- Email: [email protected]
Space-charge oriented methods for studying organic amorphous semiconductors
Journal of Non-Crystalline Solids 227–230 Ž1998. 659–663
Space-charge oriented methods for studying organic amorphous semiconductors F. Schauer
)
Faculty of Chemistry, Technical UniÕersity of Brno, Veslarska 230, 637 00 Brno, Czech Republic
Abstract The purpose of the present paper is to provide the present state of art in the knowledge of the suitability of space-charge oriented methods for the density of states ŽDOS. spectroscopy in organic amorphous semiconducting materials. In the first part we intend to describe the space-charge forming in organic materials and explain the advantage of using the differential approach in the steady-state methods. The static capacitance–voltage Ž C–V ., field effect ŽFE. and space-charge-limited currents ŽSCLC. methods are presented with the unifying approach for organic semiconductors. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Space-charge; Amorphous semiconductors; Density of states
1. Introduction The need for examining the density of localized electron states of disordered inorganic semiconducting Žmainly a-Si:H and its alloys—for references, see, e.g., Ref. w1x. and organic substances has renewed the interest in the methods based on the space-charge effects. Many experimental methods based on the space-charge-limited currents ŽSCLC. w2x, space-charge capacitance w3x and field effect ŽFE. w4x occurred. Furthermore, the need for a differential method using the numerical derivative of the measured data developed w5x. In the present paper, we want to generalize the experimental methods for the description of the lo-
)
Corresponding author. Fax: q421 5 4321 1101; e-mail: [email protected].
calized electron states based on the space-charge in steady-state using the first and higher derivatives of the experimental data suitable for organic semiconductors. The space-charge volume density, rsc , in substances with an energetically distributed density of states ŽDOS. function, hŽ E ., is in general given by the contribution from all energy states
rsc Ž x . s ye n s Ž x . y n so s ye h Ž E . f E F , E q eV Ž x .
HE
yf Ž E Fo , E . 4 d E,
Ž 1.
where n sŽ x . is the total concentration of charge carriers forming space-charge at a point x and n so Ž x . is the concentration of carriers at the thermodynami-
0022-3093r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 Ž 9 8 . 0 0 1 3 3 - 1
F. Schauerr Journal of Non-Crystalline Solids 227–230 (1998) 659–663
660
cal equilibrium, E F and E Fo are the quasi- and thermodynamical Fermi levels and V Ž x . is the potential at a point x. The total space-charge, residing in the semiconductor per unit area is then
it is straightforward to extract the DOS function, hŽ E ., using a deconvolution technique. 2.2. Static MIS methods
`
Qsc s
H0 r
sc
Ž x.d x
It is possible to write for the metal-insulator– semiconductor ŽMIS. system the general equation for the surface potential Vs s V Ž x s 0.
`
H0 HEh Ž E . f
s ye
E F , E q eV Ž x .
yf Ž E Fo , E . 4 d Ed x.
Ž 2.
These two equations, Eqs. Ž1. and Ž2., represent the computational framework of all spectroscopical methods, based on the existence of space-charge. The main idea of the corresponding experiments is changing on purpose either the quasi-Fermi level position E F Žin methods like SCLC and using Eq. Ž1. or the potential V Ž x . in FE conductivity or all sorts of space-charge capacitance measurements Žusing then Eq. Ž2.., and thus effectively scanning the DOS.
d n s Ž Vs .
s
d Ž eVs . s
HEh Ž E . f Ž E q eV .
1 y f Ž E q eVs . d E
s
´ o ´ i2
2
ed 2´s
ž
2
dVG
y1
dVs
q Ž VG y Vs .
/
d 2 VG
.
dVs 2
Ž 4.
For the capacitance–voltage Ž C–V . dependencies we have w6x 2
d n sc Ž Vs . 2. Theory
s
d Ž eVs .
´o ´ i 2 2
e d
´i
2
1
ž / ´s
2.1. Space-charge-limited currents It is possible to prove w5x that for the increment of the concentration of carriers forming space-charge at the anode, i.e., at x s L, it is possible to write using experimental data for the increment of the spacecharge
d
d n sL s Hh Ž E . f Ž E, E F . 1 y f Ž E, EF . d E EF
ž / kT
=
Ci
1 ee o U 2 m y 1 2
kT eL
m
2
Ž1qC . ,
Ž 3.
where m s dŽln j .rdŽlnU . is the slope of the current–voltage characteristic, and the parameter C represents the correction stemming from the second order derivatives of the jŽU . characteristic, ee 0 is the permittivity, e is the elementary charge, L is the sample thickness and U is the applied voltage. Then,
ž / ž /
d q VG
Ci
Cm Ci
dVs
,
Ž 5.
where Cm and Ci are the capacitance of the semiconductor and the insulator, respectively, and VG is the applied voltage. For the FE we then have w6x d ns
s
Cm
Cm
1y
d Ž eVs .
s
´o ´ i2 ´s e 2 d 2
=
eVG
2
ž / kT
2 m2
Ž m2 q mX .
2
y
mmX Ž m2 y mX .
Ž m2 q mX .
3
,
Ž 6. w h ere m s d Ž ln j . r d Ž ln V G .. an d m ’ s dŽ m.rdŽlnVG .. are the first Žsecond. derivatives of
F. Schauerr Journal of Non-Crystalline Solids 227–230 (1998) 659–663
661
the jŽ VG . characteristic, B is the correction stemming from the higher derivatives. It is interesting to notice that the necessity of two deconvolutions Žsee Eq. Ž2.. was effectively removed by the surface potential approach in the last two methods ŽEqs. Ž5. and Ž6...
3. Reconstruction of DOS To prove the feasibility of all the space-charge based methods, their sensitivity and resolution we present the simulation experiments. 3.1. Space-charge-limited currents Here, we test the ‘ visibility’ of the localized states distribution with respect to its energy position of the maximum situated within the energy interval Ec y Et e Ž0.2–0.8. eV. We further used Tt s 300 K, Nt s 1.10 14 cmy3 . In Fig. 1, there are the SCLC characteristics, i.e., the current–voltage jŽU . and activation energy–voltage EaŽU . dependencies expected in the experiment, obtained by the modeling procedure described in w5x. For the DOS reconstruc-
Fig. 2. Reconstructed DOS functions hŽ E . Žheavy lines. obtained using Eqs. Ž3. and Ž7. from the data in Fig. 1 compared with the input Žthin lines. —the peak positions of the DOS were chosen: 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7 eV.
tion we used Eq. Ž3. and for the corresponding energy axis then the expression Ea s Ec y E Fo s Ea) q q
d Ž ln mo . d Ž 1rkT .
Ž 3 y 4 m. n kT , m Ž 2 m y 1. Ž m y 1.
Ž 7.
where Ea ) is the measured activation energy of the SCLC defined Ea ) s yln jrd Ž1rkT ., n s ydŽ Ea )rkTrdŽlnU . is the slope of the activation energy–voltage characteristics. In Fig. 2 the reconstructed DOS function hŽ E . Žobtained from Eqs. Ž3. and Ž7.. is compared with the input DOS functions, all these with the position of the maximum of the DOS distribution Et as a parameter. We are able to ‘see’ and reconstruct the DOS function for the peak position Ec y Et ) 0.4 eV. For the shallower states this information cannot be extracted, as the contribution to the space-charge from the free carriers in the transport band strongly predominates. 3.2. Static MIS methods
Fig. 1. SCLC model calculations, i.e, the current–voltage jŽU . and activation energy–voltage EaŽU . characteristics for the bellshaped DOS formed by the double-exponential distributions Žsee Fig. 2. Tt s 300 K, Nt s1.10 14 cmy3 , T s100 K.
For the C–V modeling ŽFigs. 3 and 4. we used two double-exponentials peaked at Ec y Et s 0.6 and 0.9 eV, Tt s 400 K, N s 1.10 17 cmy3 . In Fig. 3 there are the C–V characteristics, obtained by the modeling procedure described in Ref. w6x, expected in the
F. Schauerr Journal of Non-Crystalline Solids 227–230 (1998) 659–663
662
Fig. 3. C – V model calculations, i.e., the dependence of the quotient of the capacitance of the semiconductor Cm and that of the insulator Ci on the applied voltage VG for two bell-shaped DOS formed by double-exponential distributions peaked at Ec y Et s 0.6 and 0.9 eV, Tt s 400 K, Nt s1.10 17 cmy3 .
experiment. For the DOS reconstruction we used Eq. Ž5. and for the corresponding energy axis Žgiven by the surface potential Vs . we used the following expression. Vs Ž VG . y Vs Ž FB . s
VG
HV Ž FB . 1
ž
1y
Cm Ci
/
dVG .
Ž 8.
In Fig. 4 is then the reconstructed DOS function hŽ E . Žobtained from Eqs. Ž5. and Ž8.. compared with the input DOS function. The results of modeling of the FE ŽFigs. 5 and 6., i.e., current–gate voltage jŽ VG . and activation en-
Fig. 4. Reconstructed DOS function d n s rdŽ eVs . obtained using Eqs. Ž5. and Ž8. from the data in Fig. 3 compared with the input DOS function.
Fig. 5. FE model calculations, i.e, the dependencies current–gate voltage jŽ VG . and activation energy–gate voltage EaŽ VG . on the applied gate voltage VG for two bell-shaped DOS formed by double-exponential distributions, Tt s 400 K.
ergy–gate voltage EaŽ VG . are in Fig. 5 for the DOS in Fig. 6. For the reconstruction, Eq. Ž6. is used and for the corresponding energy axis Žthe dominant energy, i.e., the surface potential Vs and its statistical shift T dVsrdT . then w6x, eVs q
d eVs dT
s Eam y Eao q Ea ,
Ž 9.
where Ea s ydln jrdŽ1rkT . is the activation of the FE, Eao is its value for the flat-bands, and Eam s ydln mrdŽ1rkT . is the activation energy of the slope of the current.
Fig. 6. Reconstructed DOS function d n s rdŽ eVs . obtained using Eqs. Ž6. and Ž9. from the data in Fig. 5 compared with the input DOS function hŽ E ..
F. Schauerr Journal of Non-Crystalline Solids 227–230 (1998) 659–663
4. Conclusions Ž1. The SCLC method, C–V and the FE method are spectroscopical methods based on the spacecharge existence known from the inorganic semiconductors, both crystalline and amorphous Žsee, e.g., Refs. w1–3x and the references given there.. Ž2. All these methods are after modifications highly suitable for the examination of organic semiconductors, especially for their transport and localized states elucidation. References w1x D.S. Misra, A. Kumar, S.C. Agarwal, J. Non-Cryst. Solids 76 Ž1985. 216.
663
w2x M.A. Lampert, P. Mark, Current Injection in Solids, Academic Press, New York, 1970. w3x J.D. Cohen, Semiconductors and semimetals, 21 C Hydrogenated Amorphous Silicon, Academic Press, New York, 1984. w4x G. Horowitz, P. Delannoy, J. Appl. Phys. 70 Ž1991. 469. w5x F. Schauer, R. Novotny, S. Nespurek, J. Appl. Phys. 81 Ž1997. 1244. w6x F. Schauer, R. Novotny, S. Nespurek, J. Appl. Phys, to be published.
Space-charge oriented methods for studying organic amorphous semiconductors F. Schauer
)
Faculty of Chemistry, Technical UniÕersity of Brno, Veslarska 230, 637 00 Brno, Czech Republic
Abstract The purpose of the present paper is to provide the present state of art in the knowledge of the suitability of space-charge oriented methods for the density of states ŽDOS. spectroscopy in organic amorphous semiconducting materials. In the first part we intend to describe the space-charge forming in organic materials and explain the advantage of using the differential approach in the steady-state methods. The static capacitance–voltage Ž C–V ., field effect ŽFE. and space-charge-limited currents ŽSCLC. methods are presented with the unifying approach for organic semiconductors. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Space-charge; Amorphous semiconductors; Density of states
1. Introduction The need for examining the density of localized electron states of disordered inorganic semiconducting Žmainly a-Si:H and its alloys—for references, see, e.g., Ref. w1x. and organic substances has renewed the interest in the methods based on the space-charge effects. Many experimental methods based on the space-charge-limited currents ŽSCLC. w2x, space-charge capacitance w3x and field effect ŽFE. w4x occurred. Furthermore, the need for a differential method using the numerical derivative of the measured data developed w5x. In the present paper, we want to generalize the experimental methods for the description of the lo-
)
Corresponding author. Fax: q421 5 4321 1101; e-mail: [email protected].
calized electron states based on the space-charge in steady-state using the first and higher derivatives of the experimental data suitable for organic semiconductors. The space-charge volume density, rsc , in substances with an energetically distributed density of states ŽDOS. function, hŽ E ., is in general given by the contribution from all energy states
rsc Ž x . s ye n s Ž x . y n so s ye h Ž E . f E F , E q eV Ž x .
HE
yf Ž E Fo , E . 4 d E,
Ž 1.
where n sŽ x . is the total concentration of charge carriers forming space-charge at a point x and n so Ž x . is the concentration of carriers at the thermodynami-
0022-3093r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 Ž 9 8 . 0 0 1 3 3 - 1
F. Schauerr Journal of Non-Crystalline Solids 227–230 (1998) 659–663
660
cal equilibrium, E F and E Fo are the quasi- and thermodynamical Fermi levels and V Ž x . is the potential at a point x. The total space-charge, residing in the semiconductor per unit area is then
it is straightforward to extract the DOS function, hŽ E ., using a deconvolution technique. 2.2. Static MIS methods
`
Qsc s
H0 r
sc
Ž x.d x
It is possible to write for the metal-insulator– semiconductor ŽMIS. system the general equation for the surface potential Vs s V Ž x s 0.
`
H0 HEh Ž E . f
s ye
E F , E q eV Ž x .
yf Ž E Fo , E . 4 d Ed x.
Ž 2.
These two equations, Eqs. Ž1. and Ž2., represent the computational framework of all spectroscopical methods, based on the existence of space-charge. The main idea of the corresponding experiments is changing on purpose either the quasi-Fermi level position E F Žin methods like SCLC and using Eq. Ž1. or the potential V Ž x . in FE conductivity or all sorts of space-charge capacitance measurements Žusing then Eq. Ž2.., and thus effectively scanning the DOS.
d n s Ž Vs .
s
d Ž eVs . s
HEh Ž E . f Ž E q eV .
1 y f Ž E q eVs . d E
s
´ o ´ i2
2
ed 2´s
ž
2
dVG
y1
dVs
q Ž VG y Vs .
/
d 2 VG
.
dVs 2
Ž 4.
For the capacitance–voltage Ž C–V . dependencies we have w6x 2
d n sc Ž Vs . 2. Theory
s
d Ž eVs .
´o ´ i 2 2
e d
´i
2
1
ž / ´s
2.1. Space-charge-limited currents It is possible to prove w5x that for the increment of the concentration of carriers forming space-charge at the anode, i.e., at x s L, it is possible to write using experimental data for the increment of the spacecharge
d
d n sL s Hh Ž E . f Ž E, E F . 1 y f Ž E, EF . d E EF
ž / kT
=
Ci
1 ee o U 2 m y 1 2
kT eL
m
2
Ž1qC . ,
Ž 3.
where m s dŽln j .rdŽlnU . is the slope of the current–voltage characteristic, and the parameter C represents the correction stemming from the second order derivatives of the jŽU . characteristic, ee 0 is the permittivity, e is the elementary charge, L is the sample thickness and U is the applied voltage. Then,
ž / ž /
d q VG
Ci
Cm Ci
dVs
,
Ž 5.
where Cm and Ci are the capacitance of the semiconductor and the insulator, respectively, and VG is the applied voltage. For the FE we then have w6x d ns
s
Cm
Cm
1y
d Ž eVs .
s
´o ´ i2 ´s e 2 d 2
=
eVG
2
ž / kT
2 m2
Ž m2 q mX .
2
y
mmX Ž m2 y mX .
Ž m2 q mX .
3
,
Ž 6. w h ere m s d Ž ln j . r d Ž ln V G .. an d m ’ s dŽ m.rdŽlnVG .. are the first Žsecond. derivatives of
F. Schauerr Journal of Non-Crystalline Solids 227–230 (1998) 659–663
661
the jŽ VG . characteristic, B is the correction stemming from the higher derivatives. It is interesting to notice that the necessity of two deconvolutions Žsee Eq. Ž2.. was effectively removed by the surface potential approach in the last two methods ŽEqs. Ž5. and Ž6...
3. Reconstruction of DOS To prove the feasibility of all the space-charge based methods, their sensitivity and resolution we present the simulation experiments. 3.1. Space-charge-limited currents Here, we test the ‘ visibility’ of the localized states distribution with respect to its energy position of the maximum situated within the energy interval Ec y Et e Ž0.2–0.8. eV. We further used Tt s 300 K, Nt s 1.10 14 cmy3 . In Fig. 1, there are the SCLC characteristics, i.e., the current–voltage jŽU . and activation energy–voltage EaŽU . dependencies expected in the experiment, obtained by the modeling procedure described in w5x. For the DOS reconstruc-
Fig. 2. Reconstructed DOS functions hŽ E . Žheavy lines. obtained using Eqs. Ž3. and Ž7. from the data in Fig. 1 compared with the input Žthin lines. —the peak positions of the DOS were chosen: 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7 eV.
tion we used Eq. Ž3. and for the corresponding energy axis then the expression Ea s Ec y E Fo s Ea) q q
d Ž ln mo . d Ž 1rkT .
Ž 3 y 4 m. n kT , m Ž 2 m y 1. Ž m y 1.
Ž 7.
where Ea ) is the measured activation energy of the SCLC defined Ea ) s yln jrd Ž1rkT ., n s ydŽ Ea )rkTrdŽlnU . is the slope of the activation energy–voltage characteristics. In Fig. 2 the reconstructed DOS function hŽ E . Žobtained from Eqs. Ž3. and Ž7.. is compared with the input DOS functions, all these with the position of the maximum of the DOS distribution Et as a parameter. We are able to ‘see’ and reconstruct the DOS function for the peak position Ec y Et ) 0.4 eV. For the shallower states this information cannot be extracted, as the contribution to the space-charge from the free carriers in the transport band strongly predominates. 3.2. Static MIS methods
Fig. 1. SCLC model calculations, i.e, the current–voltage jŽU . and activation energy–voltage EaŽU . characteristics for the bellshaped DOS formed by the double-exponential distributions Žsee Fig. 2. Tt s 300 K, Nt s1.10 14 cmy3 , T s100 K.
For the C–V modeling ŽFigs. 3 and 4. we used two double-exponentials peaked at Ec y Et s 0.6 and 0.9 eV, Tt s 400 K, N s 1.10 17 cmy3 . In Fig. 3 there are the C–V characteristics, obtained by the modeling procedure described in Ref. w6x, expected in the
F. Schauerr Journal of Non-Crystalline Solids 227–230 (1998) 659–663
662
Fig. 3. C – V model calculations, i.e., the dependence of the quotient of the capacitance of the semiconductor Cm and that of the insulator Ci on the applied voltage VG for two bell-shaped DOS formed by double-exponential distributions peaked at Ec y Et s 0.6 and 0.9 eV, Tt s 400 K, Nt s1.10 17 cmy3 .
experiment. For the DOS reconstruction we used Eq. Ž5. and for the corresponding energy axis Žgiven by the surface potential Vs . we used the following expression. Vs Ž VG . y Vs Ž FB . s
VG
HV Ž FB . 1
ž
1y
Cm Ci
/
dVG .
Ž 8.
In Fig. 4 is then the reconstructed DOS function hŽ E . Žobtained from Eqs. Ž5. and Ž8.. compared with the input DOS function. The results of modeling of the FE ŽFigs. 5 and 6., i.e., current–gate voltage jŽ VG . and activation en-
Fig. 4. Reconstructed DOS function d n s rdŽ eVs . obtained using Eqs. Ž5. and Ž8. from the data in Fig. 3 compared with the input DOS function.
Fig. 5. FE model calculations, i.e, the dependencies current–gate voltage jŽ VG . and activation energy–gate voltage EaŽ VG . on the applied gate voltage VG for two bell-shaped DOS formed by double-exponential distributions, Tt s 400 K.
ergy–gate voltage EaŽ VG . are in Fig. 5 for the DOS in Fig. 6. For the reconstruction, Eq. Ž6. is used and for the corresponding energy axis Žthe dominant energy, i.e., the surface potential Vs and its statistical shift T dVsrdT . then w6x, eVs q
d eVs dT
s Eam y Eao q Ea ,
Ž 9.
where Ea s ydln jrdŽ1rkT . is the activation of the FE, Eao is its value for the flat-bands, and Eam s ydln mrdŽ1rkT . is the activation energy of the slope of the current.
Fig. 6. Reconstructed DOS function d n s rdŽ eVs . obtained using Eqs. Ž6. and Ž9. from the data in Fig. 5 compared with the input DOS function hŽ E ..
F. Schauerr Journal of Non-Crystalline Solids 227–230 (1998) 659–663
4. Conclusions Ž1. The SCLC method, C–V and the FE method are spectroscopical methods based on the spacecharge existence known from the inorganic semiconductors, both crystalline and amorphous Žsee, e.g., Refs. w1–3x and the references given there.. Ž2. All these methods are after modifications highly suitable for the examination of organic semiconductors, especially for their transport and localized states elucidation. References w1x D.S. Misra, A. Kumar, S.C. Agarwal, J. Non-Cryst. Solids 76 Ž1985. 216.
663
w2x M.A. Lampert, P. Mark, Current Injection in Solids, Academic Press, New York, 1970. w3x J.D. Cohen, Semiconductors and semimetals, 21 C Hydrogenated Amorphous Silicon, Academic Press, New York, 1984. w4x G. Horowitz, P. Delannoy, J. Appl. Phys. 70 Ž1991. 469. w5x F. Schauer, R. Novotny, S. Nespurek, J. Appl. Phys. 81 Ž1997. 1244. w6x F. Schauer, R. Novotny, S. Nespurek, J. Appl. Phys, to be published.