3 Electronic Structure Theory for X-Ray Absorption and Photoemission Spectroscopy
79
for various phenomena seen in photoemission spectra, such as band narrowing and
satellite structures.
Dynamic correlation effects can, to some extend, be described by the dynamical
mean-field theory (DMFT) [27], where the self-energy is taken to be frequency
dependent but local, i.e. momentum independent, kσ (ω) → σ (ω). We note that
self-energies from static mean-field theory (such as LDA) and dynamical mean-field
theory can be, and often are, combined. Although the total self-energy is then both
momentum and frequency dependent, it is still an approximation.
The basic idea of DMFT is to map the Hubbard model with correlation (U = 0)
on all lattice sites onto the Anderson model, which describes one correlated atom (the
‘impurity’) coupled to an effective bath of band states. In the Anderson model, we
have U = 0 only at the impurity site (i = 0), and as a consequence, the self-energy
is a frequency dependent, but local, quantity i j (ω) = 0 (ω)δ i0 δ i j . In DMFT, this
local self-energy is taken as the self-energy of the lattice problem (Hubbard model).
The mapping, i.e. the definition of the effective bath, must be done in a self-consistent
manner such that the on-site matrix elements of the lattice model Green’s function
G ii (ω) coincide with those of the impurity model [27]. While the Anderson impurity
model is simpler than the Hubbard model, it is nonetheless a complex many-body
problem. Implementations of DMFT mostly differ in the approximations used for
solving the impurity problem.
DMFT has been applied to photoemission spectroscopy of correlated systems [28]
and results in improved spectra compared to independent particle approximation (HF
or LDA). In transition metal systems, for example, photoemission spectra calculated
in DMFT can account for finite temperature effects, correlation-driven band narrowing and satellite peaks [29].
3.8 Conclusions
In this chapter, I have tried to give a brief introduction to the theory of X-ray absorption and photoemission spectroscopy. Along the way, it appeared useful to present
succinctly the principles of several computational methods of electronic structure
that are used in spectroscopic calculations. Given the vast nature of the subject, this
account is necessarily very incomplete. But I hope that the reader got an idea of
the physics underlying the different theoretical methods and that it aroused his/her
curiosity to dwell deeper into the subject by reading some of the cited literature.
References
1. L.I. Schiff, Quantum Mechanics, 3rd edn. (McGraw-Hill, New York, 1968)
2. R.D. Cowan, The Theory of Atomic Structure and Spectra (University of California Press,
Berkeley, 1981)
79
for various phenomena seen in photoemission spectra, such as band narrowing and
satellite structures.
Dynamic correlation effects can, to some extend, be described by the dynamical
mean-field theory (DMFT) [27], where the self-energy is taken to be frequency
dependent but local, i.e. momentum independent, kσ (ω) → σ (ω). We note that
self-energies from static mean-field theory (such as LDA) and dynamical mean-field
theory can be, and often are, combined. Although the total self-energy is then both
momentum and frequency dependent, it is still an approximation.
The basic idea of DMFT is to map the Hubbard model with correlation (U = 0)
on all lattice sites onto the Anderson model, which describes one correlated atom (the
‘impurity’) coupled to an effective bath of band states. In the Anderson model, we
have U = 0 only at the impurity site (i = 0), and as a consequence, the self-energy
is a frequency dependent, but local, quantity i j (ω) = 0 (ω)δ i0 δ i j . In DMFT, this
local self-energy is taken as the self-energy of the lattice problem (Hubbard model).
The mapping, i.e. the definition of the effective bath, must be done in a self-consistent
manner such that the on-site matrix elements of the lattice model Green’s function
G ii (ω) coincide with those of the impurity model [27]. While the Anderson impurity
model is simpler than the Hubbard model, it is nonetheless a complex many-body
problem. Implementations of DMFT mostly differ in the approximations used for
solving the impurity problem.
DMFT has been applied to photoemission spectroscopy of correlated systems [28]
and results in improved spectra compared to independent particle approximation (HF
or LDA). In transition metal systems, for example, photoemission spectra calculated
in DMFT can account for finite temperature effects, correlation-driven band narrowing and satellite peaks [29].
3.8 Conclusions
In this chapter, I have tried to give a brief introduction to the theory of X-ray absorption and photoemission spectroscopy. Along the way, it appeared useful to present
succinctly the principles of several computational methods of electronic structure
that are used in spectroscopic calculations. Given the vast nature of the subject, this
account is necessarily very incomplete. But I hope that the reader got an idea of
the physics underlying the different theoretical methods and that it aroused his/her
curiosity to dwell deeper into the subject by reading some of the cited literature.
References
1. L.I. Schiff, Quantum Mechanics, 3rd edn. (McGraw-Hill, New York, 1968)
2. R.D. Cowan, The Theory of Atomic Structure and Spectra (University of California Press,
Berkeley, 1981)