VOLUME 86, NUMBER 20

PHYSICAL REVIEW LETTERS

14 MAY 2001

Fano Resonances as a Probe of Phase Coherence in Quantum Dots
A. A. Clerk, X. Waintal, and P. W. Brouwer
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853(Received 18 May 2000)
In the presence of direct trajectories connecting source and drain contacts, the conductance of a quantum dot may exhibit resonances of the Fano type. Since Fano resonances result from the interferenceof two transmission pathways, their line shape (as described by the Fano parameter q) is sensitive todephasing in the quantum dot. We show that under certain circumstances the dephasing time can beextracted from a measurement of q for a single resonance. We also show that q fluctuates from level tolevel, and we calculate its probability distribution for a chaotic quantum dot. Our results are relevant torecent experiments by Göres et al. [Phys. Rev. B 62, 2188 (2000)].
DOI: 10.1103/PhysRevLett.86.4636

PACS numbers: 73.23.Hk, 73.21.– b, 73.40.Gk

Perhaps one of the most fundamental issues in the fieldof mesoscopic physics is that of phase coherence: Under
what conditions are electrons able to retain a well-definedphase? This issue is of particular interest for quantum
dots in the Coulomb blockade regime, where the electricalconductance is suppressed except for points of charge degeneracy [1]. Despite the fact that interactions are strong
in these dots, the shape of the conductance peaks can bewell understood in terms of single-particle wave functions.
Unfortunately, a simple conductance measurement cannot
discriminate between coherent and incoherent (sequential)
tunneling, as both mechanisms give rise to line shapes of
the Breit-Wigner form [2]. Instead, to establish phase coherence, the quantum dot has to be embedded in an interferometer. This was first done by Yacoby et al. [3], who
included a quantum dot in one arm of an Aharonov-Bohm
ring (see also Ref. [4]).
Given this result, it is natural to question the extent towhich transport is phase coherent. This question could not
be fully addressed in Ref. [3], because dephasing in thequantum dot and in the arms of the interferometer cannot
easily be separated. An extremely promising developmentin this respect is found in a recent work by Göres et al. [5],who observed resonances with a Fano line shape in the conductance of Coulomb-blockaded dots [6], instead of theusual Breit-Wigner form. Fano resonances are caused by
the interference of two transport pathways, a resonant anda nonresonant one, and are thus sensitive to phase coherence. In the dots of Ref. [5], the direct pathway is probably direct transmission through the dot, as schematically
depicted in Fig. 1. In this sense, the dot serves as its own
interferometer.
Fano resonances have a line shape of the form
G共´兲 苷 Gd

j2´ 1 qGj2,
4´2 1 G 2

(1)pwith transmission amplitude td 苷 eibd Gd and a resonant
path with transmission amplitude tr 共´兲 苷 zr G兾共2´
p 1 iG兲,
where G共´兲 苷 jtd 1 tr j2 and q 苷 i 1 zr e2ibd 兾 Gd . The
Fano line shape (1) is for temperatures T ø G, which isappropriate for the experiments on very small quantum
dots of Refs. [3–5]. Examples of dots that could show
Fano resonances are shown in Fig. 1.
Through the complex Fano parameter q, Fano resonances contain more information than Breit-Wigner resonances. Moreover, as the direct and resonant paths are notspatially separated, no source of decoherence other thandephasing inside the quantum dot can affect the line shape.
In fact, as we show below, in the presence of time-reversal
symmetry (TRS) or for a dot of the form of Fig. 1b, where
the opening to the cavity contains at most one propagating channel, measurement of a single Fano resonance at
T ø G is sufficient to determine the dephasing time tf
in the quantum dot:ph̄
苷 G关jqj2 1 1 2 共jqj2 1 1兲2 2 4共Imq兲2 兴 .
tf(2)In this Letter, we develop a detailed description of Fano
resonances in quantum dots. In addition to the effect ofdephasing, we consider mesoscopic fluctuations — theFano parameter q and the width G fluctuate from resonance

1

1
b)

a)
2

2

where G is the conductance, measured in units of 2e 兾h,´ is the energy, set by a gate voltage, G is the resonance width, Gd is the nonresonant conductance, and q isthe (complex) “Fano parameter.” The resonance form (1)arises from the interference of a “direct” nonresonant path

FIG. 1. Schematic drawing of two quantum dot systems inwhich Fano resonances are expected. (a) A dot similar to thatused in a recent experiment by Göres et al. [5], showing a possible direct path. (b) A system where the width of Fano resonances
can be tuned without altering the direct path.

4636© 2001 The American Physical Society

20031-9007兾01兾86(20)兾4636(4)$15.00

VOLUME 86, NUMBER 20

PHYSICAL REVIEW LETTERS

to resonance. We calculate the probability distribution of q
for a set of consecutive resonances in a chaotic quantum
dot using random matrix theory (RMT). We close with acomparison of our results and the experiment of Ref. [5].
Several previous studies have treated Fano resonances inquasi-one-dimensional systems rather than quantum dots
[7] and without mesoscopic fluctuations or dephasing.
Model.—We consider a quantum dot coupled to twosingle-mode leads (1 and 2) via point contacts; see Fig. 1.
Transport through the system is characterized by the 2 3
2 scattering matrix S共´兲, which we parametrize in termsof a (unitary) scattering matrix S0 共´兲 for processes that involve ergodic exploration of the cavity, and a (nonunitary)matrix S̄ that describes scattering via the direct, nonresonant paths (both transmitting and reflecting) [8,9],
S 苷 S̄ 1 t 0

1
S0 t .
1 2 S0 r 0(3)

The auxiliary matrix t describes transmission from theleads to an ergodic dot state, while t 0 describes transmission from such a state back into the leads. Similarly, r 0
describes reflection of an electron leaving an ergodic dot
state back into the dot. Our results are independent of t 0 ,t, and r 0 , as long as the 4 3 4 matrixµ∂S̄ t 0t r0
is unitary. As the time scale of the direct processes is muchsmaller than h̄兾D, where D is the level spacing of the dot,S̄ will be constant over an energy interval spanning a large
number of distinct resonances. Also, since TRS cannot be
broken on this short time scale, S̄ is symmetric. In contrast,
the scattering matrix S0 , which describes scattering fromlong, resonant paths, depends on energy and is only symmetric in the absence of a TRS breaking magnetic field.
We use the polar decomposition of S̄ [9],pS̄ 苷 U 共1 2 T 兲 U T ,(4)where U is a 2 3 2 unitary matrix and T 苷 diag共T1 , T2 兲.
0T
Without
loss ofp generality, we may choose t 苷 t 苷p0
U T , r 苷 2 1 2 T . The numbers 0 # T1 # T2 # 1
are known as “sticking probabilities” [10], i.e., the probabilities for scattering through paths that explore the dot ergodically, instead of direct transmission via a short path.
(The inequality T1 # T2 is made for definiteness.) The
standard theory of Fano resonances [6] (see also [7]) assumes the existence of a single sticking probability only. In
our case, the existence of two sticking probabilities T1 andT2 follows because the dot is coupled to two single-mode
leads. We assume T1 , T2 ø 1, which ensures that the resonances are narrow and well separated.
For S0 , we make use of the formula [9]S0 共´兲 苷 关1 2 iK共´兲兴兾关1 1 iK共´兲兴 ,(5)where K is a 2 3 2 matrix representing the Green function of the closed cavity at the contacts [11]. A reso-

14 MAY 2001

nance occurs when K has a pole, i.e., when ´ coincides
with an energy level of the closed dot. Close to resonance, K ⯝ DCC y 兾共p´兲, where the two-component
vector C 苷 共C1 , C2 兲 represents the values of the wave
function at the leads. For convenience, we have set theresonance energy to zero. Then S共´兲 takes the formp ∂pµ
2iD T CC y T
U T,(6)
S共´兲 苷 U 1 2
4p´ 1 i2pG
where G 苷 DC y T C兾2p. By the Landauer formula,S共´兲 determines the conductance G共´兲 苷 jS共´兲12 j2 , andhence the Fano parameter q. Far from resonance, thesecond term in Eq. (6) vanishes, so that the nonresonant
contribution to the conductance reads Gd 苷 j共UU T 兲12 j2 .
Several conclusions can be drawn directly from Eq. (6).
First, writing q 苷 qx 1 iqy , unitarity of S implies that qx
and qy are bounded. Defining 共qxmax 兲2 苷 共qymax 兲2 2 1 苷1兾Gd 2 1 and changing to “normalized” real and imaginary parts of the Fano parameter q̃x 苷 qx 兾qxmax , q̃y 苷
qy 兾qymax , Eq. (6) gives the constraintq̃x2 1 q̃y2 # 1 .

(7)Second, in the presence of TRS, the wave function C can
be chosen real, from which one finds qy 苷 0. This implies
that, in the absence of dephasing, the conductance drops
to zero at ´ 苷 2qx G兾2.
Resonance-to-resonance fluctuations.—We now consider a set of consecutive resonances in a single quantum
dot, all occurring within an energy interval in which S̄ canbe considered constant. In general, the Fano parameter q
is sensitive to the resonance wave function C only throughthe ratio T1 C1 兾T2 C2 . If one of the sticking probabilities
is zero, so that the resonant state is coupled only to theoutside world via a single channel, this ratio is independent of C, and q is set solely by the direct process. This
results in q being real and the same for each resonance:q 苷 qa 苷 i共U11 U21 2 U22 U12 兲兾共U11 U21 1 U22 U12 兲 .
(8)In quantum dots, this limit can be realized, for example,
in the geometry of Fig. 1b, if the opening to the cavity
supports only one mode at the Fermi level. However, in thegeneric case (if the opening contains more than one mode,or in the geometry of Fig. 1a), both sticking probabilities
T1 and T2 are nonzero. Then q depends on the wave
function of the resonance and should exhibit mesoscopic
fluctuations from resonance to resonance. In the case ofa chaotic quantum dot, we obtain the distribution of q
for our set of resonances by using RMT for the statisticsof the wave function C [12], keeping S̄ the same samefor all resonances (and hence U and T ). According to
RMT, the elements of C are independently distributed real(complex) Gaussian random numbers with zero mean andunit variance, in the presence (absence) of TRS. In termsof the normalized real and imaginary parts q̃x and q̃y , andwith q̃a 苷 qa 兾qxmax , we find that in the presence of TRS
the distribution P is given by4637

VOLUME 86, NUMBER 20

PHYSICAL REVIEW LETTERS
1
P苷
p

s

a

1 1 2 共1 2 q̃x q̃a 兲
11ad共q̃y 兲 ,22
1 2 q̃x 1 1 a共1 2 q̃x q̃a 兲 1 a 共q̃x 2 q̃a 兲2

q.
a2
2p 1 2 q̃x2 2 q̃y2 兵1 1 4 关共q̃x 2 q̃a 兲2 1 q̃y2 2 q̃y2 q̃a2 兴 1 a共1 2 q̃x q̃a 兲其2

共qy 兲2 ! 1 2 xf 1 xf 共qx2 1 qy2 2 xf qx2 兲 .

(11a)
(11b)

In the presence of TRS, or in the extreme asymmetric limit,
where only one sticking probability is nonzero, qy 苷 0
in the absence of dephasing. Hence, measurement of anonzero qy in those cases can be used to determine tf[17]. Calculating tf from Eq. (11) with qy 苷 0 in theabsence of dephasing yields the relationship (2). If TRS
is broken and if both sticking probabilities are finite, qy isalready nonzero in the absence of dephasing, and it cannot
be used to find tf . Note that as tf ! 0, G共´兲 ! GD ,consistent with earlier work on resonant tunneling [18].
Role of Coulomb interactions. — So far we have not addressed the issue of Coulomb interactions, which are cer-

2

0.8
G
0

x

q x ! x f qx ,(9b)tainly present and important for small quantum dots. In
this respect, we note that the time needed to traverse thequantum dot via a direct trajectory is of the same order
or smaller than the inverse charging energy Ec . Hence,by the time-energy uncertainty principle, transmission via
direct paths is not forbidden by Coulomb blockade. In
fact, as was shown by Matveev and co-workers [19,20],
Coulomb interactions actually enhance the probability of
direct processes — both direct reflection and direct transmission [21]— at the cost of ergodic scattering. Hence,by virtue of Coulomb interactions, the dot is driven to
the weak coupling regime T1 , T2 ø 1. Such an interaction-induced renormalization of the coupling between the
dot and its environment may explain why in the experiment of Ref. [5] sharp resonances were observed, despite
the presence of diffraction at the point contacts. To apply
our theory to this situation, it is necessary to assume thatthe renormalization of the scattering parameters for directprocesses has already taken place. We also implicitly assumed that interactions play no further role in modifying
the resonances and that a single-particle approach is thus
valid close to resonance; see Refs. [22,23].
Two interesting observations of the experiment [5] canbe interpreted with the results of this Letter. First, in
Ref. [5], it was seen that application of a magnetic field

max
)
x

Here a 苷 T2 兾T1 2 1 $ 0. (We have averaged over the
resonance width G, which is also a random variable [13].)
In case of symmetric couplings (T1 苷 T2 ), the distribution simplifies to P共q̃x 兲 苷 p 21 共1 2 q̃x2 兲21兾2 [P共q̃x , q̃y 兲 苷
共2p兲21 共1 2 q̃x2 2 q̃y2 兲21兾2 ] with [without] TRS. In the extreme asymmetric regime a ¿ 1, Eq. (9) tends to a delta
function distribution at q 苷 qa ; cf. Eq. (8). Note howeverthat for large but finite a, the distribution still has an appreciable width (see Fig. 2).
Dephasing. —The effects of dephasing are treated phenomenologically by attaching a fictitious voltage probe to
the dot [2,14]. This approach is not limited to a particular
microscopic mechanism and can describe dephasing fromboth intrinsic sources (i.e., from electron-electron interactions) and extrinsic sources (e.g., radiation, magnetic impurities). However, similar to a golden rule calculation[15], it does not account for possible interaction effects
beyond lifetime broadening which occur at low temperatures [16]. In practice, we first replace ´ ! ´ 1 i h̄兾2tf
in Eq. (6), where tf is the phenomenological dephasing
time. The imaginary part of ´ models escape through thefictitious voltage probe. A correction term is then added
to the conductance formula to account for the incoherent
injection of electrons from the voltage probe [2,14],
关1 2 共SS y 兲11 兴 关1 2 共SS y 兲22 兴
G共´兲 苷 jS12 j2 1
. (10)
2 2 共SS y 兲11 2 共SS y 兲22
The second term corresponds to incoherent transmission
through the dot and has a Breit-Wigner line shape. As
a result, the imaginary part qy of the Fano parameter isincreased. Inclusion of dephasing also changes the resonance width G to G 1 h̄兾共2tf 兲. Writing the ratio ofresonance widths without and with dephasing as xf 苷G兾关G 1 h̄兾共2tf 兲兴, we find that the change of the Fanoparameters upon inclusion of dephasing is given by

4638

a2

a

关1 1 2 共1 2 q̃x q̃a 兲兴2 1 4 共1 2 q̃x2 2 q̃y2 兲 共1 2 q̃a2 兲

P(q /q

P苷

(9a)

4

while in the absence of TRS
11a

14 MAY 2001

0

−0.75

ε

max
x

q /q
x

0.75

FIG. 2. Distribution of qx in the presence of TRS for T2 苷 T1
(solid line) and T2 苷 100T1 (dashed line). Inset: Two Fano
resonances with the same qx , but where qy is greater for thedashed curve compared to the solid curve. (Breaking TRS causes
qy to increase on average, leading to more Breit-Wigner-like line
shapes.)VOLUME 86, NUMBER 20

PHYSICAL REVIEW LETTERS

tended to make resonances more Breit-Wigner-like. Our
calculation indicates that, for the generic case when bothsticking probabilities are nonzero, breaking of TRS generically leads to an increase of qy and thus to more BreitWigner-like resonances; cf. Eq. (9b) (in the absence ofdephasing qy was zero without a magnetic field). Thisis illustrated in the inset in Fig. 2 (compare against Fig. 6of Ref. [5]).
Another observation in Ref. [5] was that, as a functionof the gate voltage that controls the transparencies of thepoint contacts, the width of the observed resonances was
nonmonotonic. The conductance peaks started as narrow
Breit-Wigner resonances when the dot was pinched off(Gd 苷 0), then widened as the contacts were opened into
resonances exhibiting the Kondo effect. As the contacts
were opened farther, the resonances became more narrow and had the Fano form with background conductanceGd ⯝ e2 兾h. One possible explanation is that diffraction
at the contacts to the dot is strongest at intermediate point
contact transparencies, leading to large sticking probabilities. (A schematic of the dot of Ref. [5] is shown in
Fig. 1a.) To test such a scenario, we have performed numerical simulations of the system shown in Fig. 1b, using
a recursive Greens function algorithm [24]. As expected,
we find conductance resonances with a Fano line shape;
see Fig. 3. In order to simulate how the nonmonotonic
resonance width in the experiment might happen, we haveplaced an impurity near the opening of the dot and varied
its scattering strength V (V is the potential of the impurity sites in the simulation). Resonances for two valuesof V are shown in Fig. 3; the resonance width G does indeed exhibit a nonmonotonic dependence on V . Initially
increasing V from zero has the effect of deflecting more
electrons into the dot and hence increasing G; larger values of V , however, cause electrons to backscatter awayfrom the dot altogether, thus reducing G and suppressing
the background conductance Gd .

G(ε)

0.8
0.4
0.0
0.2
0.0
0.04
0.00

0

2 ε/∆

4

FIG. 3. Numerical simulation of conductance vs voltage showing Fano resonances for a quantum dot with direct transmission(right). The thin line corresponds to the background conductance(i.e., when the dot is closed off). The three plots correspond to
the same resonances for an increasing value of the scatteringstrength of an impurity near the contact.

14 MAY 2001

Conclusion.— Fano resonances provide a powerful tool
for the study of phase coherence in transmission throughquantum dots. We have shown that in certain cases, measurement of a single resonance already allows for the determination of tf . We have also calculated the distributionof Fano parameters for a chaotic quantum dot.
We thank V. Ambegaokar, D. Goldhaber-Gordon,
M. Kastner, and C. Marcus for useful discussions. A. C.
acknowledges support of the Cornell Center for Materials
Research.

[1] D. V. Averin and K. K. Likharev, in Mesoscopic Phenomena
in Solids, edited by B. L. Altshuler, P. A. Lee, and R. A.
Webb (North-Holland, Amsterdam, 1991).
[2] M. Büttiker, IBM J. Res. Dev. 32, 63 (1988).
[3] A. Yacoby et al., Phys. Rev. Lett. 74, 4047 (1995).
[4] E. Buks et al., Phys. Rev. Lett. 77, 4664 (1996); R. Schuster
et al., Nature (London) 385, 417 (1997).
[5] J. Göres et al., Phys. Rev. B 62, 2188 (2000).
[6] U. Fano, Phys. Rev. 124, 1866 (1961); J. A. Simpson andU. Fano, Phys. Rev. Lett. 11, 158 (1963).
[7] E. Tekman and P. Bagwell, Phys. Rev. B 48, 2553 (1993);
J. U. Nöckel and A. D. Stone, Phys. Rev. B 50, 17 415(1994); P. S. Deo, Solid State Commun. 107, 69 (1998);
C-M. Ryu and S. Y. Cho, Phys. Rev. B 58, 3572 (1998).
[8] P. A. Mello, in Mesoscopic Quantum Physics, edited by
E. Akkermans, G. Montambaux, J.-L. Pichard, and J. ZinnJustin (North-Holland, Amsterdam, 1995).
[9] C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
[10] S. Iida et al., Ann. Phys. (N.Y.) 200, 219 (1990).
[11] C. H. Lewenkopf and H. A. Weidenmüller, Ann. Phys.
(N.Y.) 212, 53 (1991).
[12] M. L. Mehta, Random Matrices (Academic, New York,
1991).
[13] The distribution of widths G after averaging over q
and in the presence of TRS is given by P 苷
GjG 2G j
共2G1 G2 兲21兾2 e2G兾4G1 e2G兾4G2 I0 共 4G11 G2 2 兲, where I0 is aD
modified Bessel function, and Gj 苷 2p Tj . In the absenceof TRS, one has P 苷 共G1 2 G2 兲21 共eG兾G1 2 eG兾G2 兲.
[14] P. W. Brouwer and C. W. J. Beenakker, Phys. Rev. B 55,
4695 (1997).
[15] U. Sivan et al., Europhys. Lett. 28, 115 (1994).
[16] B. L. Altshuler et al., Phys. Rev. Lett. 78, 2803 (1997).
[17] Using Eq. (2) and the data of Ref. [5], we find dephasing
times on the order of 1022 1021 ns, which corresponds
to dephasing rates smaller than D. An explicit experimental study of dephasing using Fano resonances is currently
planned [G. Granger (private communication)].
[18] A. D. Stone and P. A. Lee, Phys. Rev. Lett. 54, 1196 (1985).
[19] K. A. Matveev, Phys. Rev. B 51, 1743 (1995).
[20] A. Furusaki and K. A. Matveev, Phys. Rev. Lett. 75, 709(1995); Phys. Rev. B 52, 16 676 (1995).
[21] P. W. Brouwer and I. L. Aleiner, Phys. Rev. Lett. 82, 390(1999).
[22] G. Hackenbroich and H. A. Weidenmüller, Phys. Rev. Lett.
76, 110 (1996).
[23] Y. Oreg and Y. Gefen, Phys. Rev. B 55, 13 726 (1997).
[24] H. U. Baranger et al., Phys. Rev. B 44, 10 637 (1991).

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