Electrostatics Wei E.I. Sha (沙威) College of Information Science & Electronic Engineering Zhejiang University, Hangzhou 310027, P. R. China Email: [email protected] Website: http://www.isee.zju.edu.cn/weisha Course Overview 1. Charge and Electrostatic Field 2. Electric Displacement and Polarization Density 3. Potential and Gauss Law 4. Poisson’s Equation 5. Debye (Screening) Length 6. Electrostatic Energy and Capacitance 7. Boundary Conditions Slide 2/17 Wei SHA 1. Charge and Electrostatic Field (1) Two point charges repel each other with a force (experimental results) = QQ12 FaR 4πε r2 ε permittivity Electric field E force on a unit charge Qq = ==F Q FaR EaR 4πε r2 qr4πε 2 Slide 3/17 Ref: Xie, Section 2.1 & 2.2 Wei SHA 1. Charge and Electrostatic Field (2) r observation point ′ ri positon of Qi (source point) ′ A unit normal vector i rr− a = i pointed from source point to R − ′ ||rri observation point N 1 Q i Er()= i a 1()ρ r′ πε 2 R Er()= a dv ' 4 i=1|-rr′ | πε 2 R i 4 v' rr- ′ 1()ρ r′ ρ →−δ ′ = ()rr-'′ dv iiQ ()rri πε 3 4 v' rr- ′ charge density for point charge Slide 4/17 Wei SHA 1. Charge and Electrostatic Field (3) (a) Line Charges dl Line charge P density, ρ (C/m) L dS (b) Surface Charges Surface charge 2 density, ρS (C/m ) (c) Vo lu me Ch arg es Vo lu me ch arg e dv 3 density, ρV (C/m ) Slide 5/17 Wei SHA 1. Charge and Electrostatic Field (4) What are the charges in optoelectronic devices? Electrons, Holes, Excitons (bound e-h pair), Positive and Negative Ions …. Interplay of charges is essential to optoelectronics. Slide 6/17 Wei SHA 2. Electric Displacement and Polarization Density 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 ==εε +=+ ε εχ DE000 EPE E ε 0 vacuum permittivity (permittivity of free space) the density of the permanent and induced P electric dipole moments in the material, called the polarization density χ susceptibility ε =+χ r 1 relative permittivity (dielectric constant) Slide 7/17 Ref: Xie, Section 2.4 Wei SHA 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! Wdqd=⋅=⋅=Fl El0 ΓΓ Electrostatic field is conservative. Is there other field is conservative? El⋅=∇×⋅=−∇×∇Φ⋅=dd ES()0 d S Γ SS E =−∇Φ Φ is a scalar quantity called electrostatic potential ρ ′ ρ ′′ = 1()r 1()r dv Er() aRdv ' Φ=()r πε 2 πε ′ 4 v' rr- ′ 4-v′ rr Slide 8/17 Ref: Xie, Section 3.1 Wei SHA 3. Potential and Gauss Law (2) Gauss Law DS⋅=dQ ∇⋅D = ρ S : area element normal to an arbitrary surface : a unit normal vector to the spherical surface ⋅=QQaSR⋅d =αθθ = Q2 ϕ ESddSrddπε (cos ) ( sin ) 42rr44πε 2 πε r 2 cosα dS is an elemental area on a spherical surface of radius r Slide 9/17 Wei SHA 4. Poisson’s Equation (1) DrE= ε () E =−∇Φ ∇⋅D = ρ free charge Poisson equation: governing equation for electrostatic problem ∇⋅(()ε r ∇Φ ) =−ρ Could we put permittivity out of divergence operator? Optoelectronic device is inhomogeneous! ETL: electron transport layer HTL: hole transport layer BHJ: bulk heterojunction Red line: boundary conditions imposed Slide 10/17 Wei SHA 4. Poisson’s Equation (2) input input output output solution of Poisson’s equation in free space 1 Φ=()rrρ (′′ ) dv πε − ′ v′ 4|0 rr | Slide 11/17 Ref: Chew, Section 3.3.2 and 3.3.3 Wei SHA 4. Poisson’s Equation (3) —— Green’s function (Optional) 1 ∇⋅εδ ∇()′′ =− − ()0 g rr,() r r , = 4| − | rr≠ ′ rr= ′ singular problem ∇⋅εδ ∇1 ′′′ =− − 0 dv()rr dv 4|πε rr− ′ | c vv′′0 ∇⋅ ∇ =0 ||rr− ′ 11 ∇⋅dS′ =−1 π S′ 4 R 1 −a R ⋅=−dS′ 1 π 2 S′ 4 R −1 aS⋅=−d ′ 1 π 2 R 4 R S′ Slide 12/17 Wei SHA 5. Electrostatic Energy and Capacitance (1) The work done is equal to the change in electrostatic energy (move a positive charge from one point to the other) WQ=⋅=−∇Φ⋅=Φ−ΦEl d Q d l Q() A B ΓΓ 1. The electrostatic energy of a positive charge Q at position r in the presence of a potential Φ =Φ UQE ()r 2. The electrostatic energy stored in a system of three and N point charges ≠ 1 QQ QQ QQ 11N Nji()Q =++12 13 23 = j U E UQEi 4πε RRR πε 012132324ij==110 Rij Slide 13/17 Wei SHA 5. Electrostatic Energy and Capacitance (2) The total electrostatic energy stored in a capacitor is 11 UQVCV==2 E 22 Capacitance Energy density (energy per unit volume) of the electrostatic field 11 1 Udvdvdv=Φ=∇⋅Φ=⋅−∇Φρ DD() E vv22 v 2 11 =⋅=⋅DEdvε EE dv vv22 Slide 14/17 Wei SHA 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 constant kB is the Boltzmann’s constant T is the kelvins temperature Ndop is the net density of dopants Slide 15/17 Wei SHA 7. Boundary Conditions (1) Tangential component of E field is continuous across dielectric boundary El⋅=d 0 Γ θθ= EE1122sin sin = EE12tt Normal component of D field is continuous across dielectric boundary DS⋅=d 0 S θθ= DD1122cos cos = DD12nn Slide 16/17 Ref: Xie, Section 3.1.1 and 3.4 Wei SHA 7. Boundary Conditions (2) Dirichlet boundary condition (Red line) Φ= ()|r r→∞ 0 unbounded problem (reference potential) Φ=()|r = V electrodes (optoelectronics) rre Neumann boundary condition (Blue line) ∂Φ()r εσ = surface charge rS∈ s ∂n (semiconductor-insulator interface) ∂Φ()r non-electrode (truncation) boundaries ∈ = 0 ∂n rS (floating BCs, optoelectronics) Slide 17/17 Wei SHA.