ElectrostaticsElectrostatics

Wei E.I. Sha (沙威)

College of Information Science & Electronic EngineeringZhejiang University, Hangzhou 310027, P. R. China

Email: weisha@zju.edu.cnWebsite: http://www.isee.zju.edu.cn/weisha

Wei SHASlide 2/17

1. Charge and Electrostatic Field2. Electric Displacement and Polarization Density3. Potential and Gauss Law4. Poisson’s Equation 5. Debye (Screening) Length6. Electrostatic Energy and Capacitance7. Boundary Conditions

Course Overview

Wei SHASlide 3/17

1 224

QQrπε

= RF a

Electric field E

Two point charges repel each other with a force (experimental results)

force on a unit charge

24Qqrπε

= RF a24

Qq rπε

= = RFE a

ε permittivity

1. Charge and Electrostatic Field (1)

Ref: Xie, Section 2.1 & 2.2

Wei SHASlide 4/17

1. Charge and Electrostatic Field (2)

r′ir positon of Qi (source point)

observation point

| |i

i

′−=′−

iR

r rar r

A unit normal vector pointed from source point to

observation point

21

1( )4 | - |

N i

i

Qπε =

= ′

iR

iE r a

r r

( )i iQρ δ ′→ − ir r

charge density for point charge

( )

2'

3'

1 ( )( ) '4 -

1 ( ) - '4 -

v

v

dv

dv

ρπε

ρπε

′=

′′ ′= ′

RrE r a

r rr r r

r r

Wei SHASlide 5/17

(a) Line ChargesLine charge density, ρL (C/m)

(b) Surface ChargesSurface charge density, ρS (C/m2)

(c) Volume ChargesVolume charge density, ρV (C/m3)

Pdl

dS

dv

1. Charge and Electrostatic Field (3)

Wei SHASlide 6/17

1. Charge and Electrostatic Field (4)

Electrons, Holes, Excitons (bound e-h pair),

Positive and Negative Ions ….

What are charges in optoelectronic devices?

Wei SHASlide 7/17

the density of the permanent and induced electric dipole moments in the material, called the polarization density

vacuum permittivity (permittivity of free space)

In a dielectric material, the presence of an electrostatic field E causes the bound charges in the material (atomic nuclei and their electrons) to slightly separate, inducing a local electric dipole moment.

In a linear, homogeneous, isotropic dielectric

0 0 0ε ε ε ε χ= = + = +D E E P E E0ε

P

relative permittivity (dielectric constant)1rε χ= +

χ susceptibility

2. Electric Displacement and Polarization Density

Ref: Xie, Section 2.4

Wei SHASlide 8/17

3. Potential and Gauss Law (1)

The work done to move a point charge from one point to another is independent of the path taken!

( ) 0S S

d d dΓ

⋅ = ∇× ⋅ = − ∇×∇Φ ⋅ = E l E S S

0W d q dΓ Γ

= ⋅ = ⋅ = F l E l

Electrostatic Field is conservative. Is there other field is conservative?

Φ is a scalar quantity called electrostatic potential= −∇ ΦE

1 ( )( )4 -v

dvρπε ′

′ ′Φ = ′

rrr r2

'

1 ( )( ) '4 -v

dvρπε

′=

′R

rE r ar r

Ref: Xie, Section 3.1

Wei SHASlide 9/17

3. Potential and Gauss Law (2)

Gauss Law

S

d Q⋅ =D S ρ∇ ⋅ =D

: area element normal to an arbitrary surface

: a unit normal vector to the spherical surface

24 2 2 24 4

(cos ) ( sin )dQ Q Q

r r rd dS r d dπε πε πε

α θ θ ϕ⋅⋅ = = =a SRE S

cosα dS is an elemental area on a spherical surface of radius r

𝑑𝐒𝐚𝐑

Wei SHASlide 10/17

= −∇ΦE

( ( ) )ε ρ∇ ⋅ ∇Φ = −r

Poisson equation: governing equation for electrostatic problem

4. Poisson’s Equation (1)

( )ε=D r E ρ∇ ⋅ =D free charge

Could we put permittivity out of divergence operator?

Optoelectronic device is inhomogeneous!

ETL: electron transport layerHTL: hole transport layerBHJ: bulk heterojunction

Red line: boundary conditions imposed

Wei SHASlide 11/17

4. Poisson’s Equation (2)

input

output

input

output

𝜌𝛷

0

1( ) ( )4 | |v

dvρπε′

′ ′Φ =′−r r

r r

solution of Poisson’s equation in free space

Ref: Chew, Section 3.3.2 and 3.3.3

Wei SHASlide 12/17

( )( )0 , ( )gε δ′ ′∇ ⋅ ∇ = − −r r r r

00

1 ( )4 | |v v

dv dvε δπε′ ′

′ ′ ′∇ ⋅ ∇ = − − ′− r r

r r

1 1 14S

dRπ′

′∇ ⋅ = − S

21 1

4S

dRπ′

− ′⋅ = − Ra S

21 1

4 S

dRπ ′

− ′⋅ = − Ra S

4. Poisson’s Equation (3) —— Green’s function (Optional)

𝑔 𝐫, 𝐫 = 14𝜋𝜀 |𝐫 − 𝐫 |

Wei SHASlide 13/17

2. The electrostatic energy stored in a system of three and N point charges

1 3 2 31 2

0 12 13 23

14E

QQ Q QQQUR R Rπε

= + +

( )A BW Q d Q d QΓ Γ

= ⋅ = − ∇Φ ⋅ = Φ − Φ E l l

The work done is equal to the change in electrostatic energy(move a positive charge from one point to the other)

( )

1 1 0

1 12 4

N j iNj

E ii j ij

QU Q

Rπε

≠

= =

=

1. The electrostatic energy of a positive charge Q at position r in the presence of a potential Φ

( )EU Q= Φ r

5. Electrostatic Energy and Capacitance (1)

Wei SHASlide 14/17

5. Electrostatic Energy and Capacitance (2)

The total electrostatic energy stored in a capacitor is

21 12 2EU QV CV= =

Capacitance

Energy density (energy per unit volume) of the electrostatic field

1 1 1 ( )2 2 2

1 12 2

Ev v v

v v

U dv dv dv

dv dv

ρ

ε

= Φ = ∇ ⋅ Φ = ⋅ −∇Φ

= ⋅ = ⋅

D D

D E E E

Wei SHASlide 15/17

6. Debye (Screening) Length

Debye length (Thomas–Fermi screening length) is a measure of a charge carrier’s net electrostatic effect and how far its electrostatic effect persists.

The Debye length of semiconductors is given

ε is the dielectric constantkB is the Boltzmann’s constantT is the kelvins temperatureNdop is the net density of dopants

Wei SHASlide 16/17

7. Boundary Conditions (1)

Tangential component of E field is continuous across dielectric boundary

Normal component of D field is continuous across dielectric boundary

1 1 2 2

1 2

sin sin

t t

E EE Eθ θ

=

=

0dΓ

⋅ = E l

0S

d⋅ = D S1 1 2 2

1 2

cos cos

n n

D DD D

θ θ==

Ref: Xie, Section 3.1.1 and 3.4

Wei SHASlide 17/17

7. Boundary Conditions (2)

Dirichlet boundary condition (Red line)

( ) | 0( ) |

e

r

r r V→∞

=

Φ =Φ =

rr

Neumann boundary condition (Blue line)

( )r S s

rn

ε σ∈∂Φ =

∂

surface charge

unbounded problem (reference potential)

electrodes (optoelectronics)

( ) 0r Srn ∈

∂Φ =∂

non-electrodes (floating BC, optoelectronics)