Theory

Extraction of NMP parameters from DLTS measured \( \ln(T^2/e_n) \) vs. \( 1000/T \).

This page briefly explains why carrier capture at defects couples to lattice relaxation, how that modifies the temperature dependence of carrier capture, and how the full temperature dependence can be used to obtain defect parameters.

1. Configuration coordinate diagram

2. Capture cross section vs temperature

3. \( \ln(T^2/e_n) \) vs \( 1000/T \)

4. NMP fitting

Part I

Lattice relaxation and configuration coordinate diagram

Carrier capture at defects probed by deep-level transient spectroscopy is a nonradiative multiphonon (NMP) electronic transition. Defects capture or emit carriers and change the charge state, so the initial and final states are associated with different equilibrium configurations, with a displacement \( \Delta Q \) along the configuration coordinate diagram, which is usually referred to as lattice relaxation. Under a single-phonon-mode approximation, with a vibrational frequency \( \omega \), the potential energy surfaces are constructed under the harmonic approximation, where \( U = \frac{1}{2}\omega^2 Q^2 \). The two potential energy surfaces are then separated by \( \Delta Q \) horizontally and by \( \Delta E \) vertically, where \( \Delta E \) is the defect transition energy associated with the defect level.

References

Huang and Rhys, Proc. A 204, 406 (1950).
Henry and Lang, Phys. Rev. B 15, 989 (1977).
Huang, Contemp. Phys. 22, 599 (1981).

Schematic Plot

Configuration coordinate diagram

Open Image

Part II

Carrier capture cross section as a combined temperature-dependent quantity

In practical DLTS analysis, the carrier capture cross section \( \sigma_n \) is not governed by a single temperature trend. Instead, it reflects the combined contributions from the NMP capture coefficient \( C_n \) (\( C_n = Vr \), \( V \) is the system volume, \( r \) is the NMP transition rate), the carrier thermal velocity \( v_{\mathrm{th}} \), and the Sommerfeld factor \( f_s \). The full cross section therefore follows \( \sigma_n = C_n / v_{\mathrm{th}} * f_s \).

The four panels below are laid out as one visual sequence so one can immediately see how the individual temperature-dependent factors combine into the observed \( \sigma_n(T) \).

Component 1

NMP transition rate

\( r(T) \)

Component 2

Carrier thermal velocity

\( v_{\mathrm{th}}(T) \)

Component 3

Sommerfeld factor

\( f_s(T) \)

→

Combined Result

Carrier capture cross section

\( \sigma_n(T) \)

Part III

\( \ln(T^2/e_n) \) vs \( 1000/T \) does not remain strictly linear

In a simplified picture, one often expects a straight line when plotting \( \ln(T^2/e_n) \) against \( 1000/T \), following the constant-\( \sigma_n \) model or the Henry-Lang model. As a result, the linear fitting is often used for obtaining defect parameters.

However, nonradiative multiphonon theory predicts that \( \ln(T^2/e_n) \) vs \( 1000/T \) can deviate systematically from the linear relation. The curvature reflected from the nonlinearity is not a nuisance. It is useful information about the defect-lattice coupling.

Our goal is to make full use of this inherent information about the defect-lattice interaction, and try to find a set of multiphonon parameters that can best repreduce the observed \( \ln(T^2/e_n) \) vs \( 1000/T \) relation.

Schematic Plot

Part IV

NMP fitting

Workflow

DLTS data input

→

Coarse screening

→

Local refining