Calculation of electric field and optical transitions in InGaN∕GaN quantum wells

Abstract

We present analytical expressions for internal electric field and strain in single and multiple quantum wells, incorporating electromechanical coupling, spontaneous polarization, and periodic boundary conditions. Internal fields are typically 2% lower than the fields calculated using an uncoupled model. We point out two possible interpolation routes to calculate the piezoelectric ͑PZ͒ constants e ij of an alloy from the PZ constants of the constituent materials and show that, for an In 0.2 Ga 0.8 N / GaN quantum well system, the respective internal electric fields differ by 10%. Using an effective-mass model, we explore the effect of the uncertainty in the elastic and PZ constants of GaN on the internal field and optical transitions of InGaN / GaN quantum wells, and find that the range of published values of e ij produces an uncertainty of more than ±20% in the internal field and of more than ±30% in the blueshift in optical transition energy between zero bias and flatband conditions ͑when the applied field is equal and opposite to the internal field͒. Using the PZ constants of Shimada et al. ͓J. Appl. Phys. 84, 4951 ͑1998͔͒ in our model gives the best fit to results in the literature for internal field and optical transition energy in InGaN / GaN quantum wells. We find that a well with a smooth In gradient along the growth direction has similar optical properties to a well with constant composition, if the average In content of the two wells is the same.

Derivation of built-in polarization potentials in nitride-based semiconductor quantum dots

“…7 InN, GaN, and AlN take values ranging from 0.12 to 0.25, suggesting that electromechanical coupling is small and can safely be neglected. This has been confirmed for InGaN / GaN quantum wells by Christmas et al, 8 where the exact solution for the fully coupled problem is presented and it is shown that the effects of electromagnetic coupling are small. An alternative argument has been provided by Andreev and O'Reilly, 19 where it is shown that the corrections due to electromechanical coupling may be taken into account by renormalizing ͑or correcting͒ the material constants and these corrections of the constants are smaller than the uncertainty in their values.…”

“…We discuss these assumptions in more detail below and how any errors that they introduce should be less than or equal to the error caused by the current uncertainty in the known values of material parameters for nitrides. 8 The derived surface integral formulas are especially useful for QDs with vertical sides such as cuboidal or cylindrical dots. Analytical solutions are derived for the built-in potential along and close to the central axis of spherical, cylindrical, cuboidal, truncated-cone, and ellipsoidal dots.…”

Strain-induced polarization in wurtzite III-nitride semipolar layers

“…25 the strain calculations were performed under the assumption of an equibiaxial character of the mismatch in the layer; as a result, not all elastic and piezoelectric coefficients were included in the calculations. Recent work 19 has demonstrated the correct handling of elasticity calculations, but the detailed approach was given for only the polar case.…”

“…Key examples of fixed charges q at AlGaN / GaN interface give rise to two-dimensional electron gases and alternating sheet charge in InGaN / GaN quantum wells, which give rise to the quantum-confined Stark effect ͑e.g., see Ref. 19 and references therein͒.…”

Strain relaxation in InGaN/GaN micro-pillars evidenced by high resolution cathodoluminescence hyperspectral imaging

“…Actual strain simulations for the In x Ga 1Àx NQ W s were performed by a thermoelastic method accommodated within ANSYS. 13,14 The piezoelectric polarization tensor was then calculated by the "E-first" route preferred by Christmas et al 29 Finally, by introducing the piezoelectric potential into the one-dimensional Schrödinger equation, and neglecting the inbuilt p-n junction field and carrier screening effects, the optical transition energy of the QW was calculated with MATLAB software. In later discussion, the calculated transition energies are compared with the measured QW emission energies directly, i.e., the Stokes shift was assumed to be constant.…”