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physica status solidi (b)

Screening of the Coulomb Potential in the Athermal Stage in Highly Excited Semiconductors

E. Heiner

Published May 1, 1989• 17 citations
Publisher

Abstract

Abstract Equations for the reduced density operator are evaluated. Investigating the subdynamics of nonrelevant observables a resolvent formulation is obtained selfconsistently for the screening part of the evolution superoperator. Utilizing the diagonal representation for the time‐dependent mean field term of the Hamiltonian, a selfconsistent concrete calculation of the screened potential is possible in the high density limit of the nonequilibrium. The result is an extension of the well known RPA‐formula for nonequilibrium two‐band systems of Fermions. An extension to the semimarkovian approximation is counted for.

DOI
10.1002/pssb.2221530131

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Generalized Bloch equations for laser-excited semiconductors are derived applying quantum-mechanical projection-operator techniques. The equations include phase-space filling and the many-body Coulomb effects. The coherent part of the equations is evaluated for the regime of ultrafast-pump-probe excitation and shown to reduce to the inhomogeneously broadened two-level Bloch equations for the different momentum states if the proper Coulomb enhancement in the density of states is taken into account. For the high-excitation quasithermal regime a generalization of the Elliott formula for the absorption spectrum is derived which is valid not only for bulk semiconductors, but also for quantum-well structures and other systems with reduced spatial dimensions.

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Abstract An explicit definition for the self‐energy field operator and the self‐energy fields obtained from an average of the associated operator within the algebraic formalism of superoperators is presented. It stems from the formal expansion of the many‐body propagator equations of motion hierachy in quantum many‐body systems within the scenario of the Liouvillian decoupling scheme developed in previous works. An essential theoretical property of such fields for the complete expansion of the propagator to any order in the interaction potential is shown. This states that the interaction potential to a given order only depends on one q‐reduced density matrix. The contraction order q of the density matrix depends on both the nature of the operators defining the propagator and the actual order of the expansion. This result is rigorous regarding infinite summation of the irreducible terms of the self‐energy fields and provides a direct way to estimate the extent to which many‐body effects are involved in successive approximations, i.e., truncation of the excitation level given by the reference state throughout the reduced density matrix and the expansion of the propagator. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002

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Bound states of the hydrogen-like atomic systems in plasma environments

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quant-pharXiv:2607.19601•Jul 23, 2026

The RPA and the GW approximation for the self-energy

Richard M. Martin, Lucia Reining, David M. Ceperley

Besides the proof of a modified Luttinger–Ward–Klein variational principle and a related self-consistency idea, there is not much new in principle in this paper. L. Hedin, Phys. Rev . 139, A796–823 (1965) Summary In this chapter a set of equations is formulated that determine the self-energy and the one-body Green's function in terms of the screened Coulomb interaction between classical charges. The equations contain a correction to the classical picture in terms of a vertex function. The physical meaning of the various contributions is discussed. The simplest approximation for the vertex yields the random phase approximation for the polarizability and the GW approximation for the self-energy. Various aspects of the GWA are analyzed, with a focus on the physics that is added beyond Hartree– Fock. A brief summary of model cases illustrates the domain of validity and the limits of the GWA. In this chapter we elaborate in more detail on the question of how to calculate the onebody Green's function from a Dyson equation with a self-energy kernel. In the previous chapter a scheme was introduced to design approximations to the self-energy. However, the question of where to stop, which pieces of physics to include and which to neglect, is not yet settled. Of course, there is no unique answer, besides the exact solution, but different strategies can be more or less advantageous in practice. In a system with a few electrons, for example, different aspects will be important than in a system with many electrons. Here we are mostly interested in solids, or more generally in extended systems. In such systems, screening plays an essential role: the interaction between two charges is strongly modified, in general reduced, by the rearrangement of all the other charges. It is therefore most convenient to reformulate the equations such that screening appears explicitly. Some steps in this direction can be found in earlier chapters, in particular in Sec. 8.3, the formulation of the Ψ [ G , W ]-functional of the screened interaction W instead of the Ф[ G , v c ]-functional of the bare v c . In Sec. 10.5 the screened interaction approximation for the self-energy is derived, with the self-energy as a product of the one-body Green's function and the screened interaction W .

Cambridge University Press eBooks•May 31, 2016•0 citations
Abstract Equations for the reduced density operator are evaluated. Investigating the subdynamics of nonrelevant observables a resolvent formulation is obtained selfconsistently for the screening part of the evolution superoperator. Utilizing the diagonal representation for the time‐dependent mean field term of the Hamiltonian, a selfconsistent concrete calculation of the screened potential is possible in the high density limit of the nonequilibrium. The result is an extension of the well known RPA‐formula for nonequilibrium two‐band systems of Fermions. An extension to the semimarkovian approximation is counted for.
Evolution equations of a laser-excited direct gap two-band semiconductor in the transient stage have been derived by using the first principle Liouville-von Neumann equation and the statistical projection operator method. To be more general, the photon field is described in a quantum mechanical manner. Quantum effects of the electron-photon coupoling as well as charge carrier Coulomb screening and dissipation are involved. The evolution of the time dependent coherent photon state is investigated. It results in additionally evolution equations. Numerical solutions of these systems are cast out for realistic parameters and are compared with previous results.
Generalized Bloch equations for laser-excited semiconductors are derived applying quantum-mechanical projection-operator techniques. The equations include phase-space filling and the many-body Coulomb effects. The coherent part of the equations is evaluated for the regime of ultrafast-pump-probe excitation and shown to reduce to the inhomogeneously broadened two-level Bloch equations for the different momentum states if the proper Coulomb enhancement in the density of states is taken into account. For the high-excitation quasithermal regime a generalization of the Elliott formula for the absorption spectrum is derived which is valid not only for bulk semiconductors, but also for quantum-well structures and other systems with reduced spatial dimensions.
Abstract An explicit definition for the self‐energy field operator and the self‐energy fields obtained from an average of the associated operator within the algebraic formalism of superoperators is presented. It stems from the formal expansion of the many‐body propagator equations of motion hierachy in quantum many‐body systems within the scenario of the Liouvillian decoupling scheme developed in previous works. An essential theoretical property of such fields for the complete expansion of the propagator to any order in the interaction potential is shown. This states that the interaction potential to a given order only depends on one q‐reduced density matrix. The contraction order q of the density matrix depends on both the nature of the operators defining the propagator and the actual order of the expansion. This result is rigorous regarding infinite summation of the irreducible terms of the self‐energy fields and provides a direct way to estimate the extent to which many‐body effects are involved in successive approximations, i.e., truncation of the excitation level given by the reference state throughout the reduced density matrix and the expansion of the propagator. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002
We conduct a non-relativistic study of plasma screening effects on hydrogen-like atomic systems using the Screened Coulomb Potential (SCP i.e. Yukawa potential). The radial Schrödinger equation is first reduced to a bi-confluent Heun (BCH) equation for the Killingbeck potential which is the truncated version of the SCP, and we write the exact BCH eigenfunctions and eigenenergies. We then study the limitations of these BCH solutions and obtain an analytic description valid for weak to moderate screening using the BCH functional form. The corrected eigenfunctions are constructed order by order up to O(k^3); they are expressed in terms of the Laguerre polynomials and reduces exactly to Coulomb eigenfunctions when k=0. Using these Functions, the energy spectrum is computed via two analytic methods: (i) direct evaluation of the full Yukawa Hamiltonian expectation value, and (ii) the Hellmann-Feynman theorem, yielding integral representation. All methods presented here provide explicit analytical formulas for both wavefunctions and eigenenergies valid for different ranges of the screening. These methods establish a powerful analytic framework for studying confined quantum systems. The thermodynamic properties are also derived using the BCH formulation.
Besides the proof of a modified Luttinger–Ward–Klein variational principle and a related self-consistency idea, there is not much new in principle in this paper. L. Hedin, Phys. Rev . 139, A796–823 (1965) Summary In this chapter a set of equations is formulated that determine the self-energy and the one-body Green's function in terms of the screened Coulomb interaction between classical charges. The equations contain a correction to the classical picture in terms of a vertex function. The physical meaning of the various contributions is discussed. The simplest approximation for the vertex yields the random phase approximation for the polarizability and the GW approximation for the self-energy. Various aspects of the GWA are analyzed, with a focus on the physics that is added beyond Hartree– Fock. A brief summary of model cases illustrates the domain of validity and the limits of the GWA. In this chapter we elaborate in more detail on the question of how to calculate the onebody Green's function from a Dyson equation with a self-energy kernel. In the previous chapter a scheme was introduced to design approximations to the self-energy. However, the question of where to stop, which pieces of physics to include and which to neglect, is not yet settled. Of course, there is no unique answer, besides the exact solution, but different strategies can be more or less advantageous in practice. In a system with a few electrons, for example, different aspects will be important than in a system with many electrons. Here we are mostly interested in solids, or more generally in extended systems. In such systems, screening plays an essential role: the interaction between two charges is strongly modified, in general reduced, by the rearrangement of all the other charges. It is therefore most convenient to reformulate the equations such that screening appears explicitly. Some steps in this direction can be found in earlier chapters, in particular in Sec. 8.3, the formulation of the Ψ [ G , W ]-functional of the screened interaction W instead of the Ф[ G , v c ]-functional of the bare v c . In Sec. 10.5 the screened interaction approximation for the self-energy is derived, with the self-energy as a product of the one-body Green's function and the screened interaction W .