Dan Mendels, Nir Tessler

Sara & Moshe Zisapel Nanoelectronic CenterElectrical Engineering Dept.

Haifa 32000Israel

www.ee.technion.ac.il/nir

Mobility and Diffusion under the Premise of Solar Cells

The Role of Energy-Transport

The operation of Solar Cells is all about balancing nergyEThink “high density” or “many charges” NOT “single charge”

There is extra energy embedded in the ensemble

If you came from session P.

There is also pseudo band like behavior

The Physical Framework• Steady State I-V measurements• Steady State Qausi Equilibrium (Incl. Traps)

• Not the transient, possibly dispersive, transport where D/m may be VERY HIGH

R. Richert, L. Pautmeier, and H. Bassler, "Diffusion and drift of charge-carriers in a random potential - deviation from Einstein law," Phys. Rev. Lett., vol. 63, pp. 547-550, 1989.

[1] K. C. Kao and W. Hwang, Electrical transport in solids vol. 14. New York: Pergamon press, 1981.

[2] H. T. Nicolai, M. M. Mandoc, and P. W. M. Blom, "Electron traps in semiconducting polymers…" PRB, 83, 195204, 2011.

Original MotivationMeasure

Diodes I-V

Extract the ideality factor

The ideality factorIs the Generalized Einstein Relation

The Generalized Einstein Relation is NOT valid for

organic semiconductors

Y. Vaynzof et. al. JAP, vol. 106, p. 6, Oct 2009.

G. A. H. Wetzelaer, et. al., "Validity of the Einstein Relation in Disordered Organic Semiconductors," PRL, 107, p. 066605, 2011.

Monte-Carlo simulation of transport

0

0.01

0.02

0.03

0.04

0.05

1017 1018 1019 1020

10-4 10-3 10-2

Ein

stei

n R

elat

ion

[eV

]

Charge Density [1/cm3]

Charge Density relative to DOS

G.E.R.

Monte-Carlo

0ddx

Standard M.C. means uniform density

Y. Roichman and N. Tessler, "Generalized Einstein relation for disordered semiconductors - Implications for device performance," APL, 80, 1948, 2002.

Comparing Monte-Carlo to Drift-Diffusion & Generalized Einstein Relation

0

5 1018

1 1019

1.5 1019

2 1019

2.5 1019

0 20 40 60 80 100

Car

rier D

ensi

ty [1

/cm

3 ]

Distance from 1st lattice plane [nm]

qE

0

5 1018

1 1019

1.5 1019

2 1019

2.5 1019

3 1019

3.5 1019

4 1019

0 20 40 60 80 100

Car

rier D

ensi

ty [1

/cm

3 ]

Distance from 1st lattice plane [nm]

qE

Implement contacts as in real Devices 0ddx

GER Holds for real device Monte-Carlo Simulation

Where does most of the confusion come from

J. Bisquert, Physical Chemistry Chemical Physics, vol. 10, pp. 3175-3194, 2008.

D The intuitive Random Walk

e e eJ qn E nd

q dDx

m

The coefficient describing ddx

Generalized Einstein Relation is defined ONLY for

What is Hiding behind ddxE

X

E

X

Charges move from high density region to low density region

Charges with High Energy move from high density region to low density

There is an Energy Transport

The Energy Balance Equation

J qn Fm dnqDdx

x

n dEd

m

The operation of Solar Cells is all about balancing nergyE

DER

00.20.40.60.8

11.2 -5 -4 -3 -2 -1 0 1

-0.4 -0.3 -0.2 -0.1 0 0.1

Dis

tribu

tion

[a.u

.]

Energy []

Density Of States=3kT; T=300K

Energy [eV]

0

0.2

0.4

0.6

0.8

1

-0.4 -0.3 -0.2 -0.1 0 0.1

Dis

tribu

tion

[a.u

.]

Energy [eV]

Carriers Jump UPJumps DN

=3kTDOS = 1021cm-3

N=5x1017cm-3=5x10-4 DOSLow Electric Field

E

B. Hartenstein and H. Bassler, Journal of Non - Crystalline Solids 190, 112 (1995).

How much “Excess” energy is there?

150meV

0

0.2

0.4

0.6

0.8

1

-0.4 -0.3 -0.2 -0.1 0 0.1

Dis

tribu

tion

[a.u

.]

Energy [eV]

Carriers

The High Density PictureMobile and Immobile Carriers

Mobile Carriers

=3kTDOS = 1021cm-3

N=5x1017cm-3=5x10-4 DOSLow Electric Field

Transport is carried by high energy carriers

Is it a BAND?

Jumps distribution

Summary• Transport: Many Charges ≠ Single Charge

– Mobile and Immobile (“trapped”, “Band”) charges• Transport of energy!

– There is “excess” energy in the system.

• Where do the carriers hop in energy – Not around EF.

• Ideality factor Einstein relation?

dEd

dnJ qn F n qDx dx

m m

Seebeck EffectdE E E

dx n Tn Tx x

Recombination

Thank You

e

Mott’s Variable Range Hopping

Effective initial energy

Effective intermediate energyDE

34 13r E D For a constant density of states:For a shaped density of states:

Transport Energy (Et=?)

?E

* *

expB

R

ERk T

A D E A D D

D

InGaAs InP

DE*

N NS

*

C8H17C8H17

n ** n

C8H17C8H1 7

PFOBT PFO

DE R

R

r and DE are determined so as to maximize the hopping rate

2

0B

ErK TR e eD

E

r

e

Effective initial energy

Effective intermediate energyDE

For a shaped density of states:

Transport Energy (Et=?)

?E

1016 1017 1018 1019-0.025-0.020-0.015-0.010-0.0050.000

Et [e

V]

Charge Density [cm-3]

Transport Energy

Effective Initial Energy E

FE E

KBT

t

B

E EK TR e

1. Mobility is charge density dependent

2. FE E

3. is E ( , )E n T

There is transport of energy even in the absence of Temperature gradients

( ) ( ) ( ) dnJ qn x F x qD xdx

m

1( ) ( ) ( ) ( )dE dnJ qn x x F x q D xq dx dx

m

(a)

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

1016 1017 1018 1019 1020

Ene

rgy

[eV

]

Charge Density [1/cm3]

Average

Effective

q-Fermi

What if we analyze the standard (uniform density) Monte-Carlo

0

0.01

0.02

0.03

0.04

0.05

1017 1018 1019 1020

10-4 10-3 10-2

Ein

stei

n R

elat

ion

[eV

]

Charge Density [1/cm3]

Charge Density relative to DOS

GER

Monte-Carlo

0ddx

e- & E

Does the Generalized Einstein

Apply

Does your system obey the laws

of Thermodynamics