Defects in Functional Materials. Группа авторов

= [1 + exp(ED − EF)]⁻¹ is the Fermi-Dirac distribution, ND and ED are respectively the concentration and ionization energy of the donor. The ratio between concentrations of D⁰ and D⁺ is given by:

      if the degeneracy (g) of the donor is taken into account.

      

      Assuming reverse bias conditions in a Schottky diode fabricated on n-type semiconductor with the donor concentration much exceeding the acceptor concentration (i.e. ND ≫ NA), the diode depletion width W is given by:

      where Vbi is the built-in potential and VR is the applied bias. The corresponding junction capacitance C is given by:

      where A is the area of the metal contact. Notably,

      Thus, the ND can be found from the slope of the 1/C² against the VR plot. Similar discussions hold for the acceptor doped p-type semiconductors.

      Temperature dependent dc conductivity measurement can be used to determine the energy levels of the defect in the band gap. This method requires good ohmic contacts fabricated on the sample. The dc conductivity is given by:

      where k is the Boltzmann constant, T is the temperature, EDi is the ionization energy of the ith defect. The ionization energies of the defects can be obtained via the Arrhenius plot log σ(T) against 1/T.

      Hall effect measurement can be used to distinguish type of carriers (i.e. electron or hole), and to obtain the materials’ carrier concentration and carrier mobility. Ohmic contacts in the van-der-Pauw configuration are needed for Hall effect measurement. The Hall coefficient for n-type materials (n ≫ p) is RH = −r/en and that for p-type material (p ≫ n) is RH = r/ep, with r being the scattering factor depending on the magnetic field and temperature; conventionally r = 1. The Hall coefficient can be found experimentally by: RH = tVH/BI, where t is the sample thickness, VH is the Hall voltage, B is the magnetic field and I is the current. The Hall mobility is given by: μH = RHσH, where σH is the conductivity. The above description of Hall-effect measurement applies for homogeneous sample structures. The interpretation of the Hall-effect data for non-homogeneous and/or multi-layered samples is possible too as developed by Look [16].

      Deep level transient spectroscopy (DLTS) is used to identify deep traps in the materials and devices. A rectifying contact (for example a Schottky junction or a p–n junction) is needed for the DLTS characterization. The concentration, energy level and capture cross section of the deep trap can be obtained from the DLTS study. A forward bias is applied to fill all the traps (for simplicity assuming one deep donor trap of density of NT₀). A reverse bias is then applied to create a depletion region at the rectifying contact. The filled traps in the depletion region will emit electrons to the conduction band depending on the temperature with the emission rate of: en = σ_capvNC exp[−(EC − ET)/kT], where σ_cap is the electron capture cross section of the trap, is the effective electron mass, ET is the energy level of the deep trap, NC is the effective density of states of the conduction band exhibiting a temperature dependence of ∼ T^3/2. This implies that en ∼ T² exp[−(EC − ET)/kT]. The rate equation for the ionized deep donor is: = en(N_T0 − (t)) and the solution is: (t) = NT₀[1 − exp(−ent)] if all the traps are filled initially as the reverse bias is applied. Since the capacitance follows C ∼ if the reverse bias is fixed, the (t) evolution can be revealed by the capacitance transient, i.e. C(t) ∼ 1 − exp(−ent). Figure 1 shows the schematic capacitance transients C(t) with different emission rates en’s of 0.01, 0.1 and 1 (in arbitrary reciprocal time unit).

      The rate-window approach is usually adopted to extract the emission rate from the capacitance transient [17, 18]. The DLTS signal is defined as a difference of the capacitance taken at two consecutive times t₁ and t₂ (with t₂ > t₁):

      Figure 1. Schematics of the capacitance transient resulted from thermally excited deep electron traps in the depletion region while applying the reverse bias. Three cases with different emission rates namely en = 0.01, 0.1 and 1 (in arbitrary reciprocal time unit) are shown.

      where Δt = t₂ −t₁ is called the rate window of the measurement (see Fig. 1). It can be seen from Fig. 1 that for the transient with the high emission rate (i.e. en = 1), ΔC is effectively zero. For the transient with the low emission rate (i.e. en = 0.01), ΔC is small as compared with the one with the moderate emission rate (i.e. en = 0.1). It can be shown that maximum DLTS signal ΔC maximum occurs at:

      Accordingly, the DLTS spectrum is obtained by fixing the rate window Δt and measuring the ΔC as a function of temperature. A peak appearing in the DLTS spectrum at the temperature of T_peak associates with the maximum DLTS signal and thus the corresponding emission rate at the temperature of T_peak is given by:

      Figure 2. (a) Examples of the DLTS spectra taken from GaN Schottky diode using different rate windows; (b) The corresponding Arrhenius plot, i.e. enT⁻² against 1/T.

      Thus applying different rate windows Δt will result in peaks occurring at different temperatures (for example, Fig. 2(a) which is the DLTS spectra