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Thermal activation of the electronic transport in porous titanium dioxides
Materials Science and Engineering B69 – 70 (2000) 489 – 493 www.elsevier.com/locate/mseb
Thermal activation of the electronic transport in porous titanium dioxides Th. Dittrich a,*, J. Weidmann a, V.Yu. Timoshenko a,b, A.A. Petrov a,b, F. Koch a, M.G. Lisachenko b, E. Lebedev c a b
Technische Uni6ersita¨t Mu¨nchen, Physik Department E16, D-58748 Garching, Germany M.V. Lomonoso6 Moscow State Uni6ersity, Faculty of Physics, 119899 Moscow, Russia c A.F. Ioffe Physico Technical Institute, 194021 St. Petersburg, Russia
Abstract The thermal activation of the electronic transport in sintered nanoporous TiO2 (anatase, rutile) layers has been investigated by current voltage characteristics, impedance spectroscopy and time of flight techniques. The thermal activation energy of the electrical conductivity and drift mobility is about 0.8 eV independent of the phase of the titanium dioxide, the diameter of the TiO2 nanocrystals and of the absolute value of the electrical conductivity. We propose a model which describes the electron transport in a sintered nanoporous TiO2 network as limited by traps located in the small contact regions of the grains. Dielectric screening is discussed as a possible reason for the increase of the energy of defect levels near the conduction band which determine the Fermi-level position. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Porous semiconductors; TiO2; Electron transport
1. Introduction Porous networks of sintered TiO2 nanocrystals are important for applications in dye sensitized solar cells [1], sensors [2] or catalysis [3]. These networks of sintered nanocrystals of metal oxides can also serve as model systems for the investigation of electronic transport in highly porous semiconductors. Metal oxides have great advantages in comparison with other semiconductors as, for example, porous silicon [4]. The surface of metal oxides can be reversibly conditioned with excellent reproducibility by oxidation and reduction. Further, the density of electronic surface states in the forbidden gap is quite low for ionic crystals which become semiconductors at high temperatures [5]. Nanocrystals of metal oxides usually are produced in a sol gel process which provides a very small variation of the diameter of the nanocrystallites [6]. In the past, the electronic transport in sintered pellets [7] and in porous sintered networks [8] of TiO2 nanocrystals has been studied mostly by impedance spectroscopy (IS). Large activation energies between 0.8 [8] and 0.8–1.0 [7] eV where found in these experi* Corresponding author.
ments. This is completely different to the thermal activation of the electrical transport in bulk rutile crystals, for which the thermal activation energy decreases with increasing electrical conductivity [9]. The n-type doping of bulk TiO2 can be achieved by creating oxygen vacancies at high temperatures in an oxygen deficient atmosphere [10]. The electron drift mobility (m) in por-TiO2 (anatase) was studied by the time-of-flight (TOF) technique [11] and was found to be limited by traps [12]. However, little is known about the limiting factors of the electronic transport in porous sintered networks of TiO2 nanocrystals. This work studies the electrical conductivity (s) and the electron drift mobility (m) by current voltage characteristics (IV), IS and TOF in a wide range of temperatures for different partial pressures of oxygen. The dielectric screening for Fermi-level pinning in the surface region of TiO2 nanocrystals has been considered in the proposed model.
2. Experimental Sandwich structures of SnO2:F/por-TiO2/graphite [11] were prepared by screen printing of a paste con-
0921-5107/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 9 9 ) 0 0 2 8 1 - 0
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T. Dittrich et al. / Materials Science and Engineering B69–70 (2000) 489–493
Fig. 1. X-ray diffraction 2u curves of the investigated samples of por-TiO2 (anatase, rutile)/SnO2:F/glass.
taining TiO2 nanocrystals and terpentine oil (INAP GmbH) on the glass substrate covered with SnO2:F (TEC 15 by L.O.F.) followed by firing at 450°C in air for 30 min and subsequent screen printing and firing of the back contact. The thickness of the por-TiO2 layers was controlled with a step profiler and was varied between 5 and 20 mm. The porosity amounted to about 65% for all por-TiO2 layers. The samples were conditioned in situ at 450°C in vacuum and in ambient with pO2 =10 − 5 mbar to get rid of water bonded in the crystallites. X-ray diffraction (XRD) measurements were carried out on some SnO2:F/por-TiO2 structures. Fig. 1 shows typical XRD plots for different charges of pastes of rutile (a) and anatase (b, c). The size of the TiO2
nanocrystals (d) can be determined from the broadening of the respective peaks by using the Scherrer equation: d= Kl/b cos u, where u is the Bragg angle, l is the wavelength of the X-ray (0.154 nm), b is the FWHM of the XRD peak and K is a constant approximately to unity. The values of d varied from 6 to 60 nm for the different pastes. The dispersion in the size of the nanocrystals is about 1 and 4 nm for the pastes with the smallest and largest crystals, respectively. The IV, IS and TOF investigations were carried out in situ in a specially designed apparatus (Fig. 2). This equipment allowed measurements at temperatures (T) up to 500°C and partial pressures of oxygen (pO2) from 1 to 10 − 8 bar. The impedance and the current voltage characteristics of the nanoporous TiO2 capacitors were measured with an HP4192 impedance analyzer and a HP4140B pA meter, respectively. The value of s was obtained from IS by standard analysis of the Cole– Cole-plots [13]. An N2 laser (wavelength 337 nm, duration time 0.3 ns) was used for the excitation of the photocurrent transients in the TOF measurements. The photocurrent transients were monitored with a HP54520A oscilloscope. The area of the nanoporous TiO2 capacitors was 7× 7 and 3× 3 mm2 for the IS and TOF measurements, respectively.
3. Results and discussion Fig. 3 shows photocurrent transients of electrons in the log–log-plot of a 6 mm thick por-TiO2 (anatase) layer for different applied field voltages. An example of a current transient in the linear scale is plotted in the insert of Fig. 3. The onsets of the applied field voltage (UF) and of the laser pulse are marked. The offset current is large during the TOF measurements, i.e. the
Fig. 2. Experimental setup used for the investigation of the electrical conductivity and of the drift mobility.
T. Dittrich et al. / Materials Science and Engineering B69–70 (2000) 489–493
Fig. 3. Log – log-plot of the photocurrent transient for por-TiO2 (anatase) at different applied field voltages in the case of TOF in the space charge limited regime. The insert shows a standard photocurrent transient in the linear scale.
Fig. 4. Arrhenius plots of the drift mobility, the offset current and the electrical conductivity for different samples of por-TiO2 (anatase, rutile) at different partial pressures of oxygen.
photocurrent transients are investigated under high injection of charge carriers at the contacts. We remark that photocurrent transients could not be measured in the case of low values of UF (UF B2 V) when the injection at the contacts was negligible. We connect this behavior with the fact that a lot of traps are present in the por-TiO2. The traps are partially saturated by the injected electrons. The offset current increases with UF by potential law with a coefficient between 1.5 and 4 depending on the condition of measurement as T and pO2. This means that the electrical field is not homogeneously distributed in the sample during the measurement. Therefore, the absolute value of the drift mobility can not be determined from the TOF experiments with
491
high accuracy. However, the temperature dependence of m is not influenced by this property. The value of m can be determined from the transit time (ttr) of the electrons in the TOF measurements (m= D 2/ttrUF, where D is the thickness of the por-TiO2 layer). The peak time in the photocurrent transient (tp) for the SCLC regime gives the value of ttr (ttr =0.8tp, [14,15]). The photocurrent transients shift to shorter times with increasing UF. The value of m is practically independent of UF for UF B 10–15 V. For higher UF m starts to decrease. All temperature dependent measurements of m are carried out for UF below 10 V. We remark that photocurrent transients could be measured only for relatively high values of pO2 when the por-TiO2 is not highly reduced and the IV characteristics are not ohmic. The IS measurements were carried out for a modulation amplitude of 50 mV and without applied bias voltage, i.e. without injection of charge carriers at the contacts. The IV characteristics were ohmic for highly reduced por-TiO2 (pO2 B 0.01 mbar) when the electrical conductivity is high and no hysteresis was observed for the cyclic IV measurements. Fig. 4 summarizes the main results of the temperature dependence of m and s for the TOF, IS and IV investigations. The results are shown for different values of pO2, for anatase and rutile and for different diameters the nanocrystals. The Arrhenius plots show that the electrical transport is thermally activated with EA = 0.75–0.8 eV for m and EA = 0.8–0.9 eV for s. The activation energy is independent of the phase of the TiO2, the diameter of the nanocrystals and of the value of pO2. The value of EA is only slightly lower under high injection at the contacts. Since EA(m): EA(s) the thermal activation of the free carrier concentration is much lower than EA determined in our experiments. The exponential prefactor of m is in the range between 30 and 600 cm2 Vs − 1 depending on pO2 (see for details [12]). These are characteristic values for trap limited transport. The expression for the trap limited mobility is: m =Nt/NCm0 exp (− EA/kT) [16], where Nt, NC, m0 and k are the concentration of the limiting traps, the density of states at the transport level (conduction band in the case of TiO2), the drift mobility in the conduction band and the Boltzmann constant, respectively. For comparison, the electron mobility is of order 1–1000 for anatase single crystals [17] and 0.1–1 for rutile single crystals [18]. It is well known that the activation energy of s for bulk rutile decreases with increasing doping level [9]. In contrast, the thermal activation energy of s and m of por-TiO2 is independent of the absolute value of s, i.e. of the doping level. The constant thermal activation energy of the electronic transport in por-TiO2 should be induced by pinning of the Fermi-level in the surface region of nanocrystals. Fig. 5(b) shows a possible
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model of the electron transport. An interparticle barrier is formed due to a high doping level in the reduced bulk of the TiO2 nanocrystals and Fermi-level pinning in the surface region. The transport cannot be limited by the barrier because EA is independent of s, d and the phase of TiO2. Defects in the near surface layers of the nanoparticles can be suggested as most probable candidates for the limitation of the carrier transport in por-TiO2. These defects are of the same nature for rutile and anatase and are probably localized near the surface of the nanoparticles. TiO2 behaves as an n-type semiconductor due to oxygen vacancies in the crystal structure [19]. The structural disorder increases with increasing doping level as shown by optical transmission experiments [20]. Therefore, a common property of all investigated por-TiO2 layers under all conditions is structural disorder in the nanocrystals. Defects being related to disorder are responsible for the exponential absorption tails. The exponential absorption tails increase strongly with increasing temperature [21]. The question is how defects being related to structural disorder of the bulk TiO2 can pin the Fermi-level selectively in the subsurface and/or surface regions at 0.8 eV below the conduction band edge. The Coulomb interaction in thin semiconductor filaments [22] and in small semiconductor nanoparticles [23] increases rapidly with decreasing diameter of the semiconductor nanostructures due to surface polarization effects when the dielectric constant inside (oin) the semiconductor nanostructure is larger than oout. For example, the increase of the Coulomb potential of a charge in the center of a semiconductor sphere (8in) is given by the following expression: 8in =2e(oin − oout)/
(oinooutd), where e is the elementary charge. The potential 8in shifts the ionization energy of impurities which can be easily calculated by using the Schro¨dinger equation for a charge in the center of the sphere. The value of the ionization energies for a hydrogenic impurity in the center of the sphere are 0.075 and 0.75 eV for d= 40 and 4 nm, respectively. As a very rough estimate, the traps limiting the drift mobility are located in a subsurface region of a thickness of about 2–4 nm. The dielectric constant of the ambient should, following the expression for 8in, sensitively influence the Coulomb potential and the ionization energy of trapped carriers. Hence, the electron drift mobility and its activation energy should decrease for increasing oout. This should be studied thoroughly in future experiments. But there are two serious experimental facts which support our model. First, it was found experimentally that EA is much lower than 0.8 eV when the surface region is not free from bounded water [11]. Second, the electron drift mobility measured by TOF (m:5× 10 − 6 cm2 Vs − 1, oout = 1) is lower by about two orders of magnitude than m deduced from the diffusion coefficient of electrons in dye sensitized injection solar cells based on por-TiO2 (oout = 36.6 for acetonitrile [24]) for which m= 6× 10 − 4 cm2 Vs − 1 [25]. For comparison, oin is 31 ( ) [26] or 48 (II) [27] a for single crystal of anatase. Fig. 5(c) shows the situation for electron transport in the case when oout = oin and EA is much lower than 0.8 eV.
4. Conclusions We found a high thermal activation energy of the electrical conductivity in por-TiO2 (anatase, rutile) which is determined by the thermal activation of the drift mobility. The drift mobility is limited by traps while the responsible trap states pin the Fermi-level at about 0.8 eV below the conduction band minimum and are located in a near surface region of the TiO2 nanocrystals. We propose a model in which the ionization energy of a charge located in a semiconductor nanocrystal or at the surface of the nanocrystal is modified by dielectric screening and limits the electronic transport.
Acknowledgements
Fig. 5. (a) Model of the electron transport through a basic unit of a porous TiO2 sample. (b) Band diagram of (a), if the dielectric constant of the TiO2 nanocrystal is much larger than the dielectric constant of the ambient. (c) Band diagram of (a), if o1 = o2.
V. Yu. Timoshenko and A. A. Petrov are grateful to the Alexander von Humboldt Foundation and to the Deutsche Forschungsgemeinschaft, respectively, for support. We thank H. Schneider for carrying out the XRD measurements.
T. Dittrich et al. / Materials Science and Engineering B69–70 (2000) 489–493
References [1] O. Regan, M. Gra¨tzel, Nature 353 (1991) 49. [2] See, for example, W. Go¨pel, G. Reinhardt, in: H. Baltes, W. Go¨pel, J. Hesse (Ed.), Sensors Update, VCH, Weinheim, 1996, p. 47. [3] See, for example, M.R. Hoffmann, S.T. Martin, W. Choi, D. Bahnemann, Chem. Rev. 95 (1995) 69. [4] A.G. Cullis, L.T. Canham, P.D.J. Calcott, J. Appl. Phys. 82 (1997) 909 and Refs therein. [5] S. Kurtin, T.C. McGill, C.A. Mead, Phys. Rev. Lett. 22 (1969) 1433. [6] C. Kohrmann, D.W. Bahnemann, M.R. Hoffmann, J. Phys. Chem. 92 (1988) 5196. [7] A.M. Azad, S.A. Akbar, L.B. Younkman, M.A. Alim, J. Am. Ceram. Soc. 77 (1994) 3145. [8] Th. Dittrich, J. Weidmann, F. Koch, I. Uhlendorf, I. Lauermann, Appl. Phys. Lett., accepted. [9] D.C. Cronemeyer, Phys. Rev. 87 (1952) 876. [10] R.G. Beckenridge, W.R. Hosler, Phys. Rev. 91 (1953) 793. [11] Th. Dittrich, E.A. Lebedev, J. Weidmann, Phys. Stat. Sol. (A) 165 (1998) R5; 167 (1998) R9. [12] Th. Dittrich, J. Weidmann, E.A. Lebedev, F. Koch, 24th International Conference on Physics of Semiconductors (ICPS’24), Jerusalem, 1998, World Scientific 1999.
.
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[13] See, for example, J.R. McDonald, Impedance Spectroscopy, John Wiley & Sons, New York, 1987. [14] A. Many, S.Z. Weisz, M. Simhony, Phys. Rev. 126 (1989) 1962. [15] D.J. Gibbons, A.C. Papadakis, J. Phys. Chem. Solids 29 (1968) 115. [16] See, for example, C. Hamann, H. Burghardt, T. Frauenheim, DVW, Berlin, 1988. [17] L. Forro, O. Chauvet, D. Emin, L. Zuppiroli, H. Berger, F. Levy, J. Appl. Phys. 75 (1994) 633. [18] V.N. Bogomolov, E.K. Kudinov, Y.A. Firsov, Sov. Phys. Solid State 9 (1968) 2502. [19] W. Go¨pel, J.A. Anderson, D. Frankel, Jeahing, K. Phillips, J.A. Scha¨fer, G. Rocker, Surf. Sci. 139 (1984) 333. [20] Th. Dittrich, V.Yu. Timoshenko, unpublished. [21] H. Tang, F. Levy, H. Berger, P.E. Schmid, Phys. Rev. B52 (1995) 7771. [22] V.S. Babichenko, L.V. Keldysh, A.P. Silinv, Sov. Phys. Solid State 22 (1980) 723. [23] G. Allan, C. Delerue, M. Lannoo, E. Martin, Phys. Rev. B 52 (1995) 11 982. [24] D.R. Lide (Ed.), Handbook of Chemistry and Physics, CRC Press, Boca Raton, 78th edition, 1997 – 1998, pp. 6 –142. [25] A. Solbrandt, H. Lindstro¨m, H. Rensmo, S.-E. Lindquist, S. So¨dergren, J. Phys. Chem. B 101 (1997) 2514. [26] S. Roberts, Phys. Rev. 76 (1949) 1215. [27] A. Eucken, A. Bu¨chner, Z. Phys. Chem. B 27 (1935) 321.
Thermal activation of the electronic transport in porous titanium dioxides Th. Dittrich a,*, J. Weidmann a, V.Yu. Timoshenko a,b, A.A. Petrov a,b, F. Koch a, M.G. Lisachenko b, E. Lebedev c a b
Technische Uni6ersita¨t Mu¨nchen, Physik Department E16, D-58748 Garching, Germany M.V. Lomonoso6 Moscow State Uni6ersity, Faculty of Physics, 119899 Moscow, Russia c A.F. Ioffe Physico Technical Institute, 194021 St. Petersburg, Russia
Abstract The thermal activation of the electronic transport in sintered nanoporous TiO2 (anatase, rutile) layers has been investigated by current voltage characteristics, impedance spectroscopy and time of flight techniques. The thermal activation energy of the electrical conductivity and drift mobility is about 0.8 eV independent of the phase of the titanium dioxide, the diameter of the TiO2 nanocrystals and of the absolute value of the electrical conductivity. We propose a model which describes the electron transport in a sintered nanoporous TiO2 network as limited by traps located in the small contact regions of the grains. Dielectric screening is discussed as a possible reason for the increase of the energy of defect levels near the conduction band which determine the Fermi-level position. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Porous semiconductors; TiO2; Electron transport
1. Introduction Porous networks of sintered TiO2 nanocrystals are important for applications in dye sensitized solar cells [1], sensors [2] or catalysis [3]. These networks of sintered nanocrystals of metal oxides can also serve as model systems for the investigation of electronic transport in highly porous semiconductors. Metal oxides have great advantages in comparison with other semiconductors as, for example, porous silicon [4]. The surface of metal oxides can be reversibly conditioned with excellent reproducibility by oxidation and reduction. Further, the density of electronic surface states in the forbidden gap is quite low for ionic crystals which become semiconductors at high temperatures [5]. Nanocrystals of metal oxides usually are produced in a sol gel process which provides a very small variation of the diameter of the nanocrystallites [6]. In the past, the electronic transport in sintered pellets [7] and in porous sintered networks [8] of TiO2 nanocrystals has been studied mostly by impedance spectroscopy (IS). Large activation energies between 0.8 [8] and 0.8–1.0 [7] eV where found in these experi* Corresponding author.
ments. This is completely different to the thermal activation of the electrical transport in bulk rutile crystals, for which the thermal activation energy decreases with increasing electrical conductivity [9]. The n-type doping of bulk TiO2 can be achieved by creating oxygen vacancies at high temperatures in an oxygen deficient atmosphere [10]. The electron drift mobility (m) in por-TiO2 (anatase) was studied by the time-of-flight (TOF) technique [11] and was found to be limited by traps [12]. However, little is known about the limiting factors of the electronic transport in porous sintered networks of TiO2 nanocrystals. This work studies the electrical conductivity (s) and the electron drift mobility (m) by current voltage characteristics (IV), IS and TOF in a wide range of temperatures for different partial pressures of oxygen. The dielectric screening for Fermi-level pinning in the surface region of TiO2 nanocrystals has been considered in the proposed model.
2. Experimental Sandwich structures of SnO2:F/por-TiO2/graphite [11] were prepared by screen printing of a paste con-
0921-5107/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 9 9 ) 0 0 2 8 1 - 0
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T. Dittrich et al. / Materials Science and Engineering B69–70 (2000) 489–493
Fig. 1. X-ray diffraction 2u curves of the investigated samples of por-TiO2 (anatase, rutile)/SnO2:F/glass.
taining TiO2 nanocrystals and terpentine oil (INAP GmbH) on the glass substrate covered with SnO2:F (TEC 15 by L.O.F.) followed by firing at 450°C in air for 30 min and subsequent screen printing and firing of the back contact. The thickness of the por-TiO2 layers was controlled with a step profiler and was varied between 5 and 20 mm. The porosity amounted to about 65% for all por-TiO2 layers. The samples were conditioned in situ at 450°C in vacuum and in ambient with pO2 =10 − 5 mbar to get rid of water bonded in the crystallites. X-ray diffraction (XRD) measurements were carried out on some SnO2:F/por-TiO2 structures. Fig. 1 shows typical XRD plots for different charges of pastes of rutile (a) and anatase (b, c). The size of the TiO2
nanocrystals (d) can be determined from the broadening of the respective peaks by using the Scherrer equation: d= Kl/b cos u, where u is the Bragg angle, l is the wavelength of the X-ray (0.154 nm), b is the FWHM of the XRD peak and K is a constant approximately to unity. The values of d varied from 6 to 60 nm for the different pastes. The dispersion in the size of the nanocrystals is about 1 and 4 nm for the pastes with the smallest and largest crystals, respectively. The IV, IS and TOF investigations were carried out in situ in a specially designed apparatus (Fig. 2). This equipment allowed measurements at temperatures (T) up to 500°C and partial pressures of oxygen (pO2) from 1 to 10 − 8 bar. The impedance and the current voltage characteristics of the nanoporous TiO2 capacitors were measured with an HP4192 impedance analyzer and a HP4140B pA meter, respectively. The value of s was obtained from IS by standard analysis of the Cole– Cole-plots [13]. An N2 laser (wavelength 337 nm, duration time 0.3 ns) was used for the excitation of the photocurrent transients in the TOF measurements. The photocurrent transients were monitored with a HP54520A oscilloscope. The area of the nanoporous TiO2 capacitors was 7× 7 and 3× 3 mm2 for the IS and TOF measurements, respectively.
3. Results and discussion Fig. 3 shows photocurrent transients of electrons in the log–log-plot of a 6 mm thick por-TiO2 (anatase) layer for different applied field voltages. An example of a current transient in the linear scale is plotted in the insert of Fig. 3. The onsets of the applied field voltage (UF) and of the laser pulse are marked. The offset current is large during the TOF measurements, i.e. the
Fig. 2. Experimental setup used for the investigation of the electrical conductivity and of the drift mobility.
T. Dittrich et al. / Materials Science and Engineering B69–70 (2000) 489–493
Fig. 3. Log – log-plot of the photocurrent transient for por-TiO2 (anatase) at different applied field voltages in the case of TOF in the space charge limited regime. The insert shows a standard photocurrent transient in the linear scale.
Fig. 4. Arrhenius plots of the drift mobility, the offset current and the electrical conductivity for different samples of por-TiO2 (anatase, rutile) at different partial pressures of oxygen.
photocurrent transients are investigated under high injection of charge carriers at the contacts. We remark that photocurrent transients could not be measured in the case of low values of UF (UF B2 V) when the injection at the contacts was negligible. We connect this behavior with the fact that a lot of traps are present in the por-TiO2. The traps are partially saturated by the injected electrons. The offset current increases with UF by potential law with a coefficient between 1.5 and 4 depending on the condition of measurement as T and pO2. This means that the electrical field is not homogeneously distributed in the sample during the measurement. Therefore, the absolute value of the drift mobility can not be determined from the TOF experiments with
491
high accuracy. However, the temperature dependence of m is not influenced by this property. The value of m can be determined from the transit time (ttr) of the electrons in the TOF measurements (m= D 2/ttrUF, where D is the thickness of the por-TiO2 layer). The peak time in the photocurrent transient (tp) for the SCLC regime gives the value of ttr (ttr =0.8tp, [14,15]). The photocurrent transients shift to shorter times with increasing UF. The value of m is practically independent of UF for UF B 10–15 V. For higher UF m starts to decrease. All temperature dependent measurements of m are carried out for UF below 10 V. We remark that photocurrent transients could be measured only for relatively high values of pO2 when the por-TiO2 is not highly reduced and the IV characteristics are not ohmic. The IS measurements were carried out for a modulation amplitude of 50 mV and without applied bias voltage, i.e. without injection of charge carriers at the contacts. The IV characteristics were ohmic for highly reduced por-TiO2 (pO2 B 0.01 mbar) when the electrical conductivity is high and no hysteresis was observed for the cyclic IV measurements. Fig. 4 summarizes the main results of the temperature dependence of m and s for the TOF, IS and IV investigations. The results are shown for different values of pO2, for anatase and rutile and for different diameters the nanocrystals. The Arrhenius plots show that the electrical transport is thermally activated with EA = 0.75–0.8 eV for m and EA = 0.8–0.9 eV for s. The activation energy is independent of the phase of the TiO2, the diameter of the nanocrystals and of the value of pO2. The value of EA is only slightly lower under high injection at the contacts. Since EA(m): EA(s) the thermal activation of the free carrier concentration is much lower than EA determined in our experiments. The exponential prefactor of m is in the range between 30 and 600 cm2 Vs − 1 depending on pO2 (see for details [12]). These are characteristic values for trap limited transport. The expression for the trap limited mobility is: m =Nt/NCm0 exp (− EA/kT) [16], where Nt, NC, m0 and k are the concentration of the limiting traps, the density of states at the transport level (conduction band in the case of TiO2), the drift mobility in the conduction band and the Boltzmann constant, respectively. For comparison, the electron mobility is of order 1–1000 for anatase single crystals [17] and 0.1–1 for rutile single crystals [18]. It is well known that the activation energy of s for bulk rutile decreases with increasing doping level [9]. In contrast, the thermal activation energy of s and m of por-TiO2 is independent of the absolute value of s, i.e. of the doping level. The constant thermal activation energy of the electronic transport in por-TiO2 should be induced by pinning of the Fermi-level in the surface region of nanocrystals. Fig. 5(b) shows a possible
492
T. Dittrich et al. / Materials Science and Engineering B69–70 (2000) 489–493
model of the electron transport. An interparticle barrier is formed due to a high doping level in the reduced bulk of the TiO2 nanocrystals and Fermi-level pinning in the surface region. The transport cannot be limited by the barrier because EA is independent of s, d and the phase of TiO2. Defects in the near surface layers of the nanoparticles can be suggested as most probable candidates for the limitation of the carrier transport in por-TiO2. These defects are of the same nature for rutile and anatase and are probably localized near the surface of the nanoparticles. TiO2 behaves as an n-type semiconductor due to oxygen vacancies in the crystal structure [19]. The structural disorder increases with increasing doping level as shown by optical transmission experiments [20]. Therefore, a common property of all investigated por-TiO2 layers under all conditions is structural disorder in the nanocrystals. Defects being related to disorder are responsible for the exponential absorption tails. The exponential absorption tails increase strongly with increasing temperature [21]. The question is how defects being related to structural disorder of the bulk TiO2 can pin the Fermi-level selectively in the subsurface and/or surface regions at 0.8 eV below the conduction band edge. The Coulomb interaction in thin semiconductor filaments [22] and in small semiconductor nanoparticles [23] increases rapidly with decreasing diameter of the semiconductor nanostructures due to surface polarization effects when the dielectric constant inside (oin) the semiconductor nanostructure is larger than oout. For example, the increase of the Coulomb potential of a charge in the center of a semiconductor sphere (8in) is given by the following expression: 8in =2e(oin − oout)/
(oinooutd), where e is the elementary charge. The potential 8in shifts the ionization energy of impurities which can be easily calculated by using the Schro¨dinger equation for a charge in the center of the sphere. The value of the ionization energies for a hydrogenic impurity in the center of the sphere are 0.075 and 0.75 eV for d= 40 and 4 nm, respectively. As a very rough estimate, the traps limiting the drift mobility are located in a subsurface region of a thickness of about 2–4 nm. The dielectric constant of the ambient should, following the expression for 8in, sensitively influence the Coulomb potential and the ionization energy of trapped carriers. Hence, the electron drift mobility and its activation energy should decrease for increasing oout. This should be studied thoroughly in future experiments. But there are two serious experimental facts which support our model. First, it was found experimentally that EA is much lower than 0.8 eV when the surface region is not free from bounded water [11]. Second, the electron drift mobility measured by TOF (m:5× 10 − 6 cm2 Vs − 1, oout = 1) is lower by about two orders of magnitude than m deduced from the diffusion coefficient of electrons in dye sensitized injection solar cells based on por-TiO2 (oout = 36.6 for acetonitrile [24]) for which m= 6× 10 − 4 cm2 Vs − 1 [25]. For comparison, oin is 31 ( ) [26] or 48 (II) [27] a for single crystal of anatase. Fig. 5(c) shows the situation for electron transport in the case when oout = oin and EA is much lower than 0.8 eV.
4. Conclusions We found a high thermal activation energy of the electrical conductivity in por-TiO2 (anatase, rutile) which is determined by the thermal activation of the drift mobility. The drift mobility is limited by traps while the responsible trap states pin the Fermi-level at about 0.8 eV below the conduction band minimum and are located in a near surface region of the TiO2 nanocrystals. We propose a model in which the ionization energy of a charge located in a semiconductor nanocrystal or at the surface of the nanocrystal is modified by dielectric screening and limits the electronic transport.
Acknowledgements
Fig. 5. (a) Model of the electron transport through a basic unit of a porous TiO2 sample. (b) Band diagram of (a), if the dielectric constant of the TiO2 nanocrystal is much larger than the dielectric constant of the ambient. (c) Band diagram of (a), if o1 = o2.
V. Yu. Timoshenko and A. A. Petrov are grateful to the Alexander von Humboldt Foundation and to the Deutsche Forschungsgemeinschaft, respectively, for support. We thank H. Schneider for carrying out the XRD measurements.
T. Dittrich et al. / Materials Science and Engineering B69–70 (2000) 489–493
References [1] O. Regan, M. Gra¨tzel, Nature 353 (1991) 49. [2] See, for example, W. Go¨pel, G. Reinhardt, in: H. Baltes, W. Go¨pel, J. Hesse (Ed.), Sensors Update, VCH, Weinheim, 1996, p. 47. [3] See, for example, M.R. Hoffmann, S.T. Martin, W. Choi, D. Bahnemann, Chem. Rev. 95 (1995) 69. [4] A.G. Cullis, L.T. Canham, P.D.J. Calcott, J. Appl. Phys. 82 (1997) 909 and Refs therein. [5] S. Kurtin, T.C. McGill, C.A. Mead, Phys. Rev. Lett. 22 (1969) 1433. [6] C. Kohrmann, D.W. Bahnemann, M.R. Hoffmann, J. Phys. Chem. 92 (1988) 5196. [7] A.M. Azad, S.A. Akbar, L.B. Younkman, M.A. Alim, J. Am. Ceram. Soc. 77 (1994) 3145. [8] Th. Dittrich, J. Weidmann, F. Koch, I. Uhlendorf, I. Lauermann, Appl. Phys. Lett., accepted. [9] D.C. Cronemeyer, Phys. Rev. 87 (1952) 876. [10] R.G. Beckenridge, W.R. Hosler, Phys. Rev. 91 (1953) 793. [11] Th. Dittrich, E.A. Lebedev, J. Weidmann, Phys. Stat. Sol. (A) 165 (1998) R5; 167 (1998) R9. [12] Th. Dittrich, J. Weidmann, E.A. Lebedev, F. Koch, 24th International Conference on Physics of Semiconductors (ICPS’24), Jerusalem, 1998, World Scientific 1999.
.
493
[13] See, for example, J.R. McDonald, Impedance Spectroscopy, John Wiley & Sons, New York, 1987. [14] A. Many, S.Z. Weisz, M. Simhony, Phys. Rev. 126 (1989) 1962. [15] D.J. Gibbons, A.C. Papadakis, J. Phys. Chem. Solids 29 (1968) 115. [16] See, for example, C. Hamann, H. Burghardt, T. Frauenheim, DVW, Berlin, 1988. [17] L. Forro, O. Chauvet, D. Emin, L. Zuppiroli, H. Berger, F. Levy, J. Appl. Phys. 75 (1994) 633. [18] V.N. Bogomolov, E.K. Kudinov, Y.A. Firsov, Sov. Phys. Solid State 9 (1968) 2502. [19] W. Go¨pel, J.A. Anderson, D. Frankel, Jeahing, K. Phillips, J.A. Scha¨fer, G. Rocker, Surf. Sci. 139 (1984) 333. [20] Th. Dittrich, V.Yu. Timoshenko, unpublished. [21] H. Tang, F. Levy, H. Berger, P.E. Schmid, Phys. Rev. B52 (1995) 7771. [22] V.S. Babichenko, L.V. Keldysh, A.P. Silinv, Sov. Phys. Solid State 22 (1980) 723. [23] G. Allan, C. Delerue, M. Lannoo, E. Martin, Phys. Rev. B 52 (1995) 11 982. [24] D.R. Lide (Ed.), Handbook of Chemistry and Physics, CRC Press, Boca Raton, 78th edition, 1997 – 1998, pp. 6 –142. [25] A. Solbrandt, H. Lindstro¨m, H. Rensmo, S.-E. Lindquist, S. So¨dergren, J. Phys. Chem. B 101 (1997) 2514. [26] S. Roberts, Phys. Rev. 76 (1949) 1215. [27] A. Eucken, A. Bu¨chner, Z. Phys. Chem. B 27 (1935) 321.