solar cells lect. 3

8/13/2019 solar cells lect. 3

1/9


2/9


3/9

J h = current per area from holesJ e = current per area from electrons

J = J h + J e

Current Density in Semiconductor Applied E field

Hole Electron

V a V

position

Total Current

Electrons move to left ! positivecurrent (positive charge flows to right)

Holes move to right ! positive current

(positive charge flows to right)

ConductionBand

ValenceBand

E c

E v Eg

E l e c

t r o n

E n e r g y

Unlike the water tank analogy: electrons and holes occupy different

energy bands but the same physical space in the

semiconductor

Charge= - e (electron)= + e (hole)

Electron Density of States in SemiconductorsRecall from the free electronmodelDensity of electron states

How does this relate to semiconductors where there is a gap in theallowed energy states?

Electrons in the conduction band

The lowest energy electron state has energy E c , the conduction band minimum The density of states above this lower limit follows the same energy dependence as

that for the free electron model

The effective electron mass m e is used as a parameter to adjust the free electronresult to fit the more complicated case of electrons in the conduction band.

This parameter is known as the density of states effective mass and for Si is given by:

Density of electron states in theconduction band

Electron mass

ConductionBand

ValenceBand

Energy

E c

E v Eg


4/9

Hole Density of States in Semiconductors

ConductionBand

ValenceBand

Energy

E c

E v Eg

Holes in the valence band

The lowest energy hole corresponds to anempty electron state at energy E v , the valenceband maximum

The density of states below this limit follows theopposite energy dependence as that freeelectrons

The effective hole mass m h is used as a parameter to adjust the free electron result tofit the more complicated case of holes in the valence band.

This parameter is known as the density of states effective hole mass, and for Si isgiven by:

Density of hole states in thevalence band

Electron mass

Density of States in Semiconductors

ConductionBand

ValenceBand

ElectronEnergy

E c

E v Eg

ConductionBand

ValenceBand

Electron Energy

E c E v

Eg

Density of StatesElectronHole

Density of hole states in thevalence band

Density of electron states in theconduction band


5/9

Occupation of States in Semiconductors

Recall from Unit 2

Occupancy of electron states isgiven by the Fermi-Dirac

distribution function

How does this relate tosemiconductors where there is a gapin the allowed energy states?

Can we find occupancy probability forelectron states and hole states?

ConductionBand

ValenceBand

Energy

E c E v

Eg

1.0

0.8

0.6

0.4

0.2

0.02.241.120.00-1.12

0 K 300 K

600 K 1000 K 1500 K 2000 K

Occupation of States in Semiconductors

ConductionBand

ValenceBand

Energy

E c E v

Eg

Occupancy of a given level is given by Fermi-Dirac

Hole is the absence of electron A state occupied by an electron is not

occupied by a hole A state not occupied by an electron is

occupied by a hole Therefore the occupancy of a given

hole state is:

1Electron Hole

0Density of States

ElectronHole

10 -12

0

ElectronHole


6/9

Carrier Concentration in Semiconductors

Occupancy0 1

e -

h+

0 40 0 5

The number density per energy of carriers is given by the product of the density ofstates times the occupancy

ConductionBand

ValenceBand

Energy

E c

E v E g

The occupancy probability for electron states is given by the Fermi Dirac distribution The occupancy probability for hole states is given by 1 minus the electron occupancy The number density per energy of states is given by the electron and hole density of

states

Electron Concentration in Semiconductors

ConductionBand

ValenceBand

ElectronEnergy

E c

E v Eg

Occupancy0 1

e -

h+

0 40 0 5

Effective conduction band density of statesFor Si

Number density perenergy of electrons inthe conduction band

Concentration (number density) of electrons in the conduction band

To find the numberdensity of electrons inthe conduction bandwe integrate to find thearea under this curve


7/9

Hole Concentration in Semiconductors

ConductionBand

ValenceBand

ElectronEnergy

E c

E v Eg

Occupancy0 1

e -

h+

0 40 0 5

Number density perenergy of electrons inthe conduction band

Concentration (number density) of holes in the valence band

Effective valence band density of statesFor Si

Carrier density is astrong function oftemperature!

Carrier Concentration in Semiconductors

Electron concentrationin conduction band

Hole concentrationin valence band

Raisetemperature

300 K " 600 K

Dramatic increasein carrier density!

10-2

10

1

104

107

1010

10

13

5004003002001000

Eg = 1.12 eV

ConductionBand

ValenceBand

ElectronEnergy

E c

E v Eg

Occupancy0 1

e -

h+

0 40 0 51


8/9

0 5

Fermi Level in Semiconductors

Neutrality condition ( n = p ) pins fermi level at midgap

Shifting the fermi level by 0.05 eV resultsin a large difference between electronsand holes

Since holes come frompromotion of electrons this isunphysical!

Try shifting Fermilevel up by 0.05 eVConduction

Band

ValenceBand

ElectronEnergy

E c

E v Eg

Occupancy0 1

e -

h+

0 40 0 40

Can show:

Fermi energy is very near center of gap

Fermi Level PositionElectron concentration in conduction band

Hole concentration in valence band

ConductionBand

ValenceBand

E c

E v Eg

For Si at 300 K:

-0.007 eV

Setting these equal

Using:

Electron mass

Lets look atchemical potential


9/9

Independent of electron chemical potential ! Holds for doped semiconductors too

For intrinsic (undoped) semiconductors:

Product of hole and electron concentration

Intrinsic Carrier Concentration

ConductionBand

ValenceBand

E c

E v Eg

neutrality conditionfor intrinsic semiconductors

Intrinsic refers to asemiconductor that ispure, or un-doped, sothat n = p

Allows us to calculate carrier concentration withoutworrying about Fermi Level

Does not hold for doped semiconductors

MSE 156/256 - Solar Cells, Fuel Cells and

Batteries: Materials for the Energy SolutionStanford University Autumn 2012

Unit 3: Transport and Carrier Concentration in Semiconductors Electrons and holes how they conduct electricity Density of states for semiconductors Occupation of states for semiconductors Number density of electrons and holes

Temperature dependence Fermi level in semiconductors Intrinsic carrier concentration

Solar panel and battery in front of hutnear Zimbabwe-Botswana border

Unit 4: Doping in Semiconductors

Coming Up:

solar cells lect. 3

Documents