how to find out dos in disordered organic semiconductors sergei baranovski tids15 september 1-5,...
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How to find out DOS in Disordered Organic Semiconductors
Sergei Baranovski
TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)
Organic amorphous semiconductors
https://www.google.de/search?q=Organic+semiconductors
Organic amorphous semiconductors
Computer simulations for a Gaussian DOS: :(G. Schönherr, H. Bässler, M. Silver, Phil. Mag. B 44, 369 (1981))
2
2
2exp
2)(
N
g
22
2
1exp
3
2exp
kTkT
2
0expT
T
Santos Lemus and Hirsch, Phil. Mag. B 53, 25 (1986) ; Borsenberger, J. Appl. Phys. 68, 6263 (1990); Borsenberger et al., J. Chem. Phys. 94, 5447 (1991)
Temperature dependence of the mobility
„Unfortunately, the Gaussian form of the DOS in random organic systems prevents closed form analytical solutions of the hopping transport problem.“
Review article [H. Bässler, phys. stat. sol. (b) 175, p.19 (1993)]
„Due to the asymmetry of the exchange rates, the Gaussian form of the DOS precludes closed form analytical solutions to the generalized transportequation.“ Review article [P.M. Borsenberger et al.,
phys. stat. sol. (a) 140, p.11 (1993)]
„Gaussian DOS cannot be cast into simple analytical solutions“ A.J. Mozer et al. Phys. Rev. B 71, 035214 (2005)
http://www.posterus.sk/?p=1247
a-Si:H
00
exp)(
N
g
a-Se
M.C.J.M. Vissenberg and M. Matters, PRB 57, 12964 (1998) :
00
exp)(
N
g
Mapping of 4d VRH onto 3d geometry:
00
exp)(
N
g
pss (b), 1979
PRB, 1998
PRL, 2012
OrganicOrganic disordered materials disordered materials
DOS:
2
2
2exp
2)(
N
g
B. Hartenstein & B. Bässler, J. Non-Cryst. Solids 190, 112 (1995)
R. Schmechel, Phys. Rev. B 66, 235206 (2002)
DOS: Gaussian or exponential?
DOS: Gaussian or exponential?
2
2
2exp
2)(
N
g
equal to
00
exp)(
N
g
???
Gaussian DOS versus exponential DOS
Fundamental difference !
2
2
2exp
2)(
N
g
00
exp)(
N
g
Our claim:DOS is evident by the concentration-dependent mobility (n)
Exponential DOS: most carriers are at the Fermi level
Thermal equilibrium
F
00
exp)(
N
g
0kT
1
1)( /)( kTFe
f
F
kTFef /)()(
cannot compete with )(f
F
)(g
Gaussian DOS: most carriers are at the energy
Thermal equilibrium
kT
2
F
kT
1
1)( /)( kTFe
f
F
kTFef /)()(
)(gcannot compete with )(f
2
2
2exp
2)(
N
g
kT
2
OrganicOrganic disordered materials disordered materials
2
2
2exp
2)(
N
g
Equilibrium energy level:
kTdgkT
dgkT 2
)()/exp(
)()/exp(
H. Bässler, Phys. Stat. Sol. (b) 175, 15 (1993)
Gaussian DOS:
0
Enery relaxation in Enery relaxation in inorganicinorganic disordered materialsdisordered materials
Exponential DOS:
Transition rates:
kTa
R ijijijij 2
2exp0
0kT
00
exp)(
N
g
)(g
0
Transition rates:
kTa
R ijijijij 2
2exp0
0kT
00
exp)(
N
g
)(g
Exponential DOS:
Enery relaxation in Enery relaxation in inorganicinorganic disordered materialsdisordered materials
0
Transition rates:
kTa
R ijijijij 2
2exp0
0kT
00
exp)(
N
g
)(g
Enery relaxation Enery relaxation in inorganic in inorganic disordered materialsdisordered materials
Exponential DOS:
0
Transition rates:
kTa
R ijijijij 2
2exp0
0kT
00
exp)(
N
g
)(g
Enery relaxation Enery relaxation in inorganic in inorganic disordered materialsdisordered materials
Exponential DOS:
0
Transition rates:
kTa
R ijijijij 2
2exp0
0kT
00
exp)(
N
g
)(g
Enery relaxation Enery relaxation in inorganic in inorganic disordered materialsdisordered materials
Exponential DOS:
0
Transition rates:
kTa
R ijijijij 2
2exp0
0kT
00
exp)(
N
g
)(g
Enery relaxation Enery relaxation in inorganic in inorganic disordered materialsdisordered materials
Exponential DOS:
0
Transition rates:
kTa
R ijijijij 2
2exp0
0kT
00
exp)(
N
g
)(g
Enery relaxation Enery relaxation in inorganic in inorganic disordered materialsdisordered materials
Exponential DOS:
0
Transition rates:
kTa
R ijijijij 2
2exp0
0kT
00
exp)(
N
g
)(g
Enery relaxation Enery relaxation in inorganic in inorganic disordered materialsdisordered materials
Exponential DOS:
0
Transition rates:
kTa
R ijijijij 2
2exp0
0kT
00
exp)(
N
g
)(g
tt)(
Enery relaxation Enery relaxation in inorganic in inorganic disordered materialsdisordered materials
Exponential DOS:
0
0kT
No stable one-particle picture
Dispersive transport
00
exp)(
N
g
)(g
),( TnF
)(t
t1
t2>t1
t3>t2
t4>t3
)!()( nF
Enery relaxation Enery relaxation in inorganic in inorganic disordered materialsdisordered materials
Exponential DOS:
Exponentuial DOS:
charge carrier mobility is not well defined (time-dependent!n-dependent!)------------------------------------- Gaussian DOS:
charge carrier mobility is a well defined quantity even for a diluted system (n-independent!)
always depends on n
Gaussian DOS
2
2
2exp
2)(
N
g
00
exp)(
N
g
kT
2
kTnF
2
)(
Transport path
)(nF
Exponential DOS
does not depends on n for
)(nF
always depends on n
Gaussian DOS
2
2
2exp
2)(
N
g
00
exp)(
N
g
kT
2
kTnF
2
)(
Transport path
)(nF
Exponential DOS
does not depends on n for
)(nF
x
E
g(E) ~ exp[-E/E0]
E
00
exp)(
N
gExponential DOS
E
g(E) ~ exp[-E2 /2σ2]
x
E
- σ2 / kT
Gaussian DOS
2
2
2exp
2)(
N
g
0
0kT
No stable one-particle picture
Dispersive transport
00
exp)(
N
g
)(g
),( TnF
)(t
t1
t2>t1
t3>t2
t4>t3
)!()( nF
Enery relaxation Enery relaxation in inorganic in inorganic disordered materialsdisordered materials
Exponential DOS:
(n) in organic amorphous materials
nn
n
)(nF
kT
2
t
B-i et al., phys. stat. sol. (b) 230, 281 (2002);J. Non-Cryst. Solids 299-302, 416 (2002)
)( 3cmn
kT
2
t
B-i et al., phys. stat. sol. (b) 230, 281 (2002);J. Non-Cryst. Solids 299-302, 416 (2002)
)(nFn )( 3cm
kT
2
t
B-i et al., phys. stat. sol. (b) 230, 281 (2002);J. Non-Cryst. Solids 299-302, 416 (2002)
)(nF
n )( 3cm
kT
2
t
B-i et al., phys. stat. sol. (b) 230, 281 (2002);J. Non-Cryst. Solids 299-302, 416 (2002)
)(nF
n )( 3cm
kT
2
t
B-i et al., phys. stat. sol. (b) 230, 281 (2002);J. Non-Cryst. Solids 299-302, 416 (2002)
)(nF
n )( 3cm
DOS in organic disordered materials is not (!) DOS in organic disordered materials is not (!) exponentialexponential
Gaussian DOS: most carriers are at the energy kT
2
F
kT
1
1)( /)( kTFe
f
p
C
Ng
exp)(
kT
2
t
Is there such an equilibration energy for DOS ?
yes for p>1
0
0
)()(
)()(
dfg
dfg
Oelerich, Huemmer, B-i, PRL 108, 226403 (2012)
0
0
)()(
)()(
dfg
dfg
ndfg F
0
),()(
)(),( ppncF
cn
cn
p
C
Ng
exp)(
Oelerich, Huemmer, B-i, PRL 108, 226403 (2012)
)( 3cmn
How to find p:
)( cF n )6(
p
C
Ng
exp)(
2.26.1 p
DOS is close to Gaussian!
Oelerich, Huemmer, B-i, PRL 108, 226403 (2012)
How to find p:
22
2
1exp
3
2exp
kTkT
General theory for :
Oelerich, Huemmer, B-i, PRL 108, 226403 (2012)
General theory for (n):
p
C
Ng
exp)(
2.28.1 p
DOS is close to Gaussian!
Oelerich, Huemmer, B-i, PRL 108, 226403 (2012)
How to find p:
Saturation effects
kT/2
:/2 kTF
F
2
expkT
C
)(nF
B-i et al., phys. stat. sol. (b) 230, 281 (2002);J. Non-Cryst. Solids 299-302, 416 (2002)
:/2 kTF
kTexp
Ft
t
Conclusions
Physics in an exponential DOS is completelydifferent to that in a Gaussian DOS
DOS can be determined by measuring (n) viajust one number nc
for GDM and CDM
ncn
Measuring (?) Exponential DOS in Organics:
00
exp)(
N
g
75exp)77(00664.0
)61.0(18.0
48.0
1.22exp
2exp 00
**
0
KeV
eVeV
nm
nm
kT
EE
a
R Fij
kT
EE
a
R Fij
**
0
2exp
1120 10 s time of experiment should be larger than
!!!10101010 122132127510 yearssset
Myths and Legends about Charge Transport in Organics
From Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki
i
t
kT
r
2exp0
B-i et al., J. Non-Cryst. Sol. 190, 283 (1995);J. Phys.: Condens. Matter 9, 2699 (1997);Phys. Rev. B 62, 13081 (2000)
The Concept of Transport Energy
0)(2
exp
kT
r i
Maximum of the hopping rate:
3/13
3/42/
22
29)exp(2
exp
N
kTdtt
xx
Equation for the transport energy:
/,3 kTNxtt
3/1
)(3
4)(
i gdr i
i
t
kT
r
2exp0
B-i et al., J. Non-Cryst. Sol. 190, 283 (1995);J. Phys.: Condens. Matter 9, 2699 (1997);Phys. Rev. B 62, 13081 (2000)
The Concept of Transport Energy
Concentration dependence of the mobility
W.D. Gill, J. Appl. Phys. 43, 5033 (1972)
kTa
R ijijijij 2
2exp0
Santos Lemus and Hirsch, Phil. Mag. B 53, 25 (1986)
Abkowitz et al. J. Appl. Phys. 53, 3453 (1981)