Lapse of transmission phase and electron molecules in quantum dots
Lapse of transmission phase and electron molecules in quantum dots
S.A. Gurvitz∗
Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel
(Dated: October 25, 2018)
The puzzling behavior of the transition phase through a quantum dot can be understood in a
natural way via formation of the electron molecule in the quantum dot. In this case, the resonance
tunneling takes place through the quasistationary (doorway) state, which emerges when the number
of electrons occupying the dot reaches a certain “critical” value, Ncr. Our estimation of this quantity
agrees with the experimental data. The dependence of Ncr on the dot’s size is predicted as well.
PACS numbers: 73.23.Hk, 73.43.Jn, 73.50.Bk, 73.63.Kv
One of the challenging problems in mesoscopic physics
is the puzzling behavior of the transmission phase
through a quantum dot, embedded in an Aharonov-Bohm
ring. It was found in series of experiments performed
by the Weizmann group1,2,3 that all transmission am-
plitudes through different resonant levels of a quantum
dot are in phase. This necessarily implies an unexpected
lapse in the evolution of the transmission phase between
different resonant levels. In addition, it was found in re-
cent measurements3 that this phenomenon takes place
when the number of electrons inside the dot reaches a
certain “critical” value (Ncr & 15)
3. In spite of many
publications addressed to these experiments no fully sat-
isfactory understanding has been found yet4,5.
In this Rapid Communication we demonstrate that the
observed phase-lapse behavior of the transmission am-
plitude can be naturally explained by implying the for-
mation of electron (Wigner) molecules inside quantum
dots, proposed in recent publications6,7,8,9. Moreover,
this framework allows us to estimate Ncr and then to
determine how it is varied with a size of the dot. In
order to explain our model in a proper way, we first elab-
orate the physical nature of the transmission phase in
the case of noninteracting and interacting electrons. In
particular, we concentrate on the role of the Pauli prin-
ciple that prevents different conductance resonances to
carry essentially the same internal wave function. This
point represents a formidable obstacle for resolving the
puzzling behavior of the transmission phase for different
models of the quantum dot. We demonstrate, however,
that this difficulty can be overcame in the context of the
Wigner-molecule when an unstable state is developed in
the middle of the dot.
Let us consider the resonant tunneling through a quan-
tum dot, represented by a potential UD(x), Fig. 1. The
bottom of this potential can be moved by the plunger
electrode, so that one observes the current sweeping
trough different resonant states (Eλ) of the dot. We
would treat this problem in the framework of a tunnel
Hamiltonian approach. This approach is more trans-
parent for evaluation of the transmission phase than the
standard scattering theory, in particular, when the Pauli
principle and the electron-electron interaction are taken
into account. We introduce therefore the following tun-
neling Hamiltonian: H = HL +HR +HD +HT , where
xxl xr
U  (x)D
Lµ Ωr
(λ)Ωl
U  (x)D
FIG. 1: (Color online) Resonant tunneling trough a quantum
dot. µL(R) are Fermi energies in the left (right) reservoir. The
dotted lines show the potential ŪD(x), needed for evaluation
of the bound state wave functions in the Bardeen formula.
HL(R) =
El(r)a
al(r) , HD =
kdk +HC ,
kal + l ↔ r
+H.c. (1)
Here, a
l,r(al,r) is the creation (annihilation) operator of
an electron in the reservoirs and d
k(dk) is the same op-
erator for an electron inside the dot. For simplicity, we
consider electrons as spin-less fermions. The term HC
denotes the Coulomb interaction between electrons in
the dot and Ω
r ] is the coupling between the states
El(Er) and Ek of the reservoir and the dot, respectively.
In the absence of magnetic field, all couplings Ω are real.
All parameters of the tunneling Hamiltonian (1) are re-
lated to the initial microscopic description of the system
in the configuration space. For instance, the coupling
is given by the Bardeen formula10
= − ~
x∈Σl(r)
φk(x)
∇n χl(r)(x)dσ , (2)
where φk(x) and χl(r)(x) are the electron wave functions
inside the dot and the reservoir, respectively, and Σl(r)
is a surface inside the left (right) barrier that separates
the dot from the corresponding reservoir. It is impor-
tant to point out that φk(x) in Eq. (2) is a bound state
wave function for the “inner” potential. The latter co-
incides with the original potential inside the surface Σ
http://arxiv.org/abs/0704.1260v3
and a constant outside this region. On the other hand,
χl(r)(x) is a non-resonant scattering wave function in the
“outer” potential, which coincides with the original po-
tential outside the surface Σ and a constant inside this
region11.
In one-dimensional case (Fig. 1), the separation surface
Σ becomes the separation point, x̄, inside the barrier,
Fig. 1. Then Eq. (2) can be rewritten as12
= −(κk/m)φk(x̄l(r))χl(r)(x̄l(r)) , (3)
where κk =
2m[UD(x̄l,r)− Ek] and φk(x) is the bound
state wave function in the potential ŪD(x) (Fig. 1). The
separation points x̄l,r are to be taken inside the left
(right) barrier as indicated in Fig. 1 and far away from
the classical turning points13.
We start with non-interacting electrons, HC = 0 in
Eq. (1). Then the electron transport through the level
Eλ can be described by the time-dependent Schrödinger
equation i~∂t|Ψ(t)〉 = H |Ψ(t)〉 for a single electron. Tak-
ing the stationary limit we obtain the Landauer formula
for the total current, with the transmission amplitude
given by the Bright-Wigner formula
tλ(E) = N
E − Eλ + i(Γ(λ)L + Γ
R )/2
, (4)
where N = −2π(̺L̺R)1/2 and Γ(λ)L,R = 2π(Ω
2̺L,R
are the partial widths, and ̺L(R) is the density of states in
the left (right) reservoir. We assumed that Ω
l,r ≡ Ω
are weakly dependent on El,r.
The corresponding evolution of the resonance trans-
mission phase for different states |λ〉 is determined by
sign of the product of Ω
R . Since the reservoir states
χl,r are not affected by the plunger voltage, one finds
from Eq. (3) that the evolution of the sign [Ω
R ] is
given by the sign of the product φλ(x̄l)φλ(x̄r). The lat-
ter is positive or negative, depending on the number of
nodes of φλ(x). Hence, it is clear that the non-interacting
electron model cannot explain the same sign for all reso-
nances, observed in Ref.2 (see also Refs.4,14).
Consider N interacting electrons trapped inside the
dot. Despite the electron-electron interaction, the cou-
pling amplitudes ΩL,R can still be evaluated by us-
ing the same multi-dimensional overlapping formula
(2), as in the non-interacting case. Indeed, the
many-body tunneling can be considered as one-body
tunneling, but in the many-dimensional space. In
this case, the wave-function χl(r)(x) is replaced by
χl(r)(xN+1)Φ
N (x1, . . . , xN ), where χl(r) is the wave
function of tunneling electron in the left (right) reser-
voir and Φ
N is the ground state wave function of N
electrons inside the dot. The wave-function φk(x) cor-
responds to Φ
N+1(x1, . . . , xN+1), which is the lowest en-
ergy state (ground state) of N + 1 electrons in the inner
potential of the dot (ŪD in Fig. 1).
Taking n along a coordinate of the tunneling electron,
xN+1, we can integrate over x1, . . . , xN in Eq. (2) thus
reducing this equation to Eq. (3) with φn being replaced
by the overlap function
ϕN (xN+1) = 〈xN+1,Φ(0)N |Φ
N+1〉 . (5)
Therefore, the sign of Ω
R is determined by the sign
of ϕN (x̄l)ϕN (x̄r).
By applying the mean-field approximation, we can
write |Φ(0)N 〉 and |Φ
N+1〉 as a product of one-electron
states (orbitals) in the effective single-particle potential,
ŪD + UC , where ŪD is the inner part of quantum-dot
potential (Fig. 1) and UC(x) is the mean-field describ-
ing the electron-electron interaction. As a result, the
overlap function ϕN (x) is a bound state wave function
in the potential ŪD(x) + UC(x), corresponding to one
of the orbitals. Since the lowest energy state is always
nodeless15, one might assume that ϕN (x) is also a node-
less one, so that the sign of ϕN (x̄l)ϕN (x̄r) would be the
same sign for all resonances. This, however, is not correct
because of the Pauli principle. Indeed, due to the anti-
symmetrization, any two orbitals in the product of the
wave functions representing |Φ(0)N+1〉 cannot be the same.
Since the lowest state is already occupied, the wave func-
tion ϕN (xN+1) must correspond to a higher non-occupied
orbital, and therefore it cannot be nodeless. Hence, the
Pauli principle would create serious problems in any at-
tempt to explain the same sign for all resonances2 in a
framework of the mean-field description of the electron-
electron interaction.
Note that this problem cannot be resolved even by as-
suming large coupling with reservoirs, so that the reso-
nances are overlap. Indeed, the problem is related only
to the inner component of the resonant state, Eqs. (2)
and (3). The latter is eventually brought by the plunger
below the Fermi level, µR, blocking an appearance of
the resonance above the Fermi level with a similar inner
component.
The same situation holds in a more general case, when
the interaction term UC varies with each new electron
trapped inside the dot, UC → U (N)C (Koopman‘s theo-
rem is violated). One finds that due to the central sym-
metry of the self-consistent potential such a variation of
UC with N would not affect the number of nodes in the
overlap function ϕN (xN+1). As a result, the sign of the
transmission amplitude would fluctuate between ±1 for
different resonances.
We illustrate this point by evaluating the overlap func-
tion ϕN (xN+1), Eq. (5), for N = 0 and N = 1. In
the first case, ϕ0(x1) coincides with the wave function
of the lowest energy state, Φ
1 (x1) ≡ φ̃0(x1), in the in-
ner potential ŪD, Fig. 1. This wave function is nodeless.
The second overlap function is ϕ1(x2) = 〈x2,Φ(0)1 |Φ
where Φ
2 (x1, x2) = [φ0(x1)φ1(x2) − φ0(x2)φ1(x1)]/
is the lowest energy state of two electrons in the potential
ŪD + U
C . Here φ0,1 represent the two first orbitals in
this potential. One easily finds that
ϕ1(x2) =
1 (x1)Φ
2 (x1, x2)dx1 = c0φ1(x2) , (6)
where c0 =
φ̃0(x1)φ0(x1)dx1/
2. (The second term is
zero, since φ̃0 and φ1 are orthogonal due to the opposite
parities). Therefore, ϕ1 contains one node, so that the
corresponding transition amplitude changes its sign.
The same behavior of the overlap function would per-
sist for any N . For instance, one easily obtains for N = 2
that ϕ2(x3) ∝ c0φ3(x3)−c13φ1(x3), where the coefficients
c0 = 〈φ̃1|φ1〉, c13 = 〈φ̃0|φ2〉 and φ̃, φ are the orbitals in
the potentials, ŪD(x)+U
C (x) and ŪD(x)+U
C (x), re-
spectively. Since c13 ≪ c0, the overlap function ϕ2 would
contain an additional node in a comparison to ϕ1. Thus,
by assuming the N dependence of the mean-field effec-
tive potential, we are still not able to explain the puzzling
behavior of the transmission phase.
The above consideration was based on symmetry argu-
ments applied to electrons moving in a spherically sym-
metric mean-field central potential. In fact, the central
mean-field picture for two-dimensional quantum dots was
challenged in recent publications6,7,8,9. It was suggested
that due to the strong inter-electron repulsion inside the
dot, spontaneous symmetry breaking takes place lead-
ing to the formation of electron molecules. As a re-
sult, the electrons appear on the ring (rings) around the
dot’s center. This idea was substantiated by unrestricted
Hartree-Fock calculations or by using other computa-
tional techniques6,7,8,9.
In principle, if the symmetry is broken, the overlap
function ϕN (xN+1), Eq. (5), could be very different from
the corresponding orbital φN (xN+1) in the spherical sym-
metric potential. Therefore, it is desirable to investigate
the evolution of the transmission phase in this case. Con-
sider again the overlap function ϕ1(x2) = 〈x2,Φ(0)1 |Φ
but now without the mean-field approximation, as in
Eq. (6). In fact, by taking the parabolic confining poten-
tial, the two-electron wave function |Φ(0)2 〉 can be exactly
calculated16, since relative and center-of-mass coordi-
nates of two electrons are decoupled in the total Hamilto-
nian. As a result, Φ
2 (x1, x2) = φcm(x1+x2)φr(x2−x1),
where φr(−x) = −φr(x) due to the Pauli principle. Such
a wave function peaks for x1 = −x2 and therefore it
would bear the features of a two-electron molecule9. One
finds from Eq. (5),
ϕ1(x2) =
φ̃0(x1)φcm(x1 + x2)φr(x2 − x1)dx1 . (7)
Taking into account that the values of x1 which mainly
contribute to the integral (7) are localized inside the dot
and that the wave function φr(x) is the odd one, we find
that the overlap function changes its sign when the ar-
gument varies from x̄l to x̄r, Fig. 1. Hence, ϕ1 displays
one node, as in the spherically symmetric mean-field po-
tential, Eq. (6).
One can continue with the same arguments for the
three and more electron molecules, where the electrons
are placed on the ring. The corresponding ground
state wave functions |Φ(0)N 〉 would represent a fully
anti-symmetrized product of the original (site) nodeless
orbitals7,8. Yet, the overlap function Eq. (5) cannot be
nodeless. As a result, the sign of ϕN (x̄l)ϕN (x̄r) would
fluctuate with N . One can demonstrate it rather easily
for N = 3, 4. Although it would be hard to extend such
a demonstration for large N , there is no reason to expect
that the sign of ϕN (x̄l)ϕN (x̄r) ceases to fluctuate when
N increases.
It seems from the above arguments that the rota-
tional symmetry breaking (the electron-molecule forma-
tion) cannot explain the evolution of the transmission
phase observed in the experiments2,3. Nevertheless, there
is an additional feature of the electron molecule, which
has not been yet utilized. That is due to the electrons
located on the ring (rings) would develop an additional
(inner) electrostatic trap inside the dot when their num-
ber (N) is large enough. As an example, we display in
Fig. 2a such a potential, VC(x) =
2/ǫ|x− xj |, pro-
duced by 14 electrons equally distributed on the ring,
where ǫ = 13.6 is the dielectric constant of the medium.
The radius of the ring (R = 50 nm) is taken close to the
dot’s size in Ref.3. The radial profile of this potential
along the angle θ = π/N , where the potential height on
the ring is minimal, is shown in Fig. 2b for two values
of N . It appears that the trap is not well developed for
N = 6, but it is already pronounced for N = 14.
A minimum number of electrons in the dot sufficient
to develop the trap with one bound state inside it can
be estimated from the condition that the barrier height,
hN in the Fig. 2b, reaches the ground state energy ε0.
We estimate the latter as π2~2/m∗R2, where m∗ is the
effective electron mass (m∗/m0 = 0.067). For instance,
one finds ε0 = 4.5 meV for R = 50 nm. Then the condi-
tion hN = ǫ0 corresponds to N ≃ 10, which is a minimal
(“critical”) number of electrons, Ncr, enabled to hold a
resonance state. This value is an approximate agreement
with that found in3. In fact, a more elaborate, semi-
classical estimations of Ncr approximately produce the
same number [hN ≃ 10 meV for N = 14, Fig. 2b]17.
The state |ε0〉 in the inner part of the trap, V̄C , Fig. 2b,
is not stable due to the symmetry breaking, leading to
formation of the electron molecule. Nevertheless, this
state is important in formation of the (N + 1)-electron
molecule by adding an additional electron to the N -
electron system. Indeed, one expects that the overlap
function (5) for the electron states on the ring is sup-
pressed in comparison to the same overlap for the cen-
tral mean-field potential. The reason is that all electrons
are shifted from their positions whenever an additional
electron is placed on the ring. This is in contrast to the
mean-field description, where the N -electron core is not
modified. On the other hand, if the electron is placed in
the center of the ring, it distorts the remaining N elec-
trons in a minimal way. We expect therefore that the
10 20 30 40 50 60
[meV]V
V (              )
Nθ=π/ρ,C
VCε0 hN
FIG. 2: (Color online) (a) Electrostatic trap generated by 14
electrons placed on the ring of the radius 50 nm. (b) The
radial profile of the Coulomb potential along the potential
valley. ε0 is the ground state energy in the potential V̄C ,
representing the inner part of the trap VC .
corresponding overlap function is large, as in the case of
the central mean-field potential. Hence, such an unstable
state |ε0〉 in the middle of the dot would play a role of
a “doorway” state in formation of the (N + 1)-electron
molecule.
It follows from the same arguments that the electron
transport would proceed through such an unstable state
when the quantum dot coupled with the reservoirs. Since
this doorway state is of the lowest energy in the inner
trap, V̄C (Fig. 2), it is nodeless. The crucial point here
is that this state is eventually not occupied, when it is
brought by the plunger below the Fermi levels of the
reservoirs. Indeed, it is not turned to a stable state be-
low the Fermi levels due to the symmetry breaking, but
it always decays to the ring states. Therefore, this state
is never blocked by the Pauli principle to carry the res-
onant transport through it, when it is above the Fermi
level µR, Fig. 1. As a result, all transmission amplitudes
for any N > Ncr would be in phase.
In fact, by taking into account the electron spin, one
finds that two electrons with the same spatial (nodeless)
wave functions are allowed to occupy the lowest energy
states. Therefore, even if the state |ε0〉 in the center of
the dot becomes a stable one for some values of N , the
resonant transport would proceed through an unstable
state of the two electrons (with opposite spin) inside the
dot. The corresponding overlap function would be again
nodeless.
40 50 60 70 80 90
R@nmD
FIG. 3: Dependence of the “critical” number of electrons on
the dot’s radius.
Note that although the doorway-state energy is the
lowest one for the inner trap, V̄C , it exceeds the energy
of the ring states. Therefore, the ring states would ap-
pear inside the bias voltage before the doorway state. We
can assume, however, that the ring states are not well
separated in the energy from the doorway state, which
dominates the resonant current. It was also taken into
account that in the presence of the Coulomb interaction,
the shift of the resonance energy due to tunneling is dif-
ferent for different levels18. In particular, the broad reso-
nance is shifted down more than the narrow one18. As a
result the doorway state could have a lower energy than
the ring states.
One of the consequences of our model is an existence of
the critical number of electrons in the dot, which is nec-
essary for formation of the resonant state inside the dot
(Ncr). This number would vary with the dot’s size. Such
a dependence of Ncr on the radius of the dot (R), ob-
tained from our estimation, hN = ε0, is shown in Fig. 3.
One finds from this figure that this dependence is rather
weak. The critical number slightly decreases with an in-
crease of the dot’s size.
In summary, we demonstrated that the unusual be-
havior of the resonant phase, observed in interference ex-
periments, can be considered as a strong evidence for
formation of electron molecules in quantum dots. This
structure would produce an electrostatic trap, contain-
ing an unstable (doorway) state localized in the center of
the dot, whenever the number of electrons occupying the
dot is large enough, N > Ncr. Then such an unstable
state would carry the electron transport through the dot
irrespective of the value of N . This would appear as if
the different transmission amplitudes are in phase. Our
prediction for the dependence of Ncr on the dot’s radius
can be experimentally verified.
∗ Electronic address: shmuel.gurvitz@weizmann.ac.il
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mailto:shmuel.gurvitz@weizmann.ac.il
