how to find out dos in disordered organic semiconductors sergei baranovski tids15 september 1-5,...
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![Page 1: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/1.jpg)
How to find out DOS in Disordered Organic Semiconductors
Sergei Baranovski
TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)
![Page 2: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/2.jpg)
Organic amorphous semiconductors
https://www.google.de/search?q=Organic+semiconductors
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Organic amorphous semiconductors
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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
![Page 5: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/5.jpg)
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
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„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)
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http://www.posterus.sk/?p=1247
a-Si:H
00
exp)(
N
g
a-Se
![Page 9: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/9.jpg)
M.C.J.M. Vissenberg and M. Matters, PRB 57, 12964 (1998) :
00
exp)(
N
g
![Page 10: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/10.jpg)
Mapping of 4d VRH onto 3d geometry:
00
exp)(
N
g
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pss (b), 1979
PRB, 1998
PRL, 2012
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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)
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DOS: Gaussian or exponential?
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DOS: Gaussian or exponential?
![Page 16: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/16.jpg)
2
2
2exp
2)(
N
g
equal to
00
exp)(
N
g
???
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Gaussian DOS versus exponential DOS
Fundamental difference !
2
2
2exp
2)(
N
g
00
exp)(
N
g
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Our claim:DOS is evident by the concentration-dependent mobility (n)
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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
![Page 21: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/21.jpg)
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
![Page 22: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/22.jpg)
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:
![Page 23: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/23.jpg)
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
![Page 24: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/24.jpg)
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
![Page 25: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/25.jpg)
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:
![Page 26: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/26.jpg)
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:
![Page 27: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/27.jpg)
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:
![Page 28: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/28.jpg)
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:
![Page 29: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/29.jpg)
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:
![Page 30: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/30.jpg)
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:
![Page 31: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/31.jpg)
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:
![Page 32: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/32.jpg)
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:
![Page 33: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/33.jpg)
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!)
![Page 34: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/34.jpg)
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
![Page 35: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/35.jpg)
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
![Page 36: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/36.jpg)
x
E
g(E) ~ exp[-E/E0]
E
00
exp)(
N
gExponential DOS
![Page 37: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/37.jpg)
E
g(E) ~ exp[-E2 /2σ2]
x
E
- σ2 / kT
Gaussian DOS
2
2
2exp
2)(
N
g
![Page 38: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/38.jpg)
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:
![Page 39: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/39.jpg)
(n) in organic amorphous materials
nn
n
![Page 40: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/40.jpg)
)(nF
kT
2
t
B-i et al., phys. stat. sol. (b) 230, 281 (2002);J. Non-Cryst. Solids 299-302, 416 (2002)
)( 3cmn
![Page 41: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/41.jpg)
kT
2
t
B-i et al., phys. stat. sol. (b) 230, 281 (2002);J. Non-Cryst. Solids 299-302, 416 (2002)
)(nFn )( 3cm
![Page 42: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/42.jpg)
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
![Page 43: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/43.jpg)
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
![Page 44: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/44.jpg)
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
![Page 45: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/45.jpg)
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)
![Page 46: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/46.jpg)
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:
![Page 47: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/47.jpg)
)( 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:
![Page 48: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/48.jpg)
22
2
1exp
3
2exp
kTkT
General theory for :
![Page 49: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/49.jpg)
Oelerich, Huemmer, B-i, PRL 108, 226403 (2012)
General theory for (n):
![Page 50: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/50.jpg)
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:
![Page 51: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/51.jpg)
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
![Page 52: How to find out DOS in Disordered Organic Semiconductors Sergei Baranovski TIDS15 September 1-5, 2013, Eden Roc Hotel, Sant Feliu de Guíxols (Spain)](https://reader037.vdocument.in/reader037/viewer/2022110209/56649e545503460f94b4b3cf/html5/thumbnails/52.jpg)
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
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Measuring (?) Exponential DOS in Organics:
00
exp)(
N
g
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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
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Myths and Legends about Charge Transport in Organics
From Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki
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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
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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
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Concentration dependence of the mobility
W.D. Gill, J. Appl. Phys. 43, 5033 (1972)
kTa
R ijijijij 2
2exp0
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Santos Lemus and Hirsch, Phil. Mag. B 53, 25 (1986)
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Abkowitz et al. J. Appl. Phys. 53, 3453 (1981)