Abstract
In this work we derive an effective Hamiltonian for the surface states of a hollow topological insulator (TI) nanotube with finite width walls. Unlike a solid TI cylinder, a TI nanotube possesses both an inner as well as outer surface on which the states localized at each surface are coupled together. The curvature along the circumference of the nanotube leads to a spatial variation of the spin orbit interaction field experienced by the charge carriers as well as an asymmetry between the inner and outer surfaces of the nanotube. Both of these features result in terms in the effective Hamiltonian for a TI nanotube absent in that of a flat TI thin film of the same thickness. We calculate the numerical values of the parameters for a Bi2Se3 nanotube as a function of the inner and outer radius, and show that the differing relative magnitudes between the parameters result in qualitatively differing behaviour for the eigenstates of tubes of different dimensions.
Highlights
Topological insulators (TI) are an emerging class of materials which have attracted much attention due to the unique properties of their surface states[1]
Both of these manifest as the emergence of more terms in the effective surface state Hamiltonian for the surface states of a topological insulator (TI) nanotube, whose derivation will be the main focus of this paper
The parameters shown here have the largest magnitudes for the hs and μs which go with each direction of σ. hxz is, to numerical precision, equal to hx despite the differing forms of the underlying expressions. h has a rather large magnitude of around 0.185 eV for the parameter ranges shown here but is relatively unimportant as it amounts to a constant energy shift
Summary
We seek linear combinations of these exponentials which disappear simultaneously at the two surfaces at x =±W/2. Two such linearly independent combinations are f+. Substituting, for example, χ±σy into (H(4B),⊥ − E f )χ±σy = 0 gives a set of 2 equations which contain hyperbolic trigonometric functions of x but which should give 0 everywhere independent of the value of x. This indicates the coefficients in front of the various hyperbolic trigonometric functions should go to 0. Imposing the condition that these two expressions for c−,±σ y agree yields the equation
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