Post on 26-Feb-2016
description
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