[PPT] - EE529 Semiconductor Optoelectronics Semiconductor Basics 1. PowerPoint Presentation

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics

EE529 Semiconductor Optoelectronics

Semiconductor Basics

1. Semiconductor materials
2. Electron and hole distribution
3. Electron-hole generation and recombination
4. p-n junction

Reading: Liu, Chapter 12, Sec. 13.1, 13.5 Reference: Bhattacharya, Sec. 2.1-2.2, 2.5-2.6, 4.2

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 2

Energy Bands in Semiconductors

Origin: Periodic lattice structure in the crystal. Indirect bandgap Direct bandgap E-k diagram details the band structure. k: Electron wave vector k = 2π/λ

Si

2 2 *

2

c e

k E E m = + 

2 2 *

2

v h

k E E m = − 

Near the band edge: Effective masses

f electrons/holes

* , : e h

m

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics

Semiconductor Materials

Elementary: e.g., Si and Ge Binary Ternary Quaternary

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics 4

III-V Compound Semiconductors

Matching lattice constant is important when depositing one semiconductor on another. Solid curve: Direct bandgap Dashed curve: Indirect bandgap

AlxGa1-xAs closely lattice matched to GaAs In1-xGaxAsyP1-y lattice matched to InP for 0 ≤ y ≤ 1 and x = 0.47y

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics 5

Electron-Hole Generation

How do electrons get to the conduction band (and leave holes in the valence band)? Free electrons and holes can be generated by:

Thermal excitation (thermal equilibrium)
Optical excitation (quasi-equilibrium)
Current injection (quasi-equilibrium)

It’s important to understand the carrier (electrons and holes) distribution as a function of energy.

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics 6

Carrier Distribution in the Energy Band

A skyscraper without elevators

Concentration of electrons (holes) versus energy in the conduction (valence) band =

Density of states (density of allowed energy levels)

Analogy: Density of available office spaces

X

Probability of occupancy (probability that each of these levels is

ccupied)

Analogy: People’s desire of occupying these spaces

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics 7

Probability of Occupancy

Fermi-Dirac function

1 exp 1 ) ( +         − = T k E E E f

B f

f(E) = probability of occupancy by an electron 1 - f(E) = probability of occupancy by a hole (in valence band)

(Electrons like to sink to the bottom, holes like to float to the top.)

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 8

Density of States ― Bulk Material

2 2 * 3/2 1/2 2 3 * 3/2 1/2 2 3

( ) (2 ) ( ) ( ) 2 (2 ) ( ) ( ) 2

e C C h V V

k k m E E E m E E E ρ = π ρ = − π ρ = − π  

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 9

Density of States ― Quantum Wells

( )

... 2, , 1 , 2 /

2 2

= π = q m d q Eq 

Discrete energy levels

* 1 2 1 1

, ( ) 0,

e C q C C q

m E E E d E E E E  > +  π ρ =   < +  

Density of states

Nature Photonics 3, 432 - 434 (2009)

Green laser diode

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 10

Quantum Wires and Quantum Dots

Quantum wires Quantum dots

3 2 1 3 2 1

1 ) ( 2 ) ( d d d E E E E E E

q q q c c

− − − − δ = ρ

(Gudiksen et al., Nature, 2002)

* 1 2 1 2 1/2 1 2

(2 / )( / 2 ) , ( ) ( ) 0, otherwise

e C q q C C q q

d d m E E E E E E E E E  π > + +  ρ = − − −    

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics 11

Thermal-Equilibrium Carrier Concentration

1/2 1/2

( ) ( ) ( ) ( )[1 ( )] ( )

c v

F C C C E B E V F V V B

E E n E f E dE N T F k T E E p E f E dE N T F k T

∞ −∞

  − = ρ = ∫       − = ρ − = ∫    

We can simplify this more …

E g(E) fE) EF nE(E) or pE(E) E E For electrons For holes [1–f(E)] nE(E) pE(E) Area = p Area = ฀ nE(E)dE = n Ec Ev Ev Ec Ec+χ EF VB CB

(a) (b) (c) (d)

g(E) ∝ (E–Ec)1/2

(a) Energy band diagram. (b) Density of states (number of states per unit energy per unit volume). (c) Fermi-Dirac probability function (probability of occupancy of a state). (d) The product of g(E) and f(E) is the energy density of electrons in the CB (number of electrons per unit energy per unit volume). The area under nE(E) vs. E is the electron concentration.

Area = ( )

E

n E dE n =

∫

Bulk Semiconductor

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 12

Carrier Concentration and Mass Action Law

( )

3 3/2 * * 2 2

2 4 exp

g B e h i B

E k T n p m m n h k T   π   = − =        

For non-degenerate semiconductors,

* 2 3/2

( ) exp 2(2 / ) : Effective density of states at the conduction band edge

C F C B C e B

E E n N k T N m k T h   − = −     = π

* 2 3/2

( ) exp 2(2 / ) : Effective density of states at the valence band edge

F V V B V h B

E E p N k T N m k T h   − = −     = π Mass action law:

The location of the Fermi level energy EF is the key.

→ Knowing one carrier concentration, you can determine the other (no matter intrinsic or extrinsic) ( ) / 3.6

C F B

E E k T − ≥ ( ) / 3.6

F V B

E E k T − ≥

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 13

Intrinsic and Extrinsic Semiconductor

e– As+ x As+ As+ As+ As+ Ec Ed CB Ev ~0.05 eV i

6 i

Electron Energy

B–

h+

x

B–

Ev Ea

B atom sites every 106 Si atoms Distance into crystal

~0.05 eV

B– B– B– h+ VB

Ec

Electron energy

Intrinsic: n-type semiconductor p-type semiconductor

1 ln 2 2

C V C Fi B V

E E N E k T N   + = −    

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 14

Exercise: Fermi Levels in Semiconductors

a) Where is the Fermi level of intrinsic bulk Si at room temperature? b) What kind of dopants can make it n-type? c) If the donor concentration Nd is 1016 cm-3, where will the Fermi level be? d) If the wafer is compensation-doped with boron (Na = 2 x 1017 cm-3), where will the Fermi level be?

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics 15

Quasi Equilibrium

Probability of occupancy for electrons:

1 ( ) exp 1 1 ( ) exp 1

C Fc B V Fv B

f E E E k T f E E E k T =   − +     =   − +    

Probability of occupancy for holes:

― What happens to the Fermi levels during photon absorption

How to calculate the quasi Fermi levels?

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 16

Quasi-Fermi Levels

The figure below shows positions of quasi-Fermi levels as a function of photo- generated electron-hole pair density. The semiconductor is n-type GaAs with ND = 1015 cm-3. Q: Why does εfp decrease gradually with increasing density while εfn shows a sudden increase?

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics

Exercise: Carrier Lifetime and Internal Quantum Efficiency in GaAs

17

A n-type (

18 3

5 10 m n

−

= × ) GaAs is under optical excitation generating excess carrier concentration N n n p p = − = − . It has the followsing recombination coefficients:

5 1

5 10 s A

−

= × ,

17 3 1

8 10 m s B

− −

= × , and

42 6 1

5 10 m s

e h

C C C

− −

= + = × . Assume that / 2

e h

C C C = = . (1) Find the range of N where each of the three different recombination processes (Shockley-Read, bimolecular, Auger) dominates. (2) Plot the spontaneous carrier lifetime

s

τ as a function of N for

18 26 3

10 10 m N

−

≤ ≤ . (3) Assume only the bimolecular recombination process is radiative, plot the internal quantum efficiency vs N.

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 18

Bimolecular Recombination and Steady-State Concentration

Rate of recombination

3

1 ( ) cm sec R Bnp = ⋅

In thermal equilibrium, generation = recombination

3

1 ( ) cm sec G Bn p = ⋅

With electron-hole injection (by external current or photon)

Net radiative recombination rate

rad

N R Bnp G = − = τ

1 ( )

rad

B N n p τ = + +

rad

dN N G dt = − = τ

→ Determines Ν if G is known, and therefore the quasi-Fermi levels. In steady sate:

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 19

Exercise: Carrier Injection at Steady State

Electron-hole pairs are injected into n-type GaAs at a rate

23 3

10 /cm s G = . At room temperature GaAs has the following parameters: Eg = 1.42 eV, ni = 2.33× 106 cm-3,

17

4.35 10

C

N = × cm-3, and

18

9.41 10

V

N = × cm-3. The thermal equilibrium concentration

f electrons is n0 = 1016 cm-3. Assume bimolecular recombination process dominates

and the coefficient for bimolecular recombination

11 3 1

8 10 cm s B

− −

= × . Determine: (a) The thermal equilibrium concentration of holes p0. (b) The steady-state excess carrier concentration Ν. (c) The recombination lifetime

rad

τ . (d) The separation between the quasi-Fermi levels EFc – EFp.

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 20

The p-n Junction

nno

x x = 0

pno

ppo npo log(n), log(p)

eNa

eNd M x E (x) B- h+ p n M As+ e– Wp Wn Neutral n-region Neutral p-region Space charge region Vo V(x) x PE(x) Electron PE(x)

Metallurgical Junction

(a) (b) (c) (e) (f)

x –Wp Wn

(d)

eVo x

(g)

–eVo Hole PE(x)

–Eo

Eo M ρnet M Wn –Wp

ni

Build-in field

a p d n

eN W eN W E = − = − ε ε Built-in potential

2

1 ( ) 2 ln

n p a d B i

V E W W N N k T e n = − +   =     Depletion widths 2 1 2 1 2

a n d d a d p a d a n p a d a d

N W V e N N N N W V e N N N W W W N N V e N N   ε =   +     ε =   +   = + + ε =

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 21

Biased p-n Junction

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 22

p-n Junction Under Forward Bias

(0) exp (0) exp

n n B p p B

eV p p k T eV n n k T   =       =    

Law of the junction Excess minority carrier concentration

' ( ') (0)exp '' ( '') (0)exp

n n h p n e

x p x p L x n x n L   ∆ = ∆ −       ∆ = ∆    

Diffusion length

, , , , ,

: Diffusion coefficient : Minority carrier lifetime

h e h e h e h e h e

L D D = τ τ

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 23

Current in Forward-biased p-n Junction

exp 2 2

p i n recom e h B

W en W eV J k T     = +     τ τ    

0 exp

1 2

recom r B

eV J J k T     = −        

Jelec

x

n-region

J = Jelec + Jhole

SCL Minority carrier diffusion current Majority carrier diffusion and drift current Total current Jhole

Wn –Wp

p-region

J

The total current anywhere in the device is

constant. Just outside the

depletion region it is due to the diffusion of minority carriers.

Diffusion current

2 , 2 , , , 2

exp 1 exp 1 exp 1 exp 1

h i D hole h d B e i D elec e a B diff D hole D elec h e i h d e a B diff so B

eD n eV J L N k T eD n eV J L N k T J J J eD eD eV n L N L N k T eV J J k T       = −                  = −            = +       = + −                 = −        

Recombination current

A more accurate result:

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 24

I-V Characteristics of a p-n Junction

nA

I

Shockley equation Space charge layer generation.

V

mA

Reverse I-V characteristics of a pn junction (the positive and negative current axes have different scales)

I = Io[exp(eV/ηkBT) − 1]

0 exp

1 : Diode ideality factor = 1: Diffusion controlled = 2: SCL recombination controlled

B

eV I I k T     = −     η     η η η

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE 529 Semiconductor Optoelectronics – Semiconductor Basics 25

p-n Junction under Reverse Bias

: Mean thermal generation time

i gen g g

eWn J = τ τ

Current due to thermally generated EHP:

Total reverse current

2 rev s gen h e i i h d e a g

J J J eD eD eWn n L N L N ≅ +   = + +   τ  

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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Semiconductor Basics 26

Exercise: GaAs p-n Junction

Device parameters:

– Cross sectional area A = 1 mm2. – Na (p-side doping) = Nd (n-side doping) = 1023 m-3. – Coefficient of recombination Β = 7.21 x 10 -16 m3s-1 – ni = 1.8 x 1012 m-3 – εr = 13.2 – µh (in the n-side) = 250 cm2V-1s-1, µe (in the p-side) = 5000 cm2V-1s-1 – Carrier recombination time in the depletion region = 10 ns

What is the I-V characteristics? Calculate the diffusion

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