Contents
Topological Insulator Nanoparticles and Light
Marie Rider
Supervisors: DKK Lee, V Giannini, PD Haynes Word count: 2998
1 Introduction 1
2 From Insulators to Topological Insulators 2
2.1 TopologicalOrder ……………………………….. 2
2.2 Spin-orbitCoupling ………………………………. 3
2.3 AttheEdge…………………………………… 3
3 Topological Insulators in 3D 4
3.1 Bi2Se3 ……………………………………… 4
3.2 BulkBehaviour…………………………………. 4
3.3 AttheSurface …………………………………. 5
3.4 IntotheNano-regime………………………………. 6
4 Topological Insulator Nanoparticles and Light 6
4.1 Novelty …………………………………….. 6
4.2 Methodology ………………………………….. 7
4.3 Approximationsandlimitations…………………………. 8
4.4 Applicationsandoutlook ……………………………. 8
Bibliography 8 1 Introduction
Symmetries and the resulting conserved quantities underpin all observed physical laws. We are most familiar with the symmetries and phases of matter characterised by local order parameters within Landau-Ginzburg theory [1, 2], but in the last few decades the exploration of topological phases of matter has lead to many new developments in our understanding of condensed matter physics, culmi- nating in a Nobel prize for Thouless, Haldane and Kosterlitz in 2016. The notion of topology in physics was introduced by Klitzing and his discovery of the 2D quantum Hall (QH) state [3], with Thouless et al. explaining the quantization of the Hall conductance in 1982 [4]. Much progress has been made since and whereas QH states explicitly breaks time-reversal (TR) symmetry, new materials obeying TR symmetry have been explored. The first proposals of the 2D topological insulator (TI) – otherwise know as the quantum spin Hall state – were remarkably recent (Kane and Mele, 2005 [5] and Bernevig and Zhang, 2006 [6]). The 3D generalisation came soon after in 2007 [7] and experiments have shown that these new phases of matter are both realisable and accessible [8–10]. All TR invariant TIs in nature (without ground-state degeneracy) fall into two distinct classes, classified by a Z2 topological
1
order parameter [11, 12]. The topologically non-trivial state has a fully insulating gap in the bulk and gapless edge or surface states consisting of an odd number of Dirac fermions.
The major work effort on TIs has been to study their the electronic properties, whilst the inter- action of TIs with light has been studied less thoroughly. The presence of surface states should make interactions with an external field different to that of a topologically trivial material, and whilst some experimental steps have been made to explore the properties of the surface states [13], their coupling with light is yet to be experimentally verified. Very recent theoretical work has studied spherical TI nanoparticles (TINPs) – whose size gives both a much higher surface to volume ratio and higher curvature than regular TIs – showing that under the influence of light, the surface state excitations of a TINP can mediate interactions between the incident light and a bulk phonon giving rise to a prevously undiscovered mode. The surface state excitations can even be strong enough to screen the bulk. The study of TINPs with light is so new that there is only one paper on this topic [14], although useful results for the spherical TI have also been found [15]. This makes a literature review on the topic fairly restricted, but is highly motivating in that there is plenty of work to be done, making this a very exciting research area.
The outlook in this very new area is to better understand the interaction of light with TI nanostruc- tures, with the expectation that the shape and curvature of the nanostructure will greatly influence the interactions experienced under illumination. As will be revisited in the section devoted to applications and outlook, this research area shows promise in its potential applications to spintronics, quantum optics and quantum information [10, 16].
2 From Insulators to Topological Insulators 2.1 Topological Order
Calculating the electronic band structure of materials allows us to categorize them into groups such as metals, semiconductors or insulators. To do so, we calculate the Bloch wave functions of electrons in momentum space, and then by calculating their corresponding energies, we find the band struc- ture [17]. The chemical potential (Fermi level) of the material, E = 0, is the demarcation between occupied and unoccupied states. Regular insulating materials are characterised by the chemical poten- tial residing within an energy gap, nestled between a conduction band and valence band (see figure 1a).
The wave functions are composed of basis states defined in a Hilbert space and the Hamiltonian of the system transforms the states within that Hilbert space. Topology studies whether objects can be transformed continuously into each other. In condensed matter physics we can ask whether the Hamiltonians of two different quantum systems with an energy gap can be continuously transformed into each other without closing the gap. If that is the case, then we can say that the two systems are ‘topologically equivalent’.I
We can also introduce more specific criteria, for example that some symmetry inherent in the system such as TR symmetry should be preserved throughout the continuous path which connects two Hamil- tonians. In order to know whether there is any path that connects two Hamiltonians without closing the gap, we can count the number of levels below the chemical potential. For gapped Hamiltonians no energy level can move through E = 0, but otherwise they can move freely. Therefore continu- ous transformations only exist between Hamiltonians with the same number of energy levels below
IIf we considered all Hamiltonians without any constraint, every Hamiltonian could be continuously deformed into every other Hamiltonian, and all quantum systems would be topologically equivalent. This changes drastically if we restrict ourselves to systems with an energy gap.
2
Figure 1: (a) Energy levels for a insulating material, with a finite band gap separating the valence and conduction bands. The fermi level lies in the energy gap. (b) The energy spectrum of a topological insulator with conducting states traversing the energy gap between the valence and conduction bands. Images: Wiki.
E = 0. Since this number cannot change under continuous transformations inside the set of gapped Hamiltonians, we call it a topological invariant. Whenever an energy level crosses E = 0, this changes the topological invariant, causing a topological phase transition. If two Hamiltonians have different topological invariants, they must be separated by such a transition and it is impossible to go from one topological phase to another without closing the gap and breaking the inherent symmetry.
2.2 Spin-orbit Coupling
Spin-orbit coupling occurs via the interaction between the electron spin magnetic moment and the orbital angular momentum of the electron. It can be visualized as a magnetic field caused by the electron’s orbital motion interacting with the spin magnetic moment. In most of the common semi- conductors, the valence band-edges are formed by the p-orbits of electrons, whereas the conduction band-edges are formed by the s-orbits of electrons. This situation belongs to normal band order, i.e., a topologically trivial phase. In other cases, the relativistic effect from heavy elements can be so large that the s-orbits are pushed below the p-orbits and an inverted band order appears (see figure 2). The band inversion is a strong indication that a material falls into a topologically nontrivial phase. For 2D systems a quantum spin Hall effect analogous to integer quantum Hall will occur [6, 18, 19], and the description of the effect can be simply extended to 3D. Many TIs (such as HgTe/CdTe wells, Bi2Se3 family, and actinide compounds [20]), have spin-orbit coupling to thank for their topological propertiesII, and these are the materials on which we focus.
2.3 At the Edge
We have so far given no account of the boundary conditions of TIs. For a TI surrounded by a trivially- insulating medium (such as vacuum), we have a boundary on the interface of these two insulators with differing topological order parameters. The change in topological order parameter at the boundary requires the band gap to close, whilst remaining gapped in the bulks of both media. This results in localised boundary states, which traverse the band gap. Physically, the picture we then have of our topological insulator is of an insulating bulk, with conducting states localised on the boundary of the material (see figure 1b).
We should recall that for the TR invariant TI, the band gap is protected by demanding the preserva-
IISome of these materials require doping as well as spin-orbit coupling to exhibit TI properties, to ensure that the chemical potential lies within the band gap.
3
Figure 2: An example of band inversion due to spin-orbit coupling. A mixing of the two bands results in edge states. Image: [21]
tion of TR symmetry. This means that unless a TR breaking perturbation is applied, the edge states will be robust against imperfectionsIII. If we break the TR symmetry, the edge states will disappear and the system will turn into a trivial insulator. For the case of TR invariant TIs, we only have two types of insulators: no edge states (trivial) or 1 pair of edge states (topological). This is very different from the integer QH state [3], which can have any number of edge states. Because we only have 0 or 1 pair of edge states, these insulators are known as Z2 topological insulators [5, 8]. Other types of TIs will have different symmetries (such as chiral invariance) and thus edge states protected by different symmetries.IV
3 Topological Insulators in 3D
The first 3D TI material that was experimentally identified was Bi1−xSbx [22], predicted by Fu and Kane [7]. It turned out to be poorly suited to detailed topological surface state studies due to a compli- cated surface state structure – Rashba-split surface states are present alongside the topological surface states and predicted surface state structure has not agreed well with experimental results. Instead we focus on the Bi2Se3 family of materials, predicted by Zhang in 2009 [23], whose model Hamiltonian for this family of materials is used ubiquitously in theoretical studies alongside modifications by Liu et al [24]. With a larger band gap and simpler surface spectrum, Bi2Se3 is a good material with which to study topologically protected phenomena.
3.1 Bi2 Se3
Bi2Se3 consists of five-atom layers stacked along the z direction (shown in figure 3a). The coupling between two atomic layers within a quintuple layer is strong, whilst that between quintuple layers is predominantly due to van der Waals forces and is thus much weaker. Symmetries of the material are comprised of a three-fold rotational symmetry, an additional two-fold rotation symmetry and a reflection symmetry. Experimental confirmation of a single Dirac-cone surface state for Bi2Se3 was reported in 2009 by Xia et al. [25].
3.2 Bulk Behaviour
As well as the lattice symmetries described above, Bi2Se3 also obeys time-reversal symmetry. The important topological physics occurs near the Γ point, allowing us to write down a low-energy, effective
IIITR breaking perturbations would be for example the addition of a magnetic field-like term.
IVNote that the Hamiltonian of the surfaces states is gapless, and symmetry would have to be broken to gap the states.
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Figure 3: (a) The lattice structure of Bi2Se3, consisting of stacked quintuple layers. Each quintuple layer consists of five atoms per unit cell with two equivalent Se atoms, two equivalent Bi atoms and a third Se atom. (b) The energy spectra for the Bi2Se3 family of materials, as given by ab initio calculations. As can be seen, Sb2Se3 does not exhibit surface states, whilst the other three materials have clearly visible surface states. The Bi2Se3 spectrum also showcases the material’s relatively large band gap. Image: [23].
Hamiltonian [24],
H(k) = ε0(k)14 +
0 A2k+
A2k+ 0
M(k) −A1kz
−A1kz −M(k)
M(k)A1kz 0A2k−
A1kz −M(k) A2k−
0
(1)
where k± = kx ±iky, ε0(k) = C +D1kz2 +D2k⊥2 , and M(k) = M −B1kz2 −B2k⊥2 . The parameters in this effective model can be determined by fitting the energy spectrum of the Hamiltonian to that of ab initio calculations [23], with the results for Bi2Se3 presented by Liu et al [24].
3.3 At the Surface
The most prominent and perhaps most important property of TIs is the existence of gapless surface states. 3D TR invariant TIs such Bi2Se3 have 2D surface states. The surface effective model can be obtained by projecting the bulk Hamiltonian (3) onto the surface states, giving a 2D massless Dirac Hamiltonian,
Hsurf(kx,ky)=C+A2(σxky −σykx). (2)
The Dirac Hamiltonian gives a linear, relativistic dispersion relation and thus tells us that the surface states are described by massless Dirac fermions. Dirac fermions occur in the 2D electron gas that forms at the surface of TIs as a result of the strong spin–orbit interaction existing in the insulating bulk phase. Materials such as Bi2Se3 have a single Dirac cone, as seen in figure 3b. Note that true 2D systems will have an even number of Dirac cones, enforced by TR symmetry (e.g. Graphene, which has 4 Dirac cones [26]). 2D surface systems allow for a single Dirac cone, with the system describd as a ‘holographic metal’. The gapless nature is protected by TR symmetry in Z2 topological insulators. What makes this surface state distinct from ordinary surface states is its spin-momentum locking (due to spin-orbit coupling in the bulk), such that the surface state is spin non-degenerate and the direction of the spin is perpendicular to the momentum vector and is primarily confined in the surface plane. This results in a dissipationless spin current existing on the surface in equilibrium, because there is no net charge flow but the spin angular momentum flows in the direction perpendicular to the spin
5
direction.
For Bi2Se3 the surface states can be directly extracted from ab initio calculations by constructing maximally localized Wannier functions and calculating the local density of states on an open bound- ary [23]. The result for the Bi2Se3 family of materials is given in [23], where the single Dirac cone surface state for the three topologically nontrivial materials can be seen. A mathematical description of how the surface states emerge from the effective model can be found in the literature [23, 24, 27, 28].
3.4 Into the Nano-regime
Although the literature on TIs is extensive, the properties of spherical topological insulator nanopar- ticles (TINPs) have been studied very little, with the exception of recent work by Siroki et al [14] and a cursory mention by [15]. Previous work on optical properties of charged nanoparticles was limited to metalic nanoparticles rather than topological insulator nanoparticles [29], although some work on the optical properties of TI thin films has been done [30].
Some experimental work on Bi2Se3 exists, where n- and p-type Bi2Te3 nanoparticles are employed as nonlinear saturable absorbers to passively mode-lock erbium-doped fiber lasers (EDFLs) for sub- 400 fs pulse generations[31]. This work does not focus on the properties of Bi2Se3 nanoparticles, but rather their application in fiber lasers. Other efforts to experimentally produce TI nanstructures are underway, but as with TINPs, the theoretical study of them is limited. Efforts include the production of TIs into nanostructures such as exfoliated flakes from bulk crystals [32], nanoribbons [33, 34] and nanoplates [35].
4 Topological Insulator Nanoparticles and Light 4.1 Novelty
The main work on the interaction of TINPS and light is found in a single paper by Siroki et al [14]. As this paper single-handedly forms the basis of this topic, we discuss the methodology of this work here.
This paper gives a theoretical treatment of a spherical Bi2Se3 TINP illuminated with THz frequency light. The authors use a model in which the collective response of bulk states is described by a di- electric constant whilst the surface states are described quantum mechanically using time-dependent perturbation theory. The incident light is treated classically. Optical cross sections are calculated, illustrating that the scattering and absorption cross sections are significantly modified compared to the case of ordinary insulators. For certain radii and frequencies the absorption cross section increases by a factor of 10 [36].
The most important results from this paper are as follows. The electrons in the surface states screen the bulk of the nanoparticle at certain frequencies, suppressing absorption in the bulk. A single elec- tron in the surface state can mediate the interaction of a phonon with the incident light, giving rise to a previously undiscovered mode and the energy of the mode can be tuned by varying the shape and size of the nanoparticle. The surface Dirac cone of the TINP is quantised rather than continuous, a feature that will have implications for future work, as described in the outlook.
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Figure 4: The surface Dirac cone of the TINP consists of discrete levels symmetrically in relation to the Dirac point at E=0 and the level spacing scales inversely with particle radius R. Image: [14].
4.2 Methodology
The work begins by studying the interaction of an ordinary insular nanoparticle with light. The effective Hamiltonian given in equation 3 is used, and repeated here for convenience.
M(k)A1kz 0A2k−
H(k) = ε0(k)14 +
0 A2k+
A1kz −M(k) A2k−
0
(3)
M(k) −A1kz
A derivation of TINP surface states is performed using the envelope function approximation [15, 37] and using the resulting surface states and the Hamiltonian given in equation 3, the energy spectrum is obtained from the time-independent Schr ̈odinger equation. A very interesting result of the finite size of the particle is that surface Dirac cone is quantised, and the level spacing scales inversely with particle radius (see figure 4). If the radius of the particle were to increase, the spectrum would tend towards a continuous dispersion relation.
With the envelope functions of the surface states and their energy spectrum obtained, the au- thors proceed to investigate how the states and spectrum change in the presence of static electric and magnetic fields. Static surface states will not affect the optical properties of a TINP, as the incident field will oscillate at a non-zero frequency and so the model is modified to include time-dependence. The radius of the TINP sets the energy scale for the surface states and allows for the application of perturbation theory. The perturbation due to external light modifies both the wave function and the Hamiltonian. Note that the perturbed wavefunctions will have different surface charge distributions so that the surface of the particle will no longer be neutral. The charged surface will in turn gives rise to a potential in the Hamiltonian which should be taken into account for the solution to be self-consistent. The effect of redistributing surface charge becomes negligible for large particles but for small particles it is significant.
Solving the system in a self-consistent way, the results show that the polarisability of a TINP is different to that of an ordinary nanoparticle. The topologically protected surface states give rise to an additional term in the polarizability of the TINP, only becoming negligible when the radius of the particle is large (as then the time-dependent surface charge density is low) or when the frequency
A2k+ 0
The authors use a simplified form of this Hamiltonian to allow for an analytic solution, in which A1 =
A2, B1 = B2 and ε0(k) is neglected.
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−A1kz −M(k)
of the incoming light is very large (as the large detuning makes the transition unlikely). However, for small radii and small detuning, the polarizability is modified which in turn affects the scattering and absorption cross sections and in particular, the absorption cross-section can become an order of magnitude larger.
4.3 Approximations and limitations
The most impacting approximation that has been made is in the use of the simplified Hamiltonian from equation 3. Bi2Se3 is not an isotropic material, differentiating between the z and x-y directions of the material due to the layered structure. This structure is not encapsulated in the model Hamiltonian used. On a similar note, the dielectric function used to calculate the bulk response is taken to be a scalar rather than a tensor, neglecting anisotropy. The main features of the TINP are nonetheless captured, making the results of the paper qualitiatively useful. A possibly limitation is found by noting that the energy spacing between the surface states of the TINP is commensurate with kBT, implying that the effects described in the paper will only be observed at low temperatures.
Due to the analytical self-consistency method used to obtain the final results, extending the method to more complicated geometries than a sphere may prove difficult and different methods may need to be employed. An exciting comment on the work is that transitions between the surface states depend on the polarisation of the incident light (and thus the work could be extended to polarizations other than circular). This provides an additional degree of freedom in manipulating these states and may be of interest in the area of quantum information. In this work, light is treated classically. In order to study the low photon limit, the light will need to be treated quantum-mechanically.
4.4 Applications and outlook
The materials described by Landau symmetry-breaking theory have already a substantial impact on technology. Digital information storage media can be made from ferromagnetic materials which break spin rotation symmetry. Liquid crystals that break the rotational symmetry of molecules are widely used in display technology. Crystals that break translation symmetry lead to well defined electronic bands which in turn allow us to make semiconducting devices such as transistors. Different types of topological orders are even richer than different types of symmetry-breaking orders, suggesting their potential for novel applications in many areas such as spintronics and plasmonic [16].
TINPs with their increased absorption cross sections could find potential use in photodetectors. The effect of surface states acting as a screening layer and suppressing absorption inside the particle could be useful in the areas of plasmonics, cavity electrodynamics and quantum information.
One specific avenue of further work planned is to exploit the quantisation of the TINP surface state energies. We seek to calculate the transition matric element between the energy levels and calculate the decay rate transition and polarization properties of the photons emitted. The goal is to understand which are the most and less probable decays. In particular, we wish to study how the decay rates are affected by the nanoparticle shape and what the effect of perturbations are on the TINP. A possible application would be to build a lasing system with the levels of topological insulator nanoparticles, with direct application to quantum optics and quantum information.
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