Received 4 September 2016; revised 6 October 2016; accepted 21
December 2016
The edge state is considered in the
spectrum region where its branch splits from the bottom of a
continuous conduction band. It is shown that in this region
the electron wave function demonstrates two different scale
behaviours: slow and fast, that enabled us to construct some
simplified procedure for the analysis of the edge state. The
slow wave function part obeys a simple Schrödinger equation
the parameters of which are insensitive to the peculiarities
of the electron dynamics, while the fast part that describes
the details of electron behaviour in the primitive cell
reveals itself only at the edge. The equation for this fast
part was transformed into the boundary condition for the slow
part equation. The proposed method is illustrated considering
the simplest continuous model for a topological insulator and
a tight binding model for graphene.
Keywords: continuous
model, tight binding model, splitting point, fast-slow wave
function parts
PACS: 3.65.Ge, 73.20.-r,
73.22.-f, 73.22.Pr
Išnagrinėta kraštinė būsena spektro srityje,
kur jos energijos šaka atskyla nuo tolydinės laidumo juostos.
Parodyta, kad šioje srityje elektrono banginė funkcija
demonstruoja charakteringą dviejų skirtingų mastelių elgesį:
greitą ir lėtą kitimą koordinatei tolstant nuo krašto. Tai
įgalino sukonstruoti tam tikrą paprastą kraštinės būsenos
analizės procedūrą panaudojant mažą parametrą – energijos
nuokrypio nuo jos įsiliejimo į tolydinę juostą taško ir
draustinės juostos energijos santykį. Lėto banginės funkcijos
kitimo srityje ji tenkina paprastą Šrėdingerio lygtį, kurios
parametrai nejautrūs detaliai elektrono dinamikai primityviajame
narvelyje. Ta detali dinamika pasireiškia tik greitojo funkcijos
kitimo srityje, kuri sukoncentruota prie paties krašto. Lygtį
šioje greitojo kitimo srityje pasisekė transformuoti į kraštinę
sąlygą lėtojo funkcijos kitimo lygčiai, taip suformuluojant
dviskalę kraštinės būsenos teoriją. Metodas iliustruotas jį
pritaikant papraščiausiam tolydiniam topologinio izoliatoriaus
modeliui ir stipraus ryšio grafeno modeliui, kuriame įskaityti
sukinio-orbitos sąveika ir elektrono tuneliavimas į tolesnius
kaimyninius mazgus.
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