Kondo Physics in the Single Electron Transistor with ac Driving

4 Pages·0.19 MB·English

by  Peter Nordlander

Kondo Physics in the Single Electron Transistor with ac Driving
Peter Nordlander
Department of Physics and Rice Quantum Institute, Rice University, Houston, Texas 77251-1892
Ned S. Wingreen
8
NEC Research Institute, 4 Independence Way, Princeton, NJ 08540
9
9
Yigal Meir
1
Physics Department, Ben Gurion University, Beer Sheva, 84105, Israel
n
a
David C. Langreth
J
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019
3
2

]
l
l
a Using a time-dependentAnderson Hamiltonian, a quantum dot with an ac voltage applied to a nearby gate is investigated.
h
Arichdependenceofthelinearresponseconductanceontheexternalfrequencyanddrivingamplitudeisdemonstrated. Atlow
-
s frequenciestheacpotentialproducessidebandsoftheKondopeakinthespectraldensityofthedot,resultinginalogarithmic
e decrease in conductance over several decades of frequency. At intermediate frequencies, the conductance of the dot displays
m
an oscillatory behavior due to the appearance of Kondo resonances of the satellites of the dot level. At high frequencies, the
. conductanceof thedot can varyrapidly dueto theinterplay between photon-assisted tunneling and theKondo resonance.
t
a
m PACS numbers: 72.15.Qm, 85.30.Vw, 73.50.Mx
-
d
n Ithasbeenpredictedthat,atlowtemperatures,trans- photon-assisted tunneling at high ac frequencies.
o port through a quantum dot should be governed by the The system of interest is a semiconductor quantum
c samemany-bodyphenomenonthatenhancestheresistiv- dot, as pictured schematically in Fig. 1. An electron can
[
ity of a metal containing magnetic impurities – namely be constrained between two reservoirs by tunneling bar-

1 theKondoeffect[1]. TherecentobservationoftheKondo riersleadingtoavirtualelectroniclevelwithinthedotat
v effect by Goldhaber et al. [2] in a quantum dot operat- energy∼ǫdot (measuredfromthe Fermilevel)and width
1
4 ingasasingle-electrontransistor(SET)hasfullyverified ∼2Γdot [6]. We assume that both the charging energy
2 these predictions. In contrast to bulk metals, where the e2/C and the level spacing in the dot are much larger
1 Kondoeffectcorrespondstothescreeningofthefreespins than Γdot, so the dot will operate as a SET [3]. In this
0 of a large number of magnetic impurities, there is only work, we consider only the linear-response conductance
8 one free spin in the quantum-dot experiment. Moreover, between the two reservoirs. However, we will allow an
9
/ a combinationofbias andgate voltagesallowthe Kondo oscillating gate voltage Vg(t) = V0 +VaccosΩt of arbi-
at regime, mixed-valence regime, and empty-site regime all trary(angular)frequencyΩandarbitraryamplitudeVac,
m tobe studiedforthe samequantumdot,bothinandout which modulates the virtual-level energy ǫdot(t).
of equilibrium [2]. Suchasystemmaybedescribedbyaconstrained(U =
-
d Hereweconsideranotheropportunitypresentedbythe ∞) Anderson Hamiltonian
n
observation of the Kondo effect in a quantum dot that
co is not available in bulk metals – the application of an ǫdot(t)nσ+ ǫkσnkσ +(Vkc†kσcσ+H.c.) . (1)
: unscreened ac potential. There is already a large liter- Xσ Xkσh i
v
ature concerning the experimental application of time-
Xi dependent fields to quantum dots [3]. For a dot acting Herec†σ createsanelectronofspinσ inthequantumdot,
r as a Kondo system, the ac voltage can be used to peri- while nσ is the corresponding number operator; c†kσ cre-
a odically modify the Kondo temperature or to alternate ates a corresponding reservoir electron; k is shorthand
between the Kondo and mixed-valence regimes. Thus it forallother quantumnumbersofthe reservoirelectrons,
is natural to ask what additional phenomena occur in a including the designation of left or right reservoir, while
driven system which in steady state is dominated by the Vk isthetunnelingmatrixelementthroughtheappropri-
Kondo effect [4,5]. Our results indicate a rich range of atebarrier. Becausethechargingenergytoaddasecond
behaviorwith increasingac frequency,fromsidebands of electron, U = e2/C, is assumed large, the Fock space in
the Kondo peak at low ac frequencies, to conductance whichthe Hamiltonian(1)operatesisrestrictedto those
oscillations at intermediate frequencies, and finally to elements with zero or one electron in the dot.
ρ (ǫ) is replaced by the time-averaged spectral density
dot
hρ (ǫ,t)i [13,14]. For a given system, this average will
dot
depend on the driving amplitude V and frequency Ω.
ac
ε
dot
0.10
V0 + Va c cos Ω t Ω=0.14 Ω=0.68
0.05
FIG.1. Schematic pictureof thequantumdot SET.
0.00
At low temperatures, the Anderson Hamiltonian (1) Ω=2.0 Ω=3.4
gives rise to the Kondo effect when the level energy ǫ
dot
lies belowthe Fermienergy. Inthis regime,a singleelec- 0.05
tron occupies the dot which, in effect, turns the dot into
a magnetic impurity with a free spin. The temperature
0.00
required to observe the Kondo effect in linear response
Ω=4.8 Ω=6.1
is of order T ∼ Dexp(−π|ǫ |/Γ ), where D is the
K dot dot
0.10
energy difference between the Fermi level and the bot-
tom of the band of states. For the temperature range no ac
that is likely to be experimentally accessible in a SET,
0.00
T ∼T or higher, there exists a well tested and reliable -8.0 -4.0 0.0 -8.0 -4.0 0.0
K
ε ε
approximationknownasthenon-crossingapproximation
(NCA) [7]. The NCA has been formally generalized to FIG.2. The spectral density ρdot(ǫ,t) vs. energy ǫ for a
h i
the full time-dependent nonequilibrium case [8], and an quantum dot with level energy ǫdot(t) = 5+4cosΩt and
−
exact method for the (numerical) solution implemented T = 0.005. The non-driven case is also shown in the final
[9]. Thetime-dependentNCAhasbeenappliedtoKondo panel. Throughout this letter, energies are in unitsof Γdot.
physics in charge transfer in hyperthermal ion scatter-
In Fig. 2 we show the calculated hρ (ǫ,t)i as a func-
ing from metallic surfaces [9,10] and to energy transfer dot
tion of energy ǫ for a level with energy ǫ (t) = ǫ +
and stimulated desorption at metallic surfaces [11]. An dot dot
ǫ cosΩt at several different frequencies Ω. The corre-
independent formulation [4] has been applied to quan- ac
sponding conductance is shown by the curve labeled dot
tum dots, although the high frequency expansion used
A,T =0.005inFig.3. For the lowestΩ,the responseof
there appears limited. Here we present the exact time-
thesystemisrelativelyadiabaticandthedisplayedspec-
dependent NCA solution for a quantumdot overthe full
tral function resembles the spectral function that would
range of applied frequencies.
have resulted if the system had been in perfect equilib-
Thetime-dependentelectronicstructureofthedotcan
rium for all the dot level positions overa period of oscil-
be characterized by the time-dependent spectral density
lation of ǫ (t). The two broad peaks are the influence
dot
∞ dτ of the virtual level peaks at the two stationary points of
ρ (ǫ,t)≡ eiǫτ/h¯h{c (t+ 1τ),c†(t− 1τ)}i (2)
dot Z 2π σ 2 σ 2 this oscillation (here at ǫ=−1 and ǫ=−9). As the fre-
−∞
quencyΩisincreased,markednonadiabaticeffectsresult,
evaluated in the restricted Fock space. For the equi- the mostobviousbeing the appearanceofmultiple satel-
librium Kondo system, ρ (ǫ) is time independent, and
dot lites around the Kondo resonance [4]. These sidebands
looks like the graph in the schematic in Fig. 1. Roughly appearatenergiesequaltoh¯ times multiplesofthe driv-
speaking, ρdot(ǫ) consists of a broad peak of width ing frequency Ω [15]. As the frequency Ω is increased,
∼2Γ atthelevelpositionǫ andasharpKondopeak
dot dot spectralweightistransferredfromthe mainKondopeak
ofwidth∼TK neartheFermilevel. Wewillrefertothese to these satellites. As the conductance is dominated by
featuresasthevirtual-levelpeakandtheKondopeak,re- hρ (ǫ,t)i at the Fermi energy, this causes the slow log-
dot
spectively. In the steady-state case, the linear-response
arithmic falloff of the conductance over two decades of
conductance G through a dot symmetrically coupled to frequency, as shown in Fig. 3.
two reservoirsis given by [12]
As ¯hΩ becomes larger than Γ , inspection of Fig. 2
dot
showsthatbroadsatellitesalsoappearatenergysepara-
e2Γ ∂f(ǫ)
G= dot dǫρdot(ǫ) − , (3) tions n¯hΩ around the averagevirtual-level position ǫdot.
¯h 2 Z (cid:18) ∂ǫ (cid:19)
These satellites of the virtual level are the analogues of
thosepredictedinthenoninteractingcase[13],whichde-
where f(ǫ) is the Fermi function. The formula (3) will
creaseinmagnitudeastheordernoftheBesselfunction
still be valid in the case where the gate voltage is time
J . Here, however, the virtual-level satellites have their
dependent if G is the time-averaged conductance and n
2
own Kondo peaks; each of the latter gets strong when Ω → ∞, value for frequencies still much larger than ei-
the corresponding virtual-level satellite reaches a posi- ther the depth of the level |ǫ | or its width. This effect
dot
tion a little below the Fermi level, and then disappears is due to a rapid decline of the amplitude of the Kondo
as the broad satellite crosses the Fermi level. This effect peak in the spectral density, as illustrated in the inset
produces the oscillationsin the conductance thatare ev- of Fig. 3. We propose the following explanation for this
identin the lowercurvesinFig.3. These oscillationsare phenomenon. The energy h¯Ω excites the dot, produc-
verydifferentfromthosethat wouldoccurin anoninter- ing satellites [16,13] of the virtual level peak at energies
acting (U = 0) case: due to the Kondo peaks they are ǫ ±n¯hΩ, which, for h¯Ω≫Γ have strength roughly
dot dot
substantially stronger, their maxima occur at different given by [J (ǫ /¯hΩ)]2 as in the U = 0 case (see Fig. 2
n ac
frequenciesandtheirmagnitudesaretemperaturedepen- and the previous discussion). For largeh¯Ω, only the two
dent. As the last virtual-level satellite crosses the Fermi n = 1 satellites have any significant strength, and the
level, h¯Ω = |ǫ |, the dot level energy begins to vary higherlies abovethe Fermilevel,allowinganelectronon
dot
too fast for the system to respond and the averagespec- the dot to decay at the rate (1/¯h)Γ (ǫ +h¯Ω). The
dot dot
tral function approaches (exactly as Ω → ∞) the equi- overallelectrondecay probability per unit time Γ /¯h
decay
librium spectral function for a dot level centered at the due to this photon-assisted-tunneling mechanism (PAT)
average position ǫ . For the parameters of Fig. 2, the is therefore given by
dot
high frequency region is uninteresting, because the tem-
peratureisfarabovetheKondotemperature. Therefore, Γdecay ≈[J1(ǫac/¯hΩ)]2Γdot(ǫdot+h¯Ω). (4)
the conductance shows little temperature or frequency
The above rate carries with it an energy uncertainty,
dependence at these high frequencies.
which we speculate has roughly the same effect on the
Kondo peak as the energy smearing due to a finite tem-
0.15
perature. We can test this conjecture by calculating the
0.8 0.10 equilibrium conductance at an effective temperature Teff
given by T = T +Γ . The results of such a cal-
eff decay
dot B 0.05 culation are shown in Fig. 3 (PAT curves), where they
compare very favorably with our results for the conduc-
0.6 -0.2 0.0
ε tance in the ac-driven system.
G Returning to the behavior at low frequencies, we find
2e)
h/ that it can be best understood in terms of the Kondo
( 0.4 dot A
Hamiltonian, which, with respect to properties near the
Fermi level, is equivalent to the Anderson Hamiltonian
A T=0.005
(1) in the extreme Kondo region −ǫ ≫ Γ [17]. In
A T=0.02 dot dot
0.2 B T=0.005 this limit the dot can be replaced simply by a dynami-
B T=0.005 PAT cal Heisenberg spin S~ (S2 = 3), which scatters electrons
B T=0.02 4
both within and between reservoirs. The Kondo Hamil-
B T=0.02 PAT
0.0 tonian corresponding to the Anderson model (1) is
0.01 0.10 1.00 10.00
Ω
FIG.3. Conductance of two different quantum dots, each Jkk′(t) S~ ·~σσσ′ + 21δσσ′ c†kσck′σ′, (5)
attwodifferenttemperatures: DotA,ǫdot(t)= 5+4cosΩt; kkX′σσ′ (cid:16) (cid:17)
−
Dot B, ǫdot(t) = 2.5 + 2cosΩt. The curves at the
− where the components of ~σ are the Pauli spin matri-
high Ω end for dot B (marked “PAT”) are from our pho-
ces. For near Fermi level properties we can suppress
ton-assisted-tunneling model, while the exact high frequency
asymptotes for dot B are shown as short horizontal lines ex- the detailed k dependence of J and V and introduce a
tending from the right vertical axis. The inset shows the large energy cutoff D [18], in which case the relation-
spectral density ρdot(ǫ,t) of dot B around the Fermi level, shipbetweentheKondoandAndersonHamiltonians[17]
h i
at T = 0.005, for large frequencies, from Ω = 4.8 (lowest is J(t) = |V2/ǫ (t)| for our U = ∞ case. If we let
dot
curve),through 5.5, 6.1, 6.8 to Ω=14 (topmost curve). w (ǫ)/¯h be the total rate at which lead electrons of
leads
energy ǫ undergo intralead and interlead scattering by
The situation is quite different for the system (dot B) the dot, then w (ǫ) will have a Kondo peak for ǫ
leads
displayed in the upper two curves in Fig. 3, which dis- near the Fermi level. Furthermore, if J is modulated
plays a strong Kondo effect when the dot level is held at asJ(t)=hJi(1+αcosΩt),thenanelectronscatteredby
its average energy ǫdot. In this case the Ω→∞ conduc- the dot will be able to absorb or emit multiple quanta
tance is strongly enhanced by the Kondo effect, and is of energy h¯Ω, leading to satellites of the Kondo peak in
consequently temperature dependent as well. Note that hρ (ǫ,t)i through the exact Anderson model relation
dot
theconductancefallsoffsignificantlyfromitsasymptotic,
3
6 Kondo peak due to photon-assisted tunneling processes
7
α=0 α=0.6 accountsforthereductionofconductance. Wehopethat
Ω=0.34 our work will inspire experimental investigation of these
5
phenomena andother ramificationsofac drivingapplied
6 to Kondo systems.
The work was supported in part by NSF grants DMR
4
95-21444 (Rice) and DMR 97-08499 (Rutgers), and
by US-Israeli Binational Science Foundation grant 94-
5
3 Order J3 00277/1(BGU).
NCA Order J3
Abrikosov NCA
Order J2
2 4
-0.5 0.0 0.5 -0.5 0.0 0.5
ε ε
FIG.4. Spectraldensity ρdot(ǫ,t) times103 intheKondo
h i
model and NCA for T =0.02 and hJiρ=0.023 (ǫdot =−7). [1] L. I. Glazman and M. E. Raikh, Pis’ma Zh. Eksp. Teor.
Forthenon-drivencase (left panel) wealso showthecompa- Fiz. 47, 378 (1988) [JETP Lett. 47, 452 (1988)]; T. K.
rable result from summing all the leading logarithmic terms Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988); S.
(Abrikosov, Ref. [19]), as well as that obtained to order J2.
Hershfield, J. H. Davies, and J. W. Wilkins, Phys. Rev.
TtohoeredneerrJgy2 idnepalelntdheencKeoonfdohJHia(mRielft.on[1i7a]n)chuarsvebse.en included LLeetet,.P6h7y,s3.7R20ev(1.9L9e1t)t;.Y7.0M, 2ei6r0,1N(.1S9.9W3)i;ngNr.eeSn.,WanidngPr.eAen.
and Y.Meir, Phys. Rev.B 49, 11040 (1994).
wleads(ǫ). This then reflects back on [2] D. Goldhaber-Gordon et al.,Nature391, 156 (1998).
[3] L.P.Kouwenhovenet al.,inMesoscopic Electron Trans-
wleads(ǫ)=Γdot(ǫ)hρdot(ǫ,t)i/ρleads(ǫ), (6) port, edited by L. L. Sohn, L. P. Kouwenhoven, and G.
Sch¨on (Kluwer, Netherlands, 1997).
where ρleads(ǫ) is the state density per spin in the leads. [4] M.H.HettlerandH.Schoeller,Phys.Rev.Lett.74,4907
The above can be illustrated explicitly using pertur- (1995).
bation theory in J. Keeping all terms of order J2 and [5] A. Schiller and S. Hershfield, Phys. Rev. Lett. 77, 1821
logarithmic terms to order J3, we find, using a nonequi- (1996); T. K. Ng, Phys. Rev. Lett. 76, 487 (1996);
librium version of Abrikosov’s pseudofermion technique Y. Goldin and Y.Avishai, preprint cond-mat/9710085.
[19], that [6] WedefineΓdot(ǫ)=2π k|Vk|2δ(ǫ−ǫk),aslowlyvarying
quantity.ThenotationPΓdot withnoenergyspecifiedwill
1 always refer thevalueat theFermi level.
w (ǫ)=2πhJ2iρ 1+3hJiρ a g(ǫ+n¯hΩ) , (7) [7] N. E. Bickers, Rev.Mod. Phys. 59, 845 (1987).
leads n
(cid:20) nX=−1 (cid:21) [8] D.C.LangrethandP.Nordlander,Phys.Rev.B43,2541
(1991).
where ρ = ρ (0), a = 1, a = α2/(2+α2), hJ2i = [9] H.Shao,D.C.Langreth,andP.Nordlander,Phys.Rev.
leads 0 ±1
(1+ 1α2)hJi2, and B 49, 13929 (1994).
2 [10] H.Shao,P.Nordlander,andD.C.Langreth,Phys.Rev.
1 D 1−2f(ǫ′) D B 52, 2988 (1995), Phys. Rev.Lett. 77, 948 (1996).
g(ǫ)= dǫ′ →ln , (8) [11] T. Brunner and D. C. Langreth, Phys. Rev. B 55, 2578
2Z ǫ′−ǫ (cid:12) ǫ (cid:12)
−D (cid:12) (cid:12) (1997); M. Plihal and D. C. Langreth, Surf. Sci. Lett.
(cid:12)(cid:12) (cid:12)(cid:12) 395, 252 (1998); Phys. Rev.B (submitted).
the last limit being approached when T ≪|ǫ|. In Fig. 4 [12] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512
we compare this prediction with the full NCA theory.
(1992).
Although we are not strictly in the parameter region [13] A.-P.Jauho,N.S.Wingreen,andY.Meir,Phys.Rev.B
where the J3 theory is quantitatively valid, the quali- 50, 5528 (1994).
tative agreement is quite satisfactory. [14] For the Hamiltonian (1) the time average ρdot(ǫ,t)
h i ≡
The present results indicate rich behavior when an Im A(ǫ,t) /π, where A(ǫ,t) is the retarded and hence
− h i
external ac potential is applied to a quantum dot in causal function defined in Ref. [13], Eq.(28).
the regime where the conductance is dominated by the [15] Henceinanexperimentonemayalsoexpectpeaksspaced
by ¯hΩ in thedifferential conductance.
Kondo effect. While the time-dependent NCA method
[16] P.K.TienandJ.P.Gordon,Phys.Rev.129,647(1963).
employedspansthe fullrangeofappliedfrequency,some
[17] J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491
additionalinsighthasbeengainedintothebehaviorboth
(1966).
atverylowandveryhighfrequencies. Atlowfrequencies
[18] For thelead statedensity used in theNCA calculations,
a time-dependent Kondo model helps explain the ampli- ρleads(ǫ) = ρleads(0)[1 ǫ2/(20Γdot)2], the appropriate
tudes of sidebands of the Kondo peak in the spectral value is given byD=2−0Γdot/√e; see Ref. [9].
density of the dot. At high frequencies, a cutoff of the [19] A. A.Abrikosov, Physics 2, 5 (1965).
4