Fano Resonances in Quantum Dots: A Non-Perturbative Role for Potential-Like Scattering

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by  Robert M. Konik

Fano Resonances in Quantum Dots: A Non-Perturbative Role for Potential-Like
Scattering
Robert M. Konik1,2
1Physics Department, Brookhaven National Laboratory, Upton NY 11973
4 2Physics Department, University of Virginia, Charlottesville VA 22904
0 (Dated: February 2, 2008)
0
Weconsiderthephysicsoftransportthroughquantumdotsinthepresenceoftwotunnelingpaths.
2
The first path sees electrons hopping on and off the dot while the second path is modeled through

n a potential scattering-like term. To study the effects of potential scattering, we employ a modified
a versionoftheAndersonmodel. SuchamodelcanbeexactlysolvedthroughtheBetheansatz,thus
J
allowing a comprehensive and exact analysis of the zero temperature linear response conductance.

9 Wefind transport properties to be extremely sensitive to the introduction of a potential scattering
2 term. Indeedthepresenceofsuchascatteringterm,interalia,inducesaseriesoffirstorderquantum
phase transitions. Focusing on the Kondo regime of the quantum dot, the non-perturbative effect
] of potential-like scattering can be directly tied to both the breaking of particle-hole symmetry and
l
l the interlinking of charge and spin degrees of freedom in the Anderson model. The sensitivity to
a
potentialscattering isalso reflected in a set of complementary exact diagonalization computations.
h
- TheconsequencesofthisanalysisextendstoobservationsingeneralofFanoresonancesinquantum
s dots as well as to the physics of transport through quantum dots embedded in Aharonov-Bohm
e
rings.
m
. PACSnumbers: 73.23.Hk,85.35.Gv,72.15.Qm,71.10.Pm
t
a
m
I. INTRODUCTION Kobayashi et al.17,18 have reported Fano resonances in
-
d SETs embedded in Aharonov-Bohm rings where neces-
n sarily the two tunneling paths are clearly delineated.
Fano resonances are a commonplace phenomena aris-
o
ing from a competition between two competing scatter- AnSET(ormorecolloquially,aquantumdot)isfabri-
c
[ ing paths, one path sensitive to the energy of the scat- catedfromagatedsemiconductorheterostructuresuchas
tering state, one path not. These resonances were first GaAs/AlGaAsandis formedfromasegregatednanome-
1
studied by the eponymous U. Fano1 in the context of ter scale sized region of a two dimensional electron gas
v
7 photoionization experiments in atomic systems where a (2DEG). Such quantum dots are highly tunable. Both
1 continuum of states interacting with a single resonant the hopping between the dot and the bulk of the 2DEG
6 levelserveasthe activeionizationpathways. This work1 as well as the dot’s energy levels relative to the 2DEG’s
1 predicted that interference betweentwo scattering paths Fermi surface can be adjusted through metallic gates
0 will ingenerallead to asymmetricallyshapedresonances placed on the heterostructure. At sufficiently small tem-
4 as a function of scattering energy. Research on Fano peratures (much smaller than the dot level spacing and
0
resonances in atomic systems remains a topic of active tunnelingwidths)onlytheelectronlevelsonthedotnear-
/
t interest.2,3,4 Resonances of similar origin have long been estinenergyto the Fermienergyofthe bulk 2DEGcon-
a
m known to characterize scattering off nuclei.5 More re- tribute to transport, turning the system into a quantum
cently they have been observed in the context of inter- impurity problem. As is the case in such problems, the
-
d ference in quantum well systems,6 polymer films,7 tun- effective dimensionality of the bulk 2DEG is deff = 1 in
n neling into and between multi-wall carbon nanotubes,8 regards to questions concerning the impurity, a result of
o and STM measurements of adsorbed magnetic atoms on focusing upon low energy impurity S-wave scattering.19
c metallic substrates,9,10,11 particularly interesting from a Withdeff =1,the bulk electronscanbe treatedasliving
:
v mesoscopic viewpoint in that Kondo physics comes into in leads.
i play. The remarkable ability to control the various parame-
X
Inthisarticlewewillbeinterestedinyetanotherman- ters of a quantum dot has allowed Kondo physics to be
r
a ifestation of Fano resonances, that involving single elec- observed in such a setting.20,21,22 Tuning the dot such
tron transistors (SETs). Interference between two tun- that one electron sits on the dot level nearest the Fermi
neling paths, one path involving an SET embedded in surface gives rise to the much of the same physics as
an arm of an Aharonov-Bohm ring, was first studied an isolated magnetic impurity.23. With the introduc-
in Ref. 12. However asymmetric resonances typical of tion of a second tunneling path, Kondo physics does not
the phenomena were only first reported in the conduc- disappear,14butaswewillargue,takesonaconsiderably
tance profiles of SETs several years thereafter.14,15 In different guise.
thiswork,theexactgeometryoftheinterferingtunneling In Figure 1 we sketchout the relevantenergy scalesof
paths through the SET was left unspecified, a situation the dot with only its energy levelnearestthe bulk Fermi
again seen in the experiments of Ref. 16. More recently, surface drawn. The energy levels of the two ‘leads’ are
2
This description of an SET yields a single, resonant
tunneling path for electrons traveling from the right to
2εd+U the left lead. A second non-resonant tunneling path in
the vicinity of the dot can arise in a variety of manners.
Thispathcanoccurthroughthedotitself,14,16 aproduct
µ
L perhaps of the detailed geometry of the confining region
Γ
µ of the 2DEG.29 Alternatively, the second path can be
R
ε spatiallyseparatedasoccurswhena quantumdotisem-
d
bedded in an arm of an Aharonov-Bohm ring,12,17,18 as
pictured in Figure 2. This is in some sense preferable as
the two tunneling paths are then marked concretely by
the two arms of the ring.
Wemodelthesecondtunnelingpathinthemostmini-
Vg
malwaypossiblebyaddingtotheAndersonmodelabove
(1.1) the term,
δ = V c† c +h.c. . (1.2)
H LR Lσ Rσ (cid:12)
FIG. 1: Sketched are the relevant energy scales of the single (cid:0) (cid:1)(cid:12)x=0
(cid:12)
active dot level. Such a term has been consideredprevio(cid:12)usly in a number
of places.30,31,32,33 It allows the electrons to bypass the
dotthroughadirectlead-leadtunneling term. Inadding
marked as forming a continuum filled to levels marked
δ ,we allowthe possibilityofmultiple windingsas elec-
by two chemical potentials, µL and µR. The active dot H
trons transit successively from one lead to the other via
level is characterized by two energy scales, one marking
different paths. This modified Hamiltonian is thus ap-
theenergyofsingleoccupancy,ǫ ,theothermarkingthe
d
propriateto describing anAharonov-Bohmringin a two
energy of double occupancy, U +2ǫ . U here represents
d terminal geometry,12,17,18 not the four terminal setup of
roughly the charging energy incurred from the addition
Ref. 34. The second tunneling path is potential-like in
of the second electron into the level. Separating the ac-
structure. Indeed with the introduction of even and odd
tive level from the two leads are two tunnel barriers. As
combinations of electrons,
drawn, the hybridization of the electrons living on the
level with those in the leads gives rise to the broadening 1
c = (c c ),
oftheleveltoawidth,Γ. Representedinthecapacitively e/o √2 L± R
coupledgatevoltage,V ,isthe abilityto shiftǫ relative
g d
to µ and µ . the addition is precisely a potential scattering term.
L R
The properties ofa single levelSET arebelieved to be At the centre of this work lies the discovery that
wellcapturedbytheAndersonmodel.20,21,22 Theoretical this Hamiltonian, And. + δ , is integrable. While it
H H
computations of the finite temperature linear response has long been known that the Anderson model alone
conductanceintheKondoregime,anon-trivialquantity, is integrable,35,36 only recently has it been realized that
bothusingnumericalrenormalizationgrouptechniques24 the addition, δ , to the Hamiltonian does not spoil the
andtheAnderson’smodelunderlyingintegrability25 well integrability.37HAs will be shown here, the perturbation,
match the observations.20 Moreover qualitative predic- δ , modifies the integrable structure of the Anderson
H
tions resulting from the out-of-equilibrium Anderson model in a straightforwardway. While this modification
model,26 namely the splitting of the zero bias anomaly isminorinatechnicalsense,itwillhaveprofoundeffects
into two peaks, have been observed20,21 (although the on the transport properties of the system.
predicted zero temperature features of the peaks25,27 Attheheartofthemodel’sintegrabilityliestheknowl-
have not yet been seen28). edge of the model’s exact eigenfunctions. From the
A continuum Anderson model governing the quantum knowledge of the exact eigenfunctions, we are able to
dot is given by extract the behaviour of excitations above an interact-
ing Fermi sea. In particular we can compute the dressed
= i dx c† (x)∂ c (x) scattering matrices of the excitations both with them-
HAnd. (cid:26)− Z ασ x ασ selves and off the impurity. While the integrability of
σX=↑/↓ (cid:0) (cid:1)
α=L,R the Anderson model has been long established, it was
+V(c† d +d†c ) +ǫ n +Un n . (1.1) only recently understood25 how to wed integrable data
ασ σ σ ασ |x=0(cid:27) d σ ↑ ↓ with a Landauer-Bu¨ttikerframeworkto accesstransport
σX=↑,↓
properties. In particular, in Ref. 25 it was shown how
Herec /dareelectronslivingintheleads/dot. Hopping theknowledgeoftheexcitationsabovetheFermiseaand
σα
betweenthetwoisgovernedbyV. AsindicatedinFigure their associatedscattering phasesallow the computation
1, ǫ marks the dot single level energy while U describes of both the in-equilibrium and out-of-equilibrium con-
d
the additional cost of double occupancy. ductances.
3
second tunneling path where V = V /2. While the additive constant,
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) p LR
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)VLR t2hteant−w1o(Vexp)p,riesssiimonpsorftoarntth,ethsecactrtuecriianlgdpiffhearseenlcieesbientwtheeant
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
n is distinctfromn . AtleastforU >0,exactdiago-
imp d
nalization computations that complement the integrable
analysisindicatethatn involvesbothcontributions
imp↑/↓
fromthe dotdegreesoffreedomaswellas bulk electrons
(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) in the leads:
lead L lead R n = n + dx c† (x)c (x) ρ ;
(cid:0)(cid:1)(cid:0)(cid:1) V imp↑ d↑ Z (cid:20)h e↑ e↑ i− bulk(cid:21)
(cid:0)(cid:1)(cid:0)(cid:1)V R
(cid:0)(cid:1)(cid:0)(cid:1) L (cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1) n = n + dx c† (x)c (x) ρ .
QD imp↓ d↓ Z (cid:20)h e↓ e↓ i− bulk(cid:21)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (1.5)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)embedded quantum dot Here ρ equals the unperturbed (absent the impurity)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) bulk
density of states of the lead electrons.
It is hardly surprising that deviations of the electron
FIG.2: Drawn isa ringconnected totwoleads. In thelower
density in the leads should affect scatteringat the Fermi
armoftheringaquantumdotisembedded. Electronstransit
surface. This was specifically forewarned as a possibil-
from lead L to lead R by either hopping on and off the dot
ity in the proof of the Friedel sum rule for the Anderson
embeddedinthelowerarmoftheringorbytravelingthrough
model by Langreth.38 And we know such deviations in
theupperarmofthering. Electronsmayalsomovefromlead
L to lead R by executingmultiple windings about thering. the lead electron density are important for determining
the finite field scattering phase (and so the magnetocon-
ductance) in the Kondo model.39,40 Instead what turns
Much of the content of Ref. 25 was devoted to under- out here to be surprising is that nimp nimp↑+nimp↓ is
≡
standing how to treat both finite temperature, T, and not a continuous function of Vp.
voltage, V, transport in the context of integrability. At AsafunctionofVp,nimp seestwodiscontinuities. The
finite T and V various uncertainties plagued the discus- first discontinuity occurs at Vp non-zero, i.e. at Vp =
sion. Away from the Fermi surface, electronic excita- 0±. Theoriginofthisdiscontinuityisperhapsclearestat
tions could be constructed only in an ambiguous fash- ǫd = U/2,atwhatwouldbetheparticle-holesymmetric
−
ion. The association of a finite energy electron with point of the Anderson model if Vp = 0. Vp finite then
a particular integrable excitation was then left unfixed. breaks particle-hole symmetry. In this breaking of the
This ambiguity was overcome only in a number of cases symmetry, a two-fold degeneracy of the ground state is
with some educated guesswork. In this way Ref. 25 brokenleadingtoafirstorderquantumphasetransition.
was able to reproduce the finite temperature linear re- The two putative ground states carry different electron
sponse conductance determined from numerical renor- occupancy. Their breakingleads then to a discontinuous
malizationgroupcalculations.24Here,incontrast,wewill change in nimp by one electron. At least at the point
focus upon zero temperature, linear response transport. ǫd = U/2, this conclusion is supported by an exact
−
There is no ambiguity in identifying the necessary elec- diagonalizationstudy.
tronicexcitationsandtheirscatteringphasesatthezero- A second first order transition as a function of Vp oc-
temperatureFermisurface. Assuchtheresultscontained cursapproximatelywhenthetwotunnelingpathsbecome
herein should be considered exact and without approxi- equalin strength. The sign ofVp atwhich the transition
mation. occurs is a function of ǫd. If ǫd > U/2, transport is
−
At zero temperature, the linear response conductance naturally expressed in terms of particles (as opposed to
in the VLR = 0 Anderson model, And., is described holes) and the transition occurs at Vp =Vpcrit >0 where
H
by the Friedel sum rule. The Friedel sum rule relates Vcrit is given by (for Γ U)
p ≪
the scattering phase, δ , at the Fermi surface to the
e↑/↓
number of electrons sitting on the dot, 2Γ
Vcrit . (1.6)
p ≃ U
δ =2πn . (1.3)
e↑/↓ d↑/↓
This positive value of V marks the energy necessary to
p
With the second tunneling path activated, we find that ionize an electron from the vicinity of the impurity. If
a variant of the Friedel sum rule still holds. But now ǫd < U/2,transportis mostnaturallygivenintermsof
the scatteringat the Fermisurface is relatedto the total holes−and the second transition occurs at Vp =−Vpcrit <
number of electrons displaced by the impurity (here the 0. Vp =−Vpcrit thenmarkstheenergynecessarytoionize
impurity should be thought of as both the dot and δ ): a hole from the impurity site.
H
These two transitions in V do not persist to all ǫ .
p d
δ =2πn 2tan−1(V ). (1.4) Rather they are bounded (roughly) to fall in the range
e↑/↓ imp↑/↓ p
−
4
V
p
n ~ 0
n ~ 2 imp
imp
n ~ 0
imp
crit
V
p
nimp~ 2 nimp~ 2 nimp~ 2 nimp~ 0
ε crit
d ε
d
−U−εdcrit n ~ 0 nimp~ 0 nimp~ 0
n ~ 2 imp
imp
crit
−V
p
n ~ 2
imp
n ~ 0
imp
n ~ 2
imp
p−h symmetry:
V = 0; ε =−U/2
p d
FIG.3: AphasediagramoftheAndersonmodelasafunctionofǫd andVp. Thehorizontaldashed(blue)linesmarkfirstorder
quantum phase transitions while the vertical solid (red) lines mark both the termini of the phase transitions as a function of
ǫd and regions of rapid variation in nimp.
U ǫcrit <ǫ <ǫcrit where ǫcrit equals By making the identifications,
− − d d d d
1
ǫcrit = U/2+ U U √2UΓlog(2πeU ). (1.7) e˜ = cot 2(δe+2tan−1(Vp)) ;
d − r8Γ 2 − 2π 8Γ (cid:0) (cid:1)
q˜ = cot(2tan−1(V )), (1.9)
p
−
ǫcrit marks the point where the disturbance in the impu-
d G can be put into a Fano-like form,
rity density of states due to V occurs above the Fermi
p
surface and so is excluded from contributing to the scat- e2 4V2 (e˜+q˜)2
p
tering phase. G=2 . (1.10)
h 1+V2 e˜2+1
These transitions are encoded in Figure 3, a phase di- p
agramofthe Andersonmodel asafunction ofǫd andVp. However this form is somewhat misleading. e˜ is not be
Each area of the phase diagramis marked by its charac- identified with ǫ /Γ as is true in the U = 0 case. In
d
teristic value of nimp. The horizontal dashed (blue) line fact unlike the non-interacting case, we find the many-
segments mark the two sets of first order phase transi- body correlations introduced by finite U can lead to the
tion. On the phase transition lines themselves, nimp ∼1 non-vanishing of G as ǫd is varied with q still real. It
(not marked in Figure 3). The termini of these phase would be interesting to reconsider then the analysis of
transitionsaredelineatedbysolidvertical(red)lines. Al- theFanoresonancesfoundinRef. 18(althoughthelarge
though the phase transitions at Vp = 0 and Vp = Vpcrit magnetic fields at which the resonances were observed
are drawn so as to end both simultaneously at ǫcrit and might largely eliminate many-body effects).
d
U ǫcrit, the termini differ by a value of order Γ. The The origin of the phase transitions and their atten-
− − d
vertical lines also mark regions of rapid (although not dant consequences can be understood not only as a con-
discontinuous) variation in n . sequenceofmany-bodyinteractionsbutofmerequantum
imp
The conductance, G, is simply expressible in terms of mechanics as well. The exact solution of the Vp =0 An-
6
the scattering phase (at H =0 where δ =δ δ ), dersonmodeldictates thatthe groundstateconsistsofa
e↑ e↓ e
≡
sea of two-electron repulsively interacting bound states.
G=2eh2 sin2(12(δe−2tan−1(Vp)). (1.8) (aTndhishofledastutrreueoffotrhtehegrVopun=d0stAatnediesrsionndempeonddele.n)tWofhVilpe
5
thisfeatureresultsfromtheexacttreatmentofthemany
body physics in the problem, the origin of the discontin- +2V c† c +c† c , (1.13)
p Lσ Rσ Rσ Lσ |x=ǫ
uous behaviour in nimp can be seen in the bare bound σX=↑,↓(cid:0) (cid:1)
states themselves. If one computes the finite V scat-
p
the scattering phase reads instead
tering phase of a two-electron bound state off the im-
purity (a quantum mechanical problem), one sees that
Γ
it is characterized as a function of energy by a number δe(q)= 2tan−1( ) 2tan−1(Vp). (1.14)
− q ǫ −
of non-analyticities. The close relationship between the − d
undressedboundstatescatteringphase,theimpurityoc- We now have excluded the possibility of multiple wind-
cupancy, nimp, and so ultimately the dressed scattering ings and the limit of the bound state scattering phase as
phase at the Fermi surface results in transport proper- a function of V will be entirely well behaved.
p
ties sharing these same non-analyticities. The explicit Generically the tunneling terms will not exactly coin-
constructionofthe bound state wavefunction and a dis- cide and we will have the situation as given in (1.14).
play of the non-analyticities are carried out in Section This is reassuring. If potential scattering was always to
II.B.3. havedrasticconsequenceswe wouldneedto explainwhy
Theappearanceofnon-analyticitiesintheboundstate its presence is not always felt in experiments in mag-
scattering phase and so transport properties is in part a netic impurities in bulk metals. The effective Anderson
consequenceofthetreatmentofthetwotunnelingpaths, Hamiltonian of a magnetic impurity in a bulk metal will
generally not possess particle-hole symmetry. Thus it
Htun = V c†lσdσ+d†σclσ |x=0 will include, however small, a potential scattering term.
l=XL,R(cid:0) (cid:1) However the coincidence of dot and potential scatter-
σ=↑,↓
+2V c† c +c† c . (1.11) ing centers as given in (1.11) requires some enforcement
p Lσ Rσ Rσ Lσ |x=0 mechanism (such as perhaps the topological possibility
σX=↑,↓(cid:0) (cid:1)
of multiple windings arising for a quantum dot embed-
In , the two tunneling terms are presented as both ded in a ring). This enforcement mechanism will not be
tun
H
infinitely narrow with delta-function strength. How- present in the general case and potential scattering will
ever in any real experimental system the two tunnel- be innocuous.
ing terms will have finite, differing widths. In general What then is the likelihood that the scattering phase
unequal widths can lead to changes in the physics. If appearingin(1.12)isrelevanttounderstandingFanores-
for example the width of the lead-lead term is much onancesinquantumdots? Certainlythisphaseisgeneric
greater than the width of the dot-lead coupling, i.e. theoretically.30,31 The differences between our work and
w w , the effect of the second tunnel- otheranalysesdetailedinthisarticlearisenotbecauseof
lead−lead dot−lead
≫
ing path on the physics becomes relatively trivial. The the form of δ (q), but because of the treatment of inter-
e
results presented in this article are presumed valid only actions on the quantum dot. But what is the situation
if w <w . More is said on this in Section experimentally? While Fano resonances have been ob-
lead−lead dot−lead
II.B.1. served, their observation does not necessarily preclude
alsotreatsthe twotunnelingsasoccurringatthe the form of the scattering phase in Eqn. (1.14). One
tun
H
same point, x = 0. If in the effective theory we were to possible means of distinguishing between the two forms
displace the scattering terms by some arbitrarily small of the scattering phase lies in the effect of finite inter-
distance the non-analyticitiesdisappear. The distinction actions. As we present in Section VI, various features
canbe madeclearbyconsideringthe singleelectronU = of the observations found in Ref. 14 suggest a role for
0 scattering phase. The scattering phase of c = (c + finite interactions only consistent with (1.12). However
e L
c )/√2 at wavevector q is given by this cannot be considered definitive while the role inter-
R
actionsplayinotherreportsofFanoresonances16,17,18 is
Γ
δ (q)= 2tan−1( +V ). (1.12) undetermined.
e p
− q ǫd Thedramaticeffectpotentialscatteringpresentsisnot
−
only a consequence of the coincidence of scattering cen-
Notice that V appears within the argument of the arc-
p
ters. It reflects the entanglement of charge and spin de-
tan. In the case of a quantum dot embedded in an
grees of freedom that occurs in the Anderson model. In
Aharonov-Bohm ring, this form of the scattering phase
the Kondo model, one may add a potential scattering
results from topology, from the possibility of multiple
term with strength, V , and maintain the model’s inte-
windingsaroundthering. While weareultimatelyinter- p
grability. If the scattering phase at the Fermi surface is
ested in the scattering phase of bound states, this form
then computed one finds
ofthescatteringphaseiscrucialifnon-analyticitiesinV
p
are to arise. Now if instead we shift (say) the potential
δ =π+f(V ), (1.15)
e↑/↓ p
scattering to x=ǫ, i.e.
where π is the original contribution of the screened spin
Htun = V c†lσdσ+d†σclσ |x=0 to the scattering phase while f is a smooth function of
l=XL,R(cid:0) (cid:1) the strength, V . This scattering phase then behaves in
σ=↑,↓ p
6
thesamefashionasthescatteringphaseforanAnderson spectral weight of the impurity degrees of freedom. This
model with potential scattering but without coincident ensuresquantitiessuchasthezerotemperaturemagneto-
scattering centers. The trivial effect of potential scat- conductancevaryrapidlyontheorderoftheKondotem-
tering is a result of the complete uncoupling of spin and perature,T . AtV =0,low energyspectralweightis in
k p
6
charge degrees of freedom in the Kondo model. Poten- generalstill presentinthe Kondoregime(ǫ U/2)of
d
∼−
tialscatteringactsdirectlyonlyinthechargesectorwhile theAndersonmodelbutinmodifiedform. WithV finite,
p
allthenon-perturbativephysicsoccursinthespinsector. spectralweightpresentnear the Fermi surface is shifted.
Thus potential scattering is unable to effect the Kondo The nature of this shift can be read off from the value
physics in an interesting fashion. This limitation is re- of n in Figure 3. For V = 0 we have n 1 for
imp p imp
∼
moved however for the Anderson model where the two ǫ U/2. Taking V finite, n shifts to 0 or 2. A
d p imp
∼ − ∼
sectors can interact. shiftofn to2coincideswithlowlyingspectralweight
imp
A consequence of this behaviour is that the Ander- justabovetheFermisurfacebeingtransferredtojustbe-
son model with potential scattering cannot be treated low. Thecorrespondingabsenceofspectralweightatlow
via a Schrieffer-Wolff (SW) transformation.41 To imple- but positive energies leads to the magnetoconductance
ment the standard SW transformation on +δ , being insensitive to small changes in H. Analogously, a
And.
H H
one should first transform away the potential scattering shift of n from 1 to 0 implies that low energy weight
imp
term via has been transferred from just below the Fermi surface
to just above. Here in contrast the magnetoconductance
ce(x=0) = e−iδ/2θ( x)c˜e(x)+eiδ/2θ(x)c˜e(x); becomes all the more sensitive to small changes in H.
6 −
The magnetoconductance is discussed in detail in Sec-
ce(0) = cos(δ/2)c˜(0)+VVpcos2(δ/2)d; tion II.B.
With these comments in hand, we now outline the
δ = 2tan−1(V ). (1.16)
− p course of the paper. A preliminary, considerably more
terse, treatment of this subject has already appeared.37
This gauge transformation transforms into the ordi-
e
H In Section II we outline the exact solution of the finite
nary Anderson model with renormalized parameters, ǫ˜
d
and V˜: Vp 6= 0 Anderson model. We begin by introducing the
model followed by the development of the Bethe ansatz
ǫ˜ = ǫ V2V cos2(δ/2); solution in several steps. By constructing the single and
d d p
− two-particle eigenstates of the model we are able to ex-
V˜ = V cos(δ/2). (1.17) tract the scattering matrices/phases. The structure of
these scattering matrices demonstrate that the model is
The SW transformationtaking the Andersonmodel to a integrable. Beyondthe scattering in the system, the two
Kondomodelgoesforwardasusualbyfreezingoutcharge particle eigenstates allow us to exhibit the nature of the
excitations. The scattering of the transformed electrons boundstates. As wehavealreadyindicated,muchofthe
at the Fermi surface is the Kondo scattering phase physics is encoded in the scattering of the bound states
offtheimpurity. Byfurthersketchingthedevelopmentof
δ˜=δKondo =π, (1.18) multi-particleeigenstates,weareabletoarguethatinall
details but the impurity scattering phase, the equations
while the scattering phase of the original electrons will
describing the quantizationof momentum inthe system,
have the form
the Bethe ansatz equations, are identical to the V = 0
p
case. This allows us readily to develop the method by
δ =δ 2tan−1(V ), (1.19)
Kondo− p whichdressedscatteringphasesofelectronsattheFermi
surface are computed. In the first part of Section II we
thatis,onerestoresthescatteringphaseduetopotential
limitourselvestothecaseǫ U/2. WesoendSection
scattering that was originally transformed away. Com- d
≥−
II by extending the analysis to ǫ U/2. This exten-
paring this result with (1.14) or (1.15), suggests that an d
≤ −
sion will prove essential when we come to compute the
SW transformation performed in such a fashion implic-
conductance exactly at ǫ = U/2.
itly assumes the potential scattering term is not exactly d
−
at x = 0. Alternatively, the gauge transformation in In Section III we develop the resulting properties of
(1.16)togetherwiththeSWtransformationseemtomis- the linear response conductance. We begin by comput-
treatthecouplingbetweenthespinandchargesectorsof ing the zero field conductance. We carefully associate
the theory. We will sharpen our critique of the gauge discontinuous changes in n with similar changes in
imp
transformation in (1.16) in what follows (see in particu- the conductance. The appearance of the discontinuous
lar Section V). changes finds support in exact diagonalization compu-
While the SW transformation fails to describe the tations which show similar discontinuous changes in the
physics of the V = 0 Anderson model, this failure does ground state occupancy. Key features of G as both a
p
6
not imply that Kondo physics is absent. One hallmark function of V and ǫ are described here. In particular
p d
of Kondo physics is the presence of the Abrikosov-Suhl the ingredients going into the phase diagram of Figure
resonance. This is an enhancement in the low energy 3 are established. Having described G(H = 0), we then
7
V V
L R II. EXACT SOLUTION
x=− x=−
8 8
L R
x=
8 x=8 A. Description of System
x=0
We use a more general Hamiltonian than in (1.1) and
(1.2) to describe the lead-impurity system,
FIG. 4: The unfolding of the left (L)/right (R) leads into
chiral fermions extendingfrom x=−∞ to x=∞. = H +H1 +H2 +H ;
H leads tun tun dot
∞
H = dx ic† (x)∂ c (x) ;
leads Z − lσ x lσ
l=XL,R −∞ (cid:0) (cid:1)
σ=↑,↓
H1 = V c† d +d†c ;
tun l lσ σ σ lσ |x=0
turn to understanding the magnetoconductance. Here l=XL,R (cid:0) (cid:1)
σ=↑,↓
we are able to characterize analytically the behaviour H2 = V c† c +c† c ;
of the H = 0 conductance at ǫ = U/2. Away from tun LR Lσ Rσ Rσ Lσ |x=0
6 d − σX=↑,↓(cid:0) (cid:1)
ǫ = U/2werelyuponnumericalsolutionsoftheequa-
tidons−describing the dressed scattering phases. Href = VLLc†LσcLσ+VRRc†RσcRσ |x=0;
σX=↑,↓(cid:0) (cid:1)
H = ǫ n +Un n . (2.1)
dot d σ ↑ ↓
The role two-electron bound states play in the exact σX=↑,↓
solution of the V = 0 Anderson model is ultimately a
p
6
consequence of a finite U. With U = 0, the behaviour Here again c† /c are electron operators in the leads
of the conductance, G, as a function of V is relatively lσ lσ
p while d†/d are the corresponding operators on the dot
bland. This points to a non-perturbative role for U. To σ σ
with n = d†d . The leads in Figure 4 are semi-infinite
reinforce this, in Section IV we consider the solution of σ σ σ
and have electrons which move both to the left and the
the V = 0 Anderson model for U < 0. With U < 0, all
p 6 right. By“unfolding”theleads(inthecontinuumlimit),
non-analyticities in V disappear suggestingthat various
p eachleadcanbe thoughtto run fromx= to x=
impurity quantities do not possess well-behaved power −∞ ∞
but with chiral electrons which are solely right moving.
series expansions in U/Γ as with the V =0 model.
p
The path fromL to R taking the electronthroughthe
dot is modeled by H1 . We have allowed for the possi-
tun
bility that tunneling between the dot and the two leads
In the penultimate Section V, we analyze an alterna-
is asymmetric, i.e. V = V . The second path from L
tiveapproachtothisproblem,anapproachpredicatedon L R
6
to R (for example, the upper arm of the ring in Figure
theuseofDysonequationsanddevelopedinRef. 30and
2) is modeled by H2 , a potential scattering-like term.
Ref. 31. This approachleads to transport quantities be- tun
We have also included a term in the Hamiltonian, H ,
inganalyticinbothU andV andsotoconsiderablydif- ref
p
bywhichelectronsagainexperienceasingle-bodypoten-
ferentresultsthanpresentedinthisarticle. Wewillargue
tialbutonewhichdoes notscatterthe electronsinto the
that the differences arise because the Dyson equations
other lead. Although likely to be generically present, we
treat the impurity occupancy, n , as equivalent to n ,
imp d
here consider such a term for technical reasons alone, as
the number ofelectrons onthe dot itself. We will equate
will be seen shortly.
the Dyson equations to the results obtained through ap-
plying the gauge transformation in (1.16). This will en- Hdot describes the energies associated with the dot.
able us to argue that even at the quantum mechanical The dot single particle Hilbert space is two fold: one for
level,theDysonequationsseemtoincorrectlytreatfinite occupancybyaspin -electronandoneforoccupancyby
↑
interactions. We believe that the origin of the discrep- a spin- electron. The chemical potential of these levels
↓
ancy between the two approaches lies in that the Dyson is controlledby ǫd σnσ while Coulomb interactions on
equations presume implicitly both Vp and U are pertur- the dot are represePnted by the term Un↑n↓. This is in
bativequantitiesandsoendwithquantitieswellbehaved fact the only interacting term in the problem– the leads
in V and U. are treated as Fermi liquids. This is reasonable as the
p
one-dimensional leads ultimately arise out of a dimen-
sional reduction of electrons living in two or three di-
In Section VI, the final section, we consider the obser- mensions. Although U is the only non-trivialinteraction
vations reported in G¨ores et al.14 as these served as the in the problem,it presents the same set ofcomplications
initialmotivationforthiswork. Whilehardlydetermina- seen in a fully interacting system.
tive, some generic features present in these experiments To treat this Hamiltonian we want first to map it to a
are well explained using the formalism developed in this model involving a single electron specie. This cannot be
work. doneforthegeneralvaluesofV ,V ,V ,andV /V .
LL RR LR L R
8
However if we restrict ourselves to the parameter space, Similar relations hold with and . Thus
RL L
T R
VLL =−VRR = 2VVLLVRR(VL2−VR2), (2.2) TLR = VLV2L+VRVR2(eiδe −eiδo);
V2eiδe +V2eiδo
wecanproceed. Notethatifthedot-leadcouplingissym- = R L . (2.8)
RR V2+V2
metric (or if V = 0) then V and V must vanish. L R
LR LL RR
Withthisconstraintontheparameterspaceweintroduce
Byparity,wemusthave = and = .
LR RL R L
even and odd electrons, c , via |T | |T | |R | |R |
e/o Thezerotemperaturelinearresponseconductanceisthen
given by
c =(V c V c )/ V2+V2. (2.3)
e/o L/R L± R/L R q L R G=2e2 2 =2e2VL2VR2 sin2(δe−δo) (2.9)
The Hamiltonian in this basis then reads h|TLR| h Γ2 2
For most of the paper we will assume V = V . Only
L R
=H +H , (2.4)
e o when comparing theory with the experimental observa-
H
tions of Ref. 14 will we relax this constraint.
where
Theprescriptionatfiniteenergy/finitetemperaturefor
relating δ /δ to and is in generalfraughtwith am-
e o
He = (cid:26)Z dx−ic†eσ(x)∂xceσ(x) biguities. ThisisTdiscussRedinscholasticdetailinRef. 25.
Xσ In this paper we will focus upon T = 0 linear response
+(2Γ)1/2(c† d +d†c ) +2V c† c quantities. Taking this limit lifts these ambiguities mak-
eσ σ σ eσ |x=0 p eσ eσ|x=0(cid:27) ing the above transformation precise.
+ǫ n +Un n ;
d σ ↑ ↓
σX=↑,↓
B. Construction of Multiparticle Eigenstates for H
e
H = dx ic† (x)∂ c (x) 2V c† c .
o Z − oσ x oσ − p oσ oσ|x=0
Xσ 1. Single Particle Eigenstates
(2.5)
We first examine the single particle eigenstates of .
The parameters Γ and V are given by e
p H
This will give the bare scattering phase off the impurity.
V2+V2 We want to look for states of the form
Γ = L R;
2 ∞
ψ = dx g (x)c†(x) +e d† 0 . (2.10)
V = ΓVLR . (2.6) | σi (cid:20)Z−∞ { σ σ } σ σ(cid:21)| i
p
2V V
L R
Solving Schr¨odinger’s equation, ψ = q ψ , yields
e σ σ
H | i | i
Γrepresentsthetotalbroadeningoftheelectrondotlevel the following constraints upon g (x) and e :
σ σ
due to coupling to the leads while V representthe effec-
p
tive potential scattering seen by the even and odd elec- gσ(0)(2Γ)1/2+ǫdeσ = qeσ;
trons. The odd sector of the theory, uncoupled to the
dot, is trivial. The even sector, we will argue,is solvable i∂xgσ(x)+δ(x)((2Γ)1/2eσ+2Vpgσ(0)) = qgσ(x).
−
via Bethe ansatz. (2.11)
Having undertaken this recasting of the Hamiltonian
Taking g (x) to be of the form
we mustdescribe how to maintaincontactwiththe orig- σ
inal problem: the transport of electrons from the left
g (x) = θ(x)eiqx+iδ/2+θ( x)eiqx−iδ/2;
lead to the right lead and vice versa. We can do this σ −
as follows. Let be the transmission amplitude of an
electron from leTaldl′l to lead l′ and the reflection am- gσ(0) = cos(δ(q)/2), (2.12)
l
R
plitude of an electron in lead l back into the same lead. wecanreadilysolve(2.11)findingexpressionsfore and
σ
In the even and odd basis it will turn out we can read- δ(q),
ily compute the scattering phases, δ /δ , of aneven/odd
e o
electron off the impurity (here the scattering must be a e = (2Γ)1/2cos(δ(q)/2)
σ
pure phase as the even/odd sectors are uncoupled and q ǫ
d
−
both chiral). / can be readily expressed in terms of Γ
δ /δ :25 T R δ(q) = 2tan−1( +Vp). (2.13)
e o − q ǫ
d
−
V eiδe = V +V ; Notethatthepotentialscatteringstrengthappearsinside
R R R L LR
R T the argument of the tan−1. This, ultimately, is a reflec-
V eiδo = V V . (2.7) tion of the coupling between the spin and charge sectors
L L R R LR
R − T
9
of the Anderson model (and if appropriate, the possibil- 2. Two-Particle Eigenstates
ityofmultiple windingsaboutanAharonov-Bohmring).
It thus sets the scene for the non-trivial physics of the Theexaminationofthetwo-particleeigenstateswillal-
potential scattering term. low us to construct the S-matrix for the model. In prin-
δ(q) contains informationonthe non-interacting(U = ciplewewillthenbereadytocomputethe multi-particle
0) states of the retarded dot Greens function: eigenstates. The only non-trivial two-particle states are
ImGret(ω) = 1 ∂ δ(q) those with total Sz = 0. We thus focus on these. Such
− d 2π q |q=ω states have the form
Γ¯ ∞
= (q−ǫ˜d)2+Γ¯2 (2.14) |ψi = (cid:20)Z−∞dx1dx2g(x1,x2)c†↑(x1)c†↓(x2)
where Γ¯ and ǫ˜d are the renormalized values of the dot + ∞ dxe(x)(c†(x)d† c†(x)d†)+fd†d† 0 . (2.16)
level broadening and dot chemical potential in the pres- Z ↑ ↓− ↓ ↑ ↑ ↓(cid:21)| i
−∞
ence of a finite V
p
Solving the Schr¨odinger equation, ψ = E ψ , leaves
Γ He| i | i
Γ¯ = ; us with the following set of consistency equations,
1+V2
p
ǫ˜d = ǫd−VpΓ¯. (2.15) Eg(x1,x2) = −i(∂x1 +∂x2)g(x1,x2)
We see the presence of the potential scattering term + √2Γ(δ(x )e(x )+δ(x )e(x ))
2 1 1 2
serves to narrow the dot resonance as well as shifting
the position of the energy level. In terms of correlators
+ V g(x ,x )(δ(x )+δ(x ));
p 1 2 1 2
involving dot degrees of freedom alone, these renormal-
izations, we believe, will be the sole response to the po- Ee(x) = i∂ e(x)+√2Γg(x,0)+√2Γfδ(x)
x
tential scattering. However the correlators relevant to −
transport go beyond dot correlations. In these correla-
+ ǫ e(x)+δ(x)V e(0);
d p
torsthe effects ofpotentialscatteringwillbe non-trivial.
The resulting single particle phase in (2.13) results Ef = 2√2Γe(0)+2ǫ f +Uf, (2.17)
d
from treating the lead-lead coupling, V , as well as the
p
dot-lead coupling, Γ, as delta-function scatterers. How- together with assuming g(x ,x ) is symmetric. To solve
1 2
ever as pointed out in the introduction, in any exper- these relations, we employ the following ansatz
imental system the two couplings will have some finite
widths, w and w . It is relatively easy to g(x ,x ) = g (x )g (x )φ(x x )
dot−lead lead−lead 1 2 q 1 p 2 1 2
−
show that in order to see the physics encoded in (2.13)
we require w w . To see an indication +g (x )g (x )φ(x x );
dot−lead lead−lead p 1 q 2 2 1
≥ −
of this, suppose we broaden out the lead-lead scattering
term,replacingV δ(x)in(2.11)withV w(x)wherew(x) e(x) = e φ(x)g (x)+e φ( x)g (x);
p p p q q p
−
is some even smooth function with unit weight. The re-
sulting solution to the modified (2.11) is E = q+p. (2.18)
x
gσ(x) = exp i dx′2Vpw(x′) Here gq/p(x) and eq/p are the coefficients appearing in
(cid:0)− Z−∞ (cid:1) the one particle wavefunction corresponding to energies
q/p (i.e. (2.12) and (2.13)). φ(x) is an expression of
θ(x)eiqx+iδ/2+θ( x)eiqx−iδ/2 ;
×(cid:18) − (cid:19) the phase change incurred when two electrons are inter-
cos(δ(q)/2) changed, x1 x2. We assume it takes the form
e = e−iVp(2Γ)1/2 ↔
σ
q ǫ
− d φ(x)=1+iα(p,q)sign(x). (2.19)
Γ
δ(q) = 2tan−1( ).
− q ǫd Under this ansatz, we find
−
Thus the effective scattering phase of an electron as it
e cos(δ(q)/2)+e cos(δ(p)/2)
traverses from x= to x= is f = 2√2Γ p q ;
−∞ ∞ − 2ǫ +U (q+p)
Γ d −
δeff(q)= 2tan−1( ) 2V 1 2UΓ
p
− q ǫ − α(p,q) = . (2.20)
d
− q p2ǫ +U q p
d
Unlike with the electron scattering phase in (2.13), we − − −
would expect all finite U transport properties to be From α(q,p) we can determine the scattering matrix for
smooth functions of V . Then in order to see the effects two electrons.
p σ′σ′
predicatedupon(2.13), we (atleast)requirew This S-matrix, S 1 2(p,q), where σ and σ are the
w . dot−lead ≥ spins of ingoing elecσt1rσo2ns while σ′ and1σ′ are2the spins
lead−lead 1 2
10
)
E 3 200
(
e
hase,δbd.stat 12 VVVppp === 0−0.01.1 (E)state11505000 VVVppp === −00.01.1
p d.
g 0 δb 0
n E
eri ∂−50
t
at −1
c −100
s
e
at −2 −150
t
s
d. −200
b −3 −1.2 −1.1 −1 −0.9 −0.8
−1.2 −1.1 −1 −0.9 −0.8
energy, E
energy, E
FIG. 6: Plots of the derivative with respect to energy of the
FIG.5: Plotsoftheboundstateimpurityscatteringphaseas boundstateimpurityscatteringphase. WetakeU+2ǫd =0,
afunctionofenergyoftheboundstate. WetakeU+2ǫd =0, U =1 and Γ=.1.
U =1 and Γ=.1.
3. Two-Particle Boundstates
ofthe outgoingelectrons. This S-matrix,onthe grounds
of SU(2) invariance, must have the form In constructing the two-particle eigenstates we deter-
mined the S-matrix of scattering between opposite spins
σ′σ′ σ′σ′ σ′σ′ of momentum p and q. In particular the state in (2.16)
S 1 2 =b(p,q)I 1 2 +c(p,q)P 1 2, (2.21)
σ1σ2 σ1σ2 σ1σ2 is a singlet with an associated scattering phase,
σ′σ′ σ′ σ′ g(p) g(q)+i
where Iσ11σ22 = δσ11δσ22 is the identity matrix while Ssinglet =b(p,q) c(p,q)= − . (2.25)
σ′σ′ σ′ σ′ − g(p) q(q) i
Pσ11σ22 = δσ12δσ21 is the permutation matrix. The coef- − −
ficients, b(p,q) and c(p,q), are related to α(p,q) via ThisS-matrixhaspoles initcorrespondingto the choice
g(p)=λ+i/2; g(q)=λ i/2,
φ(x x ) 1+iα(p,q)
1 2 −
b(p,q) c(p,q) = − =
− φ(x2 x1) 1 iα(p,q) where λ is for now an arbitrary real parameter. At
− −
g(p) g(q)+i these values,the pole in S has a positive imaginary
= − , (2.22) singlet
g(p) g(q) i residue indicative of a stable bound state.
− − For U >0,the groundstate is constructedoutof such
where g(p)=(p ǫ U/2)2/(2UΓ). With this relation bound states. Much of the physics is thus contained in
d
− − the behaviourofthe boundstate scatteringfromthe im-
in hand, b and c can then be fixed by considering the
purity. As such it is worthwhile to construct the bound
scattering of two electrons with equal spin. As two such
electrons do not interact, necessarily Sσσ =1. Hence state wave function explicitly. We assume the bound
σσ
state has the same form as (2.16). The various compo-
nentsoftheeigenstate,g(x ,x ),e(x),andf,stillsatisfy
b(p,q)+c(p,q)=1. (2.23) 1 2
(2.17). However instead of (2.18), we take as an ansatz
for these functions the following
So then
g(x ,x ) = θ(x x )h(x ,x )f (x )f (x )
1 2 1 2 1 2 + 1 − 1
g(p) g(q) −
b(p,q) = − ;
g(p) g(q) i + θ(x x )h(x ,x )f (x )f (x );
2 1 2 1 + 2 − 1
− − −
i
c(p,q) = − . (2.24)
e(x) = e θ(x)h(x,0)f (x)
g(p) g(q) i − +
− −
+ e θ( x)h(0,x)f (x); (2.26)
What is to be noticed is that this is exactly the same S- + −
−
matrix as for the Andersonmodel with V =0. The sole
p where
change V finite introduces to the analysis is a change in
p
the impurity scattering phase. f (x) = θ(x)eiδ±/2+θ( x)e−iδ±/2;
±
−