plane depends, of course, on the size of the ligands surrounding the metal ion. For large ligands it is impossible for the wave function of the metal d-electron and the polymer π-d-electrons to overlap and have exchange coupling.
So we concentrate on the case of metal complexes with smaller ligands like halogen atoms like MnCl−4. In order to show the strong effect of Rd on the exchange coupling, we plot in Fig. 3.5 the system-size independent exchange integral NVk,k0 with k = k0 = 0 and the volume intersystem crossing rate wT 2S2 as a function of Rd. The exchange integral NV0,0 varies from 0.2 eV to 0.08 eV as Rd increase from 1.5 ˚A to 2 ˚A. This is close to exchange integral of 0.18 to 0.26 eV for the substitutional magnetic ion in inorganic semiconductors[91] whose bond length is around 2 ˚A. In our case the dopant is not chemically bonded to the polymer chain and the distance is larger. As Rd increases and the wave function overlap diminishes both NV0,0 and wT 2S2
decay exponentially as expected. For the small ligand complexes the size of the whole complex is around 3-4 ˚A, so in the following we take Rdto be fixed at 3 ˚A unless otherwise specified. For Rd beyond 3.5 ˚A the effect of the dopant becomes negligible. In Fig. 3.6, the thermally averaged intersystem crossing rates γS1T 1 and γT 2S2are shown as a function of temperature for var-ious ∆ES2T 2. γT 2S2(T ) increases as T increases or ∆ES2T 2 decreases because more carriers can be thermally excited above the small barrier between S2 and T 2. It is generally agreed that the energy splitting between T 2 and S2 is smaller than 0.1 eV, but there is an uncertainty on its actual value since it is difficult to be measured directly. When ∆ES2T 2 is below 0.05 eV, γT 2S2(T ) becomes higher than γS1T 1(T ). Larger γT 2S2(T ) will cause more T2 → S2 → S1 transition which increases the efficiency, as discussed in the next section.
3.4 Rate equations and the singlet formation ratio
We substitute the transition rates in section 3.3 into the rate equations for the spin-dependent exciton formation, and obtain the singlet branching ratio ηS as a function of magnetic doping density
Using the intersystem crossing rates in the last section, we can formulate a set of spin-dependent rate equations for exciton formation and calculate the singlet formation ratio ηS. The intersystem crossing rate from S2 to T 2 is much smaller than the spin and one-phonon allowed sub-picosecond relaxation from S2 to S1[92]. So S2 to T 2 transition can be neglected in the rate equations. The rate from T 1 to S1 is also negligible because S1 is
56 CHAPTER 3. HARVESTING TRIPLET EXCITONS
Figure 3.5: The effect of the metal-polymer distance Rd on the strength of the exchange coupling is shown. Both the exchange integral and the resulting intersystem crossing rate wT 2S2 decrease exponentially as Rd increases due to reduced wavefunction overlap. For wT 2S2 the temperature is 300K, there is no energy splitting between S2 and T2 levels, and the total spin S of the metal ion is 5/2.
3.4. RATE EQUATIONS 57
Figure 3.6: The volume intersystem crossing rate wS1T 1and wT 2S2for various T2/S2 energy splitting ∆ES2T 2 are plotted as functions of temperature T . The total spin quantum number S for the metal ions is assumed to be 5/2.
The actual transition rate is the volume rate times the number of dopant per repeat unit. wS1T 1 is independent of T because the T1/S1 energy splitting is much large than the thermal energy.
much higher in energy[75, 76]. Due to the fast relaxation between S2 and S1 there is no need to distinguish them in the rate equation. So the total singlet exciton density is labeled as NS. NT 2 and NT 1 are the densities for T 2 and T 1 triplet excitons.
The considerations in the previous sections are restricted to one single perfect polymer chain. In practice, the number of repeat unit Nc of a con-jugation segment is about ten[93]. Instead of delocalized in an infinite per-fect chain, the excitons hop among the conjugation segments due to the Forster or Dexter energy transfer mechanisms. The time scale τtr for the transfer is sub-picosecond[92]. For an exciton in a conjugation segment with Nc = 10 and containing a dopant, the intersystem crossing time is about 10/wT 2S2 ∼ 100 ps, which is much slower than the transfer time. In other words, the excitons experience many segments and sample an averaged dop-ing concentration. Consider a finite time ∆t which is much longer than τtr but much shorter than the intersystem crossing time. Within ∆t the excitons visit ∆t/τtr of conjugation segments. For each segment the proba-bility that there is a dopant is NcNd. If the exciton is in one such segment, the probability that it makes an intersystem crossing is τtrw/Nc. So the
58 CHAPTER 3. HARVESTING TRIPLET EXCITONS probability ∆P that an exicton makes an intersystem crossing within ∆t is (∆t/τtr)(NcNd)(τtrw/Nc) = Ndw∆t. Therefore the averaged effective inter-system crossing rate is ∆P/∆t = wNd, irrespective of the conjugation length Nc. The relation γ = wNd in the previous section between the actual in-tersystem crossing rate γ and the volume rate w is thus justified for a film of many polymer chains with finite conjugation length. The above consid-eration is not valid if the doping is so low that many excitons do not have the chance change to encounter a dopant before they decay. The exciton diffusion volume is l3ex, where lex is the exciton diffusion length and v is the primitive cell volume. The singlet exciton diffusion length is about 100 ˚A[94].
so we can take it as the lower bound for the diffusion length of the longer-lived triplet exciton. There are totally l3ex/v repeat units in the diffusion length. The probability q that at least one of the repeat units contains a dopant is 1 − exp(−Ndl3ex/v). Effectively this corresponds to a modification γT 2S2 → qγT 2S2 for the intersystem crossing rate. q is actually almost equal to one for the physically interested doping regime. Finally the rate equations for the exciton densities in a realistic polymer film are
d γS is the singlet exciton decay rate. γT is the triplet exciton decay rate.
G is the rate of the initial electron-hole capture which is assumed to be spin-independent. So one quarter of the electron-hole pairs become S2, and three quarters become T 2 initially. There are then two possible ways to go from T2. The first is T2 → T1, the second is T2 → S2 → S1. The branching ratio between the above two ways is equal to the ratio of their rates γT 2S2/γT T. Because the T2/T1 energy splitting around 1.5 eV is nine times larger than the optical phonon energy 0.17 eV, there is expected to be a phonon bottleneck[77] between T2 and T1 and the rate γT T can be as low as 2.7 × 108 s−1. There is therefore the chance to control the branching ratio by raising γT 2S2 above γT T through magnetic doping. As for the S2 exciton, there are also two possible ways to go. The first is S2 → S1 → ground state, and the second is S2 → S1 → T1. The magnetic doping enhances the second possibility. The ratio RT S between recombination through the triplet and singlet is given by [γTNT 1]/[γSNS], which can be obtained from steady state solution of the rate equation
3.4. RATE EQUATIONS 59
Figure 3.7: γT 2S2 and γS1T 1 have to be in the shaded region in order to raise the singlet formation ratio ηS. The straight lines A and B are mapped by changing the doping density Nd. Only in the case of line B there is a doping region for which the net effect of magnetic dopants is positive to the yield.
RT S ≡ γTNT 1 γSNS
=
3
4(1 + qγT 2S2/γT T)−1
n1
4 +34[1 + γT T/(qγT 2S2)]−1o(1 + γS1T 1/γS)−1 +(1 + γS/γS1T 1)−1 (1 + γS1T 1/γS)−1 . The singlet recombination branching ratio ηS is related to RT S by ηS = 1/(1 + RT S). ηS = 1/4(RT S = 3) if there is no intersystem crossing. If intersystem crossing is introduced by magnetic doping, ηS will deviate from 1/4. In order to redirect the triplet electron-hole pair into the singlet ex-citon, two requirements need to be satisfied: γT 2S2 > γT T and γS1T 1 < γS. The former condition makes the redirection from T 2 to S2 possible, while the latter condition ensures that most of the S1 excitons do not decay non-radiatively through the intersystem crossing to T 1. γT T and γS are intrinsic material parameters, while γT 2S2 and γS1T 1 are proportional to the density Nd of the magnetic doping. Increasing the doping density favors the for-mer requirement but disfavors the latter. One of the main purposes of our theoretical calculation is to decide whether these two requirements can be satisfied simultaneously or not. The picture is qualitatively illustrated in Fig. 3.7. These two requirements define a shaded region in the γT 2S2-γS1T 1 plane where the yield can be raised. As the doping density Nd changes, a
60 CHAPTER 3. HARVESTING TRIPLET EXCITONS straight line is mapped from the origin. The slope of the line γS1T 1/γT 2S2 is an intrinsic property of the d-π exchange coupling and is independent of the doping density. For larger slope we have line A and the yield can never be raised. For smaller slope we have line B and the yield can be raised in an op-timal range of doping density. The range of Nd where ηS > 1/4 corresponds to the case that the line passes through the dark region. The minimum re-quirement for the magnetic doping to work is γS1T 1/γT 2S2 < γS/γT T. Our calculation shows that it is indeed the case. ηS as a function of Ndfor various
∆ES2T 2 is plotted in Fig. 3.8. For γS = 3 × 109 s−1 and γT T = 2.7 × 108 s−1, ηS rises initially with doping density Nd due to more redirection of T 2 excitions. After reaching a maximum ηS decreases with Nd due to strong intersystem crossing from S1 to T 1. There is a range of Nd where ηS is well above 1/4. Smaller energy barrier ∆ES2T 2 implies larger γT 2S2, which in turn implies more efficient T 2 to S2 redirection (smaller slope for the line in Fig. 3.7) and stronger enhancement effect. ηS as a function of Ndfor various τT T = 1/γT T is shown in Fig. 3.9. ηS rises then decreases with Ndbecause of the same reason as in Fig. 3.8. Larger τT T (tighter triplet bottleneck) makes the redirection from T 2 to S2 easier (larger shaded region in Fig. 3.7) and the enhancement effect stronger. From Fig. 3.8 and 3.9 one can clearly see that the fast intersystem crossing between T 2 and S2 excitons due to near degeneracy and slow decay from T 2 to T 1 excitons due to large energy gap are the two key conditions for the mechanism to succeed. In Fig. 3.10, ηS is plotted for various metal-polymer vertical distance Rd. The optimum doping density shifts to higher values as Rdincreases and the strength of the dopants decreases. The physically relevant range of Rd is between 3 to 3.5 ˚A. Due to the ligand surrounding the metal ion it is impossible for have Rd < 3˚A. For Rd> 3.5 ˚Athe exchange coupling is so weak that there is no effect of doping on ηS.