Introduction to Solide State Physics · 2007-12-21 · 5 Ming-Chang Lee, Integrated Photonic Devices Density of States as a function of k and E z • The density of states near the

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Ming-Chang Lee, Integrated Photonic Devices

Introduction to Solid State Physics

Class: Integrated Photonic DevicesTime: Fri. 8:00am ~ 11:00am.

Classroom: 資電206Lecturer: Prof. 李明昌(Ming-Chang Lee)


Electrons in An Atom

2


Electrons in An Atom


Electrons in Two atoms

3


Analogy to A Coupled Two-Pendulum System

f f

spring

f1

f2


One-dimensional Kronig-Penney Model

Pote

ntia

lEn

ergy

Pote

ntia

lEn

ergy

Electron

Electron

4


Solutions of the S.E. in Kronig-Penny Model

Periodic Potential Well

( ) 0)(2

22

=Ψ−+Ψ∇− ExUm

Schrödinger’s equation

Bloch Theorem

a+b

)()exp()( xuxjkx ⋅=Ψ)()]([ xUbaxU =++If

Then

solution

E-k diagram

)()]([ xubaxu =++where


Density of States as a function of k and E

• For electrons near the bottom of the band, the band itself form a pseudo-potential well. The well bottom lies at Ec and the termination of the band at the crystal surfaces forms the walls of the well

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z

• The density of states near the band edges can therefore be equal to the density of states available to a particle of mass m*

in a three dimension box with the dimension of the crystal

2 2 22

2 2 2 0kx y zϕ ϕ ϕ ϕ∂ ∂ ∂+ + + =

∂ ∂ ∂

0 , 0 , 0 x a y b z c< < < < < <

22 /k mE≡

2 2

2kEm

=or

(a)

(Time-independent S.E.)( ) 0U x = except boundary

Boundary



( , , ) ( ) ( ) ( )x y zx y z x y zϕ ϕ ϕ ϕ=

22 22

2 2 2

1 1 1 0yx z

x y x

dd d kdx dy dz

ϕϕ ϕϕ ϕ ϕ

+ + + =

( , , ) sin sin sinx y zx y z A k x k y k zϕ = ⋅ ⋅

2 2 2 2x y zk k k k= + +

; ;yx zx y z

nn nk k ka b c

ππ π= = = , , 1, 2, 3,...x y zn n n = ± ± ±

where

22

2

1 constantxx

x

d kdx

ϕϕ

= = −

• To solve the equation, we employ the separation of variables technique; that is

(b)

• Substitute (b) to (a)

(Solution)

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- Densities of states in k-spaceA unit cell of volume:

( )( )( )a b cπ π π

• The larger the a,b and c, the smaller the unit cell volume



• How many states are within k and k+dk?

23

1( ) (4 ) 28

Energy states with k abck dk k dkbetween k and k dk

σ ππ

≡ = ⋅ ⋅ ⋅ ⋅ +

Redundancy

spin up and down

Density of states

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• How many states are within E and E+dE?

2 2

2kEm

=

From the previous equation,

( ) ( )E dE k dkσ σ= ( ) ( ) dkE kdE

σ σ=

2dE kdk m

= 2 3

2( ) ( ) m mEE abcσπ

=

Density of states per unit volume

2 3 2 3

2 2( ) ( ) / ( ) /m mE m mEE E V abc Vρ σπ π

= = =

V abc=

Therefore


Electrons and Holes in Semiconductors

Electric Field

8


Electron Distribution Functions

−+

=

kTEE

EfFexp1

1)(

The probability density function of electron residing in energy level E can be represented by Fermi-Dirac distribution

where EF: Fermi Energy Level


Carrier Density Distribution of Intrinsic Semiconductors

=×

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Metal, Insulator, and Semiconductor

Metal Metal Insulator Semiconductor


Work Function

• Work function is the minimum energy required to enable an electron to escape from the surface of solid.

• Work function φis the energy difference between Fermi level and the vacuum level.

• In semiconductors, it is more usual to use the electron affinity χ, defined as the energy difference between the bottom of the conduction band and the vacuum level.

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Extrinsic Semiconductors


Carrier Density Distribution of Extrinsic Semiconductors

n-type

p-type

=×

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p-n Junction

(a). Initially separated p-type and n-type semiconductor;

(b) the energy band distortion after the junction is formed;

(c) the space charge layers of ionized impurity atoms within the depletion region W; and

(d) the potential distribution at the junction.


Contact Potential on p-n junction

• Contact potential V0

The electron concentration in the conduction band of p-type side as

exp p pC Fp c

E En N

kT

− = −

exp n nC Fn c

E En N

kT −

= −

Similarly the electron concentration in the n-type side is

Since the Fermi level is constant. n pF F FE E E= =

0lnp n

nC C

p

nE E kT eVn

− = =

0 ln n

p

nkTVe n

=

12


Minority Carrier Distribution

• At the temperature in the range of 100K ≤ T ≤ 400K, the majority carrier concentrations are equal to the doping levels, that is and , n dn N=

p aP N=

0 ln a d

i

N NkTVe n

=

(recall ) 2

inp n=

• Carrier concentration difference between p- and n-type semiconductor

0expp neVn nkT

= −

0expn peVp pkT

= −

and


Current flow in a forward-biased p-n junction

• The junction is said to be forward biased if the p region is connected to the positive terminal of the voltage source

• The external voltage is dropped across the depletion region and lower down the potential barrier by (V0-V). So the diffusion current becomes larger than the drift current. There is a net current from the p to the n region.

• The Fermi levels are no longer aligned across the junction in light of the external voltage.

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Carrier densities

• Once the majority carriers flow across depletion region, they become minority carriers. The minority concentrations near the junction rise to new value np’ and pn’. The majority carrier concentration is almost unchanged unless a large current injection

• Due to the minority concentration gradient, the injected current diffuses away from the junction. The nonlinear gradient indicates minority holes (electrons) are recombined with electrons (holes) that are replenished by external voltage source.



• The injected carrier concentration at the edge of depletion region

0( )' expp ne V Vn nkT

− = −

0( )' expn pe V Vp pkT

− = −

0expn peVp pkT

= −

(a)

Since

−=kTeVnn np

0expand

Then

=kTeVpp nn exp'

=kTeVnn pp exp'and

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As we noted in previous slides, the excess minority carrier concentration will decrease due to recombination

( ) (0)exp( )h

xp x pL

∆ = ∆ −

where ( ) '( )n np x p x p∆ = −

Therefore, from (a)

(0) exp 1neVp pkT

∆ = −



• The diffusion current

(0)exp( )hh

h h

eD xJ pL L

= ∆ −

At x = 0

exp( ) 1hh n

h

eD eVJ pL kT

= −

Similarly, for electron diffusion current at the depletion edge

exp( ) 1ee p

e

eD eVJ nL kT

= − The total current

0 exp 1eVJ JkT

= − 0

h en p

h e

D DJ e p nL L

= +

where

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Current flow in a reverse-biased p-n junction

0 exp 1eVJ JkT

= − V is negative


I-V curves of p-n junction

I-V Curve

Reversed Bias Forward Bias

0 exp 1eVJ JkT

= −

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Zener Breakdown

•Tunnelling of carriers across the depletion region and the process is independent of temperature•Doping densities must be high to occur before avalanching •Breakdown current is independent of voltage


Metal-semiconductor Junction --- SchottkyContact

A schottky barrier formed by contacting a metal to an n-type semiconductor with the metal having the larger work function: band diagrams (a) before and (b) after the junction is formed.

sm φφ >

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Work Function of Metal


Electron Affinity of Semiconductor

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Metal-semiconductor Junction --- SchottkyContact

The metal to n-type semiconductor junction shown under (a) forward bias and (b) reverse bias.


Metal-semiconductor Junction --- OhmicContact

sm φφ <

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Summary of Metal-semiconductor Junction Contact

Schottly


p-N heterojunction

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n-P heterojunction


Double Heterojunctions

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Carriers Confined in Double Heterojuncitons

It is extremely efficient to enhance electron-hole recombination


Fundamentals of Electromagnetic Waves

• E and B are fundamental fields • D and H are derived from the response of the medium• How many variables in Electromagnetic Wave?

Electrical field

Electrical displacement

Magnetic field

Magnetic induction

),( trE

),( trD

),( trH

),( trB

V/m

C/m2

A/m

webers/m2

space time

Documents

Introduction to Solide State Physics · 2007-12-21 · 5 Ming-Chang Lee, Integrated Photonic Devices Density of States as a function of k and E z • The density of states near the