Mesoscopic to universal crossover of the transmission phase of multilevel quantum dots.

Abstract

Transmission phase alpha measurements of many-electron quantum dots (small mean level spacing delta) revealed universal phase lapses by pi between consecutive resonances. In contrast, for dots with only a few electrons (large delta), the appearance or not of a phase lapse depends on the dot parameters. We show that a model of a multilevel quantum dot with local Coulomb interactions and arbitrary level-lead couplings reproduces the generic features of the observed behavior. The universal behavior of alpha for small delta follows from Fano-type antiresonances of the renormalized single-particle levels.

Full text

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arXiv:cond-mat/0609191v3 [cond-mat.mes-hall] 22 Mar 2007
Mesoscopic to universal crossover of transmission phase of multi-level quantum dots
C. Karrasch,1T. Hecht,2A. Weichselbaum,2Y. Oreg,3J. von Delft,2and V. Meden1
1Institut f¨ ur Theoretische Physik, Universit¨ at G¨ ottingen, 37077 G¨ ottingen, Germany
2Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience,
Ludwig-Maximilians-Universit¨ at, 80333 Munich, Germany
3Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
(Dated: March 22, 2007)
Transmission phase α measurements of many-electron quantum dots (small mean level spacing δ)
revealed universal phase lapses by π between consecutive resonances. In contrast, for dots with only
a few electrons (large δ), the appearance or not of a phase lapse depends on the dot parameters.
We show that a model of a multi-level quantum dot with local Coulomb interactions and arbitrary
level-lead couplings reproduces the generic features of the observed behavior. The universal behavior
of α for small δ follows from Fano-type antiresonances of the renormalized single-particle levels.
PACS numbers: 73.23.-b, 73.63.Kv, 73.23.Hk
One of the longest-standing puzzles in mesoscopic
physics is the intriguing phase-lapse behavior observed in
a series of experiments [1, 2, 3] on Aharonov-Bohm rings
containing a quantum dot in one arm. Under suitable
conditions in linear response, both the phase and magni-
tude of the transmission amplitude T = |T|eiαof the dot
can be extracted from the Aharonov-Bohm oscillations
of the current through the ring. If this is done as func-
tion of a plunger gate voltage Vgthat linearly shifts the
dot’s single-particle energy levels downward, εj= ε0
(j = 1,2,... is a level index), a series of well-separated
transmission resonances [peaks in |T(Vg)|, to be called
“Coulomb blockade” (CB) peaks] of rather similar width
and height was observed, across which α(Vg) continu-
ously increased by π, as expected for Breit-Wigner-like
resonances. In each CB valley between any two succes-
sive CB peaks, α always jumped sharply downward by π
(“phase lapse”, PL). The PL behavior was observed to
be “universal”, occurring in a large succession of valleys
for every many-electron dot studied in [1, 2, 3]. This uni-
versality is puzzling, since naively the behavior of α(Vg)
is expected to be “mesoscopic”, i.e. to show a PL in
some CB valleys and none in others, depending on the
dot’s shape, the parity of its orbital wavefunctions, etc.
Despite a large amount of theoretical work (reviewed in
[4, 5]), no fully satisfactory framework for understanding
the universality of the PL behavior has been found yet.
j−Vg
A hint at the resolution of this puzzle is provided by
the most recent experiment [3], which also probed the
few-electron regime: as Vg was increased to successively
fill up the dot with electrons, starting from electron num-
ber Ne= 0, α(Vg) was observed to behave mesoscopically
in the few-electron regime, whereas the above-mentioned
universal PL behavior emerged only in the many-electron
regime (Ne ? 15).Now, one generic difference be-
tween few- and many-electron dots is that the latter have
smaller level spacings δj = ε0
filled levels. With increasing Ne, their δj’s should even-
tually become smaller than the respective level widths
j+1− ε0
jfor the topmost
Γj stemming from hybridization with the leads. Thus,
Ref. 3 suggested that a key element for understanding the
universal PL behavior might be that several overlapping
single-particle levels simultaneously contribute to trans-
port. Due to the dot’s Coulomb charging energy U, the
transmission peaks remain well separated nevertheless.
Previous works have studied the transmission ampli-
tude of multi-level, interacting dots [6, 7, 8, 9, 10]. How-
ever, no systematic study has yet been performed of the
interplay of level spacing, level widths, and charging en-
ergy that combines a wide range of parameter choices
with an accurate treatment of the correlation effects in-
duced by the Coulomb interaction.
ter aims to fill this gap by using two powerful methods,
the numerical (NRG) [11, 12] and functional (fRG) [13]
renormalization group approaches, to study systems with
up to 4 levels (for spinless electrons; see below). We find
that if the ratio of average level spacing δ to average
level width Γ is decreased into the regime δ ? Γ, one
of the renormalized effective single-particle levels gener-
ically becomes wider than all others, and hovers in the
vicinity of the chemical potential µ in the regime of Vg
for which the PLs occur. Upon varying Vg, the narrow
levels cross µ and the broad level, leading to Fano-type
antiresonances accompanied by universal PLs. For δ ? Γ,
α(Vg) behaves mesoscopically [14] for all U. Decreasing
δ thus causes the PL behavior to generically change from
mesoscopic to universal, as observed experimentally [3].
Model: The dot part of our model Hamiltonian is
The present Let-
Hdot=
N
?
j=1
εjnj+1
2U
?
j?=j′
?nj−1
2
??nj′ −1
2
?,
with nj = d†
less electrons, where U > 0 describes Coulomb repulsion.
The semi-infinite leads are modeled by a tight-binding
chain Hl = −t?∞
lead couplings by HT = −?
cm,lannihilates an electron on site m of lead l = L,R and
jdj and dot creation operators d†
jfor spin-
m=0(c†
m,lcm+1,l+ H.c.) and the level-
j,l(tl
jc†
0,ldj+ H.c.), where

Page 2

2
tl
ative signs for successive levels, sj= sgn(tL
are sample-dependent random variables determined by
the parity of the dot’s orbital wave functions. The ef-
fective width of level j is given by Γj = ΓL
Γl
the end of the leads, to be energy independent, choose
µ = 0, and specify our choices of tl
σ = {s1,s2,...}, γ = {ΓL
Methods: We focus on linear response transport and
unless stated otherwise, on zero temperature (τ = 0).
Then the dot produces purely elastic, potential scattering
between left and right lead, characterized by the trans-
mission matrix Tll′ = 2πρ?
is the retarded local Green function which we compute
using NRG and fRG. The NRG is a numerically exact
method that is known to produce very accurate results
[11, 12]. The fRG is a renormalization procedure for the
self-energy Σ and higher order vertex functions (see [13]
for details). We use a truncation scheme that keeps the
flow equations for Σ and for the frequency independent
part of the effective two-particle (Coulomb) interaction.
Comparisons with NRG [13] have shown this approxima-
tion to be reliable provided that the number of (almost)
degenerate levels and the interaction do not become too
large. fRG is much cheaper computationally than NRG,
enabling us to efficiently explore the vast parameter space
relevant for multi-level dot models.
At the end of the fRG flow, the full Green func-
tion at zero frequency takes the form [GR(0)]−1
−hij+ i∆ij, with an effective, noninteracting (but Vg
and U-dependent) single-particle Hamiltonian hij =
(ε0
j− Vg)δij+ Σij, whose level widths are governed by
∆ij= πρ?
eigenbasis of [GR(0)]−1
view ˜ εj and˜Γj as level positions and widths of a renor-
malized effective model (REM) describing the system.
For LR-symmetry, ΓL
used, which is much less demanding than computing the
full GR
ij(ω): the S-matrix is then diagonal in the even-odd
basis of the leads and its eigenvalues depend on the total
occupancies n±of all levels coupled to the even/odd lead
(Friedel sum rule), so that the transmission amplitude
T = TLRtakes the form T = sin[π(n+−n−)]eiπ(n++n−).
A transmission zero (TZ) and hence PL occurs when
n+= n−mod1 [Figs. 1(e,h): n±in thin dashed/dashed-
dotted line].
Results: Our results are illustrated in Figs. 1 to 3.
fRG and NRG data generally coincide rather well (com-
pare black and orange lines in Figs. 1 and 2), except
for N = 4 when both U ≫ Γ, δ < Γ, and correlations
become very strong [Fig. 2(f)]. The figures show the fol-
lowing striking qualitative features, that we found to be
generic by running the fRG for ten-thousands of parame-
ter sets, which is possible as a complete T(Vg) curve can
jare real level-lead hopping matrix elements. Their rel-
jtR
jtL
j+1tR
j+1),
j+ ΓR
j, with
j= πρ|tl
j|2. We take ρ, the local density of states at
jusing the notation
2,...}, Γ =
1,ΓR
1,ΓL
1
N
?
j,lΓl
j.
ijtl
iGR
ij(0)tl′
j, where GR
ij(ω)
ij
=
ltl
itl
j. To interpret our results, we adopt the
ij, with eigenvalues −˜ εj+ i˜Γj, and
j= ΓR
j, an NRG-shortcut can be
0
2
1
|T|
0
1
α/π
0
2
|T|
0
1
α/π
-808
Vg / Γ
0
|T|
-12
0
12
Vg / Γ
-18
0
18
Vg / Γ
0
2
α/π
fRG (γσ=−)i
NRG (γσ=−)i
fRG (γσ=+)
ι
na, NRG (γσ=−)i
nb, NRG (γσ=−)
i
U/Γ=0.4U/Γ=6U/Γ=30
δ/Γ=0.04: U N I V E R S A L
δ/Γ=1: C R O S S O V E R
δ/Γ=10: M E S O S C O P I C
τ=0 γσ=−/Γ={0.3,0.3,0.7,0.7} γσ=+/Γ={0.2,0.4,0.4,1}
(a)
(b)
(c)
(d) (e)
(f)
(g)
(h) (i)
N=2
FIG. 1: |T(Vg)| and α(Vg) for N = 2, ε0
decreasing δ/Γ produces a change from (a,b,c) mesoscopic via
(d,e,f) crossover to (g,h,i) universal behavior; increasing U/Γ
leads to increased transmission peak spacing. (b,e,h) include
the occupancies n+ (thin dashed) and n− (thin dash-dotted)
of the levels coupled to the even/odd lead in the case of LR-
symmetry. The condition n+ = n−mod1 produces a TZ and
PL. For the blip and hidden TZ near Vg = 0 in (i), see [15].
2,1= ±δ/2, and τ = 0:
be obtained within a few minutes on a standard PC:
Mesoscopic regime: For δ ? Γ [Figs. 1(a,b,c), 2(a,b,c)],
we recover behavior that is similar to the U = 0 case.
Within the REM it can be understood as transport oc-
curring through only one effective level at a time [see
Fig. 3(a,b,c)], with˜Γj ≃ Γj.
produces a Breit-Wigner-like transmission resonance of
width 2Γjand height governed by ΓL
ing the other levels are shifted upwards by U [charging
effect; Fig. 3(a)] leading to renormalized peak separa-
tions (“level spacings”) δj+ U .
α(Vg) behaves mesoscopically: depending on the sign sj
one either observes a PL (sj = +) or continuous evolu-
tion of α (sj= −) [14]. Additional PLs occur to the left
or right, beyond the last transmission peak [Fig. 2(a)].
Mesoscopic to universal crossover: As the ratio δ/Γ
is reduced, the behavior changes dramatically: the TZs
and PLs that used to be on the far outside move inward
across CB resonances [see evolution in Figs. 1(b,e,h)].
Universal regime: A universal feature [16] emerges for
δ ? Γ ? U (crossover scales are of order 1, but depend
on the chosen parameters): for all choices of the signs σ
and generic couplings γ, the N CB peaks over which α
increases by π are separated by N − 1 PLs, each accom-
panied by a TZ [Figs. 1(g,h,i), 2(d,e,f)]. This is consis-
Each ˜ εj that crosses µ
j/ΓR
j. At the cross-
Between two peaks,

Page 3

3
0
1
4
α/π
-30030
0
2
|T|
-30030 -60060
0
1
α/π
-606
Vg / Γ
0
|T|
-12012
Vg / Γ
-40040
Vg / Γ
U/Γ=4, σ={---}, τ/Γ=0.1
γ/Γ={.266,.533,.4,.8,.133,.266,.533,1.07}
(b)
U/Γ=2, σ={+-+}, τ/Γ=0.1
γ/Γ={.48,1.12,.4,1.2,.16,.24,.12,.28}
U/Γ=24, σ={--+}, τ/Γ=0.16
γ/Γ={.16,.24,.32,.48,.72,.48,.68,.92}
UNIV.
δ/Γ=0.2
(a) (c)
(d)(e) (f)
NRG (τ=0)
NRG (τ>0)
fRG (τ=0)
UNIV.
δ/Γ=0.4
MESO.
δ/Γ=20
MESO.
δ/Γ=25
N=4
FIG. 2: |T(Vg)| and α(Vg) for N = 4, with equidistant levels, δj ≡ δ. The qualitative features do not change if this assumption
is relaxed, or if U is assumed to be slightly level-dependent, U → Ujj′. Decreasing δ/Γ produces a crossover from (a,b,c)
mesoscopic to (d,e,f) universal behavior; increasing U/Γ leads to increased spacing of the transmission peaks and PLs. For
clarity the finite temperature (τ > 0) curves were shifted downwards (by 0.1 for |T|). (f) For U/Γ ≫ 1 and τ/Γ = 0.16, the
CB peak and PL shapes are strikingly similar to those observed experimentally at comparable ratios of τ/Γ (see Fig. 3 of [2]
and Fig. 6 of [3]). For small δ/Γ and U/Γ ≫ 1 fRG becomes less reliable and the results begin to differ from those of NRG.
tent with the experimentally observed trend. For small
to intermediate values of U/Γ [Figs. 1(g,h), 2(d,e)], the
transmission peaks are not well-separated, and α(Vg) has
a saw-tooth shape. As U/Γ increases so does the peak
separation and the corresponding phase rises take a more
S-like form [Figs. 1(h,i), 2(e,f)]. At finite temperatures of
order τ ? δ [17] sharp features are smeared out (Fig. 2).
For U/Γ as large as in Figs. 1(i) and 2(f), the behavior of
α(Vg) (both the S-like rises and the universal occurrence
of PLs in each valley) as well as the one of |T(Vg)| (simi-
lar width and height of all CB peaks) is very reminiscent
of that observed experimentally, in particular for τ ? δ
[Fig. 2(f)]. For U/Γ ≫ 1 the full width of the CB peaks
is of order 2NΓ (not 2Γj as in the mesoscopic regime),
indicating that several bare single-particle levels simul-
taneously contribute to transport.
of the width of the PLs is different from the behavior
τ2/(δ+U)2found in the mesoscopic regime [19] and will
be discussed in an upcomming publication. Note that for
the temperatures considered here the width of the PLs is
still much smaller than the width of the CB peaks.
For certain fine-tuned parameters (γ and σ) the behav-
ior at small δ/Γ deviates from the generic case. For N =
2 the nongeneric cases were classified in [9]. In Fig. 1 only
generic parameters are shown. For N ≥ 3 LR-symmetric
couplings produce nongeneric features. However, these
features are irrelevant to the experiments. They quickly
disappear upon switching on LR-asymmetry or τ > 0.
Interpretation: We can gain deeper insight into the ap-
pearance of the TZs and PLs in the universal regime from
the properties of the REM obtained by fRG for moderate
U/Γ [at which NRG and fRG agree well; Fig. 2(b,e)]. For
The τ dependence
N ≥ 3, δ ? Γ and U = 0, two of the effective levels are
much wider than the others, since ∆ij, being a matrix
of rank 2, has only two nonzero eigenvalues [18] (for the
N = 2 case, see [16, 19]). We found that this also holds
at U > 0: for N = 3,4 one effective level is typically
a factor of 2 to 3 wider than the second widest, while
the remaining 1 or 2 levels are very narrow [Fig. 3(f)].
At δ ? Γ [Fig. 3(d)] the interaction leads to a highly
nonmonotonic dependence of ˜ εj on Vg which is essen-
tial for our universal PL scenario: As Vg is swept, the
widest level hovers in the vicinity of µ over an extended
range of Vgvalues, whereas the narrow ones cross µ – and
therefore also the widest one – rather rapidly. This leads
to a Fano-type effect [20, 21, 22, 23, 24] whose effective
Fano parameter q is real, by time-reversal symmetry [21].
Thus, each TZ, and hence PL, can be understood as a
Fano-type antiresonance arising (irrespective of the signs
of tl
i) from destructive interference between transmission
through a wide and a narrow level. The crossings of the
narrow levels and µ, and thus the PLs, are separated
by U due to charging effects. In contrast, for U = 0,
˜ εj∝ −Vg for all renormalized levels and no levels cross
each other. Our fRG studies indicate that for the regime
δ ? Γ, the Fano-antiresonance mechanism is generic for
U ? Γ. We thus expect it to apply also for interactions
U ≫ Γ for which fRG is no longer reliable.
The fact that the combination of a wide and several
narrow levels leads to PLs was first emphasized in [8]
(without reference to Fano physics). However, whereas in
[8] a bare wide level was introduced as a model assump-
tion (backed by numerical simulations for noninteract-
ing dots of order 100 levels), in our case a renormalized

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4
-60
0
60
εj/Γ
~
0
1
1
nj
~
-40
0
40
0
2
|T|, α/4π, Γj/NΓ
-2
0
εj/Γ
~
0
1
1
nj
~
-808
Vg / Γ
0
|T|, α/2π, Γj/NΓ
δ/Γ=0.2: U N I V E R S A L
δ/Γ=20: M E S O S C O P I C
U/Γ=4, τ=0, σ={---}, γ/Γ={.266,.533,.4,.8,.133,.266,.533,1.07}
(a)
(b)
(c)
(d)
(e)
(f)
N=4
|T|
|T|
α
α
~
~
FIG. 3: Renormalized single-particle energies ˜ εj, occupancies
˜ nj and level widths˜Γj (dashed) of the REM, and the resulting
|T| and α (all as functions of Vg), for N = 4 and δj ≡ δ at
τ = 0. The parameters are the same as in Fig. 2(b,e).
wide level is generated for generic couplings if δ ? Γ.
Also, whereas in [8] the wide level repeatedly empties
into narrow ones as Vgis swept (because Γwide≪ U was
assumed), this strong occupation inversion [25] is not re-
quired in our scenario. In Fig. 3(e), e.g., the wide level
remains roughly half-occupied for a large range of Vg,
but TZs and PLs occur nevertheless. We thus view oc-
cupation inversion, if it occurs, as a side effect, instead
of being the cause of PLs [26].
Conclusions: The most striking feature of our results,
based on exhaustive scans through parameter space for
N = 2,3,4, is that for any given, generic choice of cou-
plings (γ and σ), the experimentally observed crossover
[3] from mesoscopic to universal α(Vg)-behavior can be
achieved within our model by simply changing the ratio
δ/Γ from ? 1 to ? 1, provided that U ? Γ. The univer-
sal π PLs result from Fano-type antiresonances of effec-
tive, renormalized levels, which arise because interactions
cause a broad level (occuring, if δ ? Γ, already for U = 0)
to be repeatedly crossed by narrow levels. A quantitative
description requires correlations to be treated accurately.
We expect that the main features of this mechanism carry
over to the case of spinful electrons, since for δ ? Γ spin
correlation physics (such as the Kondo effect) does not
play a prominent role [27].
We thank P. Brouwer, Y. Gefen, L. Glazman, D.
Golosov, M. Heiblum, J. Imry, J. K¨ onig, F. Marquardt,
M. Pustilnik, H. Schoeller, K. Sch¨ onhammer, and A.
Silva for valuable discussions. This work was supported
in part by the DFG, for VM by SFB602; for TH and JvD
by SFB631, Spintronics RTN (HPRN-CT-2002-00302),
NSF (PHY99-07949) and DIP-H.2.1; and for YO by DIP-
H.2.1, BSF and the Humboldt foundation.
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[26] In a model with a weakly Vg-dependent, bare wide level
and several narrow levels with δ > Γ, Fano-like inter-
ference produces TZs and PLs even at U = 0, for which
interaction-induced occupation inversions are absent [24].
[27] We have recently shown this explicitly for N = 2 [19].
J.vonDelft,submitted,
j and N = 2, the universal PL
LR(ε)