Ferromagnetism vs. Antiferromagnetism in Double Quantum Dots: Results from the Bethe Ansatz - PowerPoint PPT Presentation

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Ferromagnetism vs. Antiferromagnetism in Double Quantum Dots: Results from the Bethe Ansatz

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Title: Ferromagnetism vs. Antiferromagnetism in Double Quantum Dots: Results from the Bethe Ansatz


1
Ferromagnetism vs. Antiferromagnetism in Double
Quantum Dots Results from the Bethe Ansatz
  • Robert Konik,
  • Brookhaven National Laboratory
  • Hubert Saleur,
  • Saclay/University of Southern California
  • Andreas Ludwig,
  • University of California, Santa Barbara
  • Galileo Galilei Institute for Theoretical Physics
  • Arcetri, Firenze
  • October 1, 2008

2
Overview
Themes
1. Show classic Tsvelik-Wiegmann exact
solution of single level Anderson model can be
extended to more generalized dot systems
2 Show that from these exact solutions,
interesting (Kondo) physics arise that is
non-perturbative in nature
3. Discuss prospects of extending exact
solvability to non-equilibrium problems
3
Outline
  • Brief introduction to quantum dots
  • Kondo physics
  • Bethe ansatz solution of double quantum dots
  • Exact linear response conductance at T0
  • Friedel sum rule
  • - RKKY mediated Kondo effect
  • Beyond T0 linear response
  • - Finite temperature
  • - Non-equilibrium

4
Quantum Dots What are they?
  • Small groups (N102-103) of electrons confined to
  • a small enough region such that
  • - Electronic levels are discrete
  • - Dot Coulombic effects are important
  • Dots interact with large electron reservoirs or
    leads
  • Dot-lead coupling is origin of all the
  • physics I will discuss

5
Semiconductor Dots
quantum dot
Built from gated heterostructures
6
Fabrication of Semiconductor Quantum Dots
quantum dot
Gates on top of GaAs/AlGaAs heterostructure
segregate a portion of the two
dimensional electron gas
two dimensional electron gas (2DEG)
Gates can arranged so that a double dot structure
is obtained (here, in parallel)
in parallel Chang et al. PRL 92, 176801
(2004) in series Petta et al. PRL 93, 186802
(2004)
7
Double Nanotube Quantum Dots
Gates allow chemical potentials of dots to be
tuned independently
N. Mason, M. Biercuk, C. Marcus, Science 303
(2004) 655
8
Regime of Interest Dots with Single Active Level
At low temperatures and/or large level
spacing, only level nearest Fermi energy is
relevant
EF
dE
lead R
lead L
V
V
dot
lead R
lead L
9
Energy Scales of a Dot with a Single Level
10
How Does Kondo Physics Arise in a Single Level
Dot
At symmetric point of dot (U 2ed 0), one
electron sits on the dot level.
U/2
EF
EF
U/2
Isolated electron acts like magnetic
impurity Kondo physics
11
What is Kondo Physics?
Two regimes separated by TK, the Kondo
temperature
T ltlt TK low temperature behaviour
lead L
lead R
Dot electron forms a singlet with electrons in
leads
T gtgt TK high temperature behaviour
Dot electron interacts only weakly with electrons
in leads
12
Signature of Kondo Physics, TltltTK
impurity density of states, ?(?)
Dynamically generated spectral weight at the
Fermi level (Abrikosov-Suhl resonance)
?
-TK
TK
13
Kondo Physics in Double Quantum Dots
Possibilities
Suppose we two electrons on the two dots and
effective channel of electrons coupling to the
dots
Ferromagnetic RKKY interaction binds electrons
into triplet underscreened spin-1 Kondo effect
(standard picture)
i)
S1
Direct singlet formation between dot electrons
no Kondo effect
ii)
Singlet formation as mediated by electrons in
leads RKKY Kondo effect
iii)
14
Double Quantum Dots in Parallel
Two dots are not capacitively or tunnel coupled
15
Anderson-like Hamiltonian of Double Dots
lead electrons
dot-lead tunneling
chemical potential of dots
Coulomb repulsion
  • Model is exactly solvable via Bethe ansatz
  • -RMK (PRL 99, 076602 (2007), New J. Phys. 9, 257
    (2007))
  • Must explicitly check, however, that the
    multi-particle wavefunctions are of Bethe ansatz
    form

16
Constraints on Parameter Space
  • i) Model is integrable if only one channel
    couples
  • for consistency need VL1/VR1 VL2/VR2
  • ii) Energy of double occupancy is the same on
    the
  • two dots
  • U1 2ed1 U2 2ed2
  • ed1 and ed2, cannot be adjusted independently
  • iii) U1(V1L2V1R2) U2 (V2L2V2R2)
  • Small perturbations on these constraints do not
    affect the physics

17
Landauer-Büttiker Transport Theory
RMK, H. Saleur, A. Ludwig PRL 87 (2001)
236801 PRB 66 (2002) 125304
- With µLgtµR charge flows from left to right
  • L-B requires knowledge of transmission
    probabilities, T(e),
  • for electrons with energies, e, in the range
    µLgtegtµR current J is then
  • Integrability allows one to both determine the
    correct scattering
  • states to use as well as their exact
    scattering matrices (i.e. T(e))

18
Computation of Scattering Amplitudes
Exact solvability means one can construct exact
many-body eigenfunctions
The momenta are quantized according to the Bethe
ansatz equations
gives scattering amplitude N. Andrei, Phys. Lett.
87A, 299 (1982)
These quantization conditions can be recast as
T(p) sin2((dimp(p))
19
T0 Linear Response Conductance in Double Quantum
Dots
one electron on dots forming a singlet with
conduction electrons standard Kondo effect
L
R
Linear response conductance
Rapid variation due to interference between dots
decreasing dots chemical potential with
ed1-ed2 kept fixed
two electrons on dots forming singlet system is
p-h symmetric
RKKY Kondo effect
20
Evidence for Kondo Physics at the Particle-Hole
Symmetric Point
Impurity Density of States
Spin-Spin Correlation Function
Slave boson mft RMK M. Kulkarni
Ferromagnetic like correlations exist between
dots although over all ground state is a singlet
An Abrikosov-Suhl resonance exists
¼ for ferro
-¾ for antiferro
21
The Friedel Sum Rule in Double Dots
Friedel sum rule states
Scattering phase of electrons at Fermi energy is
proportional to the number of electrons
displaced by impurity, Ndisplaced d
pNdisplaced G 2e2/h sin2(d)
Ndisplaced has two contributions
  • Electrons on dots
  • deviations in electron
  • density in leads

Langreth, Phys. Rev. 150, 516 (1966)
22
Beyond Linear Response at T0 Finite Temperature
and Out-of-Equilibrium
Challenge Compute transmission probability,
T(e), at finite energies
Two difficulties in doing so
2) Does this construction of the electron behave
well under map from original (two leads 1,2) to
integrable basis (even, odd) Yes, maybe.
1) Non-unique construction of electron
charge excitations
one parameter family used reproduces Kondo phys
ics
spin excitations
23
Finite Temperature Linear Response Conductance in
Single Dots Universal Scaling Curve
no parameter fit
Kaminski et al.
numerical renormalization group - Costi et al.
24
Finite Temperature Linear Response Conductance in
Double Quantum Dots
Dot levels well separated
Dot levels close together
25
Out-of-Equilibrium Conductance in a Single Dot
In zero magnetic field, ansatz fails
  • At large voltages, the computed
  • conductance does not behave as
  • G 1/log2(V/TK)
  • At small voltages, produces incorrect
  • coefficient, a, of second order term in voltage

Likely culprit Wrong choice of one parameter
family of scattering states Why? Ansatz does work
at qualitative level in large magnetic fields
where ambiguity of choice is lifted
26
Out-of-Equilibrium Magnetoconductance
Observations
Bethe Ansatz Computations
T0 peak value occurs at ?µltH agrees with
experiments on carbon nanotube dots
Kogan et al. Phys. Rev. Lett. 93, 166602 (2004)
(µR-µL)/H
We can make similar computations for the noise at
large fields and bias (Gogolin, RMK, Ludwig, and
Saleur, Ann. Phys. (Leipzig) 16, 678 (2007))
27
Other Strategies for Out-of-Equilibrium
Transport Open Bethe Ansatz
Our approach Equilibrium scattering data
out of equilibrium results
Open Bethe ansatz attempt to construct
wavefunctions of Bethe type
with non-equilibrium boundary conditions
Mehta and Andrei, PRL 96, 216802 (2006)
Wavefunctions encode the different number of
electrons in the two leads
28
Nature of Wavefunctions in Open Bethe Ansatz
In the interacting resonant level model, to
construct such wavefunctions, single particle
wavefunctions with removable singularities must
be used
?(x)2
?(0)2
?(0)2 ? ?(0)2? ?(0-)2
?(x)2
Probabilitistic interpretation of
wavefunction? Can physics be changed by changing
wavefunction on a set of measure zero?
29
Conclusions
Exactly solvable variants of the Anderson
model applicable to multi-dot systems exist -
double dots in series (RMK, PRL 99, 076602
(2007)) - dots exhibiting Fano resonances
(RMK, J. Stat. Mech L11001 (2004)) These
solutions can exhibit unexpected physics -
singlet formation in a double dot where a
triplet might be expected Out-of-equilibrium no
generic approach employing exact solvability
seems to be available
30
Finite Temperature Conductance Through a Single
Dot
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