Carrier Generation Process on Photoconductive Polymer Films as

Apr 12, 2005 - The boundary site number for the ion pairs in the one-dimensional model is estimated by the best fit to the experimental results. The i...
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J. Phys. Chem. B 2005, 109, 8707-8717

8707

Carrier Generation Process on Photoconductive Polymer Films as Studied by Magnetic Field Effects on the Charge-Transfer Fluorescence and Photocurrent Fuyuki Ito, Tadaaki Ikoma, Kimio Akiyama, Akira Watanabe, and Shozo Tero-Kubota* Institute of Multidisciplinary Research for AdVanced Materials, Tohoku UniVersity, Sendai 980-8577, Japan ReceiVed: October 13, 2004; In Final Form: January 5, 2005

We have studied the magnetic field effects (MFEs) on the charge-transfer fluorescence and transient photocurrent of a 1,2,4,5-tetracyanobenzene-doped poly(N-vinylcarbazole) film, which reflect the recombination and escape yields of the carriers, respectively. The recombination yield dependence of the external magnetic field (B) clearly shows two types of the MFEs, growth with increasing B due to the hyperfine mechanism (HFM) and a negative dip due to the level-crossing mechanism (LCM). On the other hand, the escape yield indicates complementary MFEs with a sharp decrease in yield with increasing B and then a positive dip. Simultaneous observation of the HFM- and LCM-MFEs proves the stepwise hole-hopping mechanism rather the long-range hole-jumping one. The quantitative analysis of the recombination and escape MFEs is performed using the stochastic Liouville equations (SLE) for a one-dimensional lattice model in which the stepwise hole hops take place between the nearest neighbor carbazole units with spin conservation. The SLE analysis provides the recombination and hole transfer rate constants of 7.0 × 107 and 4.5 × 108 s-1, respectively. The boundary site number for the ion pairs in the one-dimensional model is estimated by the best fit to the experimental results. The interionic distance of the boundary ion pair in the one-dimensional model including eight sites agrees with the thermalization distance in the Onsager model. Hence, it is concluded that the elementary processes in the Onsager model applied to molecular amorphous solids are the stepwise hole hops rather than a long-range hole jump.

Introduction Much attention has been paid to photoinduced electrontransfer reactions in nature and in artificial systems because of the fundamental interest in photochemistry and photobiology as well as more effective application of solar energy conversion. In green plants, the transport of electrons initiated by photoexcitation gives rise to a charge-separated state with a surprisingly high quantum yield, resulting in efficient conversion to chemical energy. In artificial systems, on the other hand, the discovery of photoconductivity in organic polymers and significant modification of their character by doping has created a new class of materials that combine the electronic and optical properties of semiconductors. For example, the photoconductive aromatic vinyl and π-conjugated polymers are used as photoreceptors in copy machines1 and as hole-transporting/electronblocking layers in electroluminescence devices.2 The photoconductivity in organic polymers is viewed as a two-step process consisting of generation and transportation of carriers. The carrier transport may be well-rationalized with the molecular and electronic structures of the polymers and the meso- or macroscopic morphology of materials. On carrier generation, over the past few decades, a great number of studies have been done, but the argument about the mechanism has not been settled. The initial studies on the electric field dependency of the carrier generation yield in amorphous polymer solids such as poly(N-vinylcarbazole) (PVCz, Chart 1) were reported from the late 1960s to the early 1970s.3 The * Author to whom correspondence should be addressed. Phone: +8122-217-5612. Fax: +81-22-217-5612. E-mail: [email protected].

CHART 1: Molecular Structures of Poly(N-vinylcarbazole) and 1,2,4,5-Tetracyanobenzene

SCHEME 1: Schematic Representation of the Onsager Model

observed electric field dependence was successfully interpreted in terms of the Onsager theory by Melz,4,5 who deals with the electric field-dependent separation probability (φ(E)) from an electron (e)-hole (h) pair with diffusion motion in the mutual potential of their Coulomb interaction (Scheme 1). The electric field dependence of the charge recombination yield (φ0(1 - φ(E))) was also studied and interpreted in terms of the model based on the Onsager theory.6 The separation between the electron and hole in the e-h pair (rt) is estimated to be 2-3 nm for the PVCz film, which is much longer than the usual scale of a few Å in the intermolecular electron transfer via the low-lying excited states. The e-h pair is equivalent to the ion pair (IP), and rt is sometimes called the thermalization

10.1021/jp0453212 CCC: $30.25 © 2005 American Chemical Society Published on Web 04/12/2005

8708 J. Phys. Chem. B, Vol. 109, No. 18, 2005 distance. To explain the long separation of the initial IP pair, several concepts such as thermalization, autoionization, and tunneling were proposed.4,7-9 However, some theoretical studies indicate a questionable point in the standard Onsager model and that a finite recombination rate constant results in shortening rt below 1 nm.7,10,11 Recent time-resolved spectroscopic studies on the carrier dynamics in polymeric solids have provided us new facts and rigorous discussion from the viewpoint of the molecular level. The time-resolved electron spin resonance study on the PVCz films, in which 1,2,4,5-tetracyanobenzene (TCNB) is doped as an electron acceptor (A) (Chart 1), in the nano- to microsecond region verified the presence of separated IPs that may be captured in trap sites.12 Through use of a dichroism technique in the picosecond region, Miyasaka et al. observed a relatively slow motion with a time constant of 0.5-2 ns of the hole in the acceptor-doped PVCz films and proposed a stepwise hole-hopping model rather than a long-range jumping during the thermalization.13 However, Abramavicius et al. found the fast relaxation dynamics of a hole with a few tens of picoseconds as well as the slow motion in the film of poly(Nepoxypropylcarbazole).14 On the basis of this fact, they proposed the presence of both separated and nonseparated IPs in the femtosecond region. In this way, the creation mechanism of the separated IPs originating from the Onsager model is still unclear even using modern ultrafast optical techniques. To solve this problem, therefore, it is desirable to perform further investigations offering more information with respect to the structure of the IPs. The negative and positive ions generated by photolysis of photoconductive films, namely, carriers, have a half-spin as well as a charge. Thus, the IP dynamics in these films is replaced with the dynamics of radical pairs that possess the singlet and triplet spin states. It is well-known that the radical pair dynamics including the spin-selective pair reaction such as the charge recombination are affected by an external magnetic field.15,16 In the external magnetic field effect (MFE) on the radical pair dynamics, the magnetic interaction between the radical species, which is closely related to the structure of the pair, plays an essential role. The MFE experiment is, therefore, one of the most powerful techniques to understand the IP dynamics not only in fluid solutions but also in polymeric solids.17 Concerning PVCz films, Okamoto et al. first found the MFE on the exciplex fluorescence and photocurrent due to the change in the intersystem crossing rate between the singlet and triplet states of the IPs,18 but no quantitative analysis had been done yet. In a recent communication, we have suggested two types of distant IPs generated by photolysis of the TCNB-doped PVCz film.19 The MFE measurements on the charge-transfer (CT) fluorescence were interpreted in terms of the stepwise hole hops in the film. In this paper, we have studied the MFEs not only on the CT fluorescence but also on the transient photocurrent of the TCNB-doped PVCz film because these effects reflect the geminate recombination and the escape from geminate IPs, respectively. The observed MFEs on the recombination and escape yields show a complimentary relationship. Recombination and hole-transfer rate constants are determined by the simulations of the observed MFEs using the stochastic Liouville equations (SLE), leading to the stepwise hole-hopping mechanism among the nearest carbazole (Cz) units. The Onsager model is reinterpreted based on the stepwise hole hops. Experimental Section PVCz (the mean unit number ∼5.7 × 103, Aldrich) was purified before use by reprecipitation using toluene and ethanol.

Ito et al.

Figure 1. Cross-sectional view of the film samples coated on the quartz glass bases for the measurements of the absorption and emission spectra (a) and the photocurrent (b).

Figure 2. Absorption (a) and emission (b) spectra of the TCNB-doped PVCz film observed at room temperature.

TCNB (Tokyo Kasei) recrystallized from ethanol was employed. TCNB-doped (3 mol %) PVCz films with a thickness of 5 µm were prepared by a spin-coat method on quartz glass plates for the measurements of emission and absorption (Figure 1a). For the experiments on the photocurrent, the TCNB-doped PVCz films were cast on a quartz substrate coated with a transparent electrode (indium tin oxide, ITO). To form a sandwiched type of cell as shown in Figure 1b, the surface of the TCNB-doped PVCz film was coated with a semitransparent circular Au electrode of 0.13 cm2 using a vacuum vapor deposition method. This sandwich cell has a configuration of blocking electrodes for the TCNB-doped PVCz sample. The thicknesses of the ITO electrode, the TCNB-doped PVCz film, and the Au electrode were 62 nm, 4.7, and 6.8 µm, respectively. The light to excite the sample was irradiated from the side of the quartz glass. The absorption spectrum was observed using a spectrometer (Hitachi, U-3210). To apply the magnetic field, the film samples fixed by a holder made of nonmagnetic materials were placed between the pole pieces of an electromagnet. The magnetic field strength was controlled by two variable power supplies (Takasago, GP060-30R and GP050-2) and measured with a Hall-type gaussmeter (F. W. BEL, 4048). A nanosecond Nd3+: YAG laser (Spectra-Physics, INDI-40-20) with 23 ns of the full width at half-maximum (fwhm, Figure 3c) was utilized as the excitation light for the fluorescence and photocurrent experiments. The intensity and size of the laser beam were controlled by neutral density (ND) filters, and a pinhole was placed before the sample to ensure proper illumination. Fluorescence spectra were detected using a multichannel analyzer system (Nippon Roper, ST-121) equipped with a monochromator (McPherson, 2035) and a detector diode array (Princeton Instruments, IRY700) for the UV and visible light. The decay curves of fluorescence were measured using a photomultiplier tube (Hamamatsu Photonics, R 636-10) and a digital oscilloscope (Tektronix, TDS-520D). The transient photocurrent was measured with an RC series circuit, which comprised the sample

Magnetic Field Effects on Photocurrent Generation

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Figure 4. Laser power dependence of the transient photocurrent under 2 × 104 V/cm of the TCNB-doped PVCz film observed by excitation at λ ) 532 nm. The powers of the incident light are (a) 0.5, (b) 2, (c) 5, and (d) 10 mJ/pulse. Inset indicates a plot of the current at τ ) 30 ns vs the power of the incident light.

Figure 3. Time profiles of the CT fluorescence (a) and the photocurrent under 1.3 × 105 V/cm (b) observed by illuminating the TCNB-doped PVCz film with a pulse light of λ ) 532 nm having an excitation profile in part c. The solid lines in parts a and b indicate the single-exponential functions fitting the observed profiles of τ > 30 ns by a nonlinear leastsquares method.

cell (CS), an external resistance of 50-200 Ω (RL), and a direct current source of (60 V (Matsusada Precision, PLE-1600.45).11,20 The electric field applied to the sample capacitance was 1.3 × 105 V/cm. The voltage signal at the external resistance as a function of the delay time after the laser flash (∆VL(t)) was fed into the digital oscilloscope through a low-noise preamplifier (NF, SA-230F5). The time profiles of the photocurrent were obtained from the proportional relationship of IC(t) ) ∆VL(t)/RL. The intensity fluctuation of the fluorescence and photocurrent resulting from the power stagger of the excitation light was corrected by monitoring every shot from the laser used. All measurements were performed at room temperature. Results CT Fluorescence and Photocurrent. Figure 2a shows the absorption spectrum of the TCNB-doped PVCz films. The absorption band with some vibronic structures from 311 to 360 nm is attributed to the first absorption band of π* r π of the Cz pendant molecules. By being doped with TCNB, an additional broad absorption band appears in the region longer than 380 nm. This broad CT band clearly suggests formation of the CT complex between the doped TCNB and the contact Cz, in which the charges are partly separated.6,18,21 The light irradiation at the CT band generates the contact pair of the TCNB anion (TCNB-) and the Cz cation (Cz+), which means that the singlet IP is initially generated by the selective CT excitation. Figure 2b shows the fluorescence spectrum of the doped film. The fluorescence signals, which peaked at 420 nm with the shoulder at 370 nm and peaked at 620 nm, are assigned to the emissions from the Cz excimers and the contact IPs, respectively.22-24 The emission from the contact IP, the so-called CT fluorescence, is accompanied by a charge recombination between TCNB- and Cz+. As shown in Figure 3a, the CT fluorescence shows a major fast decay within the time width of the excitation pulse and a minor slow-decay component. The latter can be fitted by an exponential function with the time constant of 50 ns. On the basis of the fact that 80% of the CT fluorescence intensity collapses within 20 ns after the peak of

Figure 5. Electrode polarity dependence of the transient photocurrent under 2 × 104 V/cm of the TCNB-doped PVCz film observed by irradiation of a pulse laser with a power of 0.7 mJ/pulse and a wavelength of λ ) 532 nm.

the excitation pulse, it is considered that the total intensity of the CT fluorescence nearly reflects the charge recombination yield of the geminate IP rather than the free IP. We also carried out the fluorescence measurements of a model system involving TCNB and ethylcarbazole in poly(methyl methacrylate) amorphous films. We detected a broad fluorescence from the contact IP by irradiating a CT complex. This fluorescence showed a single-exponential decay with a time constant of 4.7 × 107 s-1. The observed fluorescence decay may be regarded as an intrinsic recombination rate constant of the contact IP. Figure 3b shows the time profile of the transient photocurrent for the TCNB-doped PVCz film observed by the selective CT excitation at 532 nm. Rapid and slow decays followed by a plateau were observed. The photocurrent increases with the increasing incident light flux as depicted in Figure 4. The plot of the photocurrent versus the laser power shows a linear relationship with a slope of 1, indicating the one-photon process for creating carriers. As shown in Figure 5, when the polarity of the electrode was changed, the amplitude of photocurrent remained though the direction of the photocurrent was inverted. No electrode polarity dependence of the current amplitude indicates that the penetration length of the pulse light is deep enough to uniformly generate the carriers along the direction of the film thickness. This is because of the low absorbance at λ ) 532 nm of the CT complex. The experimental condition of uniform excitation excludes the possibility of the rapid diffusion of the Cz+ cation due to the gradient of the carrier density for the origin of the quick decay of the transient photocurrent in the early time.25 The wave form of the photocurrent up to 30 ns is substantially distorted by the relatively slow response time of the circuit (RLCS ≈ 15-50 ns) and the pulse width of the laser. Approximately half of the photocurrent is diminished with

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Figure 6. Magnetic field effects on the CT fluorescence spectrum (a and a′) and the transient photocurrent profile (b and b′) under 1.3 × 105 V/cm of the TCNB-doped PVCz film observed by irradiation of a pulse laser with a wavelength of λ ) 532 nm.

Ito et al. the selective CT excitation decreased in intensity by applying 10 mT (Figure 6b). As shown in Figure 6b′, this MFE disappeared after τ ) 500 ns and was independent of the mutual angle between the directions of the electric and magnetic fields. The present results clearly suggest that the decrease in the photocurrent in the early period corresponds to the reduction of the charge escape probability from the geminate IPs rather than the Hall effect. Figure 7a shows a plot of the MFE of the CT fluorescence intensity at λ ) 620 nm as a function of the magnetic field strength.29 The MFE is defined by [IB - I0]/I0. The MFE steeply increases with increasing B from the zero field, is suddenly suppressed around 40 mT, and then becomes nearly constant in the high-field region of more than 80 mT. A similar magnetic field strength dependence was also observed in the case of the CT fluorescence of 620 nm < λ < 720 nm. From the MFE observed in the low field, the B1/2 value at which the MFE reaches one-half of the maximum was estimated to be 3 mT. The B1/2 due to hyperfine interaction is theoretically approximated by the following equation.30

(

)

B2D + B2A B1/2 ) 2 BD + B A

Figure 7. Magnetic field strength dependence of the CT-fluorescence intensity observed at λ ) 620 nm (a) and the integrated intensity of the photocurrent from τ ) 20 ns to τ ) 30 ns under 1.3 × 105 V/cm (b) induced by CT excitation.

a time constant of less than 12 ns. The decay time profile in the 30-500 ns region is fitted by an exponential function with a decay constant of about 60 ns, which is convoluted by a response function with a time constant of 15 ns. This decay constant of the photocurrent is almost the same as that of the slow component of the CT fluorescence decay that is attributed to the recombination emission of a slowly formed contact IP from the free IPs.26 From a comparison with the time profiles of the CT fluorescence, the observed photocurrent before 30 ns and after 30 ns can be assigned to the decrease in the carrier number due to the charge recombination within the geminate IPs and free IPs, respectively. Therefore, the sum of the initial current within 30 ns is proportional to the escape yield of ions from the geminate IP. The remaining current signal after 500 ns, which is the plateau region, may be related to detrapping of the carrier from deep traps, because the PVCz polymers have various types of stacking conformations among the Cz pendants that can act as deep trap sites.27,28 Magnetic Field Effects. As shown in Figures 6a and 6a′, when the external magnetic field (B) of 10 mT was applied, the CT fluorescence in the selective CT excitation at λ ) 532 nm increased in intensity by ca. 12%, indicating that the geminate recombination yield was enhanced by the external magnetic field. On the other hand, the initial photocurrent in

(1)

Here, BD and BA refer to the energies of the hyperfine interactions between the nuclear spins and the unpaired electron spins on the donor and acceptor, respectively. According to the above equation, the B1/2 value of the TCNB-doped PVCz film sample is estimated to be 2.4 mT using a BD of 1.46 mT and a BA of 0.76 mT, which were determined from the EPR line widths of the PVCz cation and TCNB anion in the rigid matrix.12 From the comparison between the observed and the theoretical B1/2 values, the MFE in the low field is ascribable to the hyperfine mechanism (HFM),15,16 which arises from the longdistance IPs having negligible exchange interaction (|J|) rather than the contact IPs. On the other hand, the dip around 40 mT is interpreted as the level-crossing mechanism (LCM),15,16 which is caused by the middle-distance IPs that have an exchange interaction (|J|) of about 20 mT. Figure 7b illustrates the magnetic field dependence of the charge quantity escaping from the geminate IPs under an electric field of 1.3 × 105 V/cm, which was obtained by a numerical integration of the initial photocurrent. The escape yield decreases in the low-field region of less than 10 mT, where B1/2 is 3 mT, and a positive shallow dip appears in the middle-field range around 40 mT. Because an electric field of the 105 V/cm order increases the escape yield by only 1%,6c we can consider that the observed magnetic field dependence on the initial photocurrent is almost equivalent to that under zero electric field. No reliable magnetic field strength dependence of the photocurrent was obtained under the condition of lower electric fields because of the poor signal-to-noise ratio of the current. Although the magnitude of the MFE due to the escape yield is smaller than that of the recombination yield, the observed complimentary MFE of the charge recombination and escape yields in the present film can be qualitatively understood by the same mechanisms based on the spin dynamics within the singlet-born geminate IPs. Simulations Model. Both the HFM and LCM were observed in the MFEs upon the recombination and escape yields of the geminate IPs

Magnetic Field Effects on Photocurrent Generation

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Figure 8. Magnetic field dependence of the energy of the spin sublevels of the contact (a), middle-distance (b), and long-distance (c) IPs. The sign of the exchange interaction constant J0 was assumed to be negative.

generated in the TCNB-doped PVCz film. The MFEs due to these two mechanisms are closely related to the carrier dynamics. Radical IPs have the four spin sublevels of S, T0, T+, and T-, and the spin Hamiltonian of the uth radical IP is written by

H ˆ (u) ) -Ju

(

1

2

)

+ 2Sˆ 1Sˆ 2 +

Ai,jSˆ iIˆi,j + ∑ giµBBSˆ iz ∑ i,j i

(2)

where Sˆ i and Iˆi,j denote the electron spin operator of the ith component radical (i ) 1, 2) and the nuclear spin operator of the jth nucleus in the ith component radical, respectively. J is the exchange interaction, and Ai,j is the hyperfine coupling constant of the jth nucleus in the ith radical. The g-factor and Bohr magneton are represented by g and µB, respectively. When the distance between the component radicals in the uth IP is written by ru, the site dependence of J can be generally approximated by a simple function as follows

Ju ) J0 exp[-R(ru - r0)]

(3)

where J0 is the value at r0, the contact distance of the IPs, and R is an exponential falloff parameter. From the diagonal matrix elements of eq 2 represented using the basis set of |S〉, |T+〉, |T0〉, and |T-〉, the energy of the spin sublevels of IP can be schematically described as shown in Figure 8. The energy gap between the singlet state and triplet manifolds corresponds to 2|Ju| depending on ru. The external magnetic field makes the triplet manifolds split into three by the Zeeman interaction of gµBB. The singlet IPs can recombine to the singlet ground state, but the triplet IPs are impossible because of the strict spinselection rule. This spin selectivity for the recombination process is the origin of the observed MFEs upon the recombination and escape yields. Moreover, the spin relaxation is long enough to hold the spin conservation rule during the early period of the hole dynamics, because the interaction related to the electron spins is very small compared with the other molecular energies such as the electronic, vibrational, and librational energies. However, the spin conversion between the singlet and triplet states within the IPs can take place by the hyperfine interaction

and becomes efficient between the degenerate states. The contact IP has a large J value, and the singlet and triplet states are separated from each other in both the absence and presence of the magnetic field (Figure 8a). Hence, any MFE due to the contact IP cannot be detected in the experiments below 100 mT. On the other hand, the S-T conversion in the middledistance and long-distance IPs can be affected by the external magnetic field as shown in Figures 8b and 8c. In the middledistance IP, no singlet-triplet degeneracy exists in the absence of the magnetic field, but the field splits the triplet sublevels, and the |T+〉 or |T-〉 state, depending on the sign of J0, can meet the singlet state only at a particular field (BLC), where the Zeeman energy of gµBBLC is identical to 2|Ju|. This level crossing results in an effective S-T conversion at BLC, leading to a dip signal in the magnetic field dependence around the middle field. This MFE at BLC is called the LCM. In the longdistance IPs, the singlet and triplet states nearly degenerate at the zero field because of the negligible exchange interaction. Thus, the S-T conversion effectively occurs in the absence of B through hyperfine interactions. The Zeeman interaction removes the degeneracy of |T+〉 and |T-〉 in the triplet states, leading to a decrease in the S-T conversion. This HFM gives a characteristic MFE on the recombination and escape yield in the low-field region. The selective CT excitation first produces the contact IP. As mentioned above, the MFEs due to the LCM and HFM are caused by the middle-distance and long-distance IPs, respectively. Consequently, the observed MFEs upon the recombination and escape yields clearly suggest that at least three kinds of IPs are involved in the carrier generation and are connected with each other by the stepwise hole hop. Therefore, we made a simple model for the stepwise hole hops as shown in Scheme 2. In this one-dimensional lattice model, it is assumed that the hole hops to the nearest Cz units with the same rate constant of kH and that the recombination occurs only from the contact IP at site 1 by a rate constant of kf based on the distance dependence of the electron-transfer rate.31 We defined a boundary condition between site n and site (n + 1), by which the holes in the sites above n + 1 no longer come back to the geminate IP. It should be noted that the hopping and recombination processes maintain the electron spin quantum number. Although the TCNB-doped PVCz film is an amorphous solid, the one dimensionality in the model arises from the helically structured Cz ensemble along the methylene polymer chain that tightly binds them in a nanoscale area centered at the CT complex.32 This assumption means that the rn distance increases and the Jn value decreases with the increasing site number n. Stochastic Liouville Equation. We have quantitatively analyzed the observed MFEs with the one-dimensional lattice model based on the spin-conservative stepwise hole hops by using the SLE.19,33-35 When we take into account n sites of the radical IPs, the SLE for their dynamics is expressed using n + 1 density operators (F(n,t))

{

F˘(0,t) ) kfF(1,t)SS F˘(1,t) ) -i[H ˆ (1),F(1,t)] - kHF(1,t) + kHF(2,t) - kfFSS(1,t) F˘(2,t) ) -i[H ˆ (2),F(2,t)] + kHF(1,t) - 2kHF(2,t) + kHF(3,t) l F˘(n,t) ) -i[H ˆ (n),F(n,t)] + kHF(n - 1,t) - 2kHF(n,t) (4)

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Ito et al.

SCHEME 2: IP Dynamic Model of the Stepwise Hole

Through use of the superoperators for the spin Hamiltonian H ˆ and of the reaction operator K ˆ , the above SLE is converted to a simple form of

F3(t) ) -iH ˆ ׂF(t) + K ˆ ‚F(t)

(5)

H ˆ × is the commutator associated with the spin Hamiltonian H ˆ. In the Liouville space, H ˆ × and K ˆ have a supermatrix form of the (n + 1, n + 1)-type, in which the each matrix elements can be represented by 16 × 16 submatrixes expanded by the basis set of electron spin (see Appendix). The H ˆ × term describes the coherent singlet-triplet interconversion of the pair driven by the hyperfine and Zeeman interactions and hindered by the exchange interaction. The K ˆ term accounts for the spinconservative transfers between the nearest neighbor sites with the first-order rate constants of kf and kH. The MFEs upon the recombination (R(B)) and escape (E(B)) yields are represented as follows

{

R(B) ) E(B) )

F(0,∞)B - F(0,∞)0 F(0,∞)0 Fesc(∞)B - Fesc(∞)0 Fesc(∞)0 ‚‚ ‚

for recombination (6) for escape

Fesc(∞) ) 1 - F(0,∞)

It is sufficient to determine only the final density operators at t ) ∞ rather than their full time evolution in the absence and presence of the magnetic field. To avoid tedious numerical calculations of the simultaneous differential equations, we performed the Laplace transformation of F(n,t) on both sides of eq 5.

sF(s) - F(t ) 0) ) -iH ˆ ׂF(s) + K ˆ ‚F(s) ‚ ‚ F(s) ‚

)

(7)

∫0∞ F(t) e-st dt

On arrangement of the above equations, the SLE is consequently simplified to the simultaneous algebraic equations

(s1 + iH ˆ× - K ˆ )F(s) ) F(t ) 0)

(8-1)

ˆ )-1‚F(t ) 0) F(s) ) (s1 + iH ˆ× - K

(8-2)

Here, 1 is an identity matrix. These simultaneous linear equations for F(s) can be solved by a simple numerical diagonalization of the coefficient matrix and the initial condition of F(t ) 0). On the other hand, we need just the value of F(0,t ) ∞) to estimate R(B) and E(B). According to a well-known theorem in Laplace transformation, F(0,t ) ∞) corresponds to a limiting value of sF(0,s) as s f 0, which is lim F(0,t) ) lim sF(0,s) tf∞

sf0

(9)

There are a few magnetic constants in the spin Hamiltonian, such as the g factors, A constants, and J values. The effective g factors of the constituent ion radicals have been estimated to

be 2.003 from the EPR measurements of the individual monoradicals of PVCz+ and TCNB-.12 It is, on the other hand, hard to estimate the explicit A constants from the EPR spectra because of the lack of clear hyperfine splittings due to many magnetic atoms. Thus, we used a composite nuclear spin ˆIc and an effective hyperfine coupling constant A (Appendix). Because the spectral width of PVCz+ was much broader than that of TCNB-,12 we employed a set of A ) 0.8 mT (1.4 × 108 rad s-1) and ˆIc ) 1 for a hole, by which the statistically weighted hyperfine lines reproduce the observed EPR spectral shape of PVCz+. On the other hand, the J values for each IP site can be roughly estimated from eq 3. The |J1| value of the contact IP located at site 1 seems to be larger than the order of 1010 rad s-1 at least, although the parameters of r0, J0, and R used in the literature vary widely. Also, it can be safely assumed that the long-distance IPs with r0 g 1 nm correspond to the sites with n g 3-4 and have |J| values smaller than the hyperfine splittings (e107 rad s-1).36 The |J| value for the middle-distance IP was estimated to be 18-23 mT (3.2-4.0 × 109 rad s-1) from the observed LCM effect. Because the |J| value is between those for the contact and long-distance IPs, the detected middledistance IP is assigned to site 2. From this assignment, we used a series of the J values that were |J1| ) 1 × 1012 rad s-1, |J2| ) 3.2 or 4.0 × 109 rad s-1 and |Jng3| ≈ 0 rad s-1 for the simulations described later. Concerning the |J2| value, the better choice was made to fit the field position of the observed LCM effect. The reaction operator also has some parameters that are closely related to the hole dynamics. The rate constants (kf and kH), the IP site number (n), and the initial position of the hole just after the photoexcitation are clarified by the following simulations. kf Dependence. To minimize the interactive influence among many IP sites, we examined the rate constant dependence of the MFE based on the simplest three-site model (n ) 3), consisting of a contact IP (site 1), a middle-distance IP (site 2), and a long-distance IP (site 3). The selective excitation of the CT complex at site 0 results in the hole dynamics beginning from site 1. Because the CT complex in the ground state has a singlet character, we set F(1,0)SS ) 1 for the initial condition in the SLE. Figure 9a shows typical simulation curves of the MFEs on the recombination and escape yields calculated by using various kf values. These MFE curves of the recombination yield rapidly increase in the low field due to the HFM of the long-distance IP at site 3 and show a dip signal due to the LCM of the middle-distance IP at site 2 in the middle field. The curves remain nearly constant in the high field. The MFE simulations for the escape yield basically indicate an opposite phase that is almost a mirror image of the recombination yield MFE. The difference in the dip position comes from the difference in the adopted |J2| value. The simulations roughly reproduce the observed MFE tendency, but the calculated MFE levels are much weaker than those of the observed ones. As the recombination rate constant (kf) increases, the recombination MFE decreases while the escape MFE increases. The simulations using any kf values did not simultaneously reach the observed MFE levels for both recombination and escape. On the other hand, the calculations with kf on the order of 107 s-1 reproduced the relative levels between the recombination and escape MFEs. kH Dependence. Figure 9b shows the simulations for the kH dependence of the recombination and escape MFEs using the three-site model. In the curve of kH ) 2.0 × 108 s-1, the HFMand LCM-MFEs clearly appear in the low- and middle-field regions, respectively. Similar strong MFEs were obtained, even when we made kH a few orders smaller than 108 s-1. In the

Magnetic Field Effects on Photocurrent Generation

J. Phys. Chem. B, Vol. 109, No. 18, 2005 8713

Figure 9. Simulations of the MFE on the recombination (R(B), solid lines) and excape (E(B), broken lines) yields calculated using the SLE. The closed and open circles indicate the observed MFE on the recombination and escape yields, respectively. (a) kf dependence of the MFE simulationis using three IP sites and kH ) 4.5 × 108 s-1. (b) kH dependence of the MFE simulations using three IP sites and kf ) 7.0 × 107 s-1. (c) n dependence of the MFE simulations using kf ) 7.0 × 107 s-1 and kH ) 4.5 × 108 s-1. (d) Initial condition dependence of the MFE simulations using eight IP sites, kf ) 7.0 × 107 s-1 and kH ) 4.5 × 108 s-1.

case of kH e 2.0 × 108 s-1, the holes stay sufficiently long at each site, resulting in an efficient interaction with the external magnetic field and a relatively high level of MFEs. On the other hand, both the MFE curves of the recombination and escape were drastically changed in the range of 2.0 × 108 s-1 < kH < 1.0 × 109 s-1. In the calculation with kH ) 4.5 × 108 s-1, the HFM- and LCM-MFEs decrease and become broad. Because kH is larger than 1.0 × 109 s-1, not only the recombination MFE but also the escape MFE becomes smaller in intensity, which is a different behavior from the kf depenence. Shortening of the length of a hole’s stay at an individual site leads to the suppression of S-T conversion within the IP site and a decrease in the MFE. Also, homogeneous broadening of the spin sublevels of the IPs due to the lifetime uncertainty effect results in the broadening of the MFE curve.37 The additional simulations with various sets of the parameters clarified that the kH sensitive area, where the MFE changes considerably, is determined by the competitive kinetic balance with the recombination rate. The simulation curve using the hopping rate constant of kH ) 4.5 × 108 s-1 best reproduced the rising curvature of the HFM and the dip width of the LCM in the observed MFE. This estimated kH value coincides with the hopping rate constant obtained from the orientation relaxation of the hole in the PVCz film measured by picosecond optical spectroscopy.13 In comparison with the observed MFE, on the other hand, the levels of R(20 mT) ) 0.007 and E(20 mT) ) -0.003 that can be regarded as pure HFM-MFE are very low. This disagreement could not be settled by changing only the two kinetic parameters of kH and kf. n Dependence. To solve the problem of the underestimation of MFE, we increased the number of the long-distance IP sites giving HFM-MFE. We consecutively added the IP sites with J ) 0 next to the last site in the model. Figure 9c depicts the calculated n dependence and observed MFEs upon the recombination and escape. Both the MFEs calculated for the recombination and escape yields increase with increasing n. By

TABLE 1: Optimized Recombination and Hopping-Rate constants (kf and kH) from the Analysis Based on the SLE Calculation Using the One-Dimensional Lattice Model, the Boundary Interionic Distance (rb), the Recombination (Grec) and Escape (Gesc) Quantum Yields Derived from Those Parameters kf/s-1 (7.0 ( 1.0) ×

kH/s-1 107

(4.5 ( 1.5) ×

rb/nm 108

2-3

Frec

Fesc

0.48 ( 0.11 0.52 ( 0.11

elongation of the IP site chain in the one-dimensional lattice model, the mean lifetime of the IP is expanded, and the IP has a greater chance of being the long-distance IP at site n with n g 3. The extension of the IP lifetime results in the growth of the MFEs. Especially, the increase in the long-distance IP causes a prominent enhancement of HFM-MFE. The simulations in the case of n ) 7-8 relatively fit well with the observed MFEs of the recombination and escape. Assuming that the distance between the IP sites in the one-dimensional model is identical to that between the neighboring Cz units in the PVCz polymer chain,36 the interionic separation of the long-distance IP at site 7 or 8 is calculated to be about 2-3 nm. This estimated interionic separation at the longest-distance IP site is comparable to the thermalization distance for PVCz films determined by the Onsager analysis.4,6,38-41 Here, we reexamined the kf and kH dependences of the MFE simulations using an eight-site model. Even in the eight-site model, the tendencies of the kf and kH dependences were the same as in the case of the three-site model. Only the parameter combination of kf ) (7.0 ( 1.0) × 107 s-1, kH ) (4.5 ( 1.5) × 108 s-1, and n ) 7-8 gave good simulation curves to reasonably fit with both the observed recombination and escape MFEs. Table 1 summarizes the parameters optimized by the simulations. After all, these kinetic parameters offered the recombination and escape quantum yields under neither electric nor magnetic fields for the case of CT excitation of the TCNBdoped PVCz film.

8714 J. Phys. Chem. B, Vol. 109, No. 18, 2005 Initial Condition Dependence. On the basis of the simulations mentioned above, it was clarified that the spin-conservative stepwise hole hops give an account of the observed MFEs upon the recombination and escape yields. Finally, to investigate how the forward long-range hole jump at the first stage affects the MFE, we performed SLE calculations based on different initial conditions. The Onsager analysis of the electric field dependence of recombination in the TCNB-doped PVCz film irradiated at λ ) 540 nm suggests the generation of a long-distance IP separated by 3 nm with a quantum yield of 0.5.42 The predicted long-distance IP approximately corresponds to the IP at the site 8 with the initial condition of FSS(8,0) ) 0.5 in our onedimensional lattice model. Figure 9d illustrates the recombination and escape MFE curves calculated using the initial condition of FSS(1,0) ) 0.5 and FSS(8,0) ) 0.5, which suggests that half of the holes in the photogenerated contact IPs at site 1 remain and half of them jump very quickly by ca. 3 nm, the so-called thermalization. The holes have to undergo the stepwise hops after the initial long-range jump to contribute the MFE. In comparison with the case of the F(1,0)SS ) 1, the recombination MFE of the pseudo-thermalization condition increased while the escape MFE decreased. These curves deviate from the experimental data. The recombination and escape yields calculated under the pseudo-thermalization condition are Frec ) 0.24 and Fesc ) 0.76, respectively. In this condition, 4% of the initial long-distance IP comes back to site 0 through the stepwise hops, but more than 80% of the long-distance IPs immediately dissociate to free ions. The increment in the recombination MFE is interpreted in terms of the extension of the effective time that a hole stays in the distant IP sites until recombination. The reduction of the escape MFE is attributed to the increase in free ions. Simulations in the pseudo-thermalization condition using any other kf and kH were not in agreement with the experimental MFEs. Discussion Two types of MFEs, the HFM- and LCM-MFEs, of the recombination and escape yields were clearly observed in the selective CT excitation of the TCNB-doped PVCz film. The SLE simulations based on the one-dimensional lattice model reproduced well the observed MFEs, suggesting the stepwise hole-hopping mechanism. The obtained kinetic parameters of kf and kH are nearly in agreement with those determined by the transient dichroism measurements.13 The calculation based on the model assuming the forward long-range jump did not simulate the experimental data very well. However, the optimized one-dimensional lattice model does not exclude all of the concepts in the Onsager model that have been accepted in many studies on photoconductive polymer materials so far. The interionic separation in the longest-distance IP in the present one-dimensional lattice model coincides with the typical thermalization distance of the Onsager model reported for nonpolar organic solids. In the one-dimensional model, the longestdistance IP is located at a boundary position, where the IP undergoes the geminate recombination or free-ion production. The distant IP in the Onsager model is also a critical state that determines statistically the destination of charged particles, recombination or escape. To analyze the electric field dependence of the carrier generation efficiency in molecular amorphous solids, Meltz used the Onsager theory.4 According to Brownian motion, the Onsager theory has been developed to calculate the escape probability of a charged particle from an e-h pair in an isotropic medium under an applied electric field. The theory does not deal with how the initial e-h pair is created.

Ito et al. Originally, there is no connection between the long-range jump and the Onsager theory. The long-range jump due to the thermalization or autoionization is a possibility that has been considered in the photoconductivity in molecular solids together with the ideas of the Onsager theory. Because most of the electric field dependence of the recombination and escape in many organic amorphous solids reported after the first study can be well understood by the presence of the long-distance IP, the postulated thermalization has also been accepted for a long time without verification. The present MFE study on the dynamics on nanosecond and nanometer scales has clarified that the long-distance IP is generated by the stepwise hops rather than the long-range jump and is consistent with the Onsager model except for the thermalization process. Moreover, the SLE analysis of the MFEs has afforded the quantum yields of Frec ) 0.48 ( 0.11 and Fesc ) 0.52 ( 0.11 (Table 1) for the CT excitation of the TCNB-doped PVCz film. The slight difference between these values is the cause of the observed MFE level difference between the recombination and escape. The experiments on the electric field dependence of MFE would make it possible to compare with the related values. As far as the LCM-MFE is concerned, the observed dip position of the recombination MFE is slightly larger than that of the escape MFE. The field position of the LCM dip slightly shifts depending on the position of the film. Lack of uniformity in the amorphous films would cause the position dependence. A small conformational change in the Cz groups also induces a small difference in the interionic distance. For example, a 10 mT difference in the LCM dip position is predicted by only a 0.014 nm difference around r ) 0.83 nm, if |J0| ) 1 × 1012 rad s-1, R ) 1.7 × 10-10 m-1, and r0 ) 5 × 10-10 m are used as the parameters in eq 3. The electric field may affect the interionic distance as proposed by Ohta.43 The underlying observed MFE level around 90 mT compared with the simulated MFE would also suggest the presence of another broad LCM dip that stems from a middle-distance IP with a short ionic distance due to a conformation difference. On the other hand, the CT interaction between the IP and nearby recombined states can influence the J value in the electron-transfer system.35 The size of J due to the CT interaction depends on the free-energy difference between the IP and nearby charge-recombined states. If the applied external and inhomogeneous internal electric fields change the energy of the IP state, then the J value may vary, resulting in a shift of the LCM dip position. A study of the electric field dependence of the MFEs is now underway. Conclusion In the MFEs upon the recombination and escape yields measured by the CT excitation of the TCNB-doped PVCz film, two kinds of effects due to the HFM and LCM, which are attributed to the long- and middle-distance IPs, respectively, were simultaneously observed. The recombination and escape yields indicated the complimentary magnetic field strength dependencies. The simultaneous contributions from both the HFM- and LCM-MFEs prove the stepwise hole hops among the IPs. The observed MFEs were analyzed using the SLE based on the one-dimensional lattice model in which the hole transfers to the next Cz units while retaining the spin multiplicity of the IPs. The SLE simulations clarified how the recombination and escape MFEs depend on the kinetic rate constants and the number of the IP sites. The kinetic rate constants optimized by the comparison between the experimental and calculated data

Magnetic Field Effects on Photocurrent Generation

[

|S,I,m〉

J. Phys. Chem. B, Vol. 109, No. 18, 2005 8715

|T-,I,m+1〉

-Ju

A xI(I + 1) - m(m + 1) 2x2 m A 2 A xI(I + 1) - m(m - 1) 2x2

|T0,I,m〉

|T+,I,m-1〉

A xI(I + 1) - m(m + 1) 2x2 m+1 -gµBB A + Ju 2 A xI(I + 1) - m(m + 1) 2x2

m A 2 A xI(I + 1) - m(m + 1) 2x2

0

2x2

A xI(I + 1) - m(m - 1) 2x2 0 A xI(I + 1) - m(m - 1) 2x2 m-1 -gµBB A + Ju 2

Ju A

xI(I + 1) - m(m - 1)

]

(A2)

agreed with those reported in the literature that analyzed the ultrafast dynamics of a stepwise hopping model. On the other hand, the interionic distance of the boundary IP in the onedimensional lattice model coincides with that of the distant IP postulated in the Onsager analysis of the electric field effects upon the recombination and escape efficiencies. The present MFE experiments, therefore, led us to the conclusion that the distant IP showing the electric field dependence is generated by spin-conservative stepwise hops of a hole rather than a longrange jump. Acknowledgment. F. I. thanks Dr. Y. Kobori (University of Chicago) for his guidance on the numerical simulations using the SLE at the beginning of this study. We also thank Professor H. Miyasaka (Osaka University), Professor N. Ohta (Hokkaido University), Professor M. Tachiya (AIST), and Mr. T. Ishii (Fuji Xerox Co. Ltd.) for their stimulating discussions and valuable comments. This study was financially supported by a Grant-in-Aid for Scientific Research of No. 15310069 and on Priority Areas (417) from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of the Japanese Government, the Murata Science Foundation, and the Showa Shell Sekiyu Foundation for Promotion of Environmental Research.

after some algebra with the nuclear basis functions specified by the quantum numbers of Ic and mc for the composite nuclear spin Iˆc. Here, we neglect the superscript c in the numbers for simplicity. The reminders of matrix elements of H ˆ (u) become zero. The first term in the right-hand side of the eq 5 is defined by the next equation.H ˆ × not only means the spin Hamiltonian but

×

-iH ˆ ‚F(t) ) -i

(

H ˆ × (0) 0

(

)( )

F(0,t) ) l × F(n,t) H ˆ (n)

0 ‚‚‚

[H ˆ (0),F(0,t)]

-i

0 ‚‚‚ [H ˆ (n),F(n,t)]

0

)

(A3)

also includes a commute operation with it. F and H× are supermatrices in which each elements are represented by a vector of 16 or matrix of 16 × 16. In Liouville space, the density matrix for the site u is written by a column vector

Appendix If a molecule contains a number of equivalent nuclei (n), then their resultant nuclear spins may be treated as a composite spin Iˆc44 n

Iˆc )

∑j Iˆj

(A1)

For a set of Iˆ ) 1/2 nuclei, Iˆc takes the range of values 2n, 2n - 1, ..., 0 with the corresponding measurable z components, Iˆcz , of the values mc, mc - 1, ..., -mc. Through the use of this composite spin approximation, the nuclear spins in the molecules with many magnetic nuclei in the present paper can be also treated with one spin operator. Iˆc ) 1 was employed for the e-h pair system in the present paper. The hyperfine interactions of the electron and hole were observed in the EPR spectrum. They showed broad Gaussian spectra, which were different from the hyperfine spectrum with three equivalent lines due to I ) 1. In simulations of MFE, therefore, we weighted the contributions due to three hyperfine interactions with a 1:2:1 ratio to adjust the calculated EPR spectrum of Iˆc ) 1 to the observed Gaussian spectra, which reduces the roughness of the composite spin treatment. As shown in eq A2, the important matrix elements of the spin Hamiltonian H ˆ (u) of eq 2 that associate with the electron spin mixing are obtained described

Through the use of the matrix elements of eq A2, H ˆ × (u) in Liouville space are represented by the following matrices

( )

0 ‚‚‚ 0 H ˆ × (0) ) l ‚‚‚ l ) 0 0 ‚‚‚ 0

(A5)

8716 J. Phys. Chem. B, Vol. 109, No. 18, 2005

in the case of u * 0. X, Y, and Z are

{

X( )

A

xI(I + 1) - m(m ( 1)

2x2

m(1 ) -gµBB A + Ju, 2 m Z) A 2 Y( u

Ito et al.

K ˆ n0 ) K ˆ n1 ) ‚‚‚ ) K ˆ n(n-2) ) 0 K ˆ n(n-1) )

}

K ˆ nn ) (A7)

)( )

ˆ 0n F(0,t) K ˆ 00 ‚‚‚ K ‚‚‚ l l K ˆ ‚F(t) ) l K ˆ n0 ‚‚‚ K ˆ nn F(n,t)

(A8)

All elements in K ˆ can be expressed by the diagonal matrices as follows.

( ) ( )

( )

0 ‚‚‚ 0 ˆ 01 ) K ˆ 00 ) l ‚‚‚ l ) 0 K 0 ‚‚‚ 0

kf

0

0

‚‚‚

0

0

K ˆ 02 ) K ˆ 03 ) ‚‚‚ ) K ˆ 0n ) 0 (A9)

-kH - kf

K ˆ 10 ) 0 K ˆ 11 )

K ˆ 12 )

(

0

kH 0

ˆ 21 ) K ˆ 20 ) 0 K

K ˆ 23 )

(

kH 0

0 ‚‚‚

(

kH

kH

)

0 kH

)

‚‚‚

K ˆ 13 ) K ˆ 14 ) ‚‚‚ ) K ˆ 1n ) 0 (A10) 0

‚‚‚

0

-kH

‚‚‚

0

-kH

kH

) ( K ˆ 22 )

-2kH 0

0

0 ‚‚‚ kH

0 0

‚‚‚ -2kH

)

) (A12)

References and Notes

respectively. On the other hand, the second term in the right-hand side of eq 5 is defined by

(

(

-2kH

(

kH

0 ‚‚‚ -2kH

)

K ˆ 24 ) K ˆ 25 ) ‚‚‚ ) K ˆ 2n ) 0 (A11)

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