LITERATURE REVIEW By
Rahul Kommineni (UIN:658260831)
Electron- Phonon Interactions and Excitonic Dephasing in Semiconductor Nanocrystals
T.Takagahara
NTT Basic Research Laboratories, Musashino-shi, Tokyo 180, Japan
(Received 24 May 1993)
Semiconductor nanocrystals for electron-phonon coupling mechanisms are classified based upon contribution of the size dependence to excitonic dephasing rate. Based on these dependencies, the proportionality of its magnitude to the inverse square of the nanocrystal size and the commonly observed linearly temperature-dependent term of the excitonic dephasing rate are ascribed to pure dephasing due to deformation-potential coupling. Semiconductor nanocrystals of a size similar to or smaller than the exciton Bohr radius in bulk material are drawing in much consideration from the fundamental physics viewpoint and from the enthusiasm for the application to practical gadgets.
In semiconductor nanocrystals, not just the electronic energy levels but also the lattice vibrational modes get to be distinctly discrete because of the three dimensional confinement and its consequences namely, the phonon confinement are now being studied extensively. The longitudinal optical phonons were observed by the resonance Raman scattering and the excitonic dephasing in various semiconductor nanocrystals has been measured as a function of temperature and crystal size. The excitonic dephasing constant in nanocrystals of II-VI compounds are measured by spectral hole burning and four wave mixing.
This paper derives electron-phonon interactions with acoustic phonons in semiconductor nanocrystals and clarify the size dependence of the contribution to the excitonic dephasing rate for various electron-phonon coupling mechanisms like piezoelectric coupling and deformation-potential coupling. Based on these results, the origin of the T-linear term of the excitonic dephasing rate and the proportionality of its magnitude to the inverse square of the nanocrystal size are identified.
As long as the nanocrystal size is not too small, its acoustic properties can be described in terms of elastic vibration of a homogeneous particle and the shape of a nanocrystal is assumed to be spherical and the anisotrophy of the elastic constants is neglected for simplicity of the arguments. The vibrations of an elastically isotropic sphere is described by the following equation
where u is the lattice displacement vector, ρ is the mass density, λ and μ are the Lames’s constants. Two kinds of eigenmodes exists namely torsional and spheroidal, where these modes are studies from the above equation. The electron-phonon interaction with acoustic phonon modes arises mainly through the deformation-potential coupling and the piezoelectric coupling.
In polar semiconductors the lattice polarization is produced by the lattice strain and this polarization interacts with an electron. The electron-lattice interaction is given by the potential energy of the lattice polarization in the electric field induced by an electron and it represented as follows
where (re, Ωe) denotes electron position in the spherical coordinates, -e the e- charge, ϵ dielectric constant, Yl,m spherical harmonic and b is the annihilation operator of the phonon mode with angular momentum indices (l,m) and the radial quantum number j. To understand the mechanisms of the excitonic dephasing it is very important to examine the size dependence of electron-phonon interactions. The size dependence arises from divu and can be estimated as following
where R is the radius of a spherical nanocrystal and the first factor comes from the operation of div, the middle one from normalization of phonon mode and the third one from the quantization of the phonon mode. The eigenfrequency ω of (1) is given by (λ+2μ) h^2= ρω^2 or μk^2= ρω^2 , where h and k are determined by the boundary conditions and scale as 1/R. Thus the eigen frequency ω scales are 1/R and we obtain the dependence in (3). The strain tensor has the same size dependence as divu in the case of piezoelectric coupling, namely 1/R2 and we obtain the following equation
where the first factor comes from the volume integral, the middle one from ∇(1/|r-r_e | ), and the third one comes from the piezoelectric polarization. From these size dependencies, we see that in small size region the deformation potential coupling is dominant, whereas in large size region the piezoelectric coupling is dominant.
The electron-phonon interactions in semiconductor nanocrystals are derived, so now we can calculate the excitonic dephasing constant. Excitonic dephasing rate is nothing but the decay rate of the Excitonic polarization which is a combination of a pure dephasing constant and a longitudinal decay constant. The diagonal and off-diagonal matrix elements of the electron-phonon interaction with respect to the excitonic states are responsible to former and latter parts of the dephasing rate. The diagonal part of the relevant Hamiltonian in the Franck-Condon approximation can be written as following when specified to the electronic ground state |g> and the lowest excitonic state |ex> ,
where the subscript j is the phonon mode index, and ω_j is the mode frequency. The coupling constant γ_j is the matrix element of the electron-phonon interaction Hamiltonian taken between exciton wave functions. In the case of the piezoelectric coupling the relevant Hamiltonian is given by (2) and the coupling constant is given by
where φ(r_e,r_h) is the sxciton wave function. The homogeneous line width due to this fluctuation can be evaluated as
where N is the phonon occupation number. In the summation over the phonon modes in (7) only the acoustic modes are included because the LO modes have large frequencies and are giving rise to the absorption or emission sidebands rather than causing the energy fluctuation of the excitonic level.
The excitonic dephasing rate in a CdSe nano-crystal with 11 A radius was measured as a function of the temperature and a typical T-linear dependence was observed, namely Th = T0+AT+…, over a wide range of temperature. The theoretically estimated pure dephasing rate of the lowest excitonic state for this nano-crystal is shown in Fig.1 as a function of temperature with the experimental results.
The coefficient A of the T-linear term of Th is plotted in Fig.2 as a function of 1/R2. The linearity to 1/R2 is clearly seen, indicating that the deformation-potential coupling is dominantly determining the T-linear term. The experimental value of 0.136 meV/K for an 11 A radius nano-crystal is reproduced fairly well by the theory. The relative ratio between contributions from the deformation-potential coupling and the piezoelectric coupling, we decompose (7) as T2h(ac) = T2h(DF) + T2h(PZ), where the first term is the contribution from the deformation-potential coupling. These contributions are plotted in Fig. 3 as a function of the radius for CdSe nanocrystals at 80K. As expected before, in the small size region T2h(DF) is overwhelming, whereas in the large size region T2h(PZ) is dominant. The crossover between the two components occurs around R = 70 A.
The size dependence of the coefficient A of the T-linear term of Th was found to be well described by 1/R2. The coefficient A is calculated employing the material parameters of bulk CuCl and is shown in Fig. 4 as a function of 1/R2 with the experimental data. In this case also the characteristic dependence of A on 1/R2 due to the deformation-potential coupling can be confirmed both theoretically and experimentally.
Conclusion
The electron-phonon interactions in semiconductor nanocrystals, especially concerning the acoustic phonon modes, are derived and the size dependence of the contribution to the excitonic dephasing rate has been clarified for various electron-phonon coupling mechanisms. On the basis of these results, the commonly observed linearly temperature-dependent term of the excitonic dephasing rate and the proportionality of its magnitude to the inverse square of the nanocrystal size are attributed to the pure dephasing due to the deformation potential coupling.