Lattice Relaxation Dynamics in Two-Level System

Let us discuss the relaxation process, i.e., the time evolution of the lattice within a classical treatment. Assuming that before the photoabsorption (t < 0), the electron is in the ground state 'Pg and the lattice does not move at all from the equilibrium position to=0} ({

• (1) At time t = 0 the electronic system is excited by a photon absorption from the ground state 'Pg to the excited state Ч'ех with an energy hv = £_ex - £_g. Since the duration of an optical transition is very short (~10"¹⁵ sec), and much shorter than the typical lattice oscillation periods (~10"¹³ sec), we can assume that the lattice does not move at all during the electronic transition (Franck-Condon principle).
• (2) Just after the photoabsorption, the system is in the excited adiabatic potential surface U_ex{{q_k}) and the induced force F^ek = -dU_ex/dq_k makes the lattice to move from q_k= 0 toward a new equilibrium q_k = q_k, starting with q_k[t = 0) = 0. Since all the normal modes q_k are independent with each other, the time dependence is simply given by

Here 0(t) is a step function: 0(t) = 0 for t < 0, and 9[t] = 1 for t > 0. Then the trajectory of the total motion {q_k[t)} is a Lissajous in {q_k} space, while in {;} space, the time dependence of the interaction mode int(t) is given by

If the distribution of the phonon frequency {co_k} has a finite width A со, the phase in each cos w_kt term will be quickly out of phase, resulting (?_int(f) shows a damping oscillation around Q_int (Fig. 3.3). During the dephasing time, г ~ 2л/A со, a part of the initial excess electronic energy c_ex - e_g, a quantity

is transformed into the lattice kinetic energy and dissipated over the crystal. This is why the quantity (3.8) is called the lattice relaxation energy. Concurrently the same amount of energy has been stored around the defect as a static elastic energy U_L (=£_Lr)- It should be noted that in the normal mode {q_k} space, each q_k(t) simply oscillates independently and there is no energy dissipation at all. The dimensionless quantity 5 = E_LR/ha>_k, where w_k is the average frequency, is called the Huang-Rhys factor [6], representing the average number of generated phonons (in quantum treatment) during the relaxation. It can be shown that for 5 >> 1 above classical treatment of the lattice motion can be applicable.

If there is a local lattice vibration mode (=normal mode) at the defect and a condition o_locq_loc ~ 1 is satisfied, the transient lattice vibration lasts even after t > z,

Figure 3.3 (a) Dephasing among different normal modes q_k(t). [b) Time evolution of the interaction modes <2_int(t) = Q~^.1._ka_kq_k(t): assuming a Gaussian distribution of {to_k} with Auj/ш = 0.3 [4].

Except for this local mode q_loc, the amount of lattice relaxation energy £(_R = (l/2)2_tetloc(a_k/tu_fc)² is dissipated over the crystal during 0 < t< t. The rest energy, E_LR - E_LR, which is once stored in the local vibration mode, is gradually dissipated through unharmonic terms which has not been considered in Eq. (3.2).

Let us next discuss a photoemission process from to W_g which occurs at time T_rad(:»T). The radiative electronic lifetime r_rad is usually the order of ~10"⁹ much longer than the lattice relaxation time т (~10'¹² sec). Then before the photoemission, the transient lattice vibration generated by the photoabsorption has been completely damped out. According to the Franck-Condon principle as same as above, a photon is emitted with an energy ^£ex ~^£g ~ (£lr ⁺ £4)- The energy difference between the absorption and emission is (E_LR + Urf, called the Stokes shift. Just after the photoemission, the lattice begins to move in the U_g({q_k}J space from q_k = q_k toward a new equilibrium q_k = 0. The time evolution of the lattice is shown to be

The first term has been generated by the photoabsorption at t = 0, and hence has almost damped out for t» r. The second term generated by the photoemission at t = r_rad is the main term for t > r_rad, which shows a damping oscillation around Q_int = 0

(Fig. 3.3). During the damping oscillation, the restored elastic energy U_L (=£_LR) is dissipated as a kinetic energy of the atomic motion in a period г (=2л/Дсо). After all, among the initial excess electronic energy e_ex - e_g, the amount of ^lr- is dissipated during the relaxation triggered by the first transition (absorption) and the amount of U_L (=£_LR) is dissipated during the relaxation after the second transition (emission).

Photoabsorption and Photoemission (Classical and Statistical Treatment)

So far, we have discussed the lattice dynamics induced by photoabsorption and photoemission at absolute zero temperature T = 0. Let us next discuss these optical processes at finite temperature T* OK with a statistical and classical treatment of the lattice, which is applicable for a situation £_ex - s_g » k_BT [3].

Photoabsorption

Starting from a situation where the electronic system is in the ground state 'Pg, while the lattice vibrations {q_k} are excited thermally (randomly) around {q_k = 0}: £_ex - e_g » k_BT (Fig. 3.4). Since the optical transition takes place (~fsec) much shorter than the typical time constant of the lattice motion (~2л/о>_к), it would be acceptable to assume that the lattice {q_k} does not move during the electronic transition: the Franck-Condon principle. The value of {q_k} before the photoabsorption distributes randomly and could be given by the Boltzmann distribution as

where the normalization condition is

Figure 3.4 (a) adiabatic potentials of a two-level system, (b) photoabsorption and photoemission spectrum.

Using the Fermi's golden rule, the optical transition probability for a single system absorbing a photon hv is given by

Here, ip_g) denotes the electronic ground state and |y>_ex) denotes the electronic excited state. <5(x) is the Dirac d-function representing the energy conservation, and

is the matrix element of the interaction between electron and the radiation field.

Optical measurements are usually done on a sample that contains with many similar subsystems, then multiplying the distribution f_g{{q_k}) with Eq. (3.14) and integrating over all the mode {q_k}, the photoabsorption spectrum is given by

Substituting Eqs. (3.1) and (3.2) into (3.15) we obtain

Next, move from {co_kq_k}-space to {(?,}-space with a Jacobian Ylk^h

The absorption spectrum is thus turned out to be a Gaussian centered at e_ex - £_g. The spectrum width is given by Q_mt^j2k_BT, which is called a homogeneous broadening, originated from the thermal lattice-fluctuation, in contrast to inhomogeneous broadening usually originated from the statistical fluctuation of the central energy level, s_ex - £_g.

In analysis of the temperature dependence of the optical spectra obtained by experiments, it is often to introduce the effective temperature T_eff defined by

to fit the spectra with the Gaussian Eq. (3.17). At high temperature hco k_BT, it reduces Teff ~ T where the classical treatment is valid. At low temperature hco » k_BT, on the other hand, the broadening of the spectrum is considered to be originated from the zeropoint vibration: k_BT_eff = hco/2.

Photoemission (Classical Treatment)

After the transient lattice relaxation induced by a photoabsorption has been completed, the lattice motions in the electronic excited state would be thermally activated around the minimum {q_k}. The distribution is given by

With a similar procedure as in the photoabsorption, the photoemission spectrum is shown to be

The emission spectrum is a Gaussian centered at s_ex - e_g - with a homogeneous width Q_mt j2k_BT. The energy difference in the peaks of the absorption and the emission is given by nt = 2E_lr, which is called the Stokes shift. As have discussed, a half of the Stokes shift is due to the lattice relaxation in the excited state and the other half is stored as a lattice elastic energy around the system. Thus, the electronic energy 2£_LR is dissipated over the lattice by two relaxation steps.

It is worthy to note that the Einstein relation holds for the absorption and emission spectra in this system [3]:

This useful, and somewhat surprising, relation holds for any systems where the absorption and emission processes start from each thermal equilibrium and no correlation between them.

If we treated the radiation field classical, the matrix elements appearing in Eq. [3.21) would be the same and then cancel each other. On the other hand, if we treat the radiation field as quantum variables, each matrix element is related to so- called Einstein's A and В coefficients, then the ratio is shown to be 4/iv³/c³. A general proof is given in Appendix II with Einstein's A and В coefficients.

So far, we have assumed that the photoabsorption and photoemission processes are, respectively, a first-order optical process with respect to an electronic transition, where the two processes can be treated separately. However, if the two processes occur within a very short duration [order of fsec (10"^1S sec)), the quantum coherence should be crucial and treated quantum mechanically. This second-order optical process is discussed in detail in Ref. [3].