A theory for nonlocal electromagnetic response of nanoscale clusters

1
A theory for nonlocal electromagnetic response
of nanoscale clusters
Ilya Grigorenko CNLS
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Background

• 1908 Mie Theory models scattering of
  electromagnetic radiation by spherical particles
  within continuum field theory (Maxwells
  equations). Phenomenological description of
  dielectric permittivity function
• 1950-60s Bohm, Pines Lindhard, Ashcroft, Rice,
  Cohen, Nozieres, Stern, - calculation of the
  dielectric function within the random phase
  approximation (RPA) accounts for light-metal
  interaction in bulk materials, dominated by
  plasmon collective excitations
• 1974 Fleishmann with co-workers Surface
  Enhanced Raman Scattering (SERS) dramatic
  amplification of electric field scattered off
  rough/nano structured surfaces Kneipp et al.- up
  to 1012
• 1990s SERS uses colloidal metal particles to
  detect single molecules Kneipp et al. Nie and
  Emory
• 2005 Nanofabrication technologies control of
  metal cluster shape and size. The challenge is
  to use increased computer power for realistic
  atomistic material modeling and design
  optimization on a quantum mechanical level

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Classical theory linear dielectric response

• Definitions
• displacement vector dielectric
  permittivity
• Clausius-Mossotti relation
• for a sphere

• Induced field due to external field defines
• the dielectric response
• Dielectric response function
• 
  non-local spatial effects
  memory effects
• Time invariance (no memory effects) allows
• us to write
• and after Fourier transform system response
• is local in frequency domain
• Induced charge follows the external field

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Quantum description electron wavefunctions and
matrix elements

• Wave functions (non-interacting electrons)
• Filling factor
• Perturbation with external
• electric field
• Matrix elements coupling different states,
• using 1st order Born approximation
• Oscillations of electric field in
• long wavelength limit
• Screening is the damping of electric fields
• caused by the presence of mobile charge
• carriers

DE gt g
DE
E
f
single-particle transitions
state number, i
collective excitations
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Linear response theory and the fluctuation-dissipa
tion theorem (FDT)
Evolution of a quantum system

• The FDT (Kubo formula) relates
• the expectation value of
• in a non-equilibrium state
• to an equilibrium expectation
• value of more complicated time-
• dependent expression
• (like noise power is related to resistance)
• Beyond Kubo formula-nonlinear
• response, strong fields

Perturbation
time

• Equilibrium state

• Non-equilibrium state

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Lindhard formula and RPA response of a bulk
metal to electric field

• Bulk linear response can be understood using RPA
  (Lindhard, 1954)
• - all physical information is contained in e(q,
  w)
• Electric field E is related to displacement
  vector D via
• Perturbative treatment
• Quantum mechanics is contained in
• matrix elements ieiq.rj
• Fermi-Dirac distribution of electrons f(Ei)
  (equilibrium only)

Quantum Liouville equation
Poisson equation

bare potential
screening
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Bulk response plasmon, dominant optical response
of homogeneous and isotropic medium
Real, Imaginary parts of the Lindhard dielectric
function, and the loss function

• Local response
• Collisionless
• Simple (phenomenological) level broadening
• Loss function
• For (visible light), Lindhard
  formula gives plasmon resonance

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Induced charge distribution and the corresponding
electric field for micro- and macro-systems
Classical
Quantum
Large systems (classical) localized charge
distribution
Small systems (quantum mechanical) delocalized
charge distribution
_
_
x

Induced electron density
Induced electron density
x

partial screening of the external field
full screening of the external field
Total field intensity
Total field intensity
x
x

• Quantum effects become significant for single
  particle transitions if DE gt G,
• For collective excitations require finite number
  of particles N gt Ncrit

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Response of classical system vs. quantum
Classical
Quantum
classical density
quantum mechanical density
continues excitation spectrum
discrete excitation spectrum
Limited response (lower induced field) due to
discrete spectrum
Strong response (induced field) near the small
radius
E-is reduced and depends on the filling factor
E1/Ra
quantum levels (discrete)
all points are approximately equally accessible
points are NOT equally accessible due to the
Heisenberg uncertainty principle
With lower filling one can easier observe quantum
effects some narrow areas can not be accessible
by low-energy electrons!
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Nonlocal linear response of finite size
clusters(quantum effects might be very
important-but from which size?)

• Quantum mechanical single particle response when
  DE gt g
• Solids are made of Ntot atoms, inter-atomic
  spacing L0.3 nm
• 10x10x10 nm3 is approximately 104 atoms
• Ions treated as positive jellium background
  charge density n(r), smeared out atomic
  potential, V(r)
• Electrons are treated quantum mechanically by
  solving Schrodinger equation for unperturbed
  system of noninteracting electrons
• ,
• Non-local linear response

• r'

r'
r'
q'
FFT
r
q
r
smooth jellium
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Quantum theory linear response theory for
strongly inhomogeneous media

• Linear response theory (the induced charge
  density is proportional to the total potential)
• The induced potential satisfies Poisson equation
• Calculation of the induced charge density (within
  the linear response approximation)
• The non-local density-density electron response
  function using RPA
• Self-consistent integral equation for the induced
  potential

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Quantum theory characterization of the
collective response calculating energy of the
induced field

• Dielectric function or
  is a tensor, but it is
  better to have a scalar value to characterize
  interaction
• The total energy of the electric field in the
  nanostructure (we neglect magnetic field effects)
• One can use as a measure of the matter-field
  interaction

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Dielectric response of elliptic cylinders
b

• Elliptic cylinder with semi-axis a and b
• and periodic boundary conditions in z direction
• can be described by x2/a2 y2/b2R2
• External electric field E with x and y-components
• and field carrier frequency w
• Classical theory predicts two plasmon frequencies
  
• at w- , w corresponding to collective electron
• oscillations along x and y axis

a
Ey
External field
Ex
induced charge density creates induced field
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Size-effects for linear dielectric response of
elliptic cylinders

• We have calculated the energy of induced electric
  field Wind as a function of field carrier
  frequency w and the system size R
• Two different aspect ratios a/b were considered
• - ab11.3, and ab12
• For R gt 10L response of the system shows two-peak
  response in agreement with classical theory
• For R lt 6L quantum finite size effects control
  dielectric response
• Number of electrons Ncrit80 is enough for
  collective phenomena
• The collective mode phase velocity is larger than
  the Fermi velocity of the electrons
• Frequencies are lower for smaller sizes L due to
  anharmonicity of the jellium potential

(a)
11.3
(b)
12
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Calculated dielectric response of a cylinder with
size parameter R11 L

• For R11L we plot the energy of induced electric
  field Wind as function of the field frequency w
• In the inset we plot the corresponding induced
  charge density at peak frequencies w- , w
• It is clearly seen that these two resonances
  correspond to collective motion of electrons
  along axis of symmetry of the ellipse

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Calculated dielectric response of a cylinder with
size parameter R3 L
cylinders contour
classical turning point

• Energy of induced electric field Wind as
  function of the field frequency for R 3L
• In the inset we plot the corresponding induced
  charge density at two peak frequencies w-, w
• Opposite to the case R 11L, the induced
  electron density for both frequencies cases is
  along the applied field
• The higher frequency transition (w) corresponds
  to one-particle excitation between two
  distinctive quantum levels

(b)
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Single particle- collective mode transition with
increasing of the systems size in 1D

• In 1D scaling is different, than in 2D or 3D!
• Lcritr -1/6 in 3D
• Lcrit r 1/2 in 1D
• Because of the different scaling of the Fermi
  velocity in 1D and 3D

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Energy of the induced field (the total structure
and the tip location) External field frequency
scan
207 electrons
Field energy in the whole nanostructure
Field energy in the tip area
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Three snapshots of the induced field intesity
w10-5 E0
w0.135 E0
w0.275 E0
External field

• Note the difference in the field enhancement near
  
• the tip at different external field frequencies-
  needs optimization
• at a given frequency

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Simulation of the frequency scan of the tip

• One can model plasmon-enhanced local field effect
  around the tip and optimization of the tunneling
  structure

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Charge and field intensity for different
distances between two conducting spheres
Induced charge density
External field
D0
D8L
D10L
D9L
Induced field intensty

• Low electron density, vertical applied field,

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Energy of the induced electric field as a
function of relative distance and field frequency

• Relatively low electron density

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Energy of the induced electric field as a
function of relative distance and field frequency

• Relative high electron density

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Time-dependent optimal control problem on a
surface
Response of the system
Potential profile
We place a molecule here!
Induced field intensity
External and total field amplitude
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Outlook Why is this so exciting?

• Current state of the art
• The properties of bulk (homogeneous, isotropic)
  materials have been studied for many years
• Great deal is known about the properties of such
  materials i.e. GaAs
• Many specialized computational techniques have
  been developed fore homogeneous and isotropic
  materials. Example Fourier transformation that
  uses translational symmetry to reduce
  computational complexity
• such techniques exclude efficient exploration of
  broken symmetry configurations

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Whats exciting!

• Functionality of homogeneous medium is limited by
  its symmetry. Example Bloch waves-band gaps,
  dispersion relation, etc.
• Broken spatial symmetry can create new
  functionality
• direct manipulation of wavefunctions
• control of coherent excitations
• new inhomogeneous materials
• How do we find the desired functionality?
• the inverse problem cannot solve this - we
  are too dumb!
• use optimization techniques
• the poor mans inverse solver

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Key directions for quantum device synthesis

• Quantum mechanics has a hierarchy of degrees of
  freedom
• hierarchy of single particle excitations
• collective excitations
• Control imposed using spatial configurations
• Finite size effects
• Robust design (local robustness)
• Increased tolerance to the quality of particular
  implementation
• Exponential sensitivity of the model
• Creation of new, super-sensitive sensors
  (opposite the to robust design)
• Global optimization
• Creation of new efficient hybrid (globallocal)
  algorithms

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• Thank you!