Mobility and Diffusion under the Premise
of Solar Cells
The Role of Energy-Transport
Dan Mendels, Nir Tessler
Sara & Moshe Zisapel Nanoelectronic Center
Electrical Engineering Dept.
Haifa 32000
Israel
www.ee.technion.ac.il/nir
The operation of Solar Cells is all about balancing
Energy
Think “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)
[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.
• 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.
Original Motivation
Measure
Diodes I-V
Extract the
ideality factor
Y. Vaynzof et. al. JAP, vol. 106, p. 6, Oct 2009.
The ideality factor
Is the Generalized
Einstein Relation
The Generalized Einstein Relation
is NOT valid for
organic semiconductors
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
d
0
dx
Standard M.C. means
uniform density
Charge Density relative to DOS
0.05
10-4
10-3
10-2
Y. Roichman and N. Tessler, "Generalized
Einstein relation for disordered
semiconductors - Implications for device
performance," APL, 80, 1948, 2002.
Einstein Relation [eV]
G.E.R.
0.04
0.03
0.02
Monte-Carlo
0.01
0
17
10
18
10
19
10
Charge Density [1/cm3]
20
10
Comparing Monte-Carlo to
Drift-Diffusion & Generalized Einstein Relation
Implement contacts as in real Devices
d
0
dx
19
4 10
3
Carrier Density [1/cm ]
3
Carrier Density [1/cm ]
19
2.5 10
19
2 10
1.5 1019
qE
1 1019
5 1018
0
19
3.5 10
19
3 10
2.5 1019
2 1019
qE
1.5 1019
1 1019
GER Holds for real device5 Monte-Carlo
Simulation
1018
0
20
40
60
80
100
Distance from 1st lattice plane [nm]
0
0
20
40
60
80
100
Distance from 1st lattice plane [nm]
Where does most of the confusion
come from
D The intuitive Random Walk
d
The coefficient describing
dx
d
J e  qnme E  qDe
n
dx
Generalized Einstein Relation is defined ONLY for
J. Bisquert, Physical Chemistry Chemical Physics, vol. 10, pp. 3175-3194, 2008.
E
d
What is Hiding behind
dx
E
There is an Energy Transport
X
X
Charges move from high density region to low density region
Charges with High Energy move from high density region to low density
The Energy Balance Equation
dn
dE
J  qnm F nqD
m
ddx
x
The operation of Solar Cells is all about balancing E nergy
R
DE
How much “Excess” energy is there?
Energy []
150meV
Distribution [a.u.]
Distribution [a.u.]
1.2 -5
1
0.8
0.6
-4
-3
-2
-1
0
1
Density Of States
=3kT; T=300K
0.4
0.2
0
-0.4
1
-0.3
-0.2
-0.1
0
Jumps DN
Energy
[eV] Jump UP
Carriers
0.8
=3kT
DOS = 1021cm-3
N=5x1017cm-3=5x10-4 DOS
Low Electric Field
E
0.6
0.4
0.2
0
-0.4
-0.3
-0.2
-0.1
0.1
0
0.1
Energy [eV]
B. Hartenstein and H. Bassler, Journal of Non - Crystalline Solids 190, 112 (1995).
The High Density Picture
Mobile and Immobile Carriers
Distribution [a.u.]
1
Carriers
Jumps distribution
0.8
Is it a BAND?
0.6
0.4
0.2
0
-0.4
Mobile Carriers
-0.3
-0.2
-0.1
=3kT
DOS = 1021cm-3
N=5x1017cm-3=5x10-4 DOS
Low Electric Field
0
0.1
Energy [eV]
Transport is carried by high energy carriers
Summary
• Transport: Many Charges
Thank You
≠ Single Charge
– Mobile and Immobile (“trapped”, “Band”) charges
• Transport of energy!
– There is “excess” energy in the system.
dE
dn
J  qnm F  nm
 qD
dx
dx
dE E n E T


dx n x T x
• Where do the carriers hop in energy
– Not around EF.
• Ideality factor
Einstein relation?
Seebeck
Effect
Mott’s Variable Range Hopping
E
R

A  D*  DE 
A*  D
 DE 
Rexp 

 k BT 
R   0e
R
R
DE
*
n
C8H17
C8 H17
N
N
*
DE
*
n*
C8 H1 7
S
PFOBT

2r


e
DE
K BT
C8 H17
InGaAs
InP
PFO
r
r and DE are determined so as to maximize the hopping rate
For a shaped density of states:
For a constant density of states:
e
Transport Energy (Et=?)
E ?
DE
4 3
 r DE  1
3
Effective intermediate energy
Effective initial energy
r a shaped density of states:
1. Mobility is charge density dependent
e
Transport Energy (Et=?)
DE
E ?
Re
Effective intermediate energy

Et  E
K BT
Effective initial energy
2.
E  EF
0.000
-0.005
-0.010
-0.015
-0.020
-0.025
16
10
3.
E
is
E (n, T )
KBT
t
E [eV]
Transport Energy
17
18
10
10
There is transport of energy even in the
absence of Temperature gradients
19
10
-3
Charge Density [cm ]
Effective Initial Energy E
J  qnm ( x) F ( x)  qD( x)
(a)
E  EF
-0.1
-0.15
Effective
-0.2
-0.25
Average
-0.3
q-Fermi
10
16
10
17
10
18
10
19
10
3
Charge Density [1/cm ]
20
Energy [eV]
-0.05
dn
dx

1 dE 
dn
J  qn( x) m ( x)  F ( x) 

q
D( x)

q dx 
dx

What if we analyze the standard
(uniform density) Monte-Carlo
Charge Density relative to DOS
-4
Einstein Relation [eV]
0.05
10
-3
10
-2
10
GER
0.04
d
0
dx
0.03
Monte-Carlo
0.02
0.01
0
1017
1018
1019
Charge Density [1/cm3]
1020
e- & E
Does the
Generalized Einstein
Apply
Does your system
obey the laws
of Thermodynamics