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E= hν

λ= hp

- ℏ22m ∂2Ψ(x,t)

∂x2 +V(x)Ψ(x,t)= jℏ ∂Ψ(x,t)∂tΨ(x, t)|Ψ(x,t)|2

- 7 -

Ψ(x,t)=Aexp[j(kx-wt)]k= 2πλ λ= ℏ

2mE

ψ(x)= 2a sin ( nπxa )

p=ℏk=ℏ( nπa ), E= p 22m = 12m ( ℏnπa ) 2

E=E n= ℏ2n 2π 22ma 2

- 8 -

ψ2(x)=A2e -K2 x (x≥0)K 2= 2m(V 0-E)/ℏ2

|ψ2(x)|2

T≅16( EV 0 )(1-EV 0 ) exp (-2K2a)

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Ψ(x) =u(x)e jkx

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J=-e ∑Ni=1v i v i= 1ℏ dEdk

F total= F ext+F int=maF ext=m *a

1m * =

1ℏ2

d 2Edk 2

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g c(E)= m*n 2m*n(E-Ec)π2ℏ3 for E≥Ec

gv(E)= m*p 2m*p(Ev-E)π2ℏ3 for E≤Ev

f(E) = 11+e ( E-EF )/kT

f(E)≈e -( E-EF)/kT for E-EF≥3kT

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ddt =0

n0=⌠⌡E top

Ecgc(E)f(E)dE

=Nc 2π F1/2(ηc)F 1/2(η c )=⌠⌡

∞

0η 1/2dη1+e η- η c ,η=

E-E ckT

where η c= EF-E ckT , N c=2[ 2πm*nkTh 2 ] 3/2

p0=Nv 2π F1/2(ηv)

η v= Ev-EFkT , Nv=2[ 2πm*pkTh 2 ] 3/2

n0≈Nce ( EF-Ec)/kT for Ec-EF≥3kT

p0≈Nve ( Ev-EF)/kT for EF-Ev≥3kT

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n0=n i=Nce ( EFi -Ec)/kT

p0=n i=Nve ( Ev-EFi)/kT

n2i =N cNve -( E c-E v)/kT=N cN ve - E gkT

n 0= p 0=>E Fi= E c+E v2 + 34 kT ln (

m*pm*n )

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n0≈Nce ( EF-Ec)/kT for Ed-EF≥3kT

p0≈Nve ( Ev-EF)/kT for EF-Ea≥3kT

n 0=p 0=n i, E F= E Fin0≈n ie ( EF-EFi)/kT

p0≈n ie ( EFi -EF)/kT

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n 0p 0= n2i

N+D≈N D, N-A≈N A

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Q=q( p 0-n 0+N+D-N-A) = 0N+D≈N D, N-A≈N A

p 0-n 0+N D-N A≈0

n 0= ND-NA2 + ( ND-NA2 ) 2+n2ip 0=n2i/n 0

n0≈ND, p0=n2i/ND for n-typep0≈NA, n0=n2i/NA for p-type

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n 0=p 0=n i, E F= E FiE F=E Fi= E c+E v2 + 34 kT ln (

m*pm*n )= E c+E v2 -0.0073eV

EFi-EF≈kTln NAn i for p-type

EF-EFi≈kTln NDn i for n-type

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32 kT= 12 m*v2thv th≈107cm/s @T=300K

vthvd

J p, drift=qpvd

vd≈μpE μp

J p, drift= qμ ppEJ n, drift= qμ nnE

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For 300K low doped Si, μ n≈1360cm2/V-secμ p≈460cm2/V-sec

For 300K low doped GaAs, μ n≈8000cm2/V-secμ p≈320cm2/V-sec

μ n∝T-2.3

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J=σE= 1ρ Eρ= 1q(μnn+μpp)

ρ≈ 1qμnNDρ≈ 1qμpNA

ρ=2πs VI F

J p, diff=-qD p dpdxJ n, diff=+qD n dndx

J p= J p, drift+J p, diffJ n=J n, drift+J n, diffJ=J p+J n

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Potential Energy(P.E.) =-qV= E c-E ref

E=- dVdx = 1qdE cdx = 1q

dE vdx = 1qdE Fidx

EF-EFi=kT ln ( Nd(x)n i )E=- kTq 1Nd(x)

dNd(x)dx

J n= J p=0 or dEF/dx=0J n= nqμ nE+qDn dndx =0n= n ie ( EF-E i)/kT, E= 1q

dE idxDnμn =

kTq

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Δp<<n0, n≈n0 in n-typeΔn<<p0, p≈p0 in p-type

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∂p∂t R=-cpNTp

∂p∂t G≈∂p∂t G-equil=-

∂p∂t R-equil=cpNTp0∂p∂t i - thermal R-G

≈ ∂p∂t R+∂p∂t G=-cpNT(p-p0)=-c pNTΔp

∂p∂t i- thermal R-G=- Δpτp where τp= 1cpNT

∂n∂t i- thermal R-G=- Δnτn where τn= 1cnNT

τp∝ 1NTτp

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- ∂p∂t = 1q∂Jp∂x + Δpτp -GL

- ∂n∂t =- 1q∂J n∂x + Δnτ n -GL

dEdx =- d 2ψdx 2 =

ρK sε 0 =qK sε 0 (p-n+ND-NA)

J p= pqμ pE-qD p dpdxJ n= nqμ nE+qD n dndx- ∂p∂t = 1q

∂Jp∂x + Δpτp -GL

- ∂n∂t =- 1q∂J n∂x + Δnτ n -GL

dEdx = qK sε 0 (p-n+ND-NA)

▽⋅D= ρ(Gauss)▽⋅B=0(자기장에 대한 Gauss)▽×H= J+ ∂D∂t (Ampere)▽×E=- ∂B∂t (Faraday)

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≈

Steady State => ∂∂t =0, No Light => GL=0∴0=- 1q

∂Jn∂x + ΔnτnJ n≈+qD n dndx = qD n

d(n p0+Δn p)dx ≈qD n d(Δn p)dx0=-D n d

2Δn pdx 2 +Δn pτ n

∴Δn p(x)=Ae- x/L n+Be x/L n where L n= D nτ n

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≈

Δn p(t)=Δn p(0)e - t /τ n

≈

∴Δn= G Lτ n

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Steady State => ∂∂t =0No Light => GL=0No R-G => Δnτn0=- 1q

∂J n∂x , J n=+qD n dndx , ∴0=-D n d2Δndx 2

∴Δn( x)= A+B⋅x

Uniformly-Doped (n 0=N D= 10 15cm- 3), RoomTemperatureLight On, t> 0, G L= 10 17cm- 3, τ p= 1μs, Δp n(t)= ?- ∂p∂t = 1q

∂J p∂x + Δpτp -GLB.C. Δp(t=0)=0J p=pqμpE-qDp dpdx ≈0

Δp= G L⋅τ p+Ae- t/τ pΔp=GL⋅τ p(1-e- t/τ p)

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Uniformly-Doped (ND=10 15cm-3), Δp n0(x=0)= 10 10 cm- 3G L=0, Δp n(x)=?

- ∂p∂t = 1q∂Jp∂x + Δpτp -GL

Jp=pqμpE-qDp dpdx∂p∂t =0

E=0 GL=0∴ 0=-D p d

2p ndx 2 +Δp nτ p

Δp n(x)=Ae- x/L p+Be x/L p where L p= D pτ pΔp n(x=0)=Δp n0, Δp n(x=∞)=0

Δp n(x)=Δp n0e- x/Lp

Lp= DpτpLn= Dnτ nND=1015cm-3,τ p=1μs

μ p=458cm2/V* sec from Fig3.5D p= μ pkT/q=12cm2/ sec (Einstein Relationship)L p= D pτ p=35μm

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n=e ( E Fn -E i )/kT, p=e ( E i -E Fp)/kTpn=n2ie ΔE F/kT where ΔE F=E Fn-E Fp