Topological state transport in topological insulators under the influence of hexagonal warping and exchange coupling to in-plane magnetizations

Abstract

A hexagonal warping term has been proposed recently to explain the experimentally observed 2D equal energy contours of the surface states of the topological insulator Bi₂Te₃. Differing from the Dirac fermion Hamiltonian, the hexagonal warping term leads to the opening up of a band gap by an in-plane magnetization. We study the transmission between two Bi₂Te₃ segments subjected to different in-plane magnetizations and potentials. The opening up of a bandgap and the accompanying displacement and distortion of the constant energy surfaces from their usual circular shapes by the in-plane magnetizations, modify the transverse momentum overlap between the two Bi₂Te₃ segments and strongly modulate the transmission profile. The strong dependence of the TI surface state transport of Bi₂Te₃ on the magnetization orientation of an adjacent ferromagnetic layer may potentially be utilized in, e.g., a memory readout application.

Introduction

Topological insulators (TIs)¹ are a relatively new class of materials which have attracted much attention due to their rich physics. For example, Majorana fermions have been shown to exist in junctions between superconductors and ferromagnetic insulators deposited on top of TIs^2,3,4. Furthermore, strong spin orbit coupling leads to interesting effects when a magnetization is applied to a TI^5,6,7. In particular, Yokoyama and colleagues studied the magnetoresistance in a two dimensional junction between two ferromagnets magnetized in different directions deposited on top of a topological insulator⁸. They found that the transmission at a given energy is strongly influenced by the relative k-space displacements of the Fermi surfaces resulting from the differing magnetizations. The influence of the Fermi surfaces on the magnetoresistance motivates the study of the topological insulator Bi₂Te₃ in this present work.

The experimentally observed^9,10 2D equal energy contours (EECs) of the surface states in Bi₂Te₃ differ at high Fermi energies from the circular Dirac cone predicted by the simple Dirac fermion Hamiltonian . At low Fermi energies, the EEC takes the form of a circle (Fig. 1(a)). As the Fermi energy increases, the contour evolves from a circle to a hexagon and then to a snowflake with sharp tips along the six ΓM directions. Based on the underlying 3-fold rotational and two-fold mirror symmetry of the [111] surface of the underlying rhombohedral Bi₂Te₃ crystal structure, Fu suggested the addition of a hexagonal warping term to the Hamiltonian¹¹. The Hamiltonian then takes the form of

where x is in the ΓK direction. The Hamiltonian reproduces the experimentally measured EECs.

Figure 1

The EECs and eigenstate spin orientations for E = (a) 0.1 eV, (b) 0.3 eV and (c) 0.5 eV.

The green arrows on the EECs indicate the orientation of the 〈σ_x〉 and 〈σ_y〉 components at each point on the contour with larger arrowheads indicating higher spin polarization. The red and blue dots indicate the 〈σ_z〉 polarization with larger dots indicating larger magnitudes of spin polarization. Blue (red) dots indicate positive (negative) spin z polarization. The dotted line in panel (c) indicates one value of k_y where there are six real values of k_x.

Unlike the Dirac cone which is rotationally symmetric about the k_x, k_y plane, the dispersion relation arising from Eq. 1 has only 6 fold rotational symmetry. The anisotropy of the Fermi surface in the k_x, k_y plane can be expected to give rise to interesting directional dependent effects. For example, it has been suggested that a Bi₂Te₃ segment with a potential step may be exploited as a flat lens for focusing electron beams¹².

Another peculiarity of the Hamiltonian is that an in-plane magnetic field modeled by a Zeeman term can open up a band gap¹¹. The Zeeman term mathematically resembles that due to magnetic coupling to a proximate ferromagnetic film or the effects of magnetic doping¹³. For ease of exposition we say that the Zeeman term arises from an external magnetization. We write the Zeeman term as and collect the appropriate coupling constants into itself. The introduction of a bandgap due to an in-plane magnetization differs from the simple Dirac fermion Hamiltonian where only an out-of-plane magnetization can open up a band gap. We find that besides opening up a band gap and displacing the position of the Dirac points in reciprocal space, the in-plane magnetic field also distorts the shape of the constant energy surfaces.

Despite the interesting bandstructure properties of Bi₂Te₃, transport across Bi₂Te₃ segments has not yet been studied extensively. Ref. 14 derived the transmission between one TI segment and another with an externally applied step-edge potential V(x) = −V₀θ(−x) where θ(x) is the Heaviside step function. In the derivation it was implicitly assumed that there is a propagating state in the transmitted segment which satisfies energy and transverse momentum conservation. However, there exists more than one propagating state with real, positive k_x for some values of k_y at high |E − V| (see Fig. 1(c)). It is not evident which of these multiple values of k_x in the transmitted region should be chosen. Moreover, the derivation matched only the wavefunctions and not the probability flux, across the interface. Ref. 16 considered the transmission of a wave incident from x < 0 across a Dirac delta potential located at x = 0. The transmitted wavefunction at x > 0 contains left propagating components back into the incident region within the sin(k_xx) and cos(k_xx) terms in Eq. (4) of the paper. The presence of backwards propagating wave components in the transmitted region is counter-intuitive.

For a given value of transverse momentum k_y and Fermi energy E, the Hamiltonian Eq. 1 admits six eigenstates with real or imaginary k_x which may be real or complex and which come in three Kramers time-reversal pairs. We consider the transmission of electrons traveling in the +x direction from one Bi₂Te₃ segment on the left to another on its right. Three of the eigenstates propagate or decay in the +x direction and are physically appropriate states in the segment which a wave traveling in the +x direction is transmitted into. The other three eigenstates propagate or decay in the −x direction and are physically appropriate as the reflected components of a wavefunction in the incident segment. Both Refs. 14 and 16 assume that an incident energy eigenstate with wavevector k_x be reflected at the interface into an eigenstate with the wavevector −k_x. This imposes an artificial constraint that the incident wave is not reflected back into the other 2 eigenstates which also propagate or decay in the correct direction.

One difficulty in considering all eigenstates in studying transmission across an interface between 2 Bi₂Te₃ TI segments is the issue of boundary conditions. In each TI segment there are 6 eigenstates for a given value of E and transverse momentum k_y. In solving the transmission problem across an interface from TI region I to region II, the weightages of six of the twelve eigenstates involved are fixed by the requirements that evanescent states decay away from the interface and that the state incident on the interface from region I is given. This leaves the weightages of the remaining 6 states to be solved for. The usual practice of matching the wavefunctions and first derivatives for the two spinor components across an interface between the two TI segments gives only 4 equations. Another boundary condition besides wavefunction and first derivative continuity is required to solve for the 6 unknowns uniquely. We note that more recent works^15,17 did consider all six eigenstates explicitly. However, Ref. 15 only considered transmission across a delta potential barrier, whereas it is not clear from the paper what set of boundary conditions Ref. 17 used.

To the best of our knowledge, transport across Bi₂Te₃ segments magnetized in different directions has not yet been studied. In this work, we show that probability flux conservation across the interface requires that the matching of the second derivatives of the wavefunctions across the interface. We then calculate the transmission between two Bi₂Te₃ segments subjected to different external potentials and magnetic fields. We find that the distortion and displacement of the constant energy surfaces and the opening up of a band gap lead to a rich transmission profile as the magnetization directions and energy are varied.

Methods

Boundary conditions

We consider a generic Hamiltonian of the form

The a, b, c and d above may be numbers or Hermitian operators which commute with , for example some combinations of or and are independent of p_x. For this Hamiltonian, we have . Explicitly expanding out ∂_tρ gives, after some manipulations of the x derivatives and integration by parts,

Comparing with the continuity equation ∂_tρ + ∂_xj = 0, we recognize the terms within the large brackets in the second line as the probability flux. If c = 0 (i.e. the Hamiltonian is only quadratic in ), the flux is the real part of so that for a Hamiltonian quadratic in , matching across an interface – which matches both the real and imaginary parts – automatically matches the flux across both sides of an interface.

However, for c ≠ 0, the real part of the c term in is which is not, in general, equal to the c term in the flux, . A simple example suffices to illustrate this. Putting c = 1 for simplicity and writing ψ = exp(ikx) where k = (k_r + ik_i) is in general complex, we have at x = 0, we have which is not equal to if the wavevector has an imaginary part.

In the system we will be considering in this paper, we identify the operators , b = 0 and c = λσ_z across both sides of the interface. (They differ in the d term of Eq. 2.) In this instance, flux conservation can be achieved by matching i) ψ ii) ∂_xψ and iii) across both sides of the interface.

Results

Equal energy contours under an in-plane magnetization

The eigenenergy E for a given and magnetization reads

The high powers of k and presence of trigonometric terms involving φ and φ_M present difficulties for the derivation of compact analytic expressions for the band gap and position of the Dirac point in k-space at which the lowest energy propagating particle state with real exist. We calculate these quantities numerically. We use the numerical values of v = 25.5 eVnm and λ = 0.25 eVnm⁻³¹¹.

The in-plane magnetization leads to the opening up of a band gap and a displacement of the Dirac point, as shown in panels (a) and (b) of Fig. 2 respectively. The band gap reaches its maximum value of M at magnetization angles φ_M = ±(2n − 1)π/6, n = 1, 2, 3 and varies in a roughly sinusoidal manner with the magnetization angle. The band gap shifts the minimum energy at which propagating particle states exist upwards. This leads to the k-space area covered by the EEC at a given energy being generally larger for a magnetization angle with a smaller band gap compared to that of a magnetization angle with a larger band gap. This fact will have consequences for charge transport between Bi₂Te₃ segments magnetized in different directions.

Figure 2

(a) The band gap in eV as a function of the magnetic field strength M and angle φ_M. (b) The thick blue lines indicate the trajectory of the Dirac points in k space as φ_M is varied at fixed values of M from 0.05 eV to 0.20 eV. The black dotted lines show the constant energy surfaces at E = 0.1 eV and E = 0.2 eV.

The trajectory of the Dirac points for a given magnetization magnitude as the magnetization angle is varied follows a more complicated path than the circular trajectories of the simple Dirac fermion Hamiltonian. For the latter, the trajectories form circles in k-space. In Bi₂Te₃, the trajectories of the Dirac points resemble the shapes of the EECs. The trajectories of the Dirac points exhibit greater distortion from the circular shape at a given value of M than the EECs do for E = M.

The positions of the Dirac points differ from the given in Ref. 11. The latter holds only in the low M limit when the trajectories of the Dirac points are roughly circular. The k-space angle φ = arctan(k_y/k_x) of the Dirac point is related to the magnetization angle by φ_M = φ − π/2. This can be seen from the last term of Eq. 4 where the energy achieves its minimum value for fixed values of k and M when the sin(φ − φ_M) term is 1. At φ = φ_M + π/2 and the large k limit, the k⁶ term of Eq. 4 dominates so that E ≈ k³λ|sin(3φ_M)|. This explains the peaking of sinusoidal variation and peaking of the band gap at φ_M = ± (2n − 1)π/6, n = 1, 2, 3.

Transport between two Bi₂Te₃ segments

To study the effects of the in-plane magnetization on the transport properties of Bi₂Te₃, we consider the system outlined schematically in Fig. 3. The system consists of two Bi₂Te₃ segments of semi-infinite length along the x direction and infinite width along the y direction. Charge flows from the left source segment to the right drain segment in which a magnetization and a potential U is applied. The potential shifts the dispersion relation of the drain segment on the energy axis with respect to that of the source segment. For a given source Fermi energy E we define E₂ = E − U.

Figure 3

A schematic of the system studied.

The system consists of two semi-infinite long Bi₂Te₃ segments of infinite width. Charge flows from the left source segment into the right drain segment in which a magnetization and external potential U is applied.

Fig. 4 shows the transmission plotted as a function of magnitude M₂ and direction φ_M2 of the drain magnetization for the transmission from an unmagnetized source segment. The electron energy is chosen to be the intermediate range, i.e., E = 0.3 eV, at which the EEC takes the form of a hexagon, while the drain segment is biased at E₂ = E − U₂ = 0.1 eV, i.e., at the low energy regime where the EEC assumes a circular shape.

Figure 4

The transmission plotted against the drain magnetization direction φ_M2 and magnetization magnitude for E = 0.3 eV and E₂ = E − U₂ = 0.1 eV.

The dependence of the transmission on M₂ can be divided into three regimes. In the low M₂ regime (0 eV < M₂ < 0.07 eV) the transmission peaks weakly around φ_M2 = ±π/2, while at intermediate values of M₂ the transmission exhibits sharper peaks near φ_M2 = 0, ±π. At large values of M₂, the trend is reversed and the transmission drops to almost zero near φ_M2 = ±nπ/6, n = (1, 3, 5).

Discussion

These observations may be explained by examining the evolution of the EECs as M₂ and φ_M2 are varied (as shown in Fig. 5).

Figure 5

The EECs in the source region (thick black hexagon) at E = 0.3 eV, the trajectories of the Dirac points traced out by φ_M2 by the variation of φ_M2 (thick green lines) and the EECs in the drain region (thin blue lines) for E₂ = 0.1 eV and M₂ = (a)0.05 eV, (b) 0.15 eV and (c) 0.28 eV.

The positions of the Dirac points and the corresponding values of φ_M2 for each of the drain EECs are marked out in (b) in the same color. The k_y range spanned by the φ_M2 = 0 drain EEC is indicated in panel (b).

For low values of M₂, the EECs at the drain lie completely within the k-space region spanned by the EEC at the source. Fig. 5(a) shows that the drain EECs adopt an almost circular profile and the k_y range spanned is only weakly dependent on φ_M2. The spin polarizations (not shown) of the right-propagating eigenstates also do not exhibit much angular variation. Thus, there is little modulation of the transmission as the orientation of M₂ is varied. In the intermediate range of M₂ [shown in Fig. 5(b)], the transmission is dominated by the k_y range spanned by the drain EEC. Since the transverse momentum is conserved across the source-drain interface, a larger range of k_y subtended by the drain EEC translates to a greater number of propagating states which can transmit electrons over a wider range of incidence angle. The transmission peaks occur near φ_M2 = 0, π, which correspond to the two magnetization angles where the drain EECs cover the largest k_y range. In the large M₂ regime shown in Fig. 5(c), the energy gap induced by the in-plane magnetization eliminates the presence of propagating states at φ_M2 = ±nπ/6, n = (1, 3, 5) and reduces the k_y range covered by the drain EECs at other magnetization angles. The absence of propagating states for certain range of φ_M2 leads to a sharp modulation of the transmission about the critical angles of φ_M2 = ±nπ/6, n = (1, 3, 5), a feature which would be conducive for applications to sense the orientation of , such as memory read-out. The displacement of the drain EECs from the k-space origin also leads to some portions of the drain EECs lying outside the source EEC. This leads to the generally lower transmission seen in Fig. 5 for large values of M₂.

The effects of varying the potential in the drain segment for a fixed magnitude of magnetization provides an alternative perspective. Fig. 6 shows the transmission plotted against the drain magnetization direction and drain potential. The trends observed can also be explained by examining the EEC profiles in the drain region, shown for three representative values of E₂ in Fig. 7. The low transmission centered around φ_M2 = nπ/6, n = 1, 3, 5 at low values of E₂ corresponds to the absence of propagating states due to the large band gaps induced at these values of φ_M (Fig. 7(a)). The transmission peaks at φ_M = 0, π at intermediate values of E₂, where the drain EECs still lie largely within the k_y region spanned by the source EEC, correspond to the largest range of k_y spanned by the drain EECs as φ_M2 is varied (Fig. 7(b)). As E₂ is increased further, an increasing proportion of the k-space area spanned by the EECs starts to fall outside the k_y region spanned by the source EEC. This leads to the transmission peaks occurring close to φ_M = ±π/2 where the overlap in the k_y region spanned by both the source and drain EECs is the greatest (Fig. 7(c)).

Figure 6

The transmission plotted against the drain magnetization direction direction φ_M2 and drain energy E₂ for E = 0.3 eV and M₂ = 0.2 eV.

Figure 7

The EECs in the source region (thick black hexagon) at E = 0.3 eV, the trajectories of the Dirac points traced out by the Dirac points (thick green lines) by the variation of φ_M2 (green dots) and the EECs (thin blue lines) in the drain region for M₂ = 0.1 eV and E₂ = (a)0.05 eV, (b) 0.12 eV and (c) 0.25 eV.

We have thus far concentrated on the k_y overlap between the source and drain EECs as the primary explanation of the observed transmission profile. However, the k_x and spin mismatch in the source and drain regions does affect the transmission as well. In order to investigate the effects of these two factors, we now let the range of E₂ in Fig. 6 extend to negative values as shown in Fig. 8. Negative values of E₂ correspond to transmission from source particle states to drain hole states, which have the opposite spin configuration.

Figure 8

(a) The transmission profile for the same set of parameters as in Fig. 6 with the exception that E₂ now extends to negative values. (b) The EECs for an unmagnetized segment at E = 0.1 and no magnetization (black) and M₂ = 0.2 eV, φ_M2 = 0.4π and φ_M2 = −0.4π respectively as labeled. The arrows indicate the directions of the in-plane spin polarization on the EECs of the right propagating hole (green) and particle (red) states in the drain segment. The lengths of the arrowheads are indicative of the magnitude of the in-plane spin polarization.

The general features of the transmission profile for particle-to-particle transmission (positive E₂) are largely similar, but not identical, to those for particle-to-hole transmission. The reason for this lies in the fact that for a given magnetization and |E| the EECs for the particle and hole states with energy ±|E| have identical shapes. The right propagating states for positive and negative values of E, however, lie in different segments of the EECs as shown in panel (b) of Fig. 8. The k_y range spanned by the right propagating states for negative and positive values of E are roughly the same. This accounts for the similarities in the particle-to-particle and particle-to-hole transmission profiles in terms of the rough positions of the transmission peaks and troughs. However, the difference in the k_x values and spin orientations of the right propagating states of electrons and holes leads to the asymmetry in the particle-to-particle and particle-to-hole transmission profiles. This asymmetry is accentuated by the distortion of the constant energy surfaces from the circular shape by the hexagonal warping term. In addition, there is also an asymmetry in the transmission profile about the φ_M2 = 0 axis. This may be explained by considering the EECs for ±φ_M, for some given φ_M. As shown in Fig. 8(b), the EECs for ±φ_M are reflections of each other about the k_x = 0 axis. However, the portions of the EECs which correspond to the right propagating states lie on different regions of the corresponding EECs for ±φ_M. This leads to, for instance, the right propagating states at a given value of k_y having different spin polarizations for opposite signs of φ_M and hence an asymmetrical transmission profile about φ_M2 = 0. Thus, in summary, two factors play a role in determining the dependence of the transport on the magnetization direction and magnitude – the extent of the overlap across k_y of the the source and drain EECs and the degree of mismatch between the spin configurations of the propagating states in the two electrodes.

We note in passing that proposals for the formation of the Majorana fermion state in TI systems involve junctions between ferromagnetic insulators and superconductors deposited on top of TIs. The anisotropy in the shape of the EECs induced by the in-plane magnetization and asymmetry between the particle and hole states just discussed may lead to the presence of Majorana fermion states in Bi₂Te₃ with unusual properties compared to Majorana states resulting from TIs with the more commonly studied circular EECs. The anisotropy may also be indicative of anomalous effects which emerge as the dimensions of the TI segments are shrunk from being semi-infinite (in the x and z directions) to finite ones¹⁸. A detailed discussion of these issues are, however, beyond the scope of this paper.

Conclusion

In this work we showed the appearance of a band gap and the accompanying distortion and displacement of the constant energy curves in k space in Bi₂Te₃ with the application of an in-plane magnetic field. We then studied the transmission to a drain Bi₂Te₃ segment with an external magnetization and potential applied from a source Bi₂Te₃ segment without the magnetization and potential. The band gap and distortion and displacement of the constant energy curves by the in-plane magnetization affects the overlap of the transverse momentum k_y range spanned by the EECs in the source and drain regions. The sharp modulation of the transmission with the magnetization orientation for the high M₂ regime and low E₂ regime, as depicted in Figs. 4 and 6, respectively, suggests a possible application of Bi₂Te₃ in the read-out of magnetic memory. The two states of a memory bit can be represented by the magnetization direction of an adjoining ferromagnetic layer (which is coupled to the topological surface states of Bi₂Te₃) being in the φ_M = 0 and φ_M = π/2 directions. In the former state there is no band gap and the transmission is finite while in the latter the transmission is 0 due to the large band gap. A gate voltage can be applied on the drain segment to tune the value of E₂ so as to achieve an optimum balance between the differences in transmission between the two memory states and the robustness of the system to deviation of the magnetization from the reference φ_M = 0 and φ_M = π/2 directions.