Physics Model and Formulas: Quantum Tab

Numerically Solving 1D Schrodinger’s Equation

The time-independent Schrodinger equation has the form

\[\left[-\frac{\hbar^2}{2}\frac{\partial}{\partial x} \frac{1}{m} \frac{\partial}{\partial x} + V(x)\right]\psi_i(x) = E_i\psi_i(x)\]

The inputs of the Schrodinger equation solver include: a finite 1D array with position \(x\), the corresponding potential \(V(x)\) with the same size, the effective mass \(m\), and an eigenstate range specified by the user, \(\left[E_\text{min}, E_\text{max}\right]\). The outputs are the eigenfunction, \(\psi\), and the eigenvalue, \(E\). The difference from standard form of the mass between spatial derivative is the requirement of Hermiticity for spatial dependent mass.

We solve the 1D Schrodinger’s equation numerically. Our method combines the Newton’s method that searches for eigenvalues \(E\) and the Numerov’s method that solves for the corresponding eigenfunction \(\psi(x)\) given any specific \(E\).

Shooting Algorithm for Eigen-problem

  1. Initialize with a range for eigen energies.

  2. For each possible eigen energy, solve for the wavefunction using the Numerov’s method for second order differential equations (if the mass is constant, otherwise Euler’s method), and check whether the solution satisfies boundary condition. If so, the energy is an eigen energy.

  3. Use secant’s method to find eigen energy. Newton’s method is not chosen because the root finding converges usually ~10 interaction, and scant’s method (O(n^0.618)) compare to Newton’s (O(n^0.5)) doesn’t worth the extra wavefunction evaluation for numerical derivative.

  4. One noticeable problem for shooting algorithm is that it can miss state pairs that are almost degenerate. When running the software and seeing missing of state, it is recommended to change the global field slightly and/or rotate the layer design and try again.

An example of solving simple Schrodinger equation can be found here.

Effective Mass in Band

Band theory predicts that movement of a particle in a potential over long distance can be very different from the movement of the same particle in vacuum. Usually, the movement is complicated; however, when the electron is in the highest energies of the valence band or the lowest energies of the conduction band, it can be shown that electrons behave as free electrons except with a different mass, the effective mass \(m_\text{eff}\).

A particle’s effective mass in each band can be approximated by Taylor expanding the band structure and ignoring higher-than-second-order terms, as the band structure can be expanded locally as

\[E(k) \approx E_0 + \frac{\hbar^2 k^2}{2 m_\text{eff}}\]

where \(k\) is the wave vector, and \(E_0\) is the edge energy of the band.

For QCL simulations, because of small layer thickness, constant effective mass approximation is sometimes not enough. This can be corrected by including non-parabolic effective mass, or effective mass with energy dependence. In this package, we will offer constant effective mass as simple solver and also non-parabolic effective mass computed using \(k\cdot p\) method.

Specifically in Zincblende crystal, the effective mass has the form including non-parabolic dispersion effect:

\[\frac{m_0}{m_\text{eff}} = 1 + 2F + \frac 13 \frac{E_P}{\Delta E_c + E_g + \Delta_{\text{SO}}} + \frac 23 \frac{E_P}{\Delta E_c + E_g}\]

Where \(m_0\) is electron mass in vacuum, \(E_g\) is the bandgap at \(\Gamma\) point, \(\Delta_{\text{SO}}, E_P, F\) are parameters describing near-\(\Gamma\) behavior of the conduction band and valence band, \(\Delta E_c\) is the energy of electrons above conduction band, or effective kinetic energy. When \(\Delta E_c=0\) the model reduces to standard effective mass model without non-parabolic dispersion. See [VMRM01], [SCFS94].

For Wurtzite crystal, \(F=0\).

The computation of effective mass is implemented in File band.c (also see File band.h). The code structure is also capable of adding new crystal structures. See the material sections for details.

Scattering mechanism: LO phonon

The dominant scattering mechanism for inter-subband transition is Longitudinal optical phonon transition [FB89]. The scattering rate between state \(\psi_u\) and \(\psi_l\) is:

\[\begin{split}&\frac{1}{\tau_{ul}} = \frac{m_{\text{eff}} e^2 \omega_{\text{LO}}}{8\pi\hbar^2\epsilon_\rho} \int_0^{2\pi} \frac{I_{ul}(Q_\theta)}{Q_\theta} \mathrm{d}\theta\\ &I_{ul}(Q) = \iint \mathrm{d}z\mathrm{d}z' \psi_u(z)\psi_l(z) \mathrm{e}^{-Q\mid z-z'\mid}\psi_u(z')\psi_l(z') \\ &Q_\theta = \sqrt{k_u^2 + k_l^2 - 2k_u k_l \cos\theta} \\ &\frac{\hbar^2k_u^2}{2m_\text{eff}} = \frac{\hbar^2k_l^2}{2m_\text{eff}} + E_u - E_l - \hbar\omega_{\text{LO}} \\ &\epsilon_\rho^{-1} = \epsilon_\infty^{-1} - \epsilon_{\text{static}}^{-1}\end{split}\]

where \(k_u\) and \(k_l\) are upper and lower state electron momentum in the epitaxy layer plain, and \(Q_\theta\) is the phonon momentum. With the assumption that \(k_u = 0\), the formula reduces to:

\[\frac{1}{\tau_{ij}} = \frac{m_{\text{eff}} e^2 \omega_{\text{LO}}} {4\hbar^2 \epsilon_\rho k_l} I_{ij}(k_l)\]

(The denominator expression maybe problematic… it needs to be checked!)

Optical gain and threshold current

Using Maxwell-Bloch equation the optical gain from intersubband transition is

\[\begin{split}&g = -2\alpha = \frac{\pi\omega \eta_{\text{inj}} e J} {\hbar c\epsilon_0 nL_p} \,\text{FoM}\,\mathcal L(\omega) \\ &\text{FoM}\equiv |d_{ul}|^2\tau_u\left(1-\frac{\tau_l}{\tau_{ul}}\right)\\ &\mathcal L(\omega) \equiv \frac 1\pi\frac{\gamma_\parallel} {\gamma_\parallel^2 + (\omega - \omega_{ul})^2}\end{split}\]

where \(\eta_{\text{inf}}\) is the injection efficiency, which is depend on transitions between all other states but is assumed to be approximatly constant, The Figure of Merit (FoM) is used to characterize the performance of a structure. \(J\) is the current density into the device, and with information of the loss of the optical cavity we can estimate a threshold current, assuming an reasonable \(\eta\) or just put it 1. This estimation is much underestimated.

To couple the design of quantum wells and waveguide, we define the gain coefficient as the ratio of gain the current density, and also assume \(\eta_{\text{inj}} = 1\), \(\omega = \omega_{ul}\), \(\gamma_\parallel = 0.1\omega\).

\[g_j = \frac {\omega e \text{FoM}} {\gamma_\parallel\hbar c \varepsilon_0 n L_p}\]

Self-consistency Solver for Electron Coulomb Potential

Electron-electron Coulomb interaction can be a determinant part of electron motion in semiconductors. To first order this interaction is included by adding a Maxwell-Poisson equation to correct the potential and solve the equations self-consistently.

\[\begin{split}&V = V_0 + V_c\\ &\nabla^2 V_c = \frac{\rho(x)}{\epsilon} = \sum_i \frac{e n_i}{\epsilon} \mid\psi_i(x)\mid^2\end{split}\]

which means that the potential depends on the eigenstates as well as the corresponding occupation number \(n_i\).

An example comparing the results from solving the simple Schrodinger equation and from solving the equation with the electron-electron interaction correction can be found here.

Electron Thermal Distributions

The 1D Schrodinger’s equation solver provides the energy bands, which are useful for calculations of physical properties of the material. Here, we consider the electron density and the mean energy, predicted by the Fermi-Dirac statistics, where the occupation frequency for each eigenstate is

\[n_i = \frac{1}{\exp\big[(E_i- \mu)/k_BT\big]+1}.\]

At zero temperature, Fermi-Dirac statistics becomes

\[\begin{split}n_i \stackrel{k_BT\to 0}{=} \begin{cases} 0, & \text{ if } { E_i > \mu, } \\ 1, & \text{ if } { E_i < \mu. } \end{cases}\end{split}\]

At high temperature, Fermi-Dirac statistics approaches Maxwell-Boltzmann distribution

\[n_i \stackrel{k_BT\gg E-\mu}{=} \exp\left(-\frac{E-\mu}{k_BT}\right).\]

In this package, we provide the zero- and finite-temperature computation of the Fermi-Dirac statistics, and the high-temperature approximation with the Maxwell-Boltzmann distribution. All distributions will have two methods: given constant chemical potential \(\mu\) distribution and return total number of particles \(\sum n_i\), and given total number of particles \(\sum n_i\) and return chemical potential \(\mu\).

An example of finding the thermal distribution of electrons, given eigen energies and wavefunctions, can be found here.

FB89

R. Ferreira and G. Bastard. Evaluation of some scattering times for electrons in unbiased and biased single- and multiple-quantum-well structures. Phys. Rev. B, 40:1074–1086, Jul 1989. URL: https://link.aps.org/doi/10.1103/PhysRevB.40.1074, doi:10.1103/PhysRevB.40.1074.

SCFS94

Carlo Sirtori, Federico Capasso, Jérôme Faist, and Sandro Scandolo. Nonparabolicity and a sum rule associated with bound-to-bound and bound-to-continuum intersubband transitions in quantum wells. Phys. Rev. B, 50:8663–8674, Sep 1994. URL: https://link.aps.org/doi/10.1103/PhysRevB.50.8663, doi:10.1103/PhysRevB.50.8663.

VMRM01

I Vurgaftman, JR Meyer, and LR Ram-Mohan. Band parameters for III–V compound semiconductors and their alloys. Journal of applied physics, 89(11):5815–5875, 2001.