Interaction-induced energy-level crossing and transport phenomena in topological insulators

Abstract

A topologically insulating material, (Bi_1−xSb_x)₂Te₃ is studied in detail with respect to its resistivity measurements by evaluating these parameters—constituent chemical elements' valence states, as well as the defect types and their concentrations. Here, the ionization energy theory (IET) is exploited to unequivocally show that the electrical resistivity and carrier-type transition (n ⇌ p) for a (Bi_1−xSb_x)₂Te₃ topological insulator can be precisely captured in a consistent manner supported by experimental data. Subsequently, we expose why the insulating bulk and its topologically protected metallic surface are the two facets of the same topological insulator. In particular, these facets arise entirely from different chemical compositions and defects found in the bulk compared to its surface. Ionization energies, valence electrons and ‘special’ energy-level crossing with a finite energy gap are some of the notions that have been invoked within IET to derive the experimentally observable parameters, namely, the doping-dependent electron-ion scattering rate, resistivity and carrier-type transition. These parameters are then used to explain the experimental data obtained from (Bi_1−xSb_x)₂Te₃ and Pb_1−xSn_xSe compounds. In our final attempt, we use IET to unequivocally show that the topologically protected metallic surface states consist of special energy-level crossing where the crossed energy levels are not due to time reversal symmetry induced Kramers degeneracy.

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1. Introduction

Degenerate energy levels (gapless) can form conducting states even if they are confined to the surface (or to the edge). If their degeneracy is kept intact against external disturbances (temperature (T), pressure, electric (E) and magnetic (B) fields), such that the bulk always has a well-defined band gap, then the materials satisfying the above conditions can be grouped into a new class of solids. These new solids are known as topological insulators (TI). At least, that is the proper and generalized definition one can think of to identify TI based on the crossed energy-level notion in two-dimensional Pancharatnam metals (with quantum Hall effects), which was first calculated by Hatsugai.¹ This definition also applicable to three-dimensional TI,^2,3 and it properly rules out free-electron and Fermi-liquid metals, even in the presence of a skin effect (frequency-dependent surface conductivity).⁴ In TI, we actually need the bulk to be gapped so that the surface states form a distinct two-dimensional system that can exhibit Pancharatnam metallic properties.^5,6 This means that one may need to perform quantum Hall effect measurements^5–7 to confirm whether an insulator is indeed a topological insulator because electric transport measurements,^8,9 and other surface analysis techniques, such as the angle-resolved photoemission spectroscopy (ARPES),¹⁰ de Haas–van Alphen¹¹ and Shubnikov–de Haas¹² effects are insufficient to claim the bulk is gapped, while the surface is metallic, independent of the bulk.

On the other hand, TI are also related to topology, a mathematical notion that deals with ‘smooth’ deformations of any entity,¹³ and in our case, this entity strictly refers to two-dimensional surface metallic states or degenerate energy levels. One can think of these metallic states or energy levels as the elements (or open subsets) within an open set that comprises all degenerate energy levels. This means that the energy levels or the subsets are the topology on an open set where this open set and its subsets satisfy some precise conditions, which allow smooth deformations.¹⁴ Now, any deformation to these energy levels requires one to invoke an arbitrary right-hand side (rhs) action operator¹³ that tacks a transition function (namely, a phase factor) on the rhs of an electronic wavefunction.¹⁵ However, it has been rediscovered recently that a wavefunction picks up or drops a phase factor as a result of a physical notion known as the Pancharatnam phase retardation (due to phase acceleration or deceleration). In fact, when the phase and (or) group momenta (μm) of a particular wavefunction changes, then one has to tack the Pancharatnam–Berry (PB) phase factor on the rhs of a wavefunction, which gives rise to the Pancharatnam wavefunction transformation¹⁶ that was originally discovered (formulated and observed) by Pancharatnam using the polarized pencil beams and Poincaré sphere.^17,18 This means that, the original Berry's phase¹⁹ is a rediscovered special case (the group momentum of a wavefunction is invariant) within the generalized Pancharatnam phase.^15–18 Compared to a conducting surface, the bulk requires the wavefunction to be transformed beyond the phase factor due to a nonzero energy gap between the valence and conduction bands. Therefore, the above rhs action operator has a precise physical origin within the context of quantum mechanics due to the Pancharatnam wavefunction transformation.¹⁵

Having said that, we can now set the next course of action—to understand how the definition for a topological insulator (stated in the first paragraph) can be achieved physically, or, how to identify the relevant physical parameter(s) responsible to obtain such an insulator. In this respect, Kane and Mele,²⁰ and independently by Bernevig, Hughes and Zhang²¹ have confirmed the Hatsugai's surface energy-level crossing such that one needs a combination of physical properties to activate the required energy-level crossing at the edge or on the surface, while the bulk is of course, remains gapped. The physical phenomena that are at play to produce both gapped bulk and gapless surface states in a single system come from different magnitudes of certain crucial parameters (the bulk has different magnitudes compared to the edge or surface).²⁰ For example, the interplay among these critical parameters, namely, the spin–orbit coupling, intersite hopping matrix elements, Rashba coupling and the staggered sublattice potential give rise to a topologically insulating phase.²⁰

Note that the above critical parameters that eventually give rise to changing electronic structure in the bulk can only be different from its surface if their respective chemical composition and/or the coordination numbers are themselves different. For example, the HgTe/Hg_0.3Cd_0.7Te quantum wells⁷ for different thicknesses (d) have given rise to a quantum phase transition (QPT) from the usual gapped surface (similar to the bulk) to a gapless surface states (while the bulk is still gapped) where the critical thickness (d_c) here is 6.3 nm, which refers to the width of their device. However, the existence of odd numbers of spin-polarized edge channels is actually a questionable assumption.⁷ Despite this, the said QPT has been correctly associated with the changing electronic structure due to spin–orbit coupling (SOC)⁷ following the Bernevig–Hughes–Zhang (BHZ) model.²¹ Here, it is acceptable to not to know exactly what other critical parameter(s) (besides SOC) are responsible for the changing electronic structure (or changing energy (Landau)-level crossing). In summary, the relation between the changing Landau-level crossing and d due to SOC originates from the changing defect types and composition or more precisely, from changing chemical composition and/or the coordination numbers with changing d. These changes due to defect types and composition will give rise to a change in SOC,²² which shall be made explicit when we discuss the doping-dependent resistivity in TI.

The point on which we do not agree with König et al.⁷ and BHZ²¹ is the assumption that time reversal symmetry (TRS) is not violated for the above device for different applied gate voltages on the basis of the BHZ model.²¹ This invariance of TRS in the presence of electric current (due to spin or charge or both) in any condensed matter system should not be haphazardly assumed to be true due to a well thought-out hypothesis put forth by Messiah.²³ Please refer to our additional notes (prior to the Conclusions) for a proper summary. In particular, the TRS defined in ref. 23 leading to Kramers degeneracy²⁴ is not necessarily required to obtain crossed energy levels due to interplay among the above-stated critical physical parameters. Of course, one can construct a particular Hamiltonian to satisfy TRS if the energy levels are crossed (not necessarily Dirac points due to Kramers degeneracy) between the bulk valence and conduction bands^7,10,25 such that E₁(k) = E₂(−k) where k denotes the wavevector.²⁶ This construction, however, does not imply that TRS is always preserved by default in the presence of an internal electric current—for example, all standard (B = 0) and Hall resistance (B ≠ 0) measurements^27–29 necessarily violate TRS as a result of the nonzero internal electric current (static or time-dependent).²³ For a Hall current, one needs to consider Zeeman splitting³⁰ due to the nonzero applied magnetic field, while a static current produces a static ‘external’ magnetic (B′) field.

Therefore, our objective here is to develop the basic mechanism required to consistently explain the electronic transport phenomena and carrier-type transition in TI down to the atomic energy level. We start our analyses by invoking the properties of ionization energies and energy-level spacing within the ionization energy theory (IET), and make use of TRS results derived earlier in ref. 16. In fact, we have derived all the relevant equations required to properly understand the time reversal operation in any solid, including TI, as well as the conditions that may give rise to TRS violation in TI.¹⁶ With this knowledge as background, we will first evaluate the influence of the energy-level spacing (ξ, which is also known as the ionization energy) on resistivity and carrier-type transition with respect to different chemical compositions in (Bi_1−xSb_x)₂Te₃ and Pb_1−xSn_xSe. The band structure properties of (Bi_1−xSb_x)₂Te₃ and Pb_1−xSn_xSe relevant to understand transport phenomena have been investigated by Jinsong Zhang et al.³¹ and Dziawa et al.,³² respectively.

We will exploit the relation between ξ and the electron excitation probability to answer the microscopic origins for (i) the changes in doping-dependent resistivity data (including the carrier-density and electron-ion scattering magnitudes) in (Bi_1−xSb_x)₂Te₃ and Pb_1−xSn_xSe, and (ii) the carrier-type transition (p- to n-type or vice versa) for different chemical compositions and defects in (Bi_1−xSb_x)₂Te₃. Our research on the above-stated points ((i) and (ii)) can lead us to understand how TI fit properly and consistently within the current knowledge of atoms, metallic Fermi liquid, semiconductors, ferromagnets, Mott–Hubbard insulators, cuprate strange metals and Pancharatnam metals. In addition, these points were not addressed nor generalized in earlier reports with respect to different types of atoms, and with respect to non-Kramers degeneracy (crossed energy levels with ξ ≠ 0).

2. Ionization energy theory

Ionization energy is defined as the minimum energy required to remove a bound electron (bound to a nucleus of an atom) to infinity (r → ∞). In contrast, IET is based on the theorem that states—the minimum energy required to excite or to polarize a bound electron to a finite distance (r → r_finite) within a quantum system (atomic, molecular or solid) is proportional to the ionization energy defined above. This proportionality is precisely known as the ionization energy approximation,^33,34 which can be written as,

ξ^quantum_system ∝ ξ^constituent_atom. (1)

Within a quantum system, the energy is quantized and therefore, the ionization energy is also known as the energy-level spacing. This energy-level spacing can be used to rewrite the standard Schrödinger equation to read,^35,36

which is known as the IET–Schrödinger equation where H_IET and V_IET are the exact Hamiltonian and potential terms, respectively. The properties of electrons are represented by the true and real (not a guessed) wavefunction, Ψ(r,t), ħ is the Planck constant divided by 2π and m denotes electron mass. Here E₀ is the ground state energy for temperature (T) equals zero Kelvin, in the absence of external disturbances. The sign, ± refers to electrons and holes, respectively. The above generalized definitions for H_IET and V_IET imply that eqn (2) is too general to capture a specific quantum matter. One way to solve eqn (2) is to rewrite it in the usual form,after which, one can incorporate the relevant potential terms (V) with appropriate approximations, and make do with the guessed wavefunction, satisfying the variational principle to find the energy eigenvalues. Here, we will not follow this approach. Instead, we make use of eqn (1) and (2) to tackle a specific quantum matter. For example, ξ will furnish one with the details on the electron excitation probabilities³⁵ for a given material with different chemical compositions.^33,34 These probabilities can be properly incorporated into the relevant physical parameters (namely, carrier density, electron–ion interaction term and carrier-type transition) by means of the energy-level spacing renormalization group method.³⁶ Our renormalization procedure will renormalize the above physical parameters such that they can be used to determine the microscopic mechanisms responsible for doping-dependent electrical resistivity and carrier-type transition in TI. A specific potential term in eqn (2) has been exploited earlier in ref. 37–39. Note that one can also reprocess the energy-level spacing cut-off parameter exactly within Shankar's wavenumber-dependent renormalization technique.^40–42 For an obvious reason (see below), we prefer to work with the energy-level spacing.

2.1. Ionization energies for isolated atoms

We now list the chemical elements needed to make TI, calculate their average ionization energies and explain what these averaged values mean. For an hypothetical atom, Xⁱ⁺ (with i number of valence electrons) the averaged ξ can be calculated from,here, the subscript j identifies the different types of chemical elements (X_j) in a given TI, while the subscript, i = 1, 2, …, z, counts the number of outer electrons required to interact with its nearest and next nearest neighbors by forming bonds. We list all the essential chemical elements required to form TI in Table 1. Before averaging, the experimental ionization energy values for each outer electron (1st, 2nd, and so on) were obtained from ref. 43 and 44. This is the only external information one needs to proceed. Table 1 also lists the averaged ionization energy values calculated from eqn (4) with respect to the stated valence states.

Table 1 Averaged atomic ionization energies (ξ) for selected atoms are ordered with increasing atomic number Z. The averaging follows eqn (4) and we used the unit kJ mol⁻¹ for numerical convenience

Element	Atomic number Z	Valence state	ξ /kJ mol^−1
Se	34	1+	941
Se		2+	1493
Se		4+	2476
Sn	50	1+	709
Sn		2+	1061
Sn		4+	1688
Sb	51	1+	834
Sb		3+	1623
Te	52	1+	869
Te		2+	1330
Te		4+	2242
Pb	82	1+	716
Pb		2+	1084
Pb		4+	2333
Bi	83	1+	703
Bi		3+	1593

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We now briefly state the importance of approximating ξ^quantum_matter from ξ^constituent_atom (ξ for short) via eqn (1) and (4). A chemical element, X_j with large ξ(X_j) means its valence electrons are not easily excited or polarized, and conversely, the electron-excitation probability and the atomic polarizability of a chemical element (X_j+1) are large if its ξ(X_j+1) is small. Furthermore, one can readily use eqn (1) to claim that the above probability and polarizability for a given TI get smaller if we systematically substitute one of its constituent chemical elements (X_j, ξ(X_j)) with another atom (X_j+1) that has a larger ξ(X_j+1). The above probability can be derived from the Fermi–Dirac statistics (FDS) with ξ as an additional restrictive condition,^33,34 and is given by,

where λ = (12πε₀/e²)a_B, ε₀ is the permittivity of free space, a_B is the Bohr radius and E⁰_F denotes the Fermi level for T = 0 K and without any external disturbances. Eqn (5) is also known as the ionization energy based FDS (iFDS). The atomic polarizability (due to outer electron displacement) can be determined from,^45,46ω_0i is the ith electron frequency, Z_j is the atomic number of the jth atom, while f_i denotes the strength factor of an ith polarizable electron in a given atom. Eqn (6) is obtained by neglecting the spontaneous emission (or classically known as the damping factor).⁴⁶ It is apparent that both f(E₀,ξ) and α(ξ) are inversely proportional to ξ. However, IET is inapplicable for free-electron (due to ξ = 0) and Fermi liquid metals (because ξ is an irrelevant constant), which implies that IET is only useful for quantum systems with ‘relevant’ ξ such that their Hamiltonians can be written in the form of eqn (2).

2.2. Relevant and irrelevant energy-level spacing

A ‘relevant’ energy-level spacing means a particular physical property is directly influenced by ξ, while an ‘irrelevant’ ξ and ξ = 0 have no influence at all. Examples of trivially relevant energy-level spacings⁴⁷ are the atomic energy-level spacing,^43,44 band gaps in band insulators and semiconductors,⁴ Mott–Hubbard gaps in Mott insulators,⁴⁸ and the molecular gaps in molecules⁴⁹ (between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO)). All Fermi liquid metals⁴ (including Pancharatnam metals^28,29) require irrelevant ξ (ξ_irr ≠ 0),¹⁵ while ξ = 0 is for free electrons (Fermi gas).⁴⁷ However, there is a special situation that can give rise to a nontrivially relevant ξ, which is applicable to strange metals in cuprates,⁴⁷ and other strongly correlated metals where ξ is ‘special’ because it exists within a degenerate electronic system, and its origin is due to different wavefunctions.⁵⁰ In particular, the special energy-level spacing theorem reads^47,50—an electron needs energy supply to occupy another degenerate energy-level because the wavefunction representing this particular electron needs to be transformed (beyond the phase factor) to occupy the new energy-level.

From our previous analyses on TRS given in ref. 16, it should be clear that TRS will be violated in the presence of static electric current regardless of whether ξ = 0, or ξ is an irrelevant constant, or ξ is a finite relevant constant. This also means that any violation of TRS does not necessarily imply a given system is gapped because the internal electric current violates TRS even in gapless systems. Apart from that, even though TRS is known to be responsible for Kramers degeneracy, TI can form non-Kramers degenerate surface states due to many other complicated interactions as pointed out in the introduction, and in ref. 47 and 50.

3. Results and discussion

We show here that the metallic property in gapless TI can be associated with the above special (nontrivially relevant) energy-level spacing (ξ^non_triv). Usually, a given compound is defined to be metallic if its resistivity (ρ(T)) gets smaller with lowering temperature T. Both ρ(T,x) and the carrier-type transition (n- to p-type) have been measured by Jinsong Zhang et al.³¹ in (Bi_1−xSb_x)₂Te₃. Apart from metallic resistivity, we will also explain why the bandgap of another TI, namely, Pb_1−xSn_xSe changes with Sn substitution, which were first measured by Strauss⁵¹ and later confirmed by Dziawa et al.³² using ARPES measurements. The latter bandgap analyses are based on the trivially relevant ξ_triv. Since the energy-level spacing for both gapless and gapped systems are nonzero and relevant, we are therefore forced to identify them with ξ^non_triv and ξ_triv for the gapless and gapped systems, respectively.

3.1. Carrier density and electron–ion interaction

Changes in the magnitude of resistivity in TI or any other strongly correlated solid state systems are usually due to doping x, T and B. In the absence of B, and for constant T, the microscopic physical parameters that will significantly influence the resistivity with respect to x are the carrier density (n(ξ)) and the scattering amplitude, |f(θ)|. Here,^34,45

n(ξ) = C(T)exp[λ(ξ − E⁰_F)], (7)

where C(T) denotes a collection of fundamental constants, which actually depends on the dimensionality of the system and C(T) also contains a T-dependent parameter. The resistivity,where we have suppressed the T dependence because ρ(x) is for constant T. Next, we need to find how the scattering rate, τ(x)⁻¹ varies with doping. For free-electron and Fermi liquid metals, τ(x)⁻¹ → τ⁻¹ by definition⁴⁷ because the electrons are completely independent of the types of ions such that they interact with neutral phonons, not with the positively charged ions. This means that, τ⁻¹ does not systematically change with the type of atoms in a Fermi liquid.⁴⁷ In this case, the scattering rates are mostly contributed by the well-known electron–electron (e:e) and electron–phonon (e:ph) type collisions. On the other hand, in strongly correlated metals, including metallic TI, we have the usual contribution from τ_e:e⁻¹ and another dominant contribution from the electron–ion (e:ion) scattering rate, τ(x)_e:ion⁻¹. We show here that τ(x)_e:ion⁻¹ is proportional to the scattering amplitude, |f(θ)|.

To understand why τ(x)_e:ion⁻¹ ∝ |f(θ)|, we use the Born approximation⁵² and the renormalized screened Coulomb potential,⁴⁵

to obtain the scattering amplitude,where μ_IET is the constant of proportionality, and as required, eqn (10) reduces to the Rutherford scattering amplitude (for two-point charges) when ξ → ∞,³⁶ and it also reduces to the Yukawa scattering amplitude if ξ → 0.³⁶ Moreover, κ = |k′ − k| depicts the changes in the wavevector such that k′ and k point in the incident and scattered directions, respectively, and during the process, the momentum transfer is given by ħ(k − k′). It is obvious from eqn (10) that θ is the angle due to scattering between the directions, k and k′. In addition, eqn (10) leads us straight to |f(θ)| ∝ ξ, which implies that a large scattering amplitude is obtained for an ion (scattering center) with large ξ, which in turn justifies the correctness of this relation, τ(x)_e:ion⁻¹ ∝  |f(θ)|. As a consequence, the approximate resistivity can be written in the form (using eqn (8)),where C′′ is another collection of fundamental constants, which includes C and C′. Firstly, even though eqn (11) is an approximate one, it captures exactly the sought-after effect of changing carrier density and e:ion scattering strength on ρ(x), and secondly, the equation also shows that n(ξ) and τ⁻¹(x) do not compete with each other with opposite effects as a result of doping or changing chemical composition. This second point is important such that ρ(x) (for constant T) is guaranteed to decrease if one substitutes a chemical element with another one that has a smaller ξ because small ξ leads to large n(ξ) and τ(x) (see eqn (8)).

Thus far, we have ignored the T-dependence even though we know both ξ(T) and τ_e:ion(T) do exist simply because their explicit forms are unknown due to their complex T-dependences. Logically, τ_e:ion has to be T-dependent, while ξ has been experimentally proven to be T-dependent by Dionicio⁵³ using the results of Fukuda et al.⁵⁴ In particular, Dionicio showed that the valence state of a given multivalent chemical element is T-dependent. Despite this, eqn (11) is exact with respect to ξ-dependence, and therefore, it is sufficient for our analyses, and to show that indeed non-Kramers degeneracy (due to relevant ξ^non_triv ≠ 0) is responsible for the metallic surface states in TI.

It is also worth noting here that one can indeed use the lattice-strain parameter (that can be estimated by calculating the changes in the Raman phonon frequency and intensity) to evaluate defects and their concentrations in topological insulators or any solids, but this information is limited to the study of defects per se, which cannot be directly exploited to examine the carrier-type transition and resistivity. Moreover, by working within IET, we do not need to assess the strength of the lattice-strain and its distribution because the energy-level spacing is far more fundamental than all other macroscopic entities, which can be applied to model and theorize many physical parameters and mechanisms as proven in ref. 16. As a matter of fact, we can also exploit IET to calculate the changes to the Raman phonons as a result of defects and doping.

3.2. Doping-dependent resistivity and carrier-type transition

We have concluded the derivation of all the equations needed to explain the measured resistivity data and the carrier-type transition in Pb_1−xSn_xSe and (Bi_1−xSb_x)₂Te₃. Dziawa et al.³² have studied the narrow bandgap semiconductor, Pb_1−xSn_xSe where a topological phase transition is observed for x = 0.23 when T is reduced from 300 K to 77 K. In particular, Pb_0.77Sn_0.23Se is a gapped semiconductor for T = 300 K, and it becomes a gapless TI for T = 77 K due to band inversion observed indirectly via ARPES. Moreover, for a given T (300 K or 195 K or 77 K), Sn substitutional doping (increasing x from 0 to about 0.3) systematically reduces the bandgap of the Pb_1−xSn_xSe semiconductor from about 0.3 to approximately 0.05 eV (see Fig. 1 in ref. 32). This observation is easily understood within IET by noting that ξ now represents the bandgap, ξ_Sn²⁺ < ξ_Pb²⁺ (see Table 1), and from eqn (7) and (11), ρ(x) is predicted to decrease with increasing Sn content, provided that the valence states of the chemical elements (Pb and Sn) do not change significantly due to defects. If the valence states do change, then the analyses become tedious, which have been addressed elsewhere for other strongly correlated materials and oxides^22,47,55 where topological insulators are no exception (see below).

In the absence of defects, the changes in ρ(x) for Pb_1−xSn_xSe have been exposed in a straightforward manner within IET. However, the (Bi_1−xSb_x)₂Te₃ material is reported to have a complicated doping-dependent resistivity by Jinsong Zhang et al.³¹ For example, the measured ρ(x) first increases with x, for up to x = 0.94, and then it reduces until x = 1 (see Fig. 4 in ref. 31). With increasing Sb content (0 ≤ x ≤ 0.94), the system also becomes relatively more insulating. Further Sb substitutional doping (0.94 < x ≤ 1) leads to a weaker insulating behavior such that Sb₂Te₃ is a semi-metal with holes as the dominant charge carriers. In contrast, Bi₂Te₃ is an electron-dominant metal. The above carrier-type transition from an n-type insulator to a p-type semi-metal are of course due to defects where the valence states for Sb³⁺ and Te²⁺ are not constant with doping (for x > 0.94).

Similar to Si = [Si⁴⁺][Si⁴⁺] semiconductor, (Bi_1−xSb_x)₂Te₃ is also predominantly covalent bonded and therefore, we can also write it in the form, [(Bi_1−xSb_x)₂]⁶⁺[Te₃]⁶⁺ = [(Bi_1−x³⁺Sb_x³⁺)₂][Te₃²⁺]. Using eqn (7) and (11), and the fact that ξ_Bi³⁺ < ξ_Sb³⁺, one can deduce that ρ(x) should increase with increasing Sb content, which has been observed for 0 ≤ x ≤ 0.94. Next, to understand the carrier-type transition in this class of material, it is sufficient for us to focus on these two extreme cases, namely, for x = 0 (n-type; Bi₂³⁺Te₃²⁺) and for x = 1 (p-type; Sb₂³⁺Te₃²⁺). We will exploit the carrier-type transition theorem developed in ref. 55, which will also lead us to understand why ρ(x) has decreased for x > 0.94.

The carrier-type transition theorem states that p-type materials with relevant ξ should satisfy this condition,⁵⁵

ξ_acceptor^a+ < ξ_host^h+, a < h and x_acceptor < y_host, (12)

where a and h denote the acceptor and host valence states, respectively, while x_acceptor and y_host are the respective concentrations of the acceptor and host chemical elements. The acceptor accepts holes from the host such that the electrons from the acceptor are more easily polarizable than that of the host following eqn (6). For example, for [(Bi_1−x³⁺Sb_x³⁺)₂][Te₃²⁺], the subscripts, 2(1 − x) and 3 are the concentrations for the host chemical elements, Bi and Te for x < 0.5. Now, Bi₂³⁺ Te₃²⁺ is clearly an n-type system because it does not fulfil the above condition where Te₃ is both the host and acceptor, which is not allowed from eqn (12), and if this is the case, then eqn (6) does not allow the creation of holes.⁵⁵ In other words, the inequality, ξ_acceptor^a+ > ξ_host^h+ does not exist because ξ_{acceptor;Te²⁺} = ξ_host;Te²⁺ where the latter equality violates the condition stated in eqn (12), and therefore, eqn (6) cannot be used to create holes. Given this background, Sb₂³⁺Te₃²⁺ is also predicted to be an n-type material.

However, Sb₂³⁺Te₃²⁺ can be made to be a p-type material if we introduce a small amount of defects (with x_new ≪ 2 and y ≪ 3) to give rise to [Sb_2−xnew³⁺Sb_xnew^a′+][Te_3−y²⁺Te_y^a′′+]. Here, if x_new exists, then y should also exist to balance the defects introduced by x_new so as to maintain a proper coordination number, regardless of the crystal growth conditions because the distribution of these defects need not be homogeneous at all. We now evaluate [Sb_2−xnew³⁺Sb_xnew^a′+] and [Te_3−y²⁺Te_y^a′′+] separately. It is apparent that Sb_2−xnew³⁺ is the host and Sb_xnew^a′+ is the acceptor where ξ_Sb³⁺ > ξ_Sb^a′+ if a′ < 3+. Similarly, for [Te_3−y²⁺Te_y^a′′+], Te_3−y²⁺ is the host, while Te_y^a′′+ is the acceptor such that ξ_Te²⁺ > ξ_Te^a′′+ if a′′ < 2+. These inequalities satisfy eqn (12), and consequently, Sb_2−xnew³⁺Sb_xnew^a′+Te_3−y²⁺Te_y^a′′+ is a p-type TI in the presence of defects.

Moreover, ξ_Sb³⁺ > ξ_Sb^a′+ and ξ_Te²⁺ > ξ_Te^a′′+ give rise to decreasing ρ(x) for x > 0.94, which is in agreement with the observed data (see Fig. 4 in ref. 31 and Fig. 1(a) and (b)). The resistivity should decrease for x > 0.94 because the carrier-type transition to p-type is caused by the increasing defect density (x_new and y), namely, Sb^a′+ and Te^a′′+ for x > 0.94, which give rise to an increasing number of holes due to these inequalities, ξ_Sb³⁺ > ξ_Sb^a′+ and ξ_Te²⁺ > ξ_Te^a′′+. Using the above analyses, one can also obtain theoretically a p-type Bi₂³⁺Te₃²⁺ with appropriate defects, Bi^a′+ and Te^a′′+ where a′ < 3+ and a′′ < 2+. The above defect densities can be determined experimentally by using the chemical method to measure the valence states of various chemical elements that was first carried out by Mahendiran et al.⁵⁶

Fig. 1 (a) Sheet resistance, R_□ at 300 K for x = 0, 0.5, 0.75, 0.88, 0.94. (b) The same resistance for x = 0.94, 0.96, 0.98, 1.0. The data points in both (a) and (b) were obtained from ref. 31 for (Bi_1−xSb_x)₂Te₃ topological insulator. The calculated curves follow the doping-dependent resistivity model, R_□ = A exp[Bλξ(x)] where A and B (dimensionless) are doping-independent constants, while x and λ denote doping and IET fundamental constants, respectively and ξ(x) is the doping-dependent energy-level spacing, which is not a constant because ξ(x) varies when x varies. Hence, the fitting is trivial. This model is a simplified version from IET (see eqn (11)). The curve in (a) increases with x because ξ(x) = (1 − x)ξ(Bi³⁺) + (x)ξ(Sb³⁺) where ξ(Bi³⁺) < ξ(Sb³⁺). The curve in (b) however, reduces with doping x > 0.94 because of defects (due to the existence of Sb^a′+ and Te^a′′+) such that ξ_Sb³⁺ > ξ_Sb^a′+ and ξ_Te²⁺ > ξ_Te^a′′+. The said defects have led us to write, ξ(x) = (2 − x_new)ξ_Sb³⁺ + (x_new)ξ_Sb^a′+ + (3 − y)ξ_Te²⁺ + (y)ξ_Te^a′′+, which means, R_□ becomes smaller for x > 0.94 due to the contributions made by Sb^a′+ and Te^a′′+ ions. The above inequalities for the curve in (b) explain why R_□ reduces for x > 0.94 and also why there is a carrier-type transition from n- to p-type (see text for details).

In summary, both Sb₂³⁺Te₃²⁺ and Bi₂³⁺Te₃²⁺ can be synthesized as p-type materials for x_new ≠ 0 and y = 0 or x_new = 0 and y ≠ 0, without requiring both x_new ≠ 0 and y ≠ 0 as discussed above. We also have used the so-called relevant ξ to explain the transport properties (in (Bi_1−xSb_x)₂Te₃ and Pb_1−xSn_xSe) and the n- to p-type carrier-type transition in (Bi_1−xSb_x)₂Te₃, which unequivocally prove that Kramers degeneracy is not responsible for the metallic surface states, at least in the above stated TI.

4. Additional notes

We have justified that these topological insulators, namely, (Bi_1−xSb_x)₂Te₃ and Pb_1−xSn_xSe, in the presence of surface electric current necessarily violate TRS. Even in the absence of IET, TRS is guaranteed to be violated for any solid state system (gapless or gapped) in the presence of electric current.¹⁶ Furthermore, within IET, we have postulated that the degeneracy in topological insulators is not of the Kramers-type. To prove this postulate, we derived the temperature-independent resistivity and scattering-rate equations as functions of doping (x). The experimental results obtained from the above materials ((Bi_1−xSb_x)₂Te₃ and Pb_1−xSn_xSe) follow exactly as predicted by these equations (eqn (11) and (12)). This means that, the energy levels are crossed (in gapless systems) in such a way that there are nonzero wavefunction-induced energy gaps at the crossing points. This ‘special’ gap exists due to different orthogonalized wavefunctions.

To reinforce the above non-Kramers degeneracy in topological insulators, we also provide precise explanation on the physics of n- to p-type carrier-type transition in (Bi_1−xSb_x)₂Te₃. As a matter of fact, the existence of non-Kramers degeneracy in any system, correlated or not, is not surprising because complicated interactions and/or their interplay indeed have given rise to crossed energy levels. For example, in Fermi metals, metallic heavy fermions, conventional superconductors, strange metals (cuprates and other strongly correlated metals), metallic ferromagnets (manganites), Pancharatnam metals (2-dimensional Fermi gas) and semi-metals.

We have made use of the fact that with or without Kramers degeneracy, TRS will be violated in the presence of spinful electric current,¹⁶ which directly challenges the analysis carried out by König et al.⁷ and Bernevig, Hughes and Zhang.²¹ They^7,21 claimed that the spin-down and spin-up (spin polarized) edge currents do not violate TRS due to Kramers degeneracy. Secondly, the degeneracy in topological insulators is not of the Kramers-type, which are unambiguously supported by the doping-dependent carrier-type transition and resistivity data.

In summary, interacting topological insulators, namely, (Bi_1−xSb_x)₂Te₃ and Pb_1−xSn_xSe cannot be understood in terms of noninteracting electrons. Even though the edge states of noninteracting or interacting topological insulators can be protected by constructing a time reversal invariant (TRI) Hamiltonian, this Hamiltonian ceases to be valid for ξ^non_triv ≠ 0. In particular, it is true that a state with current breaks TRS, and this violation is not going to make the Hamiltonian break its time reversal operation. However, the gapped-degeneracy condition, ξ^non_triv ≠ 0, which is also known as the special gap, is the one that has been shown to break TRS of a TRI Hamiltonian. In other words, TRI Hamiltonian is incompatible for interacting topological insulators with ξ^non_triv ≠ 0. Note that the time reversal operation of this Hamiltonian can only stay intact if it represents a noninteracting topological insulator such that the said insulator obeys the gapless-degeneracy condition, ξ_irr ≠ 0 or ξ = 0.

5. Conclusions

We have derived the temperature-independent resistivity and scattering-rate equations as functions of doping (x) supported by the experimental results obtained from these topological insulators, (Bi_1−xSb_x)₂Te₃ and Pb_1−xSn_xSe. The experimental doping-dependent resistivity data follow exactly as predicted by eqn (11). Apart from that, the n- to p-type transition observed in (Bi_1−xSb_x)₂Te₃ has also been explained in a consistent manner using the ionization energy theory on the basis of eqn (12).

First, we have correctly determined the microscopic physics responsible for the systematic changes to the band gap and the resistivity in Pb_1−xSn_xSe where the said changes are due to changing the energy-level spacing (ξ) that controls the density of valence electrons. Subsequently, by introducing relevant defects to invoke the possibility of different valence states for a given chemical element, we then moved on to explain the physics responsible for the n- to p-type carrier-type transition and doping-dependent (temperature-independent) resistivity curves in (Bi_1−xSb_x)₂Te₃ compound.

We can extend our conclusions stated above to find the origin of the topologically protected metallic surface states from the insulating bulk material. In particular, by varying the chemical composition (x) on the surface of an insulator, we have actually introduced significant amount of defects on the surface, leading to the said metallic surface. These defects have permitted us to invoke the changes in the valence states of the constituent chemical elements on the surface of the (Bi_1−xSb_x)₂Te₃ topological insulator. We have found that these changes in the valence states for different doping, x, on the surface are responsible for (i) the n- to p-type carrier-type transition, and (ii) the existence of topologically protected metallic states on the surface such that the bulk remains insulating. Therefore, the bulk has remained insulating due to the fact that the defect and the chemical composition in the bulk have effectively stayed intact with doping compared to its surface. In other words, if the chemical composition and defects have remained unchanged with doping, then we cannot produce two physically distinctive features, namely, the insulating bulk and the topologically protected metallic surface. Conversely, to produce a topological insulator, we need to introduce sufficient amount of defects by changing the chemical composition on the surface of an insulator. Using the ionization energy theory, we have provided the precise chemico-physical analysis required to explain the observed doping-dependent resistivity and carrier-type transition for (Bi_1−xSb_x)₂Te₃.

The above conclusions that have been derived from the transport and carrier-type transition analysis automatically rule out the existence of time-reversal symmetry-induced Kramers degeneracy in (Bi_1−xSb_x)₂Te₃. For example, we have exploited the interaction induced crossed energy levels within IET to explain the transport properties and the n- to p-type carrier-type transition consistently, in the absence of Kramers degeneracy. In doing so, we have established that the nontrivially relevant ξ played a pivotal role in the metallic states of topological insulators where Kramers degeneracy is not required to capture the gapless metallic surface states due to nonzero and relevant ξ.

Acknowledgements

I am grateful to the financiers, Sebastiammal Savarimuthu, Arulsamy Innasimuthu, Amelia Das Anthony, Malcolm Anandraj and Kingston Kisshenraj for their kind support and hospitality between Aug 2011 and Aug 2013. I am also grateful to Marco Fronzi (Tyndall National Institute, Ireland) for his unconditional help in providing most of the references. I thank the referee for pointing out the missing explanation on the violation of TRI Hamiltonian.

References

1. Y. Hatsugai, Phys. Rev. Lett., 1993, 71, 3697.
2. L. Fu, C. L. Kane and E. J. Mele, Phys. Rev. Lett., 2007, 98, 106803.
3. J. E. Moore and L. Balents, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 75, 121306.
4. N. W. Ashcroft and N. D. Mermin, Solid State Physics, Rinehart and Winston, New York, Holt, 1976.
5. R. Yu, W. Zhang, H. J. Zhang, S. C. Zhang, X. Dai and Z. Fan, Science, 2010, 329, 61.
6. C. Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L. L. Wang, Z. Q Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S. C. Zhang, K. He, Y. Wang, L. Lu, X. C. Ma and Q. K. Xue, Science, 2013, 340, 167.
7. M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X. L. Qi and S. C. Zhang, Science, 2007, 318, 766.
8. S. Barua and K. P. Rajeev, AIP Adv., 2014, 4, 017135.
9. K. von Klitzing and G. Landwehr, Solid State Commun., 1971, 9, 2201.
10. Y. L. Chen, J. H. Chu, J. G. Analytis, Z. K. Liu, K. Igarashi, H. H. Kuo, X. L. Qi, S. K. Mo, R. G. Moore, D. H. Lu, M. Hashimoto, T. Sasagawa, S. C. Zhang, I. R. Fisher, Z. Hussain and Z. X. Shen, Science, 2010, 329, 659.
11. C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1976.
12. L. W. Schubnikov and W. J. de Haas, Proc. R. Neth. Acad. Arts Sci.: Nat. Sci., 1930, 33, 130.
13. M. Nakahara, Geometry, Topology and Physics, IOP Publishing, Bristol, 2003.
14. R. Geroch, Mathematical Physics, University of Chicago Press, Chicago, 1985.
15. A. D. Arulsamy, Pramana, 2015, 85, 161.
16. A. D. Arulsamy, Ionization Energy Theory: Formalism, CreateSpace, Seattle, 2016, vol. I.
17. S. Pancharatnam, Proc. - Indian Acad. Sci., Sect. A, 1956, 44, 427.
18. S. Pancharatnam, Collected Works of Shivaramakrishnan Pancharatnam, Oxford, Oxford University Press, 1975.
19. M. V. Berry, Proc. R. Soc. London, Ser. A, 1984, 392, 45.
20. C. L. Kane and E. J. Mele, Phys. Rev. Lett., 2005, 95, 146802.
21. B. A. Bernevig, T. L. Hughes and S. C. Zhang, Science, 2006, 314, 1757.
22. A. D. Arulsamy, X. Y. Cui, C. Stampfl and K. Ratnavelu, Phys. Status Solidi B, 2009, 246, 1060.
23. A. Messiah, Quantum Mechanics, Dover Pub. Inc., New York, 1999, ch. XV, p. 664.
24. H. A. Kramers, Proc. Amsterdam Acad., 1930, 33, 959.
25. M. Z. Hasan and C. L. Kane, Rev. Mod. Phys., 2010, 82, 3045.
26. R. Roy, Phys. Rev. B: Condens. Matter Mater.Phys., 2009, 79, 195321.
27. E. H. Hall, Philos. Mag., Ser. 5, 1881, 12, 157.
28. K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett., 1980, 45, 494.
29. D. C. Tsui, H. L. Stormer and A. C. Gossard, Phys. Rev. Lett., 1982, 48, 1559.
30. P. Zeeman, Philos. Mag., 1897, 43, 226.
31. J. Zhang, C. Z. Chang, Z. Zhang, J. Wen, X. Feng, K. Li, M. Liu, K. He, L. Wang, X. Chen, Q. K. Xue, X. Ma and Y. Wang, Nat. Commun., 2011, 2, 1.
32. P. Dziawa, B. J. Kowalski, K. Dybko, R. Buczko, A. Szczerbakow, M. Szot, E. Lusakowska, T. Balasubramanian, B. M. Wojek, M. H Berntsen, O. Tjernberg and T. Story, Nat. Mater., 2012, 11, 1023.
33. A. D. Arulsamy, Phys. C, 2001, 356, 62.
34. A. D. Arulsamy, Phys. Lett. A, 2002, 300, 691.
35. A. D. Arulsamy, Pramana, 2010, 74, 615.
36. A. D. Arulsamy, Ann. Phys., 2011, 326, 541.
37. A. D. Arulsamy, Indian J. Phys., 2014, 88, 609.
38. A. D. Arulsamy, RSC Adv., 2016, 6, 52082.
39. A. D. Arulsamy, J. Earth Syst. Sci., 2015, 124, 683.
40. R. Shankar, Rev. Mod. Phys., 1994, 66, 129.
41. R. Shankar, Phys. A, 1991, 177, 530.
42. R. Shankar, Philos. Trans. R. Soc. London, Ser. A, 2011, 369, 2612.
43. Y. Ralchenko, A. E. Kramida and J. Reader, 2013, http://physics.nist.gov/asd3.
44. M. J. Winter, 2013, http://www.webelements.com.
45. A. D. Arulsamy, Phys. Lett. A, 2005, 334, 413.
46. A. D. Arulsamy and K. Ostrikov, Phys. Lett. A, 2009, 373, 2267.
47. A. D. Arulsamy, J. Supercond. Novel Magn., 2014, 27, 309.
48. N. F. Mott, Rev. Mod. Phys., 1968, 40, 677.
49. I. N. Levine, Quantum Chemistry, Prentice-Hall, New Jersey, 2000.
50. A. D. Arulsamy, Prog. Theor. Phys., 2011, 126, 577.
51. A. J. Strauss, Phys. Rev., 1967, 157, 608.
52. D. J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall, New Jersey, 2005.
53. G. A. Dionicio, Electrical resistivity of YbRh₂Si₂ and EuT₂Ge₂ (T = Co, Cu) at extreme conditions of pressure and temperature, PhD thesis, Technical University of Dresden, Dresden, 2006.
54. S. Fukuda, Y. Nakanuma, J. Sakurai, A. Mitsuda, Y. Isikawa, F. Ishikawa, T. Goto and T. Yamamoto, J. Phys. Soc. Jpn., 2003, 72, 3189.
55. A. D. Arulsamy, K. Eleršič, M. Modic, U. Cvelbar and M. Mozetič, ChemPhysChem, 2010, 11, 3704.
56. R. Mahendiran, S. K. Tiwary, A. K. Raychaudhuri, T. V. Ramakrishnan, R. Mahesh, N. Rangavittal and C. N. R. Rao, Phys. Rev. B: Condens. Matter, 1996, 53, 3348.

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Footnote

† PACS no. 72.10.Bg; 72.20.Dp.

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