Summary lecture VI. with the reduced mass and the dielectric background constant
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1 Summary lecture VI Excitonic binding energy reads with the reduced mass and the dielectric background constant Δ Statistical operator (density matrix) characterizes quantum systems in a mixed state and builds the expectation value of observables To tackle the many-particle-induced hierarchy problem, we perform the correlation expansion followed by a systematic truncation resulting in semiconductor Bloch equations on Hartree-Fock level
2 Graphene Bloch equations 2. Semiconductor Bloch equations Coupled system of differential equations on Hartree-Fock level Interaction-free contribution (kinetic energy) leads to an oscillation of the microscopic polarization p k (t)
3 Graphene Bloch equations 2. Semiconductor Bloch equations Coupled system of differential equations on Hartree-Fock level Electron-light coupling is determined by the Rabi frequency giving rise to a non-equilibrium distribution of electrons after optical excitation
4 Graphene Bloch equations 2. Semiconductor Bloch equations Coupled system of differential equations on Hartree-Fock level Electron-electron interaction leads to renormalization of energy and Rabi frequency (excitons!) as well as to dephasing of the polarization γ k
5 Microscopic polarization 2. Semiconductor Bloch equations Optical excitation with a laser pulse with an excitation energy of 1.5 ev Frequency of the oscillation of the microscopic polarization changes due to the Coulomb interaction
6 Non-equilibrium carrier distribution 2. Semiconductor Bloch equations Optical excitation with a laser pulse with an excitation energy of 1.5 ev We generate a non-equilibrium carrier distribution around the excitation energy (corresponding momentum k 0 = 1.25 nm -1 ) Fermi distribution Non-equilibrium carrier distribution
7 Anisotropic carrier distribution 2. Semiconductor Bloch equations 90 o 0 o Generation of an anisotropic non-equilibrium carrier distribution Maximal occupation perpendicular to polarization of excitation pulse (90 o ) due to the anisotropy of the optical matrix element Carrier dynamics needs to be modelled by extending Bloch equations beyond the Hartree-Fock approximation (Boltzmann equation)
8 Learning outcomes lecture VII Sketch the derivation of the Boltzmann scattering equation Explain the Markov approximation Describe the carrier dynamics in graphene Recognize the potential of carrier multiplication
9 Chapter IV IV. Density matrix theory 1. Statistical operator 2. Semiconductor Bloch equations 3. Boltzmann scattering equation
10 Second-order Born approximation Extend the correlation expansion to the second-order Born approximation explicitly taking into account the dynamics of 2-particle quantities Recap of the many-particle hierarchy problem (system of equations is not closed) Systematic truncation at the second order, i.e. we calculate the dynamics of 2-particle quantities, but neglect the contribution of three-particle quantities or higher
11 Second-order Born approximation The full Coulomb contribution to the dynamics of the carrier occupation Apply again the Heisenberg equation to botain the temporal evolution of the two-particle quantities
12 Second-order Born approximation Correlation expansion of the three-particle quantities with single-particle quantities and correlation quantities Now neglect the three-particle correlation quantity expression in the equation for and plug the
13 Markov approximation For 2-dimensional materials, such as graphene, with the compound indices A = (k x, k y, λ A ), the evaluation of full equations is a numerical challenging Markov approximation is applied neglecting quantum-mechanical memory effects stemming from energy-time uncertainty principle Inhomogeneous differential equation with the inhomogeneous part containing integrals over all scattering contributions and the convergence factor γ that will be sent to 0 at the end
14 Markov approximation The formal solution of the inhomogeneous differential equation reads Neglecting memory effects we find Finally, taking the limit
15 Boltzmann scattering equation Applying the second-order Born-Markov approximation, we obtain the Boltzmann scattering equation The equation describes time- and momentumresolved electron-electron scattering dynamics Pauli-blocking energy conservation
16 Boltzmann scattering equation Applying the second-order Born-Markov approximation, we obtain the Boltzmann scattering equation The equation describes time- and momentumresolved electron-electron scattering dynamics phonon absorption phonon emission
17 Carrier dynamics in graphene Non-equilibrium distribution
18 Carrier thermalization Significant relaxation takes place already during the excitation pulse
19 Carrier thermalization Significant relaxation takes place already during the excitation pulse Coulomb-induced carrier-carrier scattering is the dominant channel
20 Carrier thermalization Significant relaxation takes place already during the excitation pulse Coulomb-induced carrier-carrier scattering is the dominant channel Thermalized Fermi distribution reached within the first 50 fs
21 Carrier cooling Carrier cooling takes place on a picosecond time scale Optical phonons (in particular ΓLO, ΓTO and K phonons) are more efficient than acoustic phonons
22 Relaxation dynamics in graphene
23 Anisotropic carrier dynamics Angle Anisotropy of the carrier-light coupling element
24 Anisotropic carrier dynamics Anisotropy of the carrier-light coupling element Scatering across the Dirac cone reduces anisotropy
25 Anisotropic carrier dynamics Anisotropy of the carrier-light coupling element Scatering across the Dirac cone reduces anisotropy Carrier distribution becomes entirely isotropic within the first 50 fs
26 Microscopic mechanism
27 Auger scattering Auger scattering changes the number of charge carriers in the system Auger recombination (AR) Impact ionization (II)
28 Impact excitation Auger scattering changes the number of charge carriers in the system Auger recombination (AR) Impact ionization (II) II
29 Impact excitation Auger scattering changes the number of charge carriers in the system Auger recombination (AR) Impact ionization (II) II
30 Carrier multiplication Auger scattering changes the number of charge carriers in the system Auger recombination (AR) Impact ionization (II) II gained electron in conduction band gained hole in valence band In conventional semiconductors (band gap) Auger scattering is inefficient due to energy and momentum conservation carrier multiplication (CM)
31 Carrier multiplication Carrier density increases during the excitation pulse
32 Carrier multiplication Carrier density increases during the excitation pulse Auger scattering leads to carrier multiplication (CM)
33 Carrier multiplication Carrier density increases during the excitation pulse Auger scattering leads to carrier multiplication (CM) Carrier-phonon scattering reduces CM on a ps time scale
34 Learning outcomes lecture VII Sketch the derivation of the Boltzmann scattering equation Explain the Markov approximation Describe the carrier dynamics in graphene Recognize the potential of carrier multiplication
Summary lecture VII. Boltzmann scattering equation reads in second-order Born-Markov approximation
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
Summary lecture IX. The electron-light Hamilton operator reads in second quantization
Summary lecture IX The electron-light Hamilton operator reads in second quantization Absorption coefficient α(ω) is given by the optical susceptibility Χ(ω) that is determined by microscopic polarization
Summary lecture II. Graphene exhibits a remarkable linear and gapless band structure
Summary lecture II Bloch theorem: eigen functions of an electron in a perfectly periodic potential have the shape of plane waves modulated with a Bloch factor that possess the periodicity of the potential
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