Quantum States and Scattering in Semiconductor Nanostructures

The following sections are included:

• Foreword
• About the Authors
• Contents

Classical mechanics states that the state of a physical system is described by the knowledge of the positions and velocities at time t plus the Newton (or Lagrange or Hamilton) law for its evolution…

The following sections are included:

• Propagating and evanescent states
• Probability current
• Boundary conditions
• Bound states
• The problem of plane waves
• Schrödinger equation, time-dependent aspects

The following sections are included:

• Variational method
• Perturbation theory
  • Non-degenerate perturbation theory
  • Degenerate perturbation theory
• Time-dependent perturbation theory
  • Static scatterers
  • Time-dependent scattering

One of the most powerful tools of investigation of the electronic states in semiconductor heterostructures is to apply a strong magnetic field to the sample. In this chapter, we investigate the energy levels of ideal materials subjected to a static and homogeneous magnetic field B…

A semiconductor heterostructure is a stacking of two or more layers of different semiconductors. The simplest is the single heterojunction (e.g., Si/SiO₂ interface which under appropriate gating leads to a Field Effect Transistor). Nowadays that the growth techniques have much improved since the 1970s, stacks of hundreds of layers with nanometric sizes are routinely grown (e.g., GaAs/(Ga,Al)As Quantum Cascade Lasers). As mentioned, the growth techniques underwent a revolution in the mid-1960s/early 1970s when, instead of growing bulk semiconductors at thermal equilibrium out of liquid phase (Czochralski, Bridgmann methods [18]), scientists made stacks of layers by sending fluxes of elemental elements (e.g., Ga, As, Al) from heated cells on heated (580°C) substrates (e.g., GaAs). Then, as surprising as it might seem, under appropriate conditions, a bidimensional growth occurs: an As plane grows, followed by a Ga plane (GaAs monolayer) and the process is iterated. Thus, by appropriately closing and opening shutters GaAs and (Ga,Al)As layers of precise thicknesses are grown (ideally at least). We refer to [19–21] to get more information on the Molecular Beam Epitaxy, Metal Organic Chemical Vapour Deposition and Epitaxial Microstructures…

The following sections are included:

• The envelope function approximation
  • Introduction
  • Electronic states in bulk semiconductors
  • Heterostructure states
• Multiple quantum wells: transfer matrix method
  • Multiple quantum wells and superlattices
  • Transfer matrix method
• Double quantum wells
  • Tight binding analysis
  • Symmetrical double quantum well
• Holes

We have so far discussed the eigenstates of ideal heterostructures: for 1D quantisation we found that the eigenstates can be written as

A gas of mobile electrons (ions) at thermal equilibrium subjected to an external potential energy $V_{\text{ext}}\left(\overrightarrow{r}\right)\equiv -e{\phi }_{\text{ext}}\left(\overrightarrow{r}\right)$ deforms itself in order to minimise its total energy (for ions −e should be changed in e_ion). For instance, a ionised donor creates a positive electrostatic potential. It is plausible (and true) that the electrons will be attracted by the ionised donor, resulting into an accumulation of negative mobile charges around the impurity compared to the uniform electron density that prevails in the absence of donor. The spatial rearrangement of the mobile carriers is called screening. The deformation induces a charge density $-e{n}_{\text{ind}}\left(\overrightarrow{r}\right)$ that creates, from Poisson equation, an induced electrostatic potential energy $-e{\phi }_{\text{ind}}\left(\overrightarrow{r}\right)$. This adds to the external potential energy $-e{\phi }_{\text{ext}}\left(\overrightarrow{r}\right)$, leading in turn to another deformation of the electron density. Ultimately, there is a total potential energy $-e{\phi }_{\text{tot}}\left(\overrightarrow{r}\right)$ and the central question is to find the relationship between ${\phi }_{\text{ext}}\left(\overrightarrow{r}\right)$ and ${\phi }_{\text{tot}}\left(\overrightarrow{r}\right)$. Often, the induced charge density (and therefore the induced electrostatic potential) is linearised with respect to the external potential energy…

The following sections are included:

• Scattering by static disorder
• Scattering of composite particles/excitons at the Born approximation
• Scattering on magnetic impurities
  • The “spin”-flip scattering of electrons
  • The “spin”-flip scattering of holes
• Three-body collisions
  • FCA in imperfect bulks and heterostructures
  • Phonon scattering in the presence of static scatterers

We give in this chapter a few results regarding the level broadening and energy loss rate due to electron–phonon interaction. In contrast with static scatterers, we shall not include screening effects. This is because the static screening model developed in Chapter II.4 is reasonably accurate for the electron–acoustical phonon interaction but is not reliable for the electron–optical phonon interaction. The LO phonons are so quickly oscillating (period T ≈ 1.2 × 10⁻¹³ s in GaAs) that the electrostatic perturbations they create require a dynamical approach of the screening action that is beyond the scope of our introductory textbook (see e.g., [64]). We therefore deal with the electron–phonon Hamiltonian (II.3.36) and (II.3.51):

In the previous chapters we have investigated the level lifetimes obtained by handling the electron defects/phonons interaction at the Born approximation (Fermi golden rule). Here, we wish to extend our analysis beyond the Born approximation because there exist situations where a more elaborate modelling is required (e.g., scattering involving bound states, scattering between Landau levels, etc.)…

The following sections are included:

• Average position and velocity
• Average velocity in a bound state
• Density of states
• Density of states of a camel back shaped dispersion relation
• Heisenberg inequality in a quantum well with infinitely high barriers
• Manipulating Slater determinants
• Pauli principle for two weakly interacting electrons in 1D
• Calculation with Pauli matrices
• Moss–Burstein shift of interband absorption
• Virial theorem
• Absence of degeneracy for the 1D bound states
• Variational method: hydrogen atom
• Variational method: electron in a triangular potential
• Variational method: anharmonic oscillator
• Screened coulombic bound states
• A two-dimensional coulombic problem
• Inter-subband transitions in cubic GaN/AlN quantum wells: information on the conduction band offset
• Asymmetrical square quantum well
• Spherical quantum dots
• Delta quantum well
• Wavefunction amplitude at the interfaces
• Interface state in HgTe/CdTe heterojunctions
• Step quantum well
• Application of the Bohr–Sommerfeld quantisation rule to 1D confining potential: digital alloying
• Transmission/reflection in a delta quantum well
• Static perturbation of a harmonic oscillator
• Static perturbation (degenerate case)
• Degenerate perturbation calculus applied to quantum dots with cylindrical symmetry
• Quantum well and a delta potential: perturbative estimate
• Quantum dot anisotropy
• Defect in a superlattice: tight binding approach
• Bound states created by two delta scatterers in a Landau level
• Time-dependent evolution in an infinitely deep quantum well
• Time-dependent problem: evolution
• A touch of interaction representation
• Time evolution if A and H commute
• Oscillator: time evolution of averages
• Time evolution of a system where one level is coupled to N degenerate levels
• Time-dependent Hamiltonian: an exactly solvable model
• Time evolution of superlattice states
• Wavepackets
• Average velocity of a wavepacket
• Time-dependent perturbation in a 2-level system
• Universal absorption probability for interband transitions in graphene
• Scattering by N random impurity dimmers
• A tractable example of selective doping by delta scatterers
• Comparison between Born and self-consistent Born approximations
• Influence of a fast emptying of the final subband on the equilibrium between two subbands
• Phonon-mediated equilibration of the electronic temperature to the lattice temperature
• Inter-subband scattering by unscreened coulombic impurities
• Evaluation of a double sum appearing in the free carrier absorption
• Energy loss rate for the in-plane polarisation T =0 K
• Inter-subband absorption versus carrier concentration in an ideal heterostructure
• Electron–LO phonon interaction: dimensionality dependence

The following sections are included:

• Bibliography
• Index