided this Hamiltonian con-
in the scattering cross section (16) vanishes. The elastic
serves the total number of electrons N and energies of
contribution becomes independent of temperature. In
themany-bodyexactstateswithN andN±1arediffer-
agreement with Ref. [21], the scattering of electrons off
ent by large value of charging energy. For the universal
the single level quantum dot becomes fully coherent.
Hamiltonian (2) the matrix elements of single-particle
For E ,E →∞ the single-level quantum dot can be
0 ↑↓ operatorsinEq. (18)canbecomputedexactlybymeans
singlyoccupiedonly,i.e. thequantumdotbehavesasthe
of the Wei-Norman-Kolokolov method [23,24] employed
spin 1/2. In this case spin-flip inelastic part of Eq. (16)
forexactevaluationofthespinsusceptibilityandtunnel-
reduces to the following expression:
ing density of states recently [25,13]. Since in this work
Aσinel,sf(ε)=πν2Js2(cid:104)p¯↓(cid:2)1−nF(ε+ωσ)(cid:3)+p¯↑nF(ε+ωσ)(cid:105). lwoewaerneerigniteesreosfteadn iantcolomwintgemelpecetrraotnu,re|εs|,(cid:28)T δ(cid:28), wδe,caanndusine
(17) the straightforward approach with Clebsch-Gordan coef-
Here ω = E −E , n (ε)= 1/[1+exp(βε)] stands for ficients used for description of conductance [26,27] and
σ σ σ¯ F
the Fermi-Dirac distribution function, and p¯ = n (ω ) shot noise [28] through a quantum dot with Heisenberg
σ F σ
is the probability of the state with spin projection σ. exchange at low temperatures.
Neglecting dependence of the tunneling amplitudes in In general, the tunneling amplitudes t are random
αk
Q on the energy, we can write the effective exchange quantitiesduetorandombehaviorofelectronwavefunc-
coupling between the spin of electrons in the reservoir tions on a quantum dot. In what follows, we are inter-
and the spin of electrons on the quantum dot as J = ested in the case when energies of an electron before (ε)
s
ν−1Q[1/E +1/E ], where ν is the average density of and after (ε(cid:48) =ε+E −E ) scattering are small in com-
0 ↑↓ i f
statesperspinprojectionattheFermilevelforelectrons parison with the Fermi energy of electrons in the reser-
in the reservoir. If we assume that there are many such voir. Thus we can neglect the energy dependence in the
quantum dots (spin 1/2 impurities) with the concentra- quantities Q . For a metallic quantum dot, g (cid:29) 1,
αγ Th
tionn anddefinethespin-fliprateforanelectroninthe the averaging of the tunneling amplitudes over disorder
s
5
realizations can be performed independently of the sin-
gleparticleenergylevels(cid:15) . Usingthefollowingrelations
α
[29]
(cid:40)
Q2, α(cid:54)=γ,
Q Q = (19)
αγ γα (2/β)Q2, α=γ,
and
Q Q =Q2, α(cid:54)=γ, (20)
αα γγ
wheretheparameterβ =1and2fortheorthogonalclass
AI and the unitary class A, respectively. Then after the a) b)
averaging of Eq. (18) over disorder we obtain
FIG. 1. Examples of low-energy eigenstates with the total
(cid:40) spin S =3/2: a) S =1/2 and b) S =3/2.
Aσ (ε)= 4πQ2 (cid:88) pi[1+e−βε] (cid:12)(cid:12)(cid:104)i|S−σ|f(cid:105)(cid:12)(cid:12)2 z z
inel Ec2 f(cid:54)=i 1+e−β(ε+Ei−Ef)
1. Inelastic scattering cross section in the absence of
(cid:18) (cid:19)
+ β2 −1 (cid:88)(cid:12)(cid:12)(cid:104)i|d†ασ(cid:48)dασ|f(cid:105)(cid:12)(cid:12)2 magnetic field
α,σ(cid:48)
(cid:41) Now let us consider the case of small electron ener-
+ (cid:88) (cid:12)(cid:12)(cid:104)i|d†γσ(cid:48)dασ|f(cid:105)(cid:12)(cid:12)2 . (21) gies ε (cid:28) δ,J. We assume that the quantum dot is
in the regime of mesoscopic Stoner instability, δ,J (cid:29)
α(cid:54)=γ,σ(cid:48)
δ − J. Also we consider the case of low temperatures
Here we take into account that the operator nˆ does not T (cid:46) δ−J (cid:28) δ,J. For simplicity, we consider the case
σ
change the many-body state and the states |i(cid:105) and |f(cid:105) of equidistant single-particle spectrum. Afterwards we
are different, (cid:104)i|nˆ |f(cid:105) = 0. The first line in Eq. (21) discuss the effect of fluctuations of single-particle levels.
σ
corresponds to the contribution to the scattering cross Theminimalenergyofthemany-bodystatewiththeto-
section due to rotation of the total spin of the quantum tal spin S is equal to
dot as a whole, i.e. the total spin in the initial and final
E =(δ−J)S2−JS. (23)
states are the same. The other terms in Eq. (21) arise S
because in the case of the quantum dot the total spin Here we omit the term proportional to the charging en-
iscomposedfromspinsofindividualelectronsoccupying ergy E since we discuss the states with the same num-
c
single-particle levels. These additional contributions in- ber of electrons. These many-body states consists of the
crease inelastic scattering cross section off the quantum three groups of levels: doubly occupied levels at the bot-
dot in comparison with a magnetic impurity with the tom, singly occupied levels in the middle, and empty
same value of the spin. levels at the top (see Fig. 1). Provided the exchange
Let us consider the case of an electron with large en- interaction is bounded to the following interval
ergy, ε (cid:29) E ,E ,T. Then the inelastic scattering cross
f i
2S−1 2S+1
section becomes <J/δ < , (24)
2S 2S+2
4πQ2 (cid:88) (cid:68) (cid:12)
Aσ = p i(cid:12)S(S+1)−S2−σS thequantumdothasthetotalspinS inthegroundstate.
inel E2 i (cid:12) z z
c i For δ−J (cid:28)δ,J its value is large, S ≈δ/[2(δ−J)](cid:29)1.
(cid:18)2 (cid:19)(cid:88)(cid:104) (cid:0) (cid:1)(cid:105) Interestingly,inthisregimetherearetwolowlyingmany-
+ −1 nˆ (1−nˆ )+nˆ 1−(cid:104)i|nˆ |i(cid:105)
β ασ¯ ασ ασ ασ bodyexcitedstateswhichcorrespondstothestateswith
α thetotalspinsS+1andS−1. ThegapsE =E −E
± S±1 S
+1 (cid:88)nˆ (2−nˆ )(cid:12)(cid:12)i(cid:69). (22) betweentheseexcitedstatesandthegroundstateismuch
2 γ α (cid:12) smallerthenthetypicallevelspacing: E =(δ−J)(2S+
γ(cid:54)=α 1)−J (cid:54)δ/S andE =−(δ−J)(2S−1)++J (cid:54)δ/(S+1).
−
We note that the last term in Eq. (22) is proportional For the case of large total spin, S (cid:29)1, the gaps E+ and
to the number K of available single-particle levels. Typ- E− aresmallincomparisonwiththemeansingle-particle
ically, the increaseofanelectronenergyon δ addsa new level splitting, E± (cid:28) δ. The next many-body excited
final state of the quantum dot which contribute in the states with the total spins S±2 have the gaps which lies
sum in Eq. (22). At zero temperature it can be es- in the following intervals, δ/(S+1) (cid:54) E++ (cid:54) 3δ/S and
timated as K ∼ ε/δ. Assuming that K (cid:29) N0,S, we δ/S (cid:54) E−− (cid:54) 3δ/(S +1) (see Fig. 2). Assuming that
obtain that the inelastic scattering cross section is pro- temperature T (cid:46)δ−J we neglect them.
portional to the electron energy, Aσ = 4πQ2N ε/E2, The operator d† d with α (cid:54)= γ has nonzero matrix
inel 0 c γσ(cid:48) ασ
for δN ,δJ/[2(δ−J)](cid:28)ε(cid:28)E . elements between the many-body states with the same
0 c
6
γ γ
6∆
S
6∆
S(cid:43)1
S
E
(cid:68)
3∆
S α α
3∆
S(cid:43)1
E(cid:45)(cid:45) E(cid:43)(cid:43)
∆
∆
S
S(cid:43)1
E(cid:45) E(cid:43)
2S(cid:45)1 2S(cid:43)1 α
α
2S 2S(cid:43)2
J(cid:144)∆
FIG.2. Energiesofthelow-lyingmany-bodyeigenstates(23)
as a function of J/δ for the case when the total spin in the γ γ
groundstateisequaltoS. ThegroundstateenergyE isset
S
to zero.
or shifted by one spin projection. Let us consider the
groundstatewiththetotalspinSandprojectionM. The
state d† d |S,M(cid:105) will have the energy equal to E FIG.3. (Coloronline)Thesketchofinelastictransitionswith
γσ(cid:48) ασ S+1 (leftcolumn)andwithout(rightcolumn)spin-flip. Thetotal
if the level α is the highest doubly occupied one whereas
spin increases (decreases) by one during the transition in the
the level γ is the lowest empty one (see Fig. 3). The
top (bottom) row (see text).
operator d† d has nonzero matrix elements between
ασ¯ ασ
thelowlyingmany-bodystateswiththesametotalspin.
Inthiscasethelevelαcanbeanyamongsinglyoccupied Now let us consider the case of higher temperatures,
levels those number is equal to 2S. The corresponding δ (cid:29) T (cid:29) δ−J. Then many low energy excited states
matrixelementscanbecalculatedinastandardwaywith withthetotalspinS±kwithk (cid:46)(cid:112)T/(δ−J)contribute
thehelpoftheClebsch-Gordancoefficients(seee.g. Ref. to the inelastic cross-section. For δ−J (cid:46)T the summa-
[30]). The necessary matrix elements are summarized in tion over discrete values of k can be substituted by an
Table I. Then for T (cid:46) δ −J and |ε|,δ −J (cid:28) δ,J we integration. Using the following result,
find the following result for the inelastic scattering cross
section: (cid:82) dS(2S+1)f(S)e−βES T
4πQ2(cid:110)(2S+1)(S+1) 1 (cid:82) dS(2S+1)e−βES =f(Sg)+ δf(cid:48)(Sg)
Aσinel(ε)= Ec2 3 + 2F(ε,E−) +4(δT−J)f(cid:48)(cid:48)(Sg), (27)
2S+3 (cid:111)
+ F(ε,E ) , (25)
2(2S+1) +
whereS =J/[2(δ−J)]andf(S)isaquadraticpolyno-
g
mialofS, weobtaintheinelasticscatteringcrosssection
where we introduce the function
for δ (cid:29)T (cid:29)δ−J at energies |ε|(cid:46)δ as follows
2cosh2(βε/2)
(cid:34) (cid:35)
F(ε,E)= . (26) 4πQ2 δ(3δ−2J) T
cosh(βε)+cosh(βE) Aσ (ε)= + . (28)
inel E2 6(δ−J)2 3(δ−J)
c
The first contribution in Eq. (25) represents the elas-
tic spin-flip scattering, the next two correspond to the We emphasize that the inelastic cross section becomes
inelastic scattering with and without spin-flip. We note larger than one can expect for the case of magnetic im-
that the contribution in Eq. (25) due to the elastic spin- purity with the spin of the order of δ/[2(δ−J)].
flipscattering,(2S+1)(S+1)/3,islargerthantheresult Above we assumed that the single-particle level spac-
for the magnetic impurity, 2S(S+1)/3. It occurs due to ing in the quantum dot is equidistant. In general, this
additional correlations between tunneling amplitudes in is not the case. Below following Ref. [13] we shall take
the case of orthogonal ensemble (β =1). into account fluctuation of the single-particle levels near
7
TABLE I. Matrix elements between low-lying many-body states. The single particle states α and γ are different, α (cid:54)= γ (see
text and Fig. 3).
√
(cid:104)S+1,m+1|d†γ↑dα↓|S,m(cid:105)=(cid:104)S+1,m+1|(|S,m(cid:105)|1,1(cid:105))= (√√S+(2mS++21))((S2S++m2+)1) m(cid:80)=S−S(cid:12)(cid:12)(cid:104)S+1,m+1|d†γ↑dα↓|S,m(cid:105)(cid:12)(cid:12)2 = 2S3+3
(cid:104)S−1,m+1|d†γ↑dα↓|S,m(cid:105)=(cid:104)S,m|(|S−1,m+1(cid:105)|1,−1(cid:105)√)= (√S−(2mS))((S2S−−m1−)1) m(cid:80)=S−S(cid:12)(cid:12)(cid:104)S−1,m+1|d†γ↑dα↓|S,m(cid:105)(cid:12)(cid:12)2 = 2S3+1
(cid:104)S+1,m|d†γ↓dα↓|S,m(cid:105)= √12(cid:104)S+1,m|(|S,m(cid:105)|1,0(cid:105))= (√√S+(2mS++11))((S2S−+m2+)1) m(cid:80)=S−S(cid:12)(cid:12)(cid:104)S+1,m|d†γ↓dα↓|S,m(cid:105)(cid:12)(cid:12)2 = 2S6+3
(cid:104)S−1,m|d†γ↓dα↓|S,m(cid:105)= √12(cid:104)S,m|(|S−1,m(cid:105)|1,0(cid:105))√= √(S(2+Sm)()2(SS−−1m)) m(cid:80)=S−S(cid:12)(cid:12)(cid:104)S−1,m|d†γ↓dα↓|S,m(cid:105)(cid:12)(cid:12)2 = 2S6+1
(cid:104)S,m+1|d†α↑dα↓|S,m(cid:105)= 21S(cid:104)S,m+1|S+|S,m(cid:105)= (S−m2)(SS−m+1) (cid:80)S (cid:12)(cid:12)(cid:104)S,m+1|d†α↑dα↓|S,m(cid:105)(cid:12)(cid:12)2 = (S+1)6(S2S+1)
m=−S
√
the Stoner instability, δ −J (cid:28) δ. For a given realiza- where erf(z) = (2/ π)(cid:82)zdtexp(−t2) denotes the error
0
tion of the single-particle levels the energy E acquires function. The result (32) is valid under the following
+
a random correction ∆E : E → E +∆E . This assumptions
2S + + 2S
randomenergycorrectionisduetofluctuationsofsingle-
(1−J/δ)2 (cid:28)(2/π2)ln(cid:2)δ/(δ−J)(cid:3)(cid:28)1. (34)
particle energy in a strip with 2S levels in average. It
can be estimated as ∆E =δ∆n where ∆n stands
2S 2S 2S This restricts the value of the total spin in the ground
forfluctuationofthenumberoflevelsintheenergystrip state to the interval 2(cid:46)S (cid:46)70. The right inequality in
with 2S levels in average. Near the Stoner instability we
Eq. (34) guaranties that fluctuations of S are small and
obtain from the condition E++∆E2S =0 that the spin Gaussian. For S (cid:29) (1/2)exp(π2/2) fluctuations of the
in the ground state is given as
totalspinbecomesnon-gaussian(seeRefs. [32]). Theleft
S = δ (cid:2)1−∆n (cid:3). (29) inequality in Eq. (34)(cid:112)guaranties that the effective tem-
2(δ−J) 2S perature Teff ∼(δ/π) 2ln[δ/(δ−J)] induced by fluctu-
ations and smearing the steps at ε=±2(δ−J) is larger
Itiswell-knownfromtherandommatrixtheory[31]that than the temperature, T (cid:29) δ−J (cid:38) T. We note that
eff
for S (cid:29)1 the fluctuations of ∆n are Gaussian and the effective temperature is low in comparison with the
2S
mean level spacing, T (cid:28) δ. All in all, fluctuations
(cid:0) (cid:1)2 2 (cid:16) (cid:17) eff
∆n =0, ∆n = ln2S+const . (30) of the single-particle levels enhances the elastic spin-flip
2S 2S βπ2
contribution (similarly to enhancement of spin suscepti-
Then with the help of Eqs. (29) and (30), for a function bility [11, 13, 32]) and smear the steps in the inelastic
f(S) which is the quadratic polynomial as in Eq. (25) spin-flip and non-spin-flip contributions.
we find
2
S 2. Inelastic scattering cross section in the presence of
f(S)=f(S)+ ln(2S)f(cid:48)(cid:48)(S), (31)
magnetic field
βπ2
where S = δ/[2(δ−J)]. Using Eq. (31) and averaging Nowweconsiderthebehaviorofinelasticcrosssection
the functions F(ε,E+ +δ∆n2S) and F(ε,E− −δ∆n2S) in the presence of magnetic field B. We assume that
in Eq. (25) over ∆nS with Gaussian distribution (30), in addition to the Zeeman splitting this magnetic field
we obtain the averaged inelastic scattering cross section produces the orbital effect and breaks the time reversal
for temperatures T (cid:46)δ−J and energies ε(cid:28)δ: symmetry. Then the parameter β becomes equal to 2,
(cid:40) β = 2. We consider the case of Zeeman splitting which
4πQ2 2δ2 (cid:104) 2 δ (cid:105) isstrongincomparisonwithtemperaturebutsmallwith
Aσ (ε)= 1+ ln
inel E2 3(δ−J)2 π2 δ−J respecttoδ,δ (cid:29)b=µ g B (cid:29)T. Thenthedegeneracy
c B L
(cid:18) (cid:19)(cid:41) of the low lying states with the total spin S is removed.
δ(cid:112) ThelowestenergystatewiththetotalspinS corresponds
+F ε,2(δ−J), 2ln[δ/(δ−J)] . (32)
π to the maximal total spin projection along the magnetic
field S = S (we assume B > 0). The energies of these
z
Hereweneglectsubleadingtermsincomparisonwiththe states become
largestoneinthefirstlineofEq. (32)whichcorresponds
to elastic spin-flip contribution. The function F(x,y,z) ES(B)=(δ−J)S2−JS−bS. (35)
is defined as follows
Hence, in the presence of Zeeman splitting the total spin
(cid:18) (cid:19) (cid:18) (cid:19)
1 x−y 1 x+y in the ground state is equal to S ≈(δ+b)/[2(δ−J)] for
F(x,y,z)=1+ erf − erf , (33)
2 z 2 z δ−J (cid:28)δ.
8
The absence of degeneracy with respect to the total (δ−J)(cid:28)b(cid:28)δ as
spin projection makes the elastic spin-flip contribution
(cid:34) (cid:32) (cid:33)
to the inelastic cross section to be exponentially small in 4πQ2 1 π[ε−2(δ−J)]
Aσ (ε)= 1+ erf
parameter βb (cid:29) 1. The same holds for the contribution inel E2 2 δ(cid:112)ln[b/(δ−J)]
c
due to inelastic scattering without spin-flip. Thus the
(cid:32) (cid:33)(cid:35)
maincontributiontotheinelasticscatteringcrosssection 1 π[ε+2(δ−J)]
− erf . (39)
comes from inelastic spin-flip scattering: 2 δ(cid:112)ln[b/(δ−J)]
Aσ (ε)= 4πQ2 (cid:88)(cid:88)(cid:48)p [1+e−βε](cid:12)(cid:12)(cid:104)i|d†γ,−σdασ|f(cid:105)(cid:12)(cid:12)2. This result is valid provided the following inequality
inel Ec2 α(cid:54)=γf(cid:54)=i i 1+e−β(ε+Ei−Ef) holds:
(36) 1 b
Here the prime sign indicates that the summation is (δ−J)2 (cid:28) ln (cid:28)1. (40)
π2 δ−J
over the low-energy many-body states i and f which
characterized by the total spin S and the maximal to- The left inequality in Eq. (40) implies that the effec-
tal spin projection along the magnetic field, Sz = S tive temperature Teff = (δ/π)(cid:112)ln[b/(δ−J)] smearing
(see Fig. 1b). The gaps between the ground state the steps in Aσ (ε) at E (B) and −E (B) is not very
inel σ σ¯
with the total spin S and the lowest many-body excited low, T (cid:29) δ − J (cid:38) T. The right inequality in Eq.
eff
states, E±(B) = E± ∓b, can be bounded from above: (40)guarantiesthatfluctuationsofthetotalspinremains
maxE+(B)(cid:54)(δ+b)/SandmaxE−(B)(cid:54)(δ+b)/(S+1). Gaussian.
We note that for S (cid:29) 1 the energy scale (δ +b)/S ≈
2(δ−J) (cid:28) δ. To calculate the matrix elements in (36)
one needs to take into account that the single-particle C. Inelastic scattering cross section in the presence
level α should be the highest doubly occupied level and of strong spin-orbit coupling
γ shouldbethelowestunoccupiedlevelorviceversa(see
transitionintheleftlowercornerofFig. 3). Usingthere- In the previous section we demonstrate that the Zee-
sultsforthematrixelementsfromthetableIwefindthe man splitting suppresses the elastic spin-flip scattering
following result for the inelastic cross section at |ε| (cid:28) δ due to lifting the 2S+1 degeneracy of the ground state.
and T (cid:46)(δ−J)(cid:28)b: Inthissectionwediscussanothermechanismofsuppres-
sionoftheelasticspin-flipscatteringonthequantumdot.
Aσ (ε)= 4πQ2(cid:104)1−n (cid:0)ε−E (B)(cid:1)+n (cid:0)ε+E (B)(cid:1)(cid:105). Weconsideraquantumdotfabricatedin2Delectrongas
inel E2 F σ F σ¯ with strong spin-orbit coupling. Such quantum dot can
c
(37) be described by the universal Hamiltonian (2) in which
As one can check the result for Aσ (ε) at B <0 can be the Heisenberg exchange is substituted by the Ising ex-
inel
obtained from the result for Aσ¯ (ε) for B >0. change: JS2 →JS2 [33,34]. In this case the statistics of
inel z
ItisinstructivetocomparetheresultsforAσ (ε)with single particle levels is described by the unitary symme-
inel
and without magnetic field. At first, the magnetic field try ensemble (class A) with β =2.
suppresses the elastic spin-flip contribution. Secondly, The low energy many-body states correspond to the
instead of four steps of height 1/2 (in case of large spin totalspinS andthemaximalorminimalspinprojection
S (cid:29)1) at energies E± and −E± in the absence of mag- Sz =±S. The energies of these states are equal to
netic field (see Eq. (37)), in the presence of the Zeeman
splitting only two steps at E (B) and −E (B) of height E =(δ−J)S2. (41)
σ σ¯ S
1 survive. We stress that, contrary to the case of mag-
netic impurity the inelastic scattering cross section off Therefore the total spin in the ground state is equal to 0
the quantum dot in the presence of the Zeeman splitting (1/2) in case of even (odd) number of electrons.
at energies |ε| (cid:29) E (B) are not exponentially small in For the even number of electrons, since S = 0, the
σ
βb(cid:29)1. Forenergies|ε|(cid:28)E (B)theinelasticscattering elastic spin-flip scattering vanishes. For T (cid:46) δ−J and
σ
cross section is zero at T =0. |ε|(cid:46)δ, the only contribution to Aσ (ε) remains due to
inel
In the presence of fluctuations of the single-particle the inelastic spin-flip:
levels the energies E (B) become random, E (B) →
σ σ
Eσ(B) + σ∆E2S. As a consequence, the spin in the Aσ,e(ε)= 4πQ2(cid:104)1−n (cid:0)ε−∆ (cid:1)+n (cid:0)ε+∆ (cid:1)(cid:105). (42)
ground state becomes fluctuating: inel E2 F e F e
c
1 (cid:104) (cid:105) Here ∆ =δ−J stands for the gap between the ground
S = δ+b−δ∆n . (38) e
2(δ−J) 2S state with S = Sz = 0 and the states with S = 1 and
S =±1.
z
Averaging the Fermi functions in Eq. (37) over ∆n In the case of the odd number of electrons, since S =
2S
over the Gaussian distribution (30) with β = 2, we find 1/2, the ground state is doubly degenerate. Then the
the inelastic scattering cross section at |ε| (cid:28) δ and T (cid:46) elasticspin-flipscatteringisthesameasforthemagnetic
9
impurity with spin 1/2. In addition, inelastic spin-flip IV. DISCUSSIONS AND CONCLUSIONS
contributes to the inelastic cross section. Then at T (cid:46)
δ−J and |ε|(cid:46)δ we find Our results for the inelastic scattering cross section
of an electron off the quantum dot at low temperatures
Aσ,o(ε)= 2πQ2(cid:104)1+1−n (cid:0)ε−∆ (cid:1)+n (cid:0)ε+∆ (cid:1)(cid:105). (43) allow us to estimate corresponding contribution to the
inel Ec2 F o F o dephasing rate. Assuming a finite concentration ns of
quantum dots we introduce the inelastic scattering rate
for an electron as follows
Here ∆ =2(δ−J) denotes the gap between the ground
o
state with S = 1/2 and the states with S = 1 and Sz = τ−1(ε,T)= ns (cid:88) Aσ (ε). (46)
±1. inel ν inel
σ=±
We note that the inelastic scattering rate at energies
|ε|(cid:29)δ−J isindependentofelectronparityinthequan- We remind that ν denotes the average density of states
tum dot, per spin for electrons in the electron liquid surrounding
quantum dots. We note that in Refs. [5, 8] the inelas-
4πQ2 tic rate at finite temperature has been related directly
Aσ,e(ε)=Aσ,o(ε)= . (44)
inel inel E2 to the difference between the imaginary part of the T-
c
matrix and the diagonal element of its square. Although
this is correct for the case of zero temperature at finite
In the case of temperatures δ (cid:29) T (cid:29) δ − J, the
temperature it is not the case in general (see discussion
low-lying many-body states with the total spin S (cid:46)
after Eq. (14)). In our case, by definition, the quantity
(cid:112)
T/(δ−J) contribute to the inelastic cross section for Aσ (ε) includes the inelastic processes only.
|ε|(cid:46)δ. Similar to low temperatures, the dominant con- inel
Using the fact that the quantity 1/τ (ε,T) repre-
inel
tribution is due to inelastic spin-flip. Then we obtain
sents the self-energy for the electron pair propagator
(cooperon) one can estimate the dephasing time τ (T)
φ
Aσinel(ε)= 8πEQ22 (cid:34) (cid:88)∞ eβ(J−δ)Sz2(cid:35)−1 (cid:88)∞ eβ(J−δ)Sz2 etinotnertiongthtehecoenxdpurecstsivioitnyf[o5r].thTehewecaokn-clroectaeliezxaptiroenssicoonrrdeec--
c Sz=−∞ Sz=−∞ pends on dimensionality. Having in mind experiments of
(cid:0) (cid:1) 8πQ2 Refs. [10] we restrict our discussion to two dimensions,
×F ε,(δ−J)(2S +1) = . (45)
z E2 d=2. In this case, one can obtain [5]:
c
(cid:20)(cid:90) (cid:21)
We note that the the inelastic cross section for |ε| ∼ δ τ−1(T)=exp dεn(cid:48) (ε)lnτ (ε,T) . (47)
φ F inel
at δ (cid:29) T (cid:29) δ −J is twice larger than at T (cid:28) δ −J.
This difference stems from the following. At high tem-
We mention that the above estimate for τ (T) is based
peratures, δ (cid:29) T (cid:29) δ−J, the following four combina- φ
on independent treatment of the inelastic scattering off
tions of initial and final states contribute to the inelastic
quantum impurities and elastic disorder scattering. As
spin-flip cross section (see Eq. (21)): (i) |i(cid:105) = |S,S(cid:105)
discussedinRef. [5],suchsimplifiedapproachisvalidfor
and |f(cid:105) = |S −1,S −1(cid:105); (ii) |i(cid:105) = |S,−S(cid:105) and |f(cid:105) =
d=2 under the following assumptions: (i) the system is
|S+1,−S−1(cid:105); (iii) |i(cid:105)=|S+1,S+1(cid:105) and |f(cid:105)=|S,S(cid:105);
very good metal: the conductance g (cid:29) (νJ )−3; (ii) the
(iv) |i(cid:105)=|S−1,−S+1(cid:105) and |f(cid:105)=|S,−S(cid:105). In the case s
density of quantum impurities is not large, n (cid:28) νT .
of low temperatures, T (cid:28) δ −J, when the state with s K
For our problem the characteristic exchange interaction
the lowest spin (S = 0 or S = 1/2) contribute only, the
between electrons and quantum impurities (quantum
transitions (i) and (iv) are not possible.
dots) can be estimated as νJ = Q/E (see Sec. IIIA).
s c
In case of the even number of electrons the gap ∆e is The Kondo temperature TK is given by the standard ex-
determinedbythedifferenceinthelevelspacingbetween pression, T ∼E exp(−1/νJ ).
K c s
the lowest singly occupied and the highest doubly occu- We start the discussion of τ (T) from the case of
φ
piedlevelsandtheexchangeenergy. Asitiswell-known, isotropicexchangeinteractiononthequantumdot. Near
the level spacing strongly fluctuates and its distribution the Stoner instability, δ−J (cid:28) δ, at temperatures T (cid:28)
can be well approximated by the Wigner Surmise (see δ−J,theinelasticscatteringrateAσ (ε)isgivenbyEq.
inel
Ref. [31]). The typical scale of this distribution is given (25). Performing expansion in small energy-dependent
by the mean level spacing. Qualitatively, averaging of terms we find from Eq. (47)
Eq. (42)overdistributionof∆ resultsinthesameform
e
of the dependence on energy but with effective temper- (cid:34)
8πn Q2 (S+1)(2S+1)
ature proportional to the mean level spacing. Similar τ−1(T)= s +βE e−βE−
φ νE2 3 −
results one obtains after averaging of Eq. (43). The in- c
elastic cross section at temperatures δ (cid:29) T (cid:29) δ−J is (cid:35)
2S+3
robustwithrespecttofluctuationssinceitisindependent + 2S+1βE+e−βE+ (48)
of particular properties of the single-particle spectrum.
10
There is a weak temperature dependence of the dephas- (which is of the order of Γ ∼ g δ) in comparison with
T
ing rate due to possibility of inelastic scattering which the intrinsic inelastic rate 1/τ due to electron-electron
ee
involves transitions to the lowest many-body levels of interaction inside the quantum dot, this nonequilibrium
the quantum dot. Also we emphasize that the elastic effect can be neglected. For the quantum dot of size
spin-flip contribution to the dephasing rate is different larger than the mean free path l, the intrinsic inelastic
from a standard one for a magnetic impurity which is ratecanbeestimatedas[35,36]: 1/τ ∼T2/(g2 δ). The
ee Th
proportional to S(S + 1)/3. We repeat that it occurs conditionΓ(cid:28)1/τ resultsinthefollowingrestrictionon
ee
due to additional correlations between tunneling ampli- temperaturesatwhichourassumptionoftheequilibrium
tudes for transitions to different levels of the quantum for the quantum dot holds:
dot. Using Eq. (28) we obtain at higher temperatures
δ−J (cid:28)T (cid:28)δ: T (cid:29)δ(g g2 )1/2. (54)
T Th
(cid:34) (cid:35)
8πn Q2 δ(3δ−2J) T Since in this work we study temperatures below δ −J,
τ−1(T)= s + . (49)
φ νE2 6(δ−J)2 3(δ−J) the following condition for the tunneling conductances
c
(or for proximity to the Stoner instability) emerges:
Wementionthatinfactbothestimates(48)and(49)hold
for dimension d = 3 as well. In the presence of Zeeman 1 (cid:18)δ−J(cid:19)2
g (cid:28) . (55)
splitting, the elastic spin-flip is suppressed. Then using T g2 δ
Th
Eq. (37), we find the following estimate for the dephas-
ing rate at low temperatures T (cid:28) δ −J and moderate Also our approach neglects the effect of the reservoir
magnetic fields, δ (cid:29)b(cid:29)δ−J: on the dynamics of the total spin in the quantum dot.
Inparticular,weneglecttherenormalizationofthevalue
τ−1(T)= 4πe2nsQ2 (cid:104)e−βE+(B)+e−βE−(B)(cid:105). (50) of the total spin due to coupling to the reservoir. Using
φ νE2 adiabatic approximation for the large total spin of the
c
quantum dot near the Stoner instability [37], one can
We note that the dephasing rate in this case is exponen-
demonstratethatthespinmovesdiffusivelyontheBloch
tially small in temperature, τ−1(T)∼exp(−2β(δ−J)).
φ sphere with the diffusive constant proportional to the
In the case of Ising exchange interaction on the quan-
tunneling coupling Q [18,19].
tum dot τ (T) depends on the parity of the number of
φ Recentexperiments[10]giveanevidencewhichmaybe
electrons at low temperatures T (cid:28) δ −J. Using Eqs.
interpreted as formation of local spin droplets in 2D dis-
(42) and (43), we obtain
ordered electron liquid at low temperatures. As known
[38], at low temperatures 2D disordered electron liquid
8πn Q2
τ−1(T)= s e−β∆e (51) tendstotheStonerinstabilitysuchthattherenormalized
φ,e νEc2 Fermi-liquid interaction constant in the triplet channel
tendsto−1,Fσ ≈−1. Thecreationofspindropletswith
for the even number of electrons, and 0
afinitespinS =1/[2(1+Fσ)](cid:29)1neartheStonerinsta-
g 0
bility in disordered electron liquid due to fluctuations in
τ−1(T)= 4πnsQ2 (cid:2)1+πe−β∆o(cid:3) (52) thetriplet(spin)channelhasbeenpredictedinRef. [12].
φ,o νE2
c ThePaulispinsusceptibilityχ∼ν/(1+Fσ)dominatesat
0
high temperatures. Due to presence of spin droplets one
fortheoddnumberofelectrons. Athighertemperatures,
expects that the spin susceptibility is dominated by the
δ (cid:29)T (cid:29)δ−J, the dephasing time becomes insensitive
Curie-like temperature dependence, χ ∼ n S2/T at low
to the parity of the number of electrons: s g
temperatures T (cid:28) T = nflS /ν. Here nfl denotes the
∗ s g s
16πn Q2 densityofspindroplets. Wenotethatintheexperiments
τ−1(T)= s . (53)
φ νE2 [10] the spin susceptibility behaves as χ ∼ T−2 suggest-
c
ingstrongtemperaturedependenceofthedropletdensity
Thus,thedephasingtimeforthetemperaturerange,δ (cid:29) nfl. The electron scattering off such spin droplets re-
s
T (cid:29) δ−J, due to scattering off the quantum dot with sults in the following contribution to the dephasing rate:
Ising exchange is similar to the magnetic impurity. nflS2/(νg) where g is the conductance of 2D disordered
s g
We note that our approach completely ignores the ef- electron liquid [12]. Comparing this contribution with
fectofelectronreservoironthequantumdot. Firstofall, the standard dephasing rate due to electron-electron in-
thecouplingtothereservoirresultsinthebroadening(Γ) teraction in the triplet channel, T/[g(1+Fσ)] [39], one
0
ofthesingle-particlelevelswhichisoftheorderofg δ. It finds that the dephasing rate should saturate below the
T
can be neglected provided temperatures are not too low, same crossover temperature T . Thus, in the presence of
∗
T (cid:29) g δ. Secondly, due to coupling to the reservoir, suchspindropletstheCurie-liketemperaturedependence
T
the probabilities p of many-body states of the quantum of the spin susceptibility should be accompanied by the
i
dot can become nonequilibrium, i.e. very different from temperatureindependentdephasingtime. Incontrast,in
theGibbsform. However, inthecaseofslowescaperate the experiments [10] the strong temperature dependence