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V. Brousseau-Couture and M. Côté
protected boundary states, since one cannot go from a bulk insulator to a topological
insulator (TI) without closing the bulk band gap.
First-principles studies have been widely used to predict and characterize new
topological materials since they do not make any assumptions on the system and
do not rely on external parameters besides the crystal structure. However, most
of these calculations are done under the assumption of the static lattice, while
experiments are inherently done at finite temperature, where atoms are necessarily in
motion. Such calculations thus disregard how an increasing population of thermally
activated phonons affects material properties.
1.1 Electron–Phonon Interaction in Semiconductors
When studying topological insulators from first-principles, one aims to identify
topologically non-trivial electronic band structures, which manifest themselves
through a local inversion of the leading orbital character of the valence and
conduction bands in the vicinity of their respective extrema (band inversion) [1]. To
accurately predict the topology of a given band structure at some finite temperature,
one must, therefore, assess how the temperature affects the electronic structure. In
analogy with the Lamb shift in QED, where the electronic energy levels are modified
by an interaction with the vacuum fluctuations, in condensed matter, electrons
interact with phonons even at T = 0 K, through the zero-point motion of the ions.
This electron–phonon interaction (EPI) gives rise to many distinctive features in the
band structure, such as a variation of the band gap energy with temperature, band
broadening induced by the finite lifetime of electronic excitations, the presence of
replica bands, and the formation of kinks near the Fermi level for metals [2].
1.2 Phonon-Induced Topological Insulation
From this point a view, since a topological phase transition requires the bulk
band gap to close and EPI governs the temperature dependence of the electronic
structure, one can naturally wonder if EPI could be strong enough to close
the gap and drive a topological phase transition (TPT). This question was first
addressed by Saha and Garate [3, 4] through model Hamiltonians, and later
investigated with first-principles calculations for BiTl(S 1−δ Se δ ) 2 [5] and Sb 2 Se 3 [6].
In those last two studies, the TPT is driven by an experimentally controllable
parameter, respectively, stoichiometric substitution and hydrostatic pressure. TPTs
can, therefore, be detected by tracking the variation of the bulk electronic band
gap, which must vanish at the critical parameter. Since temperature affects the
gap energy, the critical parameter at which the TPT takes place evolves with
temperature. Understanding how temperature influences the topological properties
of such materials is crucial for the efficient design of technological applications
V. Brousseau-Couture and M. Côté
protected boundary states, since one cannot go from a bulk insulator to a topological
insulator (TI) without closing the bulk band gap.
First-principles studies have been widely used to predict and characterize new
topological materials since they do not make any assumptions on the system and
do not rely on external parameters besides the crystal structure. However, most
of these calculations are done under the assumption of the static lattice, while
experiments are inherently done at finite temperature, where atoms are necessarily in
motion. Such calculations thus disregard how an increasing population of thermally
activated phonons affects material properties.
1.1 Electron–Phonon Interaction in Semiconductors
When studying topological insulators from first-principles, one aims to identify
topologically non-trivial electronic band structures, which manifest themselves
through a local inversion of the leading orbital character of the valence and
conduction bands in the vicinity of their respective extrema (band inversion) [1]. To
accurately predict the topology of a given band structure at some finite temperature,
one must, therefore, assess how the temperature affects the electronic structure. In
analogy with the Lamb shift in QED, where the electronic energy levels are modified
by an interaction with the vacuum fluctuations, in condensed matter, electrons
interact with phonons even at T = 0 K, through the zero-point motion of the ions.
This electron–phonon interaction (EPI) gives rise to many distinctive features in the
band structure, such as a variation of the band gap energy with temperature, band
broadening induced by the finite lifetime of electronic excitations, the presence of
replica bands, and the formation of kinks near the Fermi level for metals [2].
1.2 Phonon-Induced Topological Insulation
From this point a view, since a topological phase transition requires the bulk
band gap to close and EPI governs the temperature dependence of the electronic
structure, one can naturally wonder if EPI could be strong enough to close
the gap and drive a topological phase transition (TPT). This question was first
addressed by Saha and Garate [3, 4] through model Hamiltonians, and later
investigated with first-principles calculations for BiTl(S 1−δ Se δ ) 2 [5] and Sb 2 Se 3 [6].
In those last two studies, the TPT is driven by an experimentally controllable
parameter, respectively, stoichiometric substitution and hydrostatic pressure. TPTs
can, therefore, be detected by tracking the variation of the bulk electronic band
gap, which must vanish at the critical parameter. Since temperature affects the
gap energy, the critical parameter at which the TPT takes place evolves with
temperature. Understanding how temperature influences the topological properties
of such materials is crucial for the efficient design of technological applications