Defect Auger dissociation of exciton

2.2 Defect Auger dissociation of exciton

We start with the total Hamiltonian H = H₀ + V for the π-electrons of a conjugated polymer chain with one deep level, where the one-particle part is

H₀ =^X

µ,k

E_µ(k)a^†_µ,ka_µ,k+ E_da^†_da_d, (2.1)

and the two-body Coulomb interaction is V = 1 k is the allowed wave number in the Brillouin zone, and µ = c, v is the band index for conduction and valence bands, respectively. Eµ(k) is the band disperson. E_d is the deep level energy. ψ_µ,k(r) is the Bloch wavefunction, and ψd(r) is the deep level wavefunction. aµ,k, a^†_µ,k, and ad, a^†_d are the corresponding annihilation and creation operators. After substituting the expansion of ˆψ(r) into V , the Coulomb interaction V can be divided into two parts : V = Vf + Vd, where Vf contains only the terms with Bloch state operators, while V_d contains the terms that involve at least one defect operators. It is well known that V_f is strong in conjugated polymers and causes the large exciton binding energy of the excitons. On the other hand, the residual Coulomb interaction V_d involving scattering of Bloch states into and out of the deep level is expected to be weak. Consequently we consider the free part of the Hamiltonian as H0+ Vf, and treat Vdas the perturbation which cause transitions between degenerate eigenstates of H₀+ V_f.

2.2.1 Free carrier matrix element

Neglecting the free carrier Coulomb interaction V_f and therefore the exciton effect first. The defect Auger process is a two-body electron-hole Coulomb scattering e(k_e) + h(k_h) −→ e(d) + h(k_{f h}), in which one free electron (e) with wave number k_e drops into the deep defect level(d) while a hole (h) with wave number k_h is scattered to k_{f h}to compensate the energy lost by the elec-tron. ”f h” denotes free hole. It can be expressed by the equivalent electron-electron scattering: e_c(k_e) + e_v(−k_{f h}) −→ e(d) + e_v(−k_h), where c,v denote conduction and valence band, respectively. The transition matrix element of this process is M_e−h = hd, −k_{f h}|V_d|k_e, −k_hi, where |k⁰, ki ≡ a^†_c,k0a_v,k|gi for the initial state, and |d, ki ≡ a^†_da_v,k|gi for the final state. |gi is the ground state with filled valence band and empty conduction band. The spin indices are

28CHAPTER 2. EXCITON DISSOCIATION AND PHOTOCONDUCTIVITY omitted first, and considered afterwards. After substituting the expansion of ψ into Vˆ din Me−h, only two combinations, the direct term and the exchange term, survive. For a spin singlet initial electron-hole pair, the direct term is (Fig. 2.1(a)) and the exchange term is (Fig. 2.1(b))

ME(ke, kh, kf h) = 1 The r₁ and r₂ integrals are performed in Appendix A. After some approxi-mations, the final results are

M_D(K, k_h, k_{f h}) = √αc

K ≡ k_e+ k_h is the total momentum of the electron-hole pair in the initial state divided by ¯h. R_dis the position of the defect. U is the on-site Coulomb repulsion energy for the direct term. For the exchange term, the matrix element is reduced by an overall factor γ, as defined in Eq.(A.9). ² is the effective dielectric constant along the chain. a is the lattice constant. N is the total number of repeat unit of the chain. The expression for the overlaps α_c,v between the defect and Bloch states can be found in Eqs.(A.5) and (A.8) within the ”zero-radius potential” approximation. π/a is the wave number at the direct band gap. For a triplet pair, the result of ME is zero. We consider only the singlet pair below because it is more relevent for the PC and EL processes. Adding M_D and M_E together we get the matrix element Me−h for a electron-hole pair

2.2. DEFECT AUGER DISSOCIATION OF EXCITON 29

Due to the Coulomb attraction V_f between the electron and the hole, the elementary excitation of the free part of the Hamiltonian H₀ + V_f is no longer a free electron-hole pair but a superposition of them, i.e. the exciton state, labelled by |ex; Ki. K = k_e+ k_h is the new exciton center of mass wave number. |ex; Ki is the initial state of the dissociation process, while the final state is still |d, −k_{f h}i as in Appendix A. The exciton state |ex; Ki can be expanded as ^P_keφ(K, k_e)|k_e, k_e − Ki. The envelope function φ is approximated by a normalized Lorenzian factor[66]

φ(K, k_e) ≡ 2

W_v and W_ex are the bandwidth of the valence and exciton bands, respec-tively. a₀ is the exciton Bohr radius. In order to get the exciton matrix element, we need to multiply the matrix element for each electron-hole pair by the corresponding envelope function, and sum over all pairs with a given exciton wave number K. Matrix element M_ex^A for defect Auger dissociation of exciton through Coulomb scattering is

M_ex^A(K, k_{f h}) ≡ hd, −k_{f h}|V_d|ex; Ki =^X

The rates W^A(K) of defect Auger dissociation for initial exciton wave number K in a chain with N repeat units and one defect can be obtained by summing

30CHAPTER 2. EXCITON DISSOCIATION AND PHOTOCONDUCTIVITY over all possible final free hole momenta:

W^A(K) = 2π The δ−function imposes the energy conservation condition. Set the origin of energy at the valence band top, ∆ε is the deviation of deep level energy Ed

from the mid-gap at ¹₂ε_g. E_v(k), E_c(k) and E_ex(K) are the dispersions for the valence, conduction and exciton bands, respectively. They are approx-imated as Ev(k) = −^W₂^v − ^W₂^vcos(ka), Ec(k) = εg + ^W₂^c + ^W₂^c cos(ka), and (1/Wc+ 1/Wv)⁻¹ within the effective mass approximation. With these ex-pressions, we can change variable from k_{f h}to ε_{f h}, the final hole kinetic energy, with two-fold degeneracy at +k_{f h} and −k_{f h}. The rate W^A(K) becomes Note that when the argument of the sin function in Eq.(2.14) is zero, i.e.

k_{f h}= k_h in Eq.(2.6), m_D± meets logarithmic singularity, which is integrable in the expression for W^A(K). The rate W^Ais, however, not the most conve-nient quantity to characterize the dissociation efficiency of the defect because

2.2. DEFECT AUGER DISSOCIATION OF EXCITON 31

Parameter Value Description

a₀ 50 ˚A[67] Bohr-radius of exciton

a 6.5 ˚A Lattice constant

² 2.75[68] Single chain dielectric constant Wv 2.3 eV[68] Band width for valance band W_c 2.0 eV[68] Band width for conduction band W_ex 1.07 eV Band width for exciton band

ε_g 2.8 eV[67] Energy gap

εB 0.34 eV[67] Binding energy of exciton

U 5.1 eV[68] On-site energy

γ 0.25[68] Correction of U for exchange term τ_ph 40 fs [69] Phonon emission time

Table 2.1: All parameters, suitable for PPV, used in the calculations are listed with references given after the values.

it is inversely proportional on the chain size N. In practice, the dissociation rate 1/τ^A is equal to W^A times the number of defect in the chain, which is also proportional to N for a fixed defect density. For convenience, we define a chain size independent quantity c^A(K), the volume dissociation rate, as W^A(K)N. The actual dissociation rate 1/τ^A is therefore c^A(K) multiplied by the defect density, defined as the average number of defect per repeat unit. c^A(K) is shown in Fig. 2.2. The singularity of at K = 0 is due to the logarithmic divergence of the exchange term mE(K). Temperature(T ) dependence for the thermal averaged rate c^A(T ) can obtained by averaging c^A(K) over the exciton wavenumber K, with the Boltzman weighting factor exp −β¯h²K²/2M, where M is the sum over electron and hole masses. c^A(T ) is shown in Fig. 2.3. The values of all the parameters used in this paper are listed in Table 2.1. They are designated for PPV.

2.2.4 Capture probability for one passage

So far we suppose the center-of-mass wavefunction of the exciton is a plane wave extended all over the chain. In reality, it is more reasonable to describe the exciton as a wave packet with finite size in the real space. The wave packet diffuses randomly on the polymer chain due to thermal fluctuations.

Whether they will be captured (dissociated) by the defect they encounter depends on both the transition rate 1/τ , and the interaction time t during which the wave packet covers the defect. t is in turn determined by the

32CHAPTER 2. EXCITON DISSOCIATION AND PHOTOCONDUCTIVITY

Figure 2.2: Volume exciton dissociation rate c^A(K) (see text) for defect Auger process is plotted as a function of the exciton center-of-mass wave number K for various defect level energy ∆ε measured from the midgap. The curves for

∆ε = −0.7 and −1.0 eV stop at K ≈ ±0.5π/a and ±0.3π/a, beyond which the energy released to the free hole exceeds the valence band width.

incident group velocity v_g(K) = ∂E_ex/¯h∂K. The capture probability P^A(t) is given by P^A(t) = 1 − e^−t/τ. Note that P^A(t) = 0 for t = 0 when the exciton wave packet just starts to hit the defect, and P^A(t) ' 1 when t À τ . The interaction time t is equal to ξ/|v_g|, where ξ is the exciton wave packet size alone the chain. On the other hand, the transition rate 1/τ is equal to c^A(K)a/ξ, where c^A(K) is the volume dissociation rate, and a/ξ is the effective defect density for the wave packet. t/τ can be then replaced by c^A(K)a/|v_g(K)|, in which the unspecified exciton size ξ is cancelled. The passage capture probability is finally given by the simple result P^A(K) = 1−

exp^h−c^A(K)a/|v_g(K)|ⁱ. P^A(K) is shown in Fig. 2.4. It is close to one when the incident velocity v_g is equal to the thermal velocity of 10⁵ cm/s at room temperature. The deep levels therefore act as efficient quenching centers, which are crucial for the 1D diffusion model of PL decay dynamics.[65] P^A(K) drops for higher K because of the decrease of interaction time t for fast exciton passage.