# Presentation: 449847

## A theory for nonlocal electromagnetic  response of nanoscale clusters

- Ilya Grigorenko
- CNLS

## Background

_**1908**_**: ****Mie Theory**** – models scattering of electromagnetic radiation by spherical particles within continuum field theory (Maxwell’s 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 10****12**
_**1990’s:**_**  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**

## 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**

## Quantum description: electron wavefunctions and matrix elements

**single-particle transitions**

**collective excitations**

**Wave functions (non-interacting electrons):**
**Filling factor**
**Perturbation with external **
**electric field:**
**Matrix elements coupling different states, **
**using 1****st**** 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**

- f

******E > ******

******E**

**E**

**state number****, i**

## Linear response theory and the fluctuation-dissipation theorem (FDT)

**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**
** **

**Linear response theory and the fluctuation-dissipation theorem (FDT)**

**Perturbation**

**time**

**Equilibrium state**

**Non-equilibrium state**

**Evolution of a quantum system**

## Bulk linear response can be understood using RPA (Lindhard, 1954)
- all physical information is contained in (q, ) 
Electric field E is related to displacement vector D via

Perturbative treatment:

Quantum mechanics is contained in: 
matrix elements ‹i|eiq.r|j› 
Fermi-Dirac distribution of electrons f(Ei) (equilibrium only)

**- all physical information is contained in**** ********(q, ********) **
**Electric field E is related to displacement vector D via**
**Perturbative treatment**:
**Quantum mechanics**** is contained in:**** **
**matrix elements ****‹*****i*****|e*****i*****q.r****|*****j*****› **
**Fermi-Dirac distribution of electrons *****f*****(*****E******i*****) (equilibrium only)**

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

**Poisson equation**

**bare potential**

**screening**

- +...

**Quantum Liouville equation**

**Notes:**
J. Lindhard, Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 28, No. 8 (1954). 

## Bulk response: plasmon, dominant optical response of homogeneous and isotropic medium

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

**Real, Imaginary parts of the Lindhard dielectric **
**function, and the loss function**

## Induced charge distribution and the corresponding electric field for micro- and macro-systems

**Induced charge distribution and the corresponding electric field for micro- and macro-systems**

**+**

**_**

**+**

**Large**** systems (classical) **
**localized**** charge distribution**

**Induced **
**electron**
**density**

**x**

**x**

**x**

**x**

**Total field**
**intensity**

**Total field**
**intensity**

**Small**** systems (quantum mechanical) **
**delocalized**** charge distribution**

**_**

**full**** screening of the external field**

**partial**** screening of the external field**

**Induced **
**electron**
**density**

**Quantum effects become significant for single particle transitions if ********E > ********,**
**For collective excitations require finite number of particles ****N > N****crit**

**Classical**

**Quantum**

**Notes:**
Real calculations

## Response of classical system vs. quantum

_***Strong***_** response (induced field) near **
**the small radius **

**With **_***lower filling***_** one can easier observe **_***quantum effects***_**:**
**some narrow areas  can not be accessible by **_***low-energy***_**  electrons!**

**continues excitation  spectrum**

**classical density**

**Classical**

**E~1/R****a**

**all points are approximately equally accessible**

**discrete excitation spectrum**

_***Limited***_** response (lower induced field) due to **
**discrete spectrum**

**quantum mechanical density**

**Quantum**

**quantum levels **
**(discrete)**

**E-is reduced and depends **
**on the filling factor**

**points are NOT equally accessible due to the Heisenberg uncertainty principle**

## Nonlocal linear response of finite size clusters(quantum effects might be very important-but from which size?)

**r**′

**r**

**r**′

**Quantum mechanical single particle response when ********E > ******
**Solids are made of N****tot**** atoms, inter-atomic spacing L~0.3 nm**
**10x10x10 nm****3  ****is approximately 10****4**** 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:**

**smooth jellium**

**r**′

**r**

**q**

**q**′

**FFT**

## Quantum theory: linear response theory for strongly inhomogeneous media

**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**

**Quantum theory: linear response theory for strongly inhomogeneous media**

## 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 **

## Dielectric response of elliptic cylinders

**Dielectric response of elliptic cylinders**

**a**

**b**

**E****x**

**E****y**

**Elliptic cylinder with semi-axis ****a**** and ****b**
**and periodic boundary conditions in z direction**
**can be described by: ****x****2****/a****2 ****+y****2****/b****2****=****R****2**
**External electric field E with x and y-components**
**and field carrier frequency ******
**Classical theory predicts two plasmon frequencies  **
**at ******************** corresponding  to collective electron **
**oscillations along x and y axis **

**induced charge density**
**creates induced field**

**External field**

## Size-effects for linear dielectric response of elliptic cylinders

- (a)

- 1:1.3

- (b)

- 1:2

**Size-effects for linear dielectric response of elliptic cylinders**

**We have calculated the energy of induced electric field W****ind**** as a function of field carrier frequency ******** and the system size R**
**Two different aspect ratios a/b were considered:**
**- ****a:b=1:1.3****, and ****a:b=1:2****	**
**For R > 10L response of the system shows two-peak response in agreement with classical theory **
**For R < 6L quantum finite size effects control dielectric response**
**Number of electrons ****N****crit****~80**** 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**

**Notes:**
D. Pines, P. Nozieres, W. A. Benjamin, Inc., The Theory of Quantum
Liquids, {\bf 1}, 55, (1966).

## Calculated dielectric response of a cylinder with size parameter R=11 L

**Calculated dielectric response of a cylinder with size parameter R=11 L**

**For R=11L we plot the energy of induced electric field W****ind**** as function of the field frequency ******** **
**In the inset we plot the corresponding induced charge density at peak  frequencies ******************** **
**It is clearly seen that these two resonances correspond to collective motion of electrons along  axis of symmetry of the ellipse**

## Calculated dielectric response of a cylinder with size parameter R=3 L

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

- (b)

**classical turning point**

**cylinder’s contour**

## Single particle- collective mode transition with increasing of the system‘s size in 1D 

**In 1D scaling is different, than in 2D or 3D!**
**L****crit****~******** ****-1/6**** in 3D**
**L****crit****~ ******** ****1/2**** in 1D**
**Because of the different scaling of the Fermi velocity in 1D and 3D **

## 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

## Three snapshots of the induced field intesity

**External field**

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

******=0.135 E****0**

******=10****-5**** E****0**

******=0.275 E****0**

## 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**

## Charge and field intensity for different distances between two conducting spheres 

**Low electron density, vertical applied field, **

- D=0

- D=8L

- D=9L

- D=10L

- Induced charge density

- Induced field intensty

**External field**

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

**Relatively low electron density**

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

**Relative high electron density**

## Time-dependent optimal control problem on a surface

- Induced field intensity

- Potential profile

- Response of the system

- External and total field amplitude

- We place
- a molecule
- here!

## 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

## What’s 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 man’s inverse solver

## 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 (global+local) algorithms

## 

Thank you!

**Thank you!**