Kondo Screening Cloud and Charge Quantization in Mesoscopic Devices

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by  Rodrigo G. Pereira

Kondo Screening Cloud and Charge Quantization in Mesoscopic Devices
Rodrigo G. Pereira,1 Nicolas Laflorencie,1,2 Ian Affleck,1 and Bertrand I. Halperin3
1Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada, V6T 1Z1
2Insitute of Theoretical Physics, E´cole Polytechnique F´ed´erale de Lausanne, Switzerland
3Physics Department, Harvard University, Cambridge, MA 02138, USA
(Dated: February 3, 2008)
8
0 We propose that the finite size of the Kondo screening cloud, ξK, can be probed by measuring
the charge quantization in a one-dimensional system coupled to a small quantum dot. When the
0
chemicalpotentialinthesystemisvariedatzerotemperature,oneshouldobservechargestepsthat
2
are controlled by the Kondo effect when the system size L is comparable to ξK. We show that, if
n the standard Kondo model is used, the ratio between the widths of the Coulomb blockade valleys
a with odd or even numberof electrons is a universal scaling function of ξK/L.
J

8 PACSnumbers: 72.15.Qm,73.21.La,73.23.Hk
2

] TheKondoeffectcanbedescribedasthestrongrenor- L
l
e malizationof the exchange coupling between an electron
-
r gas and a localized spin at low energies [1]. Below some
t characteristicenergy scale,knownasthe Kondotemper-
s
t. ature TK, a non-trivial many-body state arises in which Vdw Vg
a the localized spin forms a singlet with one conduction
m
electron. IntherecentrealizationsoftheKondoeffect[2], FIG. 1: Possible experimental setup. V controls the tun-
dw
- a semiconductor quantum dot in the Coulomb blockade nelingt′ betweenthesmalldot(ontheleft)andthewireand
d
n regime plays the role of a spin S = 1/2 impurity when Vg varies the chemical potential in the wire.
o the number of electrons in the dot is odd. The Kondo
c temperatureinthesesystemsisafunctionofthecoupling
[
between the dot and the leads and can be conveniently with an adjacent dot. Therefore, the addition spectrum

3 controlled by gate voltages. As a clear signature of the should exhibit signatures of the finite size of the Kondo
v Kondoeffect,oneobservesthattheconductancethrough cloud.
5
a quantum dot with S =1/2 is a universal scaling func- We considerthe geometryshowninFig. 1. Oneofthe
3
6 tion G(T/TK) and reaches the unitary limit G = 2e2/h ends of a wire of length L with a large number of elec-
2 when T ≪ TK for symmetric coupling to the leads [3]. trons(suchthat∆≪ǫF,whereǫF istheFermienergy)is
1 Onthe other hand, we expect that in a finite systemthe weaklycoupledtoasmallquantumdotviaagatevoltage
6 infraredsingularitiesthatgiverisetotheKondoeffectare V . Boththedotandthewire(whichcanbethoughtof
dw
0
cut off not by temperature, but by the level spacing ∆ if asaverylongandthindot)areintheCoulombblockade
/
t ∆ T. For a one-dimensional (1D) system with length regime. The dot has a very large charging energy and
a ≫
m L and characteristic velocity v, the relevant dimension- a ground state with S = 1/2. We assume that the wire
- less parameter is ∆/TK ∼ ξK/L, where ξK ≡ v/TK is is very weakly connected to a reservoir and that trans-
d identifiedwiththesizeofthe Kondoscreeningcloud,the fer of a single electron between wire and reservoir could
n mesoscopic sized wave function of the electron that sur- bemeasuredbysomeCoulombblockadetechnique. This
o rounds and screens the localized spin. This large length coupling is assumed weak enough so as not to affect the
c
v: rsceanltely(ξsKtu∼die0d.1m−e1sµomsc)oipsiccodmepviacreasb,lwewhiicthhthhaessmizoetoivfactuerd- esmnearlglydoletvselasnodfltahregewdiroet-sdohtavseysbteemen. pEexrfpoerrmimeden,tfsorweitxh-
i
X several proposals that the Kondo cloud should manifest ample, in [7] and considered theoretically in [6].
itself through finite size effects in such systems [4, 5, 6].
r Here we investigate this effect using the basic Kondo
a
One familiar property of small metallic islands in the model, ignoring charge fluctuations on the dot, assum-
Coulomb blockade regime is the quantization of charge. ing the wire contains a single channel and ignoring all
Since each electron added to the system costs a finite electron-electron interactions in the wire. Within the
energy, the number of electrons changes by steps as constantinteractionapproximation[2]theseinteractions
one varies the chemicalpotential and sharpconductance arerepresentedbyasimplechargingenergywhoseeffects
peaks are observed at the charge degeneracy points [2]. on the charge steps are easily subtracted off. We post-
In AlGaAs-GaAs heterostructures,the level spacing of a pone a more complete treatment of these complications,
1D system with size L 1µm is of order ∆ 100µeV, toalaterwork. (Theeffectofshortrangeinteractionson
∼ ∼
large enough to be resolved experimentally. Clearly, the the conductance through a Kondo impurity was treated
energylevelsshouldbeaffectedbytheKondointeraction in the Luttinger liquid framework in [5].)
2
Weak coupling. – The tunneling between the wire and
5
the dot is well described by the Anderson model [5]. For N = odd
0
zero tunneling amplitude the system reduces to an open
chain and a free spin. In the continuum limit, we in- 4
troduce the fermionic field Ψ(x) for the electrons in the N0 δµo
wire, as well as its right- and left-moving components - 3
µ) δµe
Ψ(x)=eikFxψR(x)+e−ikFxψL(x). N( 2
The open boundary conditions Ψ(0) = Ψ(L) = 0 im-
ply that ψL,R are not independent. Instead, ψL can be 1 fwineiatke cξoKu/Lpling limit
regarded as the extension of ψR on the negative-x axis: strong coupling limit
ψ ( x)= ψ (x). Wecanthenworkwithrightmovers 0
R L 0 1 2
− − µ µ
only and write down an effective Kondo model (we drop 1/2 3/2 (µ−µ*)/∆
the index R hereafter) 0
L ~σ FIG.2: (Coloronline)Chargequantizationstepsforthewire
H = ivF dxψ†∂xψ+2πvFλψ†(0) ψ(0) S~, (1) coupled to a small quantum dot. The arrows indicate the
− 2 ·
Z−L direction in which the single steps move as λeff(L) grows.
whereλisthedimensionlessKondocoupling. Weassume
that λ is independent of Vg, which may require that t′ we write GS(N =odd) = c↑†c↓† 0 γ , where
and the energy of the localized state in the dot be tuned | i n≤0 kn kn| i⊗| i
γ = , is the spin state of the dot. However, for
accordingly. For λ 1, the size of the Kondo screening | i |⇑i |⇓i Q
≪ N even, there is one single electron occupying the state
cloud (defined in the thermodynamic limit) is exponen-
at the Fermi surface (with momentum k ). To lowest
tially large: ξ (v /D)e1/λ, where D ǫ is a high- F
K ∼ F ≪ F orderindegenerateperturbationtheory,thegroundstate
energy cutoff. The boundary condition in the weak cou- for λ 0 is GS(N =even) = c↑†c↓† 0 s ,
pling fixed point (L/ξ 0) reads ψ( L) = eiδψ(L), → | i n≤0 kn kn| i⊗| i
K → − where s [k k ]/√2is the singlet
where δ = 2kFL mod π. Using the mode expansion | i≡ | F ↑i⊗|⇓i−| F ↓i⊗Q|⇑i
state between the spin of the dot and the electron at
ψ(x) = (i/√2L) eikxc , we obtain the momentum
eigenvalu−es kn =(nPπk−δ)/Lk, n∈Z. wkFh.ereTS~his statech†a~σscStiostth=et0otaanldsphinS~of·tS~heel ico=ndu−c3ti/o4n,
WedenotebyN(µ)thetotalnumberofelectronsinthe el ≡ k k2 k
electrons. The singlet formation lowers E(N) for N =
system,includingtheoneinthequantumdot. Wearein-
P
even relatively to N = odd. As a result, the Kondo
terestedintheelementarystepsofN aroundsomeinitial
value N N(µ∗). At T =0, we calculate N(µ) as the interactionsplits the double steps andgivesriseto small
0 ≡ 0 plateaus with N even (Fig. 2). To O λ2 , we find
integervalueofN thatminimizesthethermodynamicpo-
stetantteiaelnΩer(gNy.)=DeEfin(Nin)g−µℓµ+N21,awshtehree Eva(luNe)oifsµthwehgeroreunNd µℓ+12∆−µ∗0 = [2ℓ] − 34(−1)ℓ λ+λ2l(cid:0)nDv(cid:1)L +... , (2)
changes from N0 +ℓ to N0 +ℓ+1, it follows from the (cid:18) F (cid:19)
chargedegeneracyconditionΩ(N0+ℓ+1)=Ω(N0+ℓ) where[ℓ] ℓ+1forℓevenand[ℓ] ℓforℓodd. Notethe
≡ ≡
twheatseµtℓµ+∗012h=alfwEa(yNb0e+twℓee+n1t)h−e hEig(hNes0t+ocℓc).upFieodraλnd=th0e, ilsotgiacroitfhtmheicKdoivnedrogeenffceecta.tWOeλre2cogansiLze→the∞e,xpchanarsaiocnteorf-
lowest unoccupied energy level of the open chain. In the effective couplingλ (L(cid:0)) (cid:1)[ln(ξ /L)]−1 inpowers
eff K
∼
this case, N0 = odd: the ground state is doubly de- of the bare λ, as expected from scaling arguments. We
generate (total spin Stot = 1/2) and consists of one have verified that the scaling holds up to O λ3 , i.e.,
ethleectFreornmiinsethae. dTohteaennderpgayirlsevoeflscomnedauscutrieodnferloemctrµo∗nsarine µℓ+21 hasnodependenceonthecutoffD butth(cid:0)eim(cid:1)plicit
0 oneintheexpansionofλeff (L). Thisisremarkablegiven
ǫ = (n 1/2)∆, where ∆ = πv /L. It is easy to see
n F that E(N) itself is cutoff dependent. Based on this, we
−
that, as we raise µ > µ , N jumps whenever µ crosses
0 conjecture that the relative width of the plateaus with
an energy level. Moreover, there are only double steps
N odd/even is a universal scaling function of ξ /L. We
K
due to the spin degeneracy of the single-particle states
define the ratio (see Fig. 2)
(µ2m+12 = µ2m+32). As a result, N(µ) − N0 takes on δµo E(N +3) 2E(N +2)+E(N +1)
only evenvalues andN is alwaysodd (see dashedline in 0 0 0
R = − . (3)
Fig. 2). ≡ δµe E(N0+2) 2E(N0+1)+E(N0)
−
We obtain the charge steps in the weak coupling limit
From Eq. (2), we have that in the weak coupling limit
by calculating the correction to the ground state energy
using perturbation theory in λ [5]. In the limit λ 0, ξ 2 ξ 1 ξ
K K K
→ R 1 ln lnln +const.. (4)
thegroundstateforN oddisstilldoublydegenerateand L ≫ ≈ 3 L −3 L
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
3
doSttfroornmgscaouspinlignlge.tw–itWhhoenne cλoenfdfu→ctio∞n,etlehcetrsopninanodf tdhee- 10 weak coupling JJJ===000...000678
J=0.1
couples from the other electrons in the chain. In this 15 J=0.125
(a) J=0.15
limit the Kondo cloud is small (ξK/L→ 0). The strong 1 10 JJ==00..23
coupling boundary conditions reflect the π/2 phase shift J=0.4
5 J=0.5
for the particle-hole symmetric case: ψ( L) = ψ(L) ) J=0.6
[8]. This implies that the k eigenvalues a−re shifte−d with ξK 0.1 0 10-9 10-6 10-3 100 JJ==00..78
/
respecttoweakcoupling: k =(nπ+π/2 δ)/L. Atthis L
fixed point we recover the snpin degenerac−y of the single- R( 1012 From collapse
0.01 0.31*exp(π/c)
particle states and double steps in N(µ) appear at the ξ 108
K
shifted energy levels (ǫ = n∆ as measured from the
oprluigsinpaalirµs∗0o).f eTlehcetrgornosuninndtshtaetweicroen,stihstussnNow(µo)fias asilnwgalyest 0.001 1100400 5 10 15 (b)20 strong co
even. Fig. 2 illustrates how the charge staircase evolves 1/J uplin
0.0001 g
monotonically from weak to strong coupling. 10-9 10-6 10-3 100 103
We explore the limit k−1 ξ L by working out L/ξ
F ≪ K ≪ K
a local Fermi liquid theory [9]. The idea is that virtual
transitions of the singlet at x=0 induce a local interac- FIG.3: (Color online) Universalratio R=δµodd/δµeven as a
tion in the spin sector of the conduction electrons. The
scaling function of ξK/L. Bethe ansatz results obtained for
leading irrelevantoperatorthat perturbs the strongcou- various systems sizes (N0 = 51, 101, 201, 501, 1001, 2001)
pling fixed point and respects SU(2) symmetry is and13differentvaluesoftheKondoexchangeJ,indicatedby
different symbols. For each value of J, the system lengths L
2π2 v2 ~σ 2 have been rescaled L→L/ξK(J) in order to obtain the best
HFL = F ψ†(0) ψ(0) , (5) collapse of the data using the strong coupling curve Eq. (7)
− 3 T 2
K (cid:20) (cid:21) (dashed red line) as a support for the rest of the collapse.
The weak coupling regime for L ≪ ξK, enlarged in the in-
where the prefactor is fixed such that the impurity sus-
set (a), is described by the weak coupling expansion Eq. (4)
ceptibility is χimp = 1/(4TK). This interaction lowers withconstant≃0.33 (continuousbluecurve). Inset(b): The
the groundstate energy when there is an odd number of Kondo length scale, extracted from the universal data col-
remaining conduction electrons (S = 1/2), thus split- lapse of the main panel (black squares), is described by the
el
ting the charge steps in the strong coupling limit. To exponential fit Eq.(8) with ξ0 ≃0.31 (dashed green line).
lowest order in 1/T , we find
K
µℓ+12 −µ∗0 = [ℓ−1]+1 +( 1)ℓπξK, (6) Tinhgetuhneivxe-rasxailsscLalingLcu/rξve(Jha)sinbeoenrdoebrttaoingedetbtyhreesbceaslt-
∆ 2 − 8 L → K
collapse of the data. The entire crossover curve is ob-
from which we obtain tained, ranging from the strong coupling L ξ to the
K
≫
weak coupling regime L ξ . Both strong coupling
K
ξ πξ ≪
R K 1 K. (7) [Eq. 7)] and weak coupling [Eq. (4)] results are perfectly
L ≪ ≈ 4 L reproduced. As displayed in the inset (a) of Fig. 3, BA
(cid:18) (cid:19)
results agree with Eq. (4) using only one fitting param-
Bethe Ansatz results. – In order to check the scal-
eter: constant 0.33. The Kondo length scale, shown in
ing property of R = δµodd/δµeven for any L/ξ we ≃
K the inset (b) of Fig.3, displays the expected exponential
use the Bethe Ansatz (BA) solution of the one-channel
behavior [10]
Kondo problem [10]. We start with a half-filled band
of N0 − 1 conduction electrons (N0 odd) coupled to a ξK(J)=ξ0eπ/c, (8)
localized impurity spin, corresponding to a system size
L = (N 1)/2 in units where the Bethe ansatz cut- with c = 2J/(1 3/4J2). Fitting ξ to this expression
0 K
− −
off parameter D [10], related to the band-width, is set gives ξ 0.31 (see inset (b) of Fig. 3) which is in very
0
≃
equal to one. Since the BA solution can be obtained good agreement with the expected value of ξ = 1/π,
0
for any filling factor, we can add particles one by one resulting [1] from the impurity susceptibility normalized
in the system (N = N , N +1, N +2,...) and com- to 1/(4T ).
0 0 0 K
pute the corresponding energies. This has been done by Flux dependence in quantum rings. – The effect we
solving numerically the coupled BA equations [10], us- have described is not unique to the wire geometry. One
ing a standard Newton-Rapson method. In Fig. 3 we interesting alternative is to suppose that the quantum
present results obtained for the ratio R = δµodd/δµeven dot is embedded in a ring with circumference L (inset of
withN =51, 101, 201, 501, 1001, 2001and13different Fig. 4). Thisgeometryoffersthepossibilityoflookingat
0
valuesoftheKondocouplingintherange0.06 J 0.8. the dependence of the charge steps on the magnetic flux
≤ ≤
4
2.5 tosplittwooutofthefourvaluesofµℓ+12 [13]. Wecalcu-
∆)/ latethe splitting δµ′n ≡µ2n+12 −µ2n−25 usingdegenerate
µ*0 µ Φ Vg perturbation theory to O(1/TK) and find (ξK ≪L)
- 7/2
+1/2 1.5 µ5/2 δµ′n πξK + α−mπ 2+ πξK 2, (11)
µl 2∆ ≈ 8 L s π 8 L
( (cid:18) (cid:19) (cid:18) (cid:19)
s µ
ep 3/2 for α ≈ mπ,m = n,n±2,... (Fig. 4). This striking
st 0.5 difference in the flux dependence of the charge steps is
ge µ1/2 consistent with the results in [5].
r
a Finally, we would like to point out that the constant
h
c interactionmodelassumedhereneglectstheexchangein-
-0.5 teraction between electrons in the wire. However, such
0 0.5 1 1.5 2
α/2π = Φ/Φ interaction could be cast into one empirical parameter
0
that gives a constant contribution to the even-odd split-
ting, even in the weak and strong coupling limits [2].
FIG. 4: (Color online) Flux dependence of the charge steps
In conclusion, we have shown that the width of the
intheringwithanembeddedquantumdot(inset),assuming
zeropotentialscattering. Dashedlines: weakcouplingregime chargestepsinamesoscopicdevicedepends onthe finite
(ξK ≫L); solid lines: strong coupling regime (ξK ≪L). size of the Kondo cloud. The signature of this effect
wouldbethebroadeningoftheCoulombblockadevalleys
with total N evenas we increase T (decrease ξ ). The
K K
Φ threading the ring, which is related to the persistent crossover between weak (ξ L) and strong (ξ
K K
current for the embedded quantum dot [5, 11]. For sim- L) coupling could be used to≫extract the characterist≪ic
plicity, we neglect potential scattering and assume that length scale of the Kondo problem.
thecouplingbetweenthedotandtheleadsissymmetric. WethankE.SørensenandJ.A.Folkforusefuldiscus-
In this case it is convenientto label the eigenstate of the sions. WeacknowledgesupportfromNSERC(RGP,NL,
open chainby their symmetry under the parity transfor- IA), CNPq (RGP, 200612/2004-2),CIAR (IA) and NSF
mation x L x. We define the even and odd fields (BIH, grant DMR-05-41988).
→ −
ψ (x) = [ψ (x) e−ikFLψ (L x)]/√2, in terms of
e,o R L
± −
which the Kondo interaction takes the form [5]
α α
H = 4πv λ cos ψ†(0) isin ψ†(0)
K F 2 e − 2 o [1] A.Hewson,TheKondo Effect to Heavy Fermions (Cam-
~σ cohsαψ (0)+isinαψ (0) Si~, (9) bridge University Press, Cambridge, England, 1993).
×2 2 e 2 o · [2] For a review, see L.I. Glazman and M. Pustilnik, in:
h i Nanophysics: Coherence and Transport. Les Houches
where α 2πΦ/Φ0 and Φ0 = hc/e is the flux quantum. Session LXXXI, edited by H. Bouchiat et al. (Elsevier,
≡
For N0 =4p+1 (p integer), the lowest unoccupied state Amsterdam, 2005). cond-mat/0501007
of the open chain (λ=0) is even under parity. Defining [3] W.G. van derWiel et al., Science 289, 2105 (2000).
δµn ≡µ2n+32 −µ2n+12 asthe splittingofthechargesteps [4] W82.,B2.1T43hi(m1,99J9.)K;rHoh.aH,ua,nGd.J-M.v.oZnhDanelgf,t,aPnhdyLs..RYeuv,.PLheytts..
in the weak coupling limit, we find
Rev.Lett.86,5558(2001);P.S.CornagliaandC.A.Bal-
δµ 3 seiro, Phys.Rev.B 66, 115303 (2002).
2∆n = 4[1+(−1)ncosα]λeff (L)+O λ2eff . (10) [5] IP..ASfflimeocnkaannddPI..ASifflmeocnk,,PPhhyyss..RReevv..LBet6t.48,608,52380584((22000011));.
Notice that for α = 2mπ only the even ch(cid:0)anne(cid:1)l couples [6] P.Simon,J.Salomez,andD.Feinberg,Phys.Rev.B73,
205325 (2006); R.K. Kaul et al., Phys. Rev. Lett. 96,
to the impurity and the steps in the odd channel (corre-
176802 (2006).
sponding to odd values of n) do not split. The opposite
[7] R.M. Potok et al., cond-mat/0610721.
happens when α=(2m+1)π (see Fig. 4).
[8] We can account for particle-hole symmetry breaking by
Inthe strongcouplinglimit, theπ/2phaseshiftmodi- adding to the Hamiltonian a marginal scattering poten-
fiestheboundaryconditionsinsuchawaythatthetrans- tialtermwithamplitude∼O(J).Thistermsimplyshifts
missionacrossthedotbecomesperfect[3,5]. Thesystem the position of thecharge steps, without affecting R.
is then equivalent to an ideal 1D ring, whose eigenstates [9] P. Nozi`eres, J. Low. Temp. Phys. 17, 31 (1974).
[10] N. Andrei, Phys. Rev. Lett. 45, 379 (1980); P.B. Wieg-
can be labeled as right or left movers. The energy lev-
man, JETP Lett., 31, 364 (1980); N.Andrei, K. Furuya
els are flux dependent and exhibit zigzag lines with level
and J.H. Lowenstein, Rev.Mod. Phys. 55, 331 (1983).
crossings at α = mπ [12]. The major effect of a Fermi [11] A.A. Zvyagin and T.V. Bandos, Low Temp. Phys. 20,
liquid interaction analogous to Eq. (5) is to lift the de- 222 (1994); K. Kang and S.-C. Shin, Phys. Rev. Lett.
generacy between right and left movers at α = mπ and 85, 5619 (2000); H.-P. Eckle, H. Johannesson, and C.A.
5
Stafford,Phys.Rev.Lett.87,016602(2001);A.A.Aligia, scatteringonehasavoidedlevelcrossingsatα=mπand
Phys.Rev.B 66, 165303 (2002). the doublesteps for small ξK/L in Fig. 4 also split.
[12] A.Fuhrer et al., Nature (London) 413, 822 (2001).
[13] In the more general case of nonzero but weak potential