page_title: High-field domain
hierarchical_section_title: High-field domain
caption_attribution_description: English: band model of the CdS/CdTe
context_page_description: A high-field domain is a band of elevated field orthogonal to the equi-current lines, and seen in photoconductive CdS and monochromatic light at the band edge as dark band was discovered by Böer, using the Franz-Keldysh effect. Such domains must appear whenever the conductivity decreases stronger than linearly. This can be caused by the field dependence of the carrier density, as observed in copper-doped CdS caused by Frenkel Poole excitation of holes, causing faster electron recombination, known as field quenching. These high-field domains, now referred to as Böer domains, or by field dependence of the mobility, caused by excitation of electrons into higher conduction bands with lower mobility as observed in GaAs, called the Gunn effect. The high-field domains can be identified by periodical field oscillations between high and low values, as shown in Fig. 1.
  Many other crystals show such domains by typical current oscillations. The high-field domains in copper doped CdS can be easily observed by the Franz-Keldysh effect as stationary, adjacent to the cathode or moving. These are analyzed as another example below.
context_section_description: A high-field domain is a band of elevated field orthogonal to the equi-current lines, and seen in photoconductive CdS and monochromatic light at the band edge as dark band was discovered by Böer, using the Franz-Keldysh effect. Such domains must appear whenever the conductivity decreases stronger than linearly. This can be caused by the field dependence of the carrier density, as observed in copper-doped CdS caused by Frenkel Poole excitation of holes, causing faster electron recombination, known as field quenching. These high-field domains, now referred to as Böer domains, or by field dependence of the mobility, caused by excitation of electrons into higher conduction bands with lower mobility as observed in GaAs, called the Gunn effect. The high-field domains can be identified by periodical field oscillations between high (the domain) and low values, as shown in Fig. 1.
  Many other crystals show such domains by typical current oscillations. The high-field domains in copper doped CdS can be easily observed by the Franz-Keldysh effect as stationary, adjacent to the cathode or moving.  These are analyzed as another example below.
  Theory: Stationary high-field domains can be analyzed  from the transport- and Poisson equations:
     and   
  The projection of any solution curves into an arbitrary nF plane can be filled with direction arrows at any point of this plane. Two auxiliary curves for which dn/dx = 0, called n₂(F) and dF/dx = 0 called n₁(F) divide this plane into four quadrants with the same type of directions. This is shown in Fig. 2(left) in a double logarithmic representation.
  Any solution of an n-type semiconductor with blocking cathode must start at the boundary density nc that is below the density in the bulk, and approaches the singular point at which dn/dx = dF/dx = 0, that is in the bulk where both n(x) and F(x) are constant. The solution curve represents a Schottky-blocking contact as shown in Fig. 2(B), curve (a).
  When n(x) decreases at higher fields due to field-quenching cause by Poole-Frenkel excitation of holes from Coulomb attractive hole traps, that consequently enhances electron recombination through recombination centers, and thereby deforms the n₁(x) curve at higher fields as shown in Fig. 2(B). When the bias is increased the current curve n₂(x) is shifted upwards and to the right, and when it crosses n₁(x) again, it produces a second singular point II. With further increased bias this singular point II reaches the value of the boundary density nc, and the solution curve changes from a monotonic increasing Schottky-solution, to a high-field domain, curve (b): that remains constant near the cathode, and then changes within a few Debye lengths to approach the constant value in the bulk, near the singular point I. The width of the domain increases with bias (Fig. 3a), while the current remains constant (Fig. 3c). The domain is visible as dark part in the transmission picture through the CdS platelett, extending from the cathode as shown in Fig. 3a. The field in the domain can be obtained from the slope of the domain that increases with bias (Fig. 3b).
  When, with further increased bias the domain fills the entire sample,then it flips to an anode adjacent high-field domain (Fig. 4b). The field at the cathode is now much higher than for the cathode adjacent domain (Fig. 4b and c), while the current still remains essentially constant (Fig. 4c).
  High-field domains to determine the work function of blocking contacts
  Since the high field domain starts at the electron density given by the work-function at the cathode and pulls the Schottky barrier open to a constant field in the domain, this work function can be determined precisely, and it can be used as a tool to determine the changes of the work function, as it varies depending on external parameters. As an example, it can be shown that it depends on the optical excitation in a photoconductor (See Fig. 5).
  High-field domains as tools to measure the electron density in the field quenched branch and of the electron mobility as a function of the temperature
  The 
caption: Fig. 11 Band model of the CdS/CdTe solar cell in operating condition at AM1.[22]
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