Principles of Semiconductor Devices-L8

8/8/2019 Principles of Semiconductor Devices-L8

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‐

Lecture 8: Density of States

Muhammad Ashraful [email protected]

Alam ECE‐606 S09 1



Outline

1) Calculation

of

density

of

states

3) Characterization of

Effective

Mass

4) Conclusions

Reference: Vol. 6, Ch. 3 (pages 88‐96)




Density of States

E

4

A single

band

has

total

of

N

‐states

n y a rac on o s a es are occup e

How many states are occupied upto E?

Or equivalently…

1

How many states per unit energy ? (DOS)




Density of States in 1‐D Semiconductors

a

E1 1States between + 2&

k E E E

ΔΔ = ×

E1+ΔE

k1+Δkk

122

k

Naπ

Δ

= ×

Na k Δ

E1

k

a

π 2

k π

δ =

1 E π Δ




1D‐DOS N atomsL

States/unit energy @ N k a

E E π

=Δ

Δ

a

( )2 20

0 2

2

2

*

*

m E E k E E k

m

−− = ⇒ =

E

( )2

02

*dk

d

m

E E E

=−

E1+ΔE

k1+Δkk

1

( )2

0

States/unit energy @2

*m

E

L E

E π =

−

E1

unit lengthStates/unit energy/ @

1 *

E

m DOS≡ =

k

a

π 2

k π

δ =


0π −



1D‐DOS

E E2

1

2

*m

DOS E E π

=−

a

π 2

k π

Δ =

DOS


NaConservation of DOS



Density of States in 2D Semiconductors

b

a

Show that 2D DOS is a constant independent of energy!




Density of States in 3D Semiconductors

( )

1 1

3 3

States between + &

4 4

E E E

k dk k π π

Δ

+ −

Macroscopic Sample

2

2 2 2 2 H

k k

L W π π π π

= = Δ

L

W

2

2unit energStates/ @

2y

V k

E E k

d π

Δ=

2π/W2π/H

( )

( )

2 20

0 2 2

0

2

2 2

* *

*

m E E mk dk E E k

m dE E E

−− = ⇒ = ⇒ =

−

k

2π/L

1States/unit energy/unit volume @

**

E

m

K+dk


02 32π −



3D‐DOS

* *

E E

3 3 0

2 3

m m

DOS π

−

=

a

π

a

π −

2k

π Δ =

DOS


NaConservation of DOS



Outline

1) Calculation of

density

of

states


Effective

Mass

4) Conclusions




Density of States of GaAs: Conduction/Valence Bands

* *

( ) 2 32

n n c

cg E π

−

=

( )2 3

2

2

* *

hh hhm m E E υ

π

⎧ −⎪

( )

2* *

lh lh

g E

m m E E

υ

υ

= ⎨⎪

−

11

2 32π ⎪⎩



Four valleys inside BZ for Germanium








Outline

1) Calculation of

density

of

states


Effective

Mass

4) Conclusions




Measurement of Effective Mass

ν0=24 GHz

(fixed)

B field variable …

IoutIin

k z

con

Bqm* =

0vπ

k xk y

I o u t - I i

n

16BBcon



Motion in Real Space and Phase Space

Energy=constant.

Liquid He temperature …x

(kx,0)0,k

k y

k

y

(0,-k y)(-kx,0) k z

k xk y



Derive the Cyclotron Formula 0

2

qm*

B

vπ

=

For an particle in (x‐y) plane with B‐field in z‐direction,

2m* B B

υ υ υ = × =

B

…

0

0 0

z zr

qB r υ =m*

02 2r m*

B

π π τ

υ = =

00

1

2

qB

m*ν

τ π ≡ =




Effective mass in Ge

B

[ ] [ ] [ ] [ ]111 111 111 111

[ ] [ ] [ ] [ ]111 111 111 111

4 angles between B field and the ellipsoids …

Recall the HW1




Derivation for the Cyclotron Formula

Show that

2 2

2 2

1

c t l t

cos sin

m m m m

θ θ = + Given three mc and three θ,

we will Find mt, and ml

The Lorentz

force

on

electrons

in

a B

‐field

[ ]d

F q B M

dt

υ υ = × =

( ) * x x y z z y t

d F q B B m

dt

υ υ υ = − =

In other words,

( ) y*

y z x x z t

d F q B B m

dt

υ υ υ = − =


( ) * z z x y y x l

F q B B mdt

υ υ υ = − =



Continued …kz

Let (B)

make

an

angle

(θ)

with

longitudinal

axis

of

the

ellipsoid (ellipsoids oriented along kz)

ky( ) ( )0 00 x y z B B cos , B , B B sin ,θ θ = = =

Differentiate (vy) and use other equations to find …

2 2 2 2 2

20 with

l

y

y t t sin cosdt

B B B

wω ω υ

υ ω ω θ θ ⎡ ⎤+ = ≡ +⎣ ⎦

0 lt * * *c t lm m mω ω ω ≡ ≡ ≡

2 21 sin cosθ θ

Alam ECE‐606 S0921( )

2 2*

l t t c

m m mm= +so that …



Measurement of Effective Mass

B=[0.61, 0.61, 0.5]

[110]

1qB

1,

2,

3Three masses mc1,mc2,mc3

Three unique angles: 7, 65, 73

2 21

t c l t

cos sinm mm m

θ θ = + 02c

vπ


Known θ and mc allows calculation of mt and ml.



Valence Band Effective Mass

HW. Which peaks relate to valence band?




Conclusions

1) Measurement of

Effective

mass

and

band

gaps

define

the energy‐band of a material.

2) Only a fraction of the available states are occupied. The

number of

available

states

change

with

energy.

DOS

captures this variation.

3 DOS is an im ortant and useful characteristic of a

material that

should

be

understood

carefully.


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Principles of Semiconductor Devices-L8