ECE606: Solid State Devices Lecture 6 - Purdue …ECE606: Solid State Devices Lecture 6 Gerhard Klimeck [email protected] 1 Klimeck –ECE606 Fall 2012 –notes adopted from Alam Reference

Klimeck – ECE606 Fall 2012 – notes adopted from Alam

ECE606: Solid State Devices

Lecture 6Gerhard Klimeck

[email protected]

1


Reference: Vol. 6, Ch. 3 & 4

Presentation Outline

•Reminder – Density of states»Possible states as a function of Energy

•Reality check - Measurements of Bandgaps•Reality check - Measurements of Effective Mass•Rules of filling electronic states •Derivation of Fermi-Dirac Statistics: three techniques

•Intrinsic carrier concentration•Conclusions

2


k

a

πa

π− 2k

Na

π∆ =

E E

DOS

( )3 3 0

2 3

2* *m m E EDOS

π−

=ℏ

Reminder: Momentum vs. DOS

Important things to remember:• Momentum k entered our thinking as a quantum number• Each quantum number is creating ONE state• Often “just” need the number of available states in an energy range

=> Density of States => appears to be solely determined by »1) band edge, »2) effective mass 3


Measurement of Band Gap

k

E

1

2

3

4

Photons are only absorbed between bands that have filled and empty states

E23

abso

rptio

n

4


Measurement of Energy Gap

k

hν

Iout

Iin

E23

I out -I in

E23+∆

23E E−∼

( )2

23 phE E E− −∼

k

E

1

2

3

4

5


Temperature-dependent Band Gap

k

E

1

2

3

4

( ) ( )2

0G G

TE T E

T

αβ

= −+

6







7


Measurements of Effective Masses

Important things to remember:• Full bands do not conduct –

electrons have no space to go• Empty bands to not conduct –

there are no electrons to go aroundQuestion:• We are interested in the top-most

valence band holes and the bottom-most electron states

• We want to figure out the slope of the bands

• How can we probe just one particular species of electrons/holes?

• We do not want to transfer them from one band to the next!

=> can we rotate the electrons around in a single band?

k

E

3

2

1

8


(kx,0)

(0,-ky)

(0,ky)

(-kx,0)

kx

Energy=constant.

kxky

kz

ky

Liquid He temperature …x

y

Motion in Real Space and Phase Space

9


Derive the Cyclotron Formula

2

0

0 0

z z

m*q B q B

r

qB r

m*

υ υ υ

υ

= × =

=

B

For an particle in (x-y) plane with B-field in z-direction, the Lorentz force is …

0

0

00

00 0

2 2

12

2

r m*

qB

qB

m*qB

m*

π πτυ

ντ π

ω πν

= =

≡ =

= =

0

02

qm*

B

vπ=

10


Measurement of Effective Mass

ν0=24 GHz(fixed)

IoutIin

B field variable …

00

0

0

2 2

q qm

B B*

vm*πν

π= =

I out -I in

BBcon

kxky

kz

Bval kxky

kz

11


Effective mass in Ge

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

111 111 111 111

111 111 111 111

4 angles between B field and the ellipsoids …Recall the HW1

B

12


Derivation for the Cyclotron Formula

B

Show that 2 2

2 2

1

c t l t

cos sin

m m m m

θ θ= +

[ ] dF q B M

dt

υυ= × =

The Lorentz force on electrons in a B-field

( )

( )

( )

* xx y z z y t

y*y z x x z t

* zz x y y x l

dF q B B m

dtd

F q B B mdt

dF q B B m

dt

υυ υ

υυ υ

υυ υ

= − =

= − =

= − =

In other words,

Given three mc and three θ,we will Find mt, and ml

13


Continued …

22 2 2 2 2

2

0 0 00

0 with l

l

yy t t

t* * *c t l

dsin cos

dtqB qB qB

m m m

wω ω

ω

υυ

ω

ω ω θ θ

ω

+ = ≡ +

≡ ≡ ≡

Differentiate (vy) and use other equations to find …

Let (B) make an angle (θ) with longitudinal axis of the ellipsoid (ellipsoids oriented along kz)

( )2 2

2 2

1*

l t tc

sin cos

m m mm

θ θ= +so that …

B0

ky

kx

kz

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

14


Measurement of Effective Mass

Three peaks B1, B2, B3Three masses mc1,mc2,mc3Three unique angles: 7, 65, 73

2 2

2 2

1

tc l t

cos sin

m mm m

θ θ= +

Known θ and mc allows calculation of mt and ml.

[110]

B=[0.61, 0.61, 0.5]

1

02cmqB

vπ=

15


Valence Band Effective Mass

Which peaks relate to valence band?Why are there two valence band peaks?

16


Conclusions

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

number of available states change with energy. DOS

captures this variation.

2) DOS is an important and useful characteristic of a

material that should be understood carefully.

3) Experimental measurements are key to making sure that

the theoretical calculations are correct. We will discuss

them in the next class.

17







18


Carrier Density

Carrier number = Number of states x filling factor

Chapters 2-3 Chapter 4

19


E-k diagram and Electronic States

ka

πa

π−

E

g(E)

E2 3

2

2π−

ℏ

*

cm m*

E E

E1

E2

E3

Energy-Band Density of States

20


Rules for filling up the States

� Pauli Principle: Only one electron per state

� Total number of electrons is conserved

� Total energy of the system is conserved

E2

E3

T iiN N=∑

T i iiE E N=∑

E1

21






In 1926, Fowler studied collapse of a star to white dwarf by F-D statistics, before Sommerfeld used the F-D statistics to develop a theory of electrons in metals in 1927. Wikipedia has a nice article on this topic. Difference between a trick and a method: A method is a trick used more than once!

22


Illustrative Example: 3 Energy Levels

E=0

E=2

E=4

T iiN N=∑ T i ii

E E N=∑ET=12

203

5!

0!5!

2!

1!2

7!

3!5

4!3

!W = • •

=

122

5!

2!3!

72!

1!1

!

5!2!!420

W = • •

=

041

5!

4!1!

2!

0!2

7!

6!5

1!3

!W = • •

=

Particle conservation Energy conservation

NT=5

23


Occupation Statistics

E=0

E=2

E=4

122 420W = 041 35W =203 35W =

W (E

)

2,0,3 1,2,2 0,4,1

Choose the most probable configuration.

24



E=0

E=2

E=4

122 420W =

*3

*2

*1

2

7

2

1

2

5

f

f

f

=

=

=W (E)

2,0,3 1,2,2 0,4,1

f(E)

E

Side note:So far everything shown here is EXACT!No approximations on the occupation probability!=> direct application to nano-scale electronics!

25


For N-states

( )!

! !i

ci i i i

SW

S N N=

−∏

lnW

2,0,3 1,2,2 0,4,1

[ ]ln ( ) ln( ) ( ) ln− − − − + − − +∑≃ i i i i i i i ii i i ii

NS S S N N NS NS N S

203

5!

0!5

2!

1 !

7!

3!4!!2!= • •WRecall.

Si

Ni

Stirling approx.

[ ]ln( ) ln ! ln( )! ln != − − −∑ i i i ii

W S S N N

[ ]ln ( ) ln( ) ln= − − − −∑ i i i i i i i ii

S S S N S N N N

( )ln( !) ln 10S S S S for S≈ − >

configurations

26


Optimization with Lagrange-Multiplier

ln 1ii i

i i

SE dN

Nα β

= − − −

∑

T iiN N=∑

T i iiE E N=∑

lnW

configurations

lnln( ) i

i i

WW dN

Nδ ∂=

∂∑

ln 1ii i i i

i i ii

SdN dN E dN

Nα β

− − −

∑ ∑ ∑≃

Choose the most

probable

configuration.[ ]ln ln ( ) ln( ) lni i i i i i i i

i

W S S S N S N N N= − − − −∑

ln 1ii

i i

SdN

N

= −

∑

See additional notes on Lagrange multiplies on ece606 page and blackboard

Particle conservation

Energy conservation

Optimization with constraints!

0=

27


Final steps …

1

1i

Ei

N

S eα β+=+

ln 1 0ii

i

SE

Nα β

− − − =

1At , ( ) 0

2FF FE f EE Eα β= ≡ ⇒ + =

f(E)

E

fmax(E)=1

1

Bk Tβ⇒ =

( )1

( )1 F

iE

iE

Nf E

S eβ −= =+

/At , ( ) BE k TBoltzmanE f E Ae−→ ∞ =

( ) /

1

1 BFE kE Te −=+

EF

ln ln 1 0ii i

i i

SW E dN

Nδ α β

= − − − =

∑

( )f E ≡

28


Derivation by Detailed Balance

� Pauli Principle, energy, and number conservation all satisfied

0 3 0 4 0 1 0 2( ) ( )[1 ( )][1 ( )]f E f E f E f E− −

E1+ E

2= E

3+ E

4 Only solution is …. 0 ( )

1( )

1 β −=+ FE E

f Ee

E=0

E=2

E=4

1E

2E

3E 4E

0 1 0 2 0 3 0 4( ) ( )[1 ( )][1 ( )]f E f E f E f E− −=

Energy conversation

Pauli Exclusion

Detailed Balance in Equilibrium

29


Derivation by Partition Function

( )

( )

0

1

000

11

0

1

β

β

− − ×

− − ×i

F

F

i i i

E

E E

state E N P

e Z

e Z

β β

β

− − − −

− −= ≡∑

F F

i i F

iiii( E ) ( E )

i ( E N E )

NE E

i

Ne eP

e Z

Ei

E=0

E=2E=4

1β = Bk T

30



( ) 1

0 1

=+P

f EP P

Probability that state is filled ….

( )

( )

0

1

000

11

0

1

β

β

− − ×

− − ×i

F

F

i i i

E

E E

state E N P

e Z

e Z

1

− −

− −=+

i F B

i F B

( E E ) / k T

( E E ) / k T

e / Z

/ Z e / Z

1

1 −=+ i F B( E E ) / k Te

1

f(E)

EF

1/2

E

1

f(E)

EF

1/2

E

1

31


Few comments on Fermi-Dirac Statistics

� Applies to all spin-1/2 particles

� Information about spin is not explicit; multiply DOS by 2. May be more complicated for magnetic semiconductors.

� Coulomb-interaction among particles is neglected,Therefore it applies to extended solids, not to small molecules

Lx32






33


Carrier Distribution

( )cg E

Eυ

FE( )g Eυ

cE

( )1 f E−

( )f E

( ) ( )cg E f E

( ) ( )1g E f Eυ −

( ) ( )top

c

E

cEn g E f E dE= ∫

( ) ( )1bot

E

Ep g E f E dE

υ

υ = − ∫

DOS F-D concentrationE E

1

34


Electron Concentration in 3D solids

( ) ( )

( )( )2 3

2 12

2 1 βπ −

=

−= ×

+

∫

∫ℏ

top

c

top

Fc

E

cE

* *E n n C

E EE

n g E f E dE

m m E EdE

e

( )3 2

2 1 2

0 1

22

*n

C

dF

e

mN

h ξ ηπ β ξ ξη

∞

−=

≡ +∫

( )( ) ( )2 3

2 1

1 β βπ − −

∞

+

−∫≃

ℏ c Fc c

*C

E

n

E EE E

*nm m

dEe e

E E

( ) ( )1 2

2 η η βπ

= ≡ −c c FC CF EN E

Assume wide bands

Include spinfactor of 2

35


Boltzmann vs. Fermi-Dirac Statistics

( ) ( )1 2

23ηη η β

π= → − ≡ − >c

C c C c C Fn N F N e if E E

( ) ( )cg E f E

( ) ( )1g E f Eυ − FE

ηce ( )1 2 ηcF

36


Effective Density of States

( ) ( )1 2

23c FE E

C c C c Fn N F N e if E Eβη βπ

− −= → − >

( ) ( )cg E f E

( ) ( )1g E f Eυ −

CN

VN

FEFE

As if all states are at a single level EC

37


Law of Mass-Action

( )

( )

β

β

− −

+ −

=

=

c F

v F

E EC

E EV

n N e

p N eFE

( )β

β

− −

−

× =

=

c v

g

E EC V

E

C V

n p N N e

N N e

Product is independent of the Fermi level!Very useful balance equation! Will use it often

38


Fermi-Level for Intrinsic Semiconductors

2β−=≡

gE

i

F

V

i

Cn e

E

N

E

NFE

( ) ( )

1

2 2

β β

β

− − + −= ⇒ =

= +

c i v iE E E E

Gi

V

C

VCn p e e

EE ln

N

N

N

N

E

k

3

2

2 β−

= =

= g

V

i

E

i C N

p n

n eN

n

39


Conclusions

• We discussed how electrons are distributed in electronic states

defined by the solution of Schrodinger equation.

• Since electrons are distributed according to their energy,

irrespective of their momentum states, the previously

developed concepts of constant energy surfaces, density of

states etc. turn out to be very useful.

=> will not discuss Schroedinger Eq. anymore

=> everything is captured in bandedges and effective masses

• We still do not know where EF is for general semiconductors … If

we did, we could calculate electron concentration.

40






41


Illustrative Example: 3 Energy Levels

E=0

E=2

E=4

T iiN N=∑ T i ii

E E N=∑ET=12

203

5!

0!5!

2!

1!2

7!

3!5

4!3

!W = • •

=

122

5!

2!3!

72!

1!1

!

5!2!!420

W = • •

=

041

5!

4!1!

2!

0!2

7!

6!5

1!3

!W = • •

=

Particle conservation Energy conservation

NT=5

42



E=0

E=2

E=4

122 420W = 041 35W =203 35W =

W (E

)

2,0,3 1,2,2 0,4,1

Choose the most probable configuration.

43



E=0

E=2

E=4

122 420W =

*3

*2

*1

2

7

2

1

2

5

f

f

f

=

=

=

W (E)

2,0,3 1,2,2 0,4,1

f(E)

E

Side note:So far everything shown here is EXACT!No approximations on the occupation probability!=> direct application to nano-scale electronics!

44


For N-states

( )!

! !i

ci i i i

SW

S N N=

−∏

lnW

2,0,3 1,2,2 0,4,1

[ ]ln ( ) ln( ) ( ) ln− − − − + − − +∑≃ i i i i i i i ii i i ii

NS S S N N NS NS N S

203

5!

0!5

2!

1 !

7!

3!4!!2!= • •WRecall.

Si

Ni

Stirling approx.

[ ]ln( ) ln ! ln( )! ln != − − −∑ i i i ii

W S S N N

[ ]ln ( ) ln( ) ln= − − − −∑ i i i i i i i ii

S S S N S N N N

( )ln( !) ln 10S S S S for S≈ − >

configurations

45


Optimization with Lagrange-Multiplier

ln 1ii i

i i

SE dN

Nα β

= − − −

∑

T iiN N=∑

T i iiE E N=∑

lnW

configurations

lnln( ) i

i i

WW dN

Nδ ∂=

∂∑

ln 1ii i i i

i i ii

SdN dN E dN

Nα β

− − −

∑ ∑ ∑≃

Choose the most

probable

configuration.[ ]ln ln ( ) ln( ) lni i i i i i i i

i

W S S S N S N N N= − − − −∑

ln 1ii

i i

SdN

N

= −

∑

See additional notes on Lagrange multiplies on ece606 page and blackboard

Particle conservation

Energy conservation

Optimization with constraints!

0=

46


Final steps …

1

1i

Ei

N

S eα β+=+

ln 1 0ii

i

SE

Nα β

− − − =

1At , ( ) 0

2FF FE f EE Eα β= ≡ ⇒ + =

f(E)

E

fmax(E)=1

1

Bk Tβ⇒ =

( )1

( )1 F

iE

iE

Nf E

S eβ −= =+

/At , ( ) BE k TBoltzmanE f E Ae−→ ∞ =

( ) /

1

1 BFE kE Te −=+

EF

ln ln 1 0ii i

i i

SW E dN

Nδ α β

= − − − =

∑

( )f E ≡

47




•Reminder - Rules of filling electronic states •Derivation of Fermi-Dirac Statistics:

»three techniques

• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Conclusions

48


Derivation by Detailed Balance

� Pauli Principle, energy, and number conservation all satisfied

0 3 0 4 0 1 0 2( ) ( )[1 ( )][1 ( )]f E f E f E f E− −

E1+ E

2= E

3+ E

4 Only solution is …. 0 ( )

1( )

1 β −=+ FE E

f Ee

E=0

E=2

E=4

1E

2E

3E 4E

0 1 0 2 0 3 0 4( ) ( )[1 ( )][1 ( )]f E f E f E f E− −=

Energy conversation

Pauli Exclusion

Detailed Balance in Equilibrium

49





»three techniques


50



( )

( )

0

1

000

11

0

1

β

β

− − ×

− − ×i

F

F

i i i

E

E E

state E N P

e Z

e Z

β β

β

− − − −

− −= ≡∑

F F

i i F

iiii( E ) ( E )

i ( E N E )

NE E

i

Ne eP

e Z

Ei

E=0

E=2E=4

1β = Bk T

51



( ) 1

0 1

=+P

f EP P

Probability that state is filled ….

( )

( )

0

1

000

11

0

1

β

β

− − ×

− − ×i

F

F

i i i

E

E E

state E N P

e Z

e Z

1

− −

− −=+

i F B

i F B

( E E ) / k T

( E E ) / k T

e / Z

/ Z e / Z

1

1 −=+ i F B( E E ) / k Te

1

f(E)

EF

1/2

E

1

f(E)

EF

1/2

E1

52


Few comments on Fermi-Dirac Statistics

� Applies to all spin-1/2 particles

� Information about spin is not explicit; multiply DOS by 2. May be more complicated for magnetic semiconductors.

� Coulomb-interaction among particles is neglected,Therefore it applies to extended solids, not to small molecules

Lx53





»three techniques


54


Carrier Distribution

( )cg E

Eυ

FE

( )g Eυ

cE

( )1 f E−

( )f E

( ) ( )cg E f E

( ) ( )1g E f Eυ −

( ) ( )top

c

E

cEn g E f E dE= ∫

( ) ( )1bot

E

Ep g E f E dE

υ

υ = − ∫

DOS F-D concentrationE E

1

55


Electron Concentration in 3D solids

( ) ( )

( )( )2 3

2 12

2 1 βπ −

=

−= ×

+

∫

∫ℏ

top

c

top

Fc

E

cE

* *E n n C

E EE

n g E f E dE

m m E EdE

e

( )3 2

2 1 2

0 1

22

*n

C

dF

e

mN

h ξ ηπ β ξ ξη

∞

−=

≡ +∫

( )( ) ( )2 3

2 1

1 β βπ − −

∞

+

−∫≃

ℏ c Fc c

*C

E

n

E EE E

*nm m

dEe e

E E

( ) ( )1 2

2 η η βπ

= ≡ −c c FC CF EN E

Assume wide bands

Include spinfactor of 2

56


Boltzmann vs. Fermi-Dirac Statistics

( ) ( )1 2

23ηη η β

π= → − ≡ − >c

C c C c C Fn N F N e if E E

( ) ( )cg E f E

( ) ( )1g E f Eυ − FE

ηce ( )1 2 ηcF

57


Effective Density of States

( ) ( )1 2

23c FE E

C c C c Fn N F N e if E Eβη βπ

− −= → − >

( ) ( )cg E f E

( ) ( )1g E f Eυ −

CN

VN

FEFE

As if all states are at a single level EC

58


Law of Mass-Action

( )

( )

β

β

− −

+ −

=

=

c F

v F

E EC

E EV

n N e

p N eFE

( )β

β

− −

−

× =

=

c v

g

E EC V

E

C V

n p N N e

N N e

Product is independent of the Fermi level!Very useful balance equation! Will use it often

59


Fermi-Level for Intrinsic Semiconductors

2β−=≡

gE

i

F

V

i

Cn e

E

N

E

NFE

( ) ( )

1

2 2

β β

β

− − + −= ⇒ =

= +

c i v iE E E E

Gi

V

C

VCn p e e

EE ln

N

N

N

N

E

k

3

2

2 β−

= =

= g

V

i

E

i C N

p n

n eN

n

60


• We discussed how electrons are distributed in electronic states

defined by the solution of Schrodinger equation.

• Since electrons are distributed according to their energy,

irrespective of their momentum states, the previously

developed concepts of constant energy surfaces, density of

states etc. turn out to be very useful.

=> will not discuss Schroedinger Eq. anymore

=> everything is captured in bandedges and effective masses

• We still do not know where EF is for general semiconductors … If

we did, we could calculate electron concentration.

Summary – DOS and Fermi Functions

61


Reference: Vol. 6, Ch. 3


•Schrodinger equation in periodic U(x)•Bloch theorem•Band structure•Properties of electronic bands•E-k diagram and constant energy surfaces •Conclusions

62

Documents

ECE606: Solid State Devices Lecture 6 - Purdue …ECE606: Solid State Devices Lecture 6 Gerhard Klimeck [email protected] 1 Klimeck –ECE606 Fall 2012 –notes adopted from Alam Reference