A New Nonlocal Force in Condensed Matter Physics:
Double Quantum Wells *
This article is a popular scientific
introduction of a new kind of nonlocal force in condensed matter
that was recently predicted by the researchers in the Institute of
Theoretical and Applied Physics (ITAP)/Turkey and Bilkent
University/Turkey. The predicted force emerges due to the condensation
of Wannier-Mott excitons in semiconductors and adds a yet new effect in
the low temperature physics of the condensed matter.
Wannier-Mott (WM) excitons was predicted by Moskalenko, Blatt, Böer
and Brandt in
1962 and elaborated later by the group lead by Keldish . WM excitons
are hydrogen-atom like bound states of interacting
holes (Fig.1.a below) itinerant within semiconductor medium and there
has been a
tremendous research to experimentally verify this proposal using bulk
semiconductor samples [4,5]. The driving interaction of the
condensation of the WM excitons is the attractive Coulomb coupling
between electrons and the holes and the repulsive dipolar interaction
between the excitons. However, the short lifetime of the
bulk WM excitons (a few nanosecond)
is insufficient for the thermal equilibrium to be reached before any
condensation can take place. Since early 90's, with the progress made
in growing semiconductor heterojunctions, the search for the
condensation of WM excitons is made by using spatially indirect
electron-hole bands confined in different quantum wells  separated
distance comparable to an exciton Bohr radius a0
~ 100 Å as depicted in Fig.1.b below (the double quantum well-DQW
geometry). In addition to this DQW geometry, a strong external electric
field is usually applied [7,8]
to keep the electrons and holes away from each other by pinning their
the outer edges of their respective quantum wells.
specifically, the authors investigate the Coulomb coupling
between the interacting and
spatially separated electron-hole
system confined to two separate
quantum wells, and due to the spin neutrality of the Coulomb
interaction, the pairs come in all possible spin configurations as
shown in Fig.1.c. There are two effects of the attractive
Coulomb coupling which is a function of the distance between the
electron and the hole wells. If this distance is on the order of an
Bohr radius a0
first effect of the Coulomb coupling is the formation of the
electron-hole bound states, i.e. the WM exciton. Below a certain
temperature (the critical temperature), the
second effect of the Coulomb interaction comes into play. In
sufficiently low densities (the critical
density), when excitons act like
they are expected to experience a phenomenon called Bose-Einstein
condensation and form a phase coherent
ground state, i.e. the exciton condensate (EC). In this new ground
state, excitons act collectively very
much like the motion of a school of small fish in the sea. This new
ground state lowers the free energy of the system with the gap in
on the strength of the Coulomb coupling. Due to the small exciton
reduced mass, the critical temperature of the EC is on the order
of a few Kelvin. As the density increases, the exciton wavefunctions
start spatially overlapping, moving the electrons and the holes into a
BCS like condensation.
A description of a Wannier-Mott exciton without the spin degree of freedom indicated. (b) Electron and hole quantum
a semiconductor DQW. Due to the external electric field the electron
and the hole wavefunctions locate at the outer surfaces of the
respective quantum wells, (c)
Dark and bright excitons.
an exciton condensate two different types of pairings are allowed
between an electron in an s-like, and a hole in a p-like orbital
states. The bright pairs are composed of opposite electron and hole
spins combined into a bright singlet and a bright triplet, whereas the
dark triplet pairs are composed of paralel electron and hole spins. In
addition to these complications, the bright and the dark states are not
totally independent. Coupling between the spin and the orbital degrees
of freedom can also be crucially important in many of the semiconductor
heterostructures. There are also fundamental symmetries such as the
time reversal symmetry, spin degeneracy, orbital rotation symmetry,
fermion exchange symmetry, and these play crucial role in the
formation of the EC .
The presence or absence of these symmetries affect the interplay
between the dark and the bright components and can even change the
topology of the condensate in the energy-spin space. A
thorough understanding of the exciton condensate (EC) has been
challenging mostly due to these and other inherently experimental
diffculties  such
as the short exciton lifetime and the momentum as well as angular
momentum dependence of the residual interactions between the fermionic
constituents. The bright states can couple to the radiation field
through the recombination and pair creation processes due to their odd
total angular momenta, whereas the dark states do not couple. However,
in reality, the dark and the bright states are mixed [11,12]. Two
dark states can also turn into to bright ones through a mechanism
Pauli exchange scattering before coupling to the radiation field .
Therefore there is always a weak bright component in the ground state
by which the photoluminescence experiments can be made.
and hole pairs can recombine to radiate a photon, and this photon with
the right energy can also create a pair, but these processes can only
the bright states as shown in Fig.2.b and 2.c. These radiative
processes cost a positive energy which is not liked much by the other
members of the condensate, i.e. the dark excitons. Therefore the bright
exciton population in
ground state is dramatically suppresed in favour of the dark states
Pauli scattering of two bright pairs can turn into to dark pairs
whereas the reverse process is energetically disfavoured in low
Fig.2. (a) Electron and hole bands in semiconductors
forming excitons. (b) & (c) Radiation field and its
coupling to the bright and dark states.
the recent experiment by High, Leonard,
Butov, Kavokin, Campman and Gossard in 2012
the photoluminescence measurements have been limited in probing all
components of the condensate due to the weakness of the bright
the names suggest, bright
condensates couple to the radiation field(or simply light) where as
their dark counterparts do not. Since photoluminescence experiments can
only probe the bright condensate, the amount of bright states in the
ground state of the coherent exciton gas is essential in these
experiments. Since bright contribution is strongly suppresed by
the dark one, this also makes the photoluminescence experiments highly
difficult. In Ref.  using
a Mach-Zhender interference measurements, a clear evidence on
the EC was established by the observation of the
interference fringes. This is a clear evidence that there is a
macroscopic and coherent
order in the ground state, hence the condensate.
should be realized that EC research with such theoretical and
experimental challenges is an outstanding resource in better
understanding the unconventional aspects of many body interacting
quantum systems in general. The exciton condensate is one of the most
difficult examples in condensed matter physics where a variety of
different mechanisms and unconventional examples of pairings play
leading role all at the same time. It is a common sense to say that, in
broad area, new effects unknown in other systems should also be
expected. In this work, the authors
demonstrated that the Coulomb interaction between electrons and holes
in the presence of the exciton condensate gives rise to a conceptually
new and nonlocal force, which they coined as the EC-force,
of which observation is expected not only to
shed light on the theoretical and experimental understanding of the EC
but also stimulate a broader research on other many body systems where
similar effects can arise.
is a fundamentally new effect in condensed matter physics:
In their earlier numerical calculations , taking into account all
manifested symmetries a well as the radiative couplings, the authors noticed
that, the change in
ground state energy of the EC with respect to the distance between the
electron and the hole quantum wells has a discontinuity at a critical
distance between the electron and the hole quantum wells (left
figure in Fig.3 below). This sharp discontinuity is actually a boundary
separates the normal excitonic liquid from the condensed
excitons. This picture is very
much like a perfectly smooth street with a huge hole in it. As
a wheel is very slowly moved towards the hole, it suddenly rushes into
the hole, the normal excitonic liquid becomes unstable as it enters
the boundary and wants to go to lower energies and condense stronger
(Fig.3 with stages A,B, C below).
As the electron and the hole wavefunctions are pinned by the strong
external electric field, the attempt to go to lower energies
results in forcing the distance to be narrowed between the electron
and the hole quantum wells. This clearly indicates a new kind of force,
purely due to the dependence of the condensate's energy on the
distance between the electron-hole quantum wells, i.e. the EC-force.
On the left is the solution of the energy gap as a function of the
distance between the layers and the exciton concentration (Adopted from
positions indicated by A,B and C correspond to different distances
between the electron and the hole quantum wells. On the
right, (A) is the absence of condensation for D > Dc
, (B) emerging condensate for D ~ Dc
, and (C) a strong
condensate for D < Dc
with Dc as the critical distance. The right arrows in green indicate the direction of lowering the free energy. The wheel
and the hole analogy used in the text is also described.
attempt to understand this, the authors succeeded in devicing tools
reproducing this effect analytically which then lead them into a
mathematical expression for the EC-force .
force is strong at the point where the discontinuity
occurs but is present and relatively constant at all distances smaller
than the critical distance (like the wheel going down a constant slope
once it enters the hole). The dependence of the energy gap as well as
the free energy on
the distance between the quantum wells is shown in Fig.4 below. The
agreement between the numerical calculations and the analytic model is
remarkable. The EC-force is proportional to the slope of the green
curve in the inset of Fig.4. The calculations show that for a typical
concentration of 1011 cm-2 the strength of this
force is tiny (but not tinier than that cannot be measured!),
i.e. approximately 10-9 Newtons.
The main figure depicts the energy gap as the electron-hole well
distance D is varied (normalized with respect to the critical distance Dc).
The inset is the same for the free energy. The red dots indicate our
earlier numerical calculations in Ref. whereas the green dots depict
the analytical calculations in Ref. (This
figure is taken from Ref. The reference and equation numbers therein are inapplicable here.)
The recently obtained clear evidence for the existence of exciton
condensate is also a promise for the experimental observation of this
new effect. Although EC-force is
reminiscent of the electromagnetic Casimir force , these two forces
are also significantly different. Standard Casimir force is caused by
existence of electromagnetic boundaries and fluctuations of the
field in the vacuum. EC-force is fully driven by the excitonic
condensate and appears due to the spatial dependence of the free energy
via the Coulomb interaction.
(a) The depiction of the proposed experimental
setup. (b) Zoomed view of the two DQWs.
(This figure is taken from Ref). Typical dimensions of the cantilever are Lx=10 µm , Ly=100 µm
experimental proposal to measure the new effect:
Fig.6. (right) Appearance of the
EC-force as a function of time follows the exciton creation and annihilation pattern indicated by green
lines. Ignoring the typically 1 µs
lifetime of the excitons, the generated EC-force follows the
pulse profile stimulated by the driving laser. The frequency of the
pulse also coincides with the mechanical resonance frequency of the cantilever (nearly 10-20 kHz range).
may be various other proposal deviced by experimental groups to measure
the EC-force in the future. Here one can discuss a basic experimental
proposal to measure this effect. Due to the existing strong Coulomb force as well as
the dielectric between the quantum wells, the direct measurement is
much more challenging than measuring the electromagnetic Casimir Force between the
two metallic plates separated by vacuum. Recently, Yamaguchi, Okamoto,
Ishihara, and Hirayama  have detected the motion of a
micromechanical oscillator with an amplitude on the order of 50nm.
Here, upon , the authors use the EC-force as the driving force of the
micromechanical oscillator as shown in Fig.6.
The growth process and the production of the sample in Fig.(4) is
paralel to . The idea is to grow two GaAs DQW structures depicted
by the green and the red lines respectively in Fig.(5) both below and
above the neutral plane of the Alx
micromechanical cantilever. The DQWs are then populated by electrons
and holes using a resonant pump laser in the presence of an external
electric field and at a temperature much less than the critical
temperature to condense the excitons. The upper DQW is in the uncondensed regime incidated by (A) in Fig.3. The lower
DQW is in the condensed regime indicated by (C) in Fig.(3), therefore the EC forms within
a few nanoseconds after thermalization of the electrons and holes with
the lattice in the lower DQW. According to their prediction, the EC-force should then
be simultaneously generated with the appearance of the condensate in
the lower DQW. The formation of the EC-force in the lower DQW follows
the same pattern as the creation and the annihilation of the
excitons as indicated in Fig.6. The internal stress created by the
EC-force becomes the driver of the mechanical oscillations at the
mechanical resonance frequency of the cantilever. The oscillations
of the resonating cantilever and the deflection angle depicted in Fig.5
can be measured by a second laser (the deflection detection-DD laser in
Fig.5). Interested reader can find more details of the experimental
proposal as well as a discussion of
the secondary effects in Ref..
a force due to an Exciton-Condensate mean?: The
EC-force is a nonlocal force like the Casimir force
between two metallic plates held in vacuum . However, this is the
only similarity between these two effects. A measurable force emerging
due to the formation of a condensate is an unique effect of its
own right and the experiments to measure this phenomenon are within reach and
possible which may find other analogies in low temperature condensed
matter physics. Bose-Einstein atomic condensates are difficult to
measure this force due to the strongly local interactions between
the atoms. However, one possible candidate is the
topological exciton condensate in which the Dirac-cone states
pinned at the top and the bottom surfaces of a topological insulator
filled with electrons and the holes are coupled through the Coulomb
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