page_title: Semiconductor Bloch equations
section_title: Interpretation and consequences
hierarchical_section_title: Semiconductor Bloch equations / Interpretation and consequences
caption_attribution_description: English: Linear absorption spectrum of bulk GaAs. Semiconductor Bloch equations used to determine the spectrum. Extremely high time dephasing constant used to make excitonic resonances visible.
context_page_description: The semiconductor Bloch equations describe the optical response of semiconductors excited by coherent classical light sources, such as lasers. They are based on a full quantum theory, and form a closed set of integro-differential equations for the quantum dynamics of microscopic polarization and charge carrier distribution. The SBEs are named after the structural analogy to the optical Bloch equations that describe the excitation dynamics in a two-level atom interacting with a classical electromagnetic field. As the major complication beyond the atomic approach, the SBEs must address the many-body interactions resulting from Coulomb force among charges and the coupling among lattice vibrations and electrons. The SBEs are one of the most sophisticated and successful approaches to describe optical properties of semiconductors originating from the classical light–matter interaction, once the many-body theory are systematically included.
context_section_description: The  dynamic shows a structure where an individual  is coupled to all other microscopic polarizations due to the Coulomb interaction . Therefore, the transition amplitude  is collectively modified by the presence of other transition amplitudes. Only if one sets  to zero, one finds isolated transitions within each  state that follow exactly the same dynamics as the optical Bloch equations predict. Therefore, already the Coulomb interaction among  produces a new solid-state effect compared with optical transitions in simple atoms.
  Conceptually,  is just a transition amplitude for exciting an electron from valence to conduction band. At the same time, the homogeneous part of  dynamics yields an eigenvalue problem that can be expressed through the generalized Wannier equation. The eigenstates of the Wannier equation is analogous to bound solutions of the hydrogen problem of quantum mechanics. These are often referred to as exciton solutions and they formally describe Coulombic binding by oppositely charged electrons and holes.
  However, a real exciton is a true two-particle correlation because one must then have a correlation between one electron to another hole. Therefore, the appearance of exciton resonances in the polarization does not signify the presence of excitons because  is a single-particle transition amplitude. The excitonic resonances are a direct consequence of Coulomb coupling among all transitions possible in the system. In other words, the single-particle transitions themselves are influenced by Coulomb interaction making it possible to detect exciton resonance in optical response even when true excitons are not present.
  Therefore, it is often customary to specify optical resonances as excitonic instead of exciton resonances. The actual role of excitons on optical response can only be deduced by quantitative changes to induce to the linewidth and energy shift of excitonic resonances.
  The solutions of the Wannier equation can produces valuable insight to the basic properties of a semiconductor's optical response. In particular, one can solve the steady-state solutions of the SBEs to predict optical absorption spectrum  analytically with the so-called Elliott formula. In this form, one can verify that an unexcited semiconductor shows several excitonic absorption resonances well below the fundamental bandgap energy. Obviously, this situation cannot be probing excitons because the initial many-body system does not contain electrons and holes to begin with. Furthermore, the probing can, in principle, be performed so gently that one essentially does not excite electron–hole pairs. This gedanken experiment illustrates nicely why one can detect excitonic resonances without having excitons in the system, all due to virtue of Coulomb coupling among transition amplitudes.
caption: Characteristic linear absorption spectrum 
    
      
        
          α
          (
          E
          )
        
      
      {\displaystyle \alpha (E)}
    
   of bulk GaAs using two-band SBEs. The decay of polarization is approximated with a decay constant 
    
      
        
          ℏ
          γ
          =
          0.13
          
          
            m
            e
            V
          
        
      
      {\displaystyle \hbar \gamma =0.13\,\mathrm {meV} }
    
   and 
    
      
        
          α
          (
          E
          )
        
      
      {\displaystyle \alpha (E)}
    
   is computed as function of the pump field's photon energy 
    
      
        
          E
        
      
      {\displaystyle E}
    
  . The energy is shifted with respect to the band-gap energy 
    
      
        
          
            E
            
              
                g
                a
                p
              
            
          
          =
          1.490
          
          
            m
            e
            V
          
        
      
      {\displaystyle E_{\mathrm {gap} }=1.490\,\mathrm {meV} }
    
   and the semiconductor is initially unexcited. Due to the small dephasing constant used, several excitonic resonances appear well below the bandgap energy. The magnitude of high-energy resonances are multiplied by 5 for visibility.
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