Modeling of tunneling through single, double and step barrier structures

Noora Tuomisto

Research output: ThesisDoctoral ThesisCollection of Articles


Modeling of electric currents through different types of tunnel junctions is important for both understanding experimental results and for designing new types of devices. In this thesis we calculate current density – voltage (J—V) curves for heterostructures forming single, double, and step barrier potentials using the quantum-mechanical Tsu-Esaki approach and the semiclassical Wigner formalism. We have implemented both methods into computer codes that solve the problem numerically. Our aim is to interpret experimental current – voltage (I—V) curves of ferroelectric tunnel junctions (FTJs) and magnetic tunnel junctions (MTJs) and to explore which barrier features affect the current characteristics the most. Different types of tunnel junctions are widely used in solid-state nanoelectronics devices. An FTJ has a ferroelectric barrier sandwiched between two electrodes. The electric polarization of the barrier can be switched using an external electric field, which affects the I—V curves. We model this phenomenon using the Tsu-Esaki approach with numerically solved transmission functions. Using a single tilted barrier model for the FTJ we show that our approach is more flexible than the commonly used analytical formulae since it is not tied to specific voltage ranges or limited barrier shapes. We also demonstrate that small changes in barrier thicknesses could be responsible for leaf-like shapes observed in experimental I—V curves. Using a step barrier model for the ferroelectric barrier we are able to reproduce experimentally measured asymmetric I—V curves. Especially, we demonstrate that steep rises in current for one bias polarity are related to resonant tunneling via quasi-bound states at the potential notch formed. Resonant tunneling diodes (RTDs) are structures with two potential barriers surrounding a quantum well (QW) exhibiting characteristic peaks in I—V curves due to resonant tunneling. We calculate self-consistently J—V curves for RTDs using the semiclassical Wigner formalism. Our results show that in the case of a QW made of a diluted magnetic semiconductor material the peak in the J—V curve splits into two as a function of increasing magnetic field or decreasing temperature. We can also reproduce the shapes of experimental I—V curves of such a magnetic RTD. This study provides insight to interpreting experimental data and gives guidelines how to obtain desired I—V characteristics by tailoring the tunnel barrier parameters. Our Tsu-Esaki approach is computationally efficient in modeling tunneling currents whereas the Wigner formalism requires much more computational resources and lacks in accuracy compared to the Green's function method. Further studies on tunneling would benefit from first-principles calculations investigating the complicated interplay of electronic and atomic structures and the tunneling process.
Translated title of the contributionTunneloitumisen mallintaminen yksittäis- ja kaksoispotentiaalivallien sekä porraspotentiaalivallien läpi
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
  • Puska, Martti, Supervising Professor
  • Puska, Martti, Thesis Advisor
Print ISBNs978-952-60-8037-6
Electronic ISBNs978-952-60-8038-3
Publication statusPublished - 2018
MoE publication typeG5 Doctoral dissertation (article)


  • quantum mechanical tunneling
  • modeling
  • Tsu-Esaki
  • Wigner
  • FTJ
  • RTD


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