principles of semiconductor devices-l5

8/8/2019 Principles of Semiconductor Devices-L5

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NCN

Lecture5:EnergyBands

MuhammadAshraful [email protected]

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Outline

1) SchrodingerequationinperiodicU(x)

2) Blochtheorem

4) Propertiesof

electronic

bands

5) Conclusions

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. , .


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GettingBacktoCrystals

Original

Problem

PeriodicStructure

Electronsinperiodic

wewanttosolve

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Finallyan(almost)RealProblem

=

E

n n n+ + +

x x

n n nC e D e = +

U(x)

a b

But N atoms have two 2N unknown constants to find .

p=a+

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For large N, isnt there a better way ?


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Outline

1) Schrodinger

equation

in

periodic

U(x)

2) Blochtheorem

3) Bandstructure

4) Propertiesofelectronicbands

5) Conclusions

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FourStepsofFindingEnergyLevelsin Crystals

d2 2 =1 Det coefficient matix =0

( ) 0x = =

dx2

2) Set2N2 equationsfor

B B x x x x

+= =

=3)

B B x x x xdx dx += =

=

Nisverylargeforcrystal,butchanging steps2and3

alittlebitwecanstillsolvethe robleminafewminutes!

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PeriodicU(x)andBlochsTheorem

notourold(k)

2 2( x ) ( x p ) = + ( ) ( ) eikp x p x + =

E U(x)=U(x+a+b)=U(x+p)

U(x)

a b

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PhasefactorforNcells

( )x ( )x p +

E U(x)

N

ikp

= 2

ikpx x e+=+

2( )

ikpx e

=

8] ( )[

ikpN

Np xx e =+


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Step2:PeriodicBoundaryCondition

E U(x)

N-1

[ ] ( ) ikpN x Np x e + =

1 2 3

21

ikpN i ne e

= 1

23

N-1 2 ..... 1,0,1,....2 2

n N Nk nNp

= =

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max min,k k

p p

= =


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Step3: BoundaryConditions

2 2

02 2 ( )mE i m U E

0 0x x += ==

sina a

A x =sinb bA x =

0 0x x

d ddx dx

+= ==

acosbB x+

a b

a b

B B

A A

=

=

ik p=sin cos

a a A a B a + =

ax a x b

ik pa b

x a x b

d de

dx dx

= =

= =

=

( )[ sin cos ]

sin cos

ik a b b b

a a

e A b B b

A a B a

+ +

=

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( )[ sin cos ]

ik a b

b be A b B b + +


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Step4: Det(matrix)=0forEnergylevels

sina aA x =sinb bA x =a b

B B=

acosbB x+

a b

A A =

( )

sin cos

[ sin cos ]

a a

ik a b

b b

A a B a

e A b B b

+

+ =

+ 0 1 0 01 aA 4)

( )

sin cos

[ sin cos ]

a a

ik a b

A a B a

e A b B b

+

=

+* *

*

0

0

a

bA

B

=

021 2 mUE

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0 2

0

............2 1 U


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Outline

1) Solution

of

Schrodinger

Equation

in

Periodic

U(x)

2) BlochTheorem

3) Bandstructure

ropert eso e ectron c an s

5) Conclusions

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GraphicalsolutiontoEnergyLevels

1 2 2 n N N............

2 1

..... , , ,....2 2

nNp

= =

k=0

k=2/Np

=E/U

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EnergyBandDiagram

k=0

= p

=E/U

0 k

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Brillouin ZoneandNumberofStates

E

2..... 1,0,1,....

n N Nk n

= =

2m mx ina

2

tates pN

band k Np

= = =

k

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p

p2

k

Np

=


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Outline

1) SolutionofSchrodingerEquationinPeriodicU(x)

2) BlochTheorem

3) Bandstructure

roper eso e ec ron c an s

5) Conclusions

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WavePacketandGroupVelocity

( )E E

i k k x iEikx i t t

Ae Aex t

+ +

= +

( )1

EE i k x i t i tikx

Ae e

= +

t1

t

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k


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GroupVelocityforaGivenBand

( )x,t

( )

1

Ei k x iikx t t

Ei

Ae e

= +

1

t21ikx i t

i const . Ae e

= +

x E

t k

E

= =

cons an .x =

1 1 d kd E d E F

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2a

t dt k dk k dt m*= = = =


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EffectiveMassforaGivenBand

1/m*vE

kk

1 E

k

=

2

2 2

1 1 d E

m* dk =

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mass or eac an


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EffectiveMassisnotEssential

1/m*vE ??

kk

1 E

k

=

2

2 2

1 1 d E

*=

kF

t

=

0

t

x x dt = + 00

t Fk k dt = +

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Conclusion

1 SolutionofSchrodin ere uationisrelativel eas fors stems

with

well

defined

periodicity.

.

andbandgapssummarizeinformationaboutpossibleelectronic

states.

3) Effectivemassisnotafundamentalconcept.Therearesystems

forwhicheffectivemasscannotbedefined.

4) KPmodelisanalyticallysolvable.Realbandstructuresaresolvedoncomputer.Suchsolutionsarerelativelyeasy wewilldoHW

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. .

principles of semiconductor devices-l5

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