Summary lecture VII. Boltzmann scattering equation reads in second-order Born-Markov approximation

Transcription

1 Summary lecture VII Boltzmann scattering equation reads in second-order Born-Markov approximation and describes time- and momentum-resolved electron scattering dynamics in non-equilibrium Markov approximation neglects quantum-mechanical memory effects stemming from energy-time uncertainty In graphene, carrier multiplication can take place due to efficient Auger scattering channels

2 Steps during relaxation dynamics Equilibrium optical excitation 50 fs thermalization & isotropy 1 ps cooling Optically generated anisotropic non-equilibrium carrier distribution Carrier-phonon scattering accounts for isotropy, while carrier-carrier scattering leads to a spectrally broad thermalized distribution in the first 50 fs Carrier-phonon scattering gives rise to carrier cooling on ps time scale

3 Chapter VI VI. Optical properties of solids 1. Electron-light interaction 3. Differential transmission spectra 4. Statistics of light

4 Learning outcomes lecture IX Describe the Hamilton operator for electron-light interaction Sketch the derivation of the absorption coefficient

5 Electron-light Hamilton operator 1. Electron-light interaction Interaction of an electron with electromagnetic light is expressed by the Hamilton operator with the vector and scalar potential A(r, t) and φ(r, t) Vector and scalar potential are gauge-dependent using the radiation gauge and assuming a small vector potential the Hamilton operator reads Dipole approximation since due to small light momentum compared to particle size

6 Gauge transformation 1. Electron-light interaction The electron-light Hamilton operator reads with the optical matrix element Since Schrödinger equation is only determined up to an arbitrary phase we can redefine the electron-light Hamiltonian with the dipole matrix element The and Hamilton operators are related via the gauge transformation

7 Absorption coefficient Bloch equations give access to microscopic quantities carrier occupation and microscopic polarization now, we relate these to macroscopic experimentally accessible observables The response of a system to a weak optical excitation is given by the optical susceptibility Χ(ω) with the refractive index The susceptibility is determined either by the macroscopic polarization P(ω) or the current density j(ω) with

8 Microscopic background The macroscopic polarization is determined by the dipole matrix element and the microscopic polarization that can be calculated via semiconductor Bloch equations The current density can be traced back to the microscopic polarization assuming full valence and empty conduction bands and neglecting intraband currents The crucial quantity is the microscopic polarization

9 Graphene Bloch equations Coupled system of differential equations on Hartree-Fock level Evaluate the graphene Bloch equations in the limit of linear optics, i.e. weak excitation resulting in a negligible change in carrier occupation

10 Graphene Bloch equations Graphene Bloch equation in the limit of linear optics Electron-electron interaction leads to a renormalization of the energy and to a renormalization of the Rabi frequency (excitons)

11 Free-particle absorption of CNTs (10,0) CNT E 11 transition Free-particle spectra of carbon nanotubes (CNTs) are characterized by van Hove singularities Here, we focus on the zig-zag (10,0) carbon nanotube and the energetically lowest transition E 11

12 Coulomb-renormalized absorption of CNTs (10,0) CNT E 11 transition Repulsive electron-electron interaction leads to the energy renormalization blue-shift of transitions energies

13 Excitonic absorption of CNTs (10,0) CNT E 11 transition 0.75 ev Attractive electron-electron interaction leads to the formation of excitons red-shifted Lorentzian-shaped peaks Excitonic binding energy is in the range of 1 ev (in vacuum)

14 Absorption spectrum of graphene The absorption of graphene is characterized by a constant absorption (2.3% of incoming light) in the linear region around the Dirac point At around 4eV at the M point in the Brillouine zone, graphene exhibits a saddle point in the band structure leading to saddle point excitons

15 Learning outcomes lecture IX Describe the Hamilton operator for electron-light interaction Sketch the derivation of the absorption coefficient