Magnetism in Topological Insulators

By Vladimir Litvinov

This book serves as a brief introduction to topological insulator physics and device applications. Particular attention is paid to the indirect exchange interaction mediated by near surface Dirac fermions and the spin texture this interaction favors. Along with useful information on semiconductor material systems, the book provides a theoretical background for most common concepts of TI physics.  Readers will benefit from up to date information and methods needed to start working in TI physics, theory, experiment and device applications.

• Discusses inter-spin interaction via massless and massive Dirac excitations;
  
• Includes coverage of near-surface spin texture of the magnetic atoms as related to their mutual positions as well to their positions with respect to top and bottom surfaces in thin TI film;
  
• Describes non-RKKY oscillating inter-spin interaction as a signature of the topological state;
  
• Explains the origin of the giant Rashba interaction at quantum phase transition in TI-conventional semiconductors.

• Language: English
• Publisher: Springer
• Release date: May 7, 2019
• ISBN: 9783030120535

Book preview

© Springer Nature Switzerland AG 2020

V. LitvinovMagnetism in Topological Insulatorshttps://doi.org/10.1007/978-3-030-12053-5_1

1. Energy Bands in Topological Insulators

Vladimir Litvinov¹ 

(1)

Sierra Nevada Corporation, Irvine, CA, USA

Keywords

Bismuth chalcogenidesSurface electron states in topological insulatorsGapless Dirac fermionsGapped Dirac fermionsSpin-momentum lockingInverse spin-galvanic effectCurrent-induced spin polarizationSpin-momentum texture in topological insulators

Topological insulators (TI) are identified by their gapless surface electronic states with the energy lying inside a bandgap of the bulk band structure. This makes the crystal conducting on the surface and insulating in the bulk. To study the magnetic and electrical properties of TI materials and to understand what features make them different from conventional semiconductors, we start by studying the energy spectrum of an example material system (Bi, Sb)2(Te, Se)3. Bismuth chalcogenides is not the only class of materials that shows a topological response to external electric and magnetic fields: we could also mention other TI materials such as II–VI compounds (Cd, Hg)Te and also IV–VI semiconductor alloys (Pb, Sn)(Te, Se) which are called crystalline topological insulators. Semiconductors from II–VI and IV–VI families are TI examples that are less convenient for experimental studies due to the small energy gaps and also because of the strongly non-stoichiometric growth that makes it difficult to pin the Fermi energy inside the energy gap of bulk crystal, where the topological effects become observable.

The electrical properties of Bismuth chalcogenides and their device applications in thermoelectricity have been intensively studied and thus all the details of their energy spectrum, electron, and phonon scattering mechanisms have long been known. What makes them special as TI solids? We start from the microscopic approach, analyzing the band structure of bulk Bi2Se3 and the specific features of the surface electron states which differentiate them from conventional solids. A phenomenological approach that relates TI general properties to band topology and symmetry will be discussed in the next chapters.

1.1 Spin–Orbit Interaction in Bismuth Chalcogenides

The bismuth chalcogenides family crystallizes in a rhombohedral structure with point group D3d and space group $$ R\overline{3}m $$ . The structure consists of the set of quintuple layers illustrated in Fig. 1.1. Quintuple layers are coupled by the weak Van der Waals interaction that results in easy cleavage along planes {0001} perpendicular to axis z.

../images/455071_1_En_1_Chapter/455071_1_En_1_Fig1_HTML.png

Fig. 1.1

Bismuth chalcogenide crystal structure. QL stands for a quintuple layer

Relevant electron energy bands have energy close to the Fermi level, they stem from Bi- and Se- p-orbitals and have been calculated in [1, 2]. The origin of conduction and valence energy levels are shown in the left panel of Fig. 1.2, where wave functions ∣P1 ↑ ↓⟩ and ∣P2 ↑ ↓⟩ stand for double spin degenerate linear combinations of the atomic Bi- and Te- p-orbitals, respectively. Wave functions have parity (±) with respect to inversion symmetry and also account for the energy splitting between Px, y and Pz levels caused by the strongly anisotropic crystal field. Spin–orbit coupling (SOC) in the lattice potential does not cause the spin-splitting as far as both the time reversal and the inversion symmetries hold (right panel in Fig. 1.2). Conduction and valence band edges exchange their positions if the SOC is large enough. That is the band inversion induced by the strong SOC that creates the non-trivial band topology and turns semiconductors Bi2Te3, Bi2Se3, and Sb2Te3 into TI.

../images/455071_1_En_1_Chapter/455071_1_En_1_Fig2_HTML.png

Fig. 1.2

Spin–orbit coupling inverted energy levels corresponding to hybridized Bi (| P1⟩) and Se(Te) (∣P2⟩) p-orbitals

1.2 Electron Spectrum and Band Inversion

In solids, the Schrodinger equation for Bloch functions Ψnk = unk exp (−ikr) can be reduced to the equation for Bloch amplitudes as follows:

$$ {H}_{\boldsymbol{k}}{u}_{n\boldsymbol{k}}={E}_{n\boldsymbol{k}}{u}_{n\boldsymbol{k}}, $$$$ {H}_{\boldsymbol{k}}={H}_0+\frac{{\mathrm{\hslash}}^2{k}^2}{2{m}_0}+\frac{{\mathrm{\hslash}}^2\boldsymbol{kp}}{m_0}+{H}_1+{H}_2, $$$$ H=\frac{p^2}{2{m}_0}+V\left(\boldsymbol{r}\right),{H}_1=\frac{\mathrm{\hslash}}{4{m}_0^2\ {c}^2}\boldsymbol{\sigma} \cdot \mathbf{\nabla}V\left(\boldsymbol{r}\right)\times \boldsymbol{p},{H}_2=\frac{\mathrm{\hslash}}{4{m}_0^2\ {c}^2}\boldsymbol{p}\cdot \boldsymbol{\sigma} \times \mathbf{\nabla}V\left(\boldsymbol{r}\right), $$

(1.1)

where σx, y, z are the Pauli matrices, e, m0 are the free electron charge and mass, respectively, p =  − i∇ is the momentum operator, and V(r) is the electron potential energy in the periodic crystal field. The fourth and fifth terms in the Hamiltonian (1.1) represent spin–orbit interaction. In order to find the eigenvalues Enk, one has to choose the full set of known orthogonal functions that create the initial basis on which we can expand the unknown Bloch amplitudes unk. In the vicinity of the Γ-point, the set of band edge amplitudes un0 normalized on the unit cell volume Ω can serve as the basis wave functions (Luttinger-Kohn representation). Within kp-perturbation theory the third, fourth, and fifth terms in the Hamiltonian (1.1) are treated as a perturbation.

Conduction and valence bands stem from two double spin degenerate levels which present the band edges in the vicinity of the Γ-point in the Brillouin zone (Fig. 1.2, right panel), so the basis functions un0(r) can be taken as

$$ \left|P,{1}_z^{+},\uparrow, \downarrow \right\rangle $$

and

$$ \left|P,{2}_z^{-},\uparrow, \downarrow \right\rangle $$

. The matrix Hamiltonian in un0 - representation is written below:

$$ {H}_{n{n}^{\prime }}\left(\boldsymbol{k}\right)=\int \left\langle {u}_{n0}\left(\boldsymbol{r}\right)|{H}_{\boldsymbol{k}}|{u}_{n^{\prime }0}\left(\boldsymbol{r}\right)\right\rangle d\varOmega, $$

(1.2)

where the integration is over the volume of the unit cell.

Energy corrections from remote bands enter in diagonal elements of $$ {H}_{n{n}^{\prime }}\left(\boldsymbol{k}\right) $$ . They are proportional to $$ {k}_i^2 $$ and contribute to effective masses turning initial levels into energy bands. Taking the basis functions in the sequence,

$$ \left|1,\Big\rangle \right.=\left|P{1}_z^{+}\uparrow, \Big\rangle \right. $$

,

$$ \left|2,\Big\rangle \right.=\left|P{1}_z^{-}\uparrow, \Big\rangle \right. $$

,

$$ \left|3,\Big\rangle \right.=\left|P{1}_z^{+}\downarrow, \Big\rangle \right. $$

,

$$ \left|4,\Big\rangle \right.=\left|P{1}_z^{-}\downarrow, \Big\rangle \right. $$

, one obtains the Hamiltonian matrix with $$ {k}_i^2 $$ -accuracy:

$$ {H}_{\mathbf{k}}=\left[\begin{array}{cccc}\varepsilon \left(\boldsymbol{k}\right)+M\left(\boldsymbol{k}\right)& {A}_1{k}_z& 0& {A}_2{k}_{-}\\ {}{A}_1{k}_z& \varepsilon \left(\boldsymbol{k}\right)-M\left(\boldsymbol{k}\right)& {A}_2{k}_{-}& 0\\ {}0& {A}_2{k}_{+}& \varepsilon \left(\boldsymbol{k}\right)+M\left(\boldsymbol{k}\right)& -{A}_1{k}_z\\ {}{A}_2{k}_{+}& 0& -{A}_1{k}_z& \varepsilon \left(\boldsymbol{k}\right)-M\left(\boldsymbol{k}\right)\end{array}\right], $$

(1.3)

where the reference energy is the middle of the bandgap,

$$ \varepsilon \left(\boldsymbol{k}\right)={D}_1{k}_z^2+{D}_2{k}^2 $$

,

$$ M\left(\boldsymbol{k}\right)=-\varDelta -{B}_1{k}_z^2-{B}_2{k}^2 $$

,

$$ {k}_{\pm }={k}_x\pm i{k}_y,{k}^2={k}_x^2+{k}_y^2 $$

. Parameters A, B, D can be determined from comparison to experimental results . Eigenvalues of the Hamiltonian (1.3) describe double spin degenerate conduction and valence bands:

$$ {E}_{\mathrm{c},\mathrm{v}}=\varepsilon \left(\boldsymbol{k}\right)\pm \sqrt{M{\left(\boldsymbol{k}\right)}^2+{A}_1^2{k}_z^2+{A}_2^2{k}^2}. $$

(1.4)

The symmetry of the Hamiltonian comprises the operations of the D3d point group as well as the time reversal transformation. In the basis assumed in Hamiltonian (1.3), the representation of the time reversal (TR) and the spatial inversion (SI) operators are given by matrices

$$ \mathrm{TR}=K\left(\begin{array}{cccc}0& 0& 1& 0\\ {}0& 0& 0& 1\\ {}-1& 0& 0& 0\\ {}0& -1& 0& 0\end{array}\right),\kern1em \mathrm{SI}=\left(\begin{array}{cccc}1& 0& 0& 0\\ {}0& -1& 0& 0\\ {}0& 0& 1& 0\\ {}0& 0& 0& -1\end{array}\right), $$

(1.5)

where K stands for the complex conjugate. It is easy to check that the transformed Hamiltonians (UHU+, U = TR, SI) have the same eigenvalues given in (1.4). Electron spectrum and band inversion are illustrated in Fig. 1.3.

../images/455071_1_En_1_Chapter/455071_1_En_1_Fig3_HTML.png

Fig. 1.3

Formation of the band inverted spectrum (1.4) at kz = 0, B2 > 0

The sign reversal of the gap parameter Δ corresponds to SOC-induced level crossing shown in Fig. 1.2. Band spectrum formation as shown in Fig. 1.3 relies on the nonzero momentum matrix element

$$ {A}_2\sim \left\langle P{1}_z^{+}\uparrow \left|{\nabla}_{x,y}\right|P{2}_z^{-}\downarrow \right\rangle =\left\langle P{1}_z^{+}\downarrow \left|{\nabla}_{x,y}\right|P{2}_z^{-}\uparrow \right\rangle $$

which lifts degeneracy and results in the camelback shape of the inverted energy bands illustrated in Fig. 1.3c. This matrix element also stems from SOC and may exist only if wave functions have opposite parities. Inverted bands also take place in the z-direction if ΔB1 < 0. The band inversion creates the non-trivial topology of the wave functions that determines the main characteristics of TI solids. As discussed in the next section, the surface normal to the z-axis carries topologically non-trivial electron states if the kz-bands are inverted.

1.3 Surface States

We start with the Hamiltonian (1.3) at D1 = D2 = 0. This simplification implies equal masses of electrons and holes that is not essential in the study of topological surface states. We account for the finite size of the sample in the z-direction, making the substitution, kz →  − i∇z. For the moment, let us restrict our consideration to the surface level at k = 0 and deal with the Hamiltonian

$$ {H}_0=\left(\begin{array}{cccc}-\varDelta +{B}_1\frac{\partial^2}{\partial {z}^2}&amp; -{iA}_1\frac{\partial }{\partial z}&amp; 0&amp; 0\\ {}-{iA}_1\frac{\partial }{\partial z}&amp; \varDelta -{B}_1\frac{\partial^2}{\partial {z}^2}&amp; 0&amp; 0\\ {}0&amp; 0&amp; -\varDelta +{B}_1\frac{\partial^2}{\partial {z}^2}&amp; {iA}_1\frac{\partial }{\partial z}\\ {}0&amp; 0&amp; {iA}_1\frac{\partial }{\partial z}&amp; \varDelta -{B}_1\frac{\partial^2}{\partial {z}^2}\end{array}\right). $$

(1.6)

Hamiltonian H0 is a block-diagonal matrix with upper and lower blocks corresponding to spin-up and spin-down electrons, respectively. It is sufficient to consider the upper block only and then use the resulting wave functions substituting A1 →  − A1 to get the characteristics of the lower block. So, we will be dealing with a 2 × 2 Hamiltonian that describes spin-up electrons and holes:

$$ {H}_0=\left(\begin{array}{cc}-\varDelta +{B}_1\frac{\partial^2}{\partial {z}^2}&amp; -{iA}_1\frac{\partial }{\partial z}\\ {}-{iA}_1\frac{\partial }{\partial z}&amp; \varDelta -{B}_1\frac{\partial^2}{\partial {z}^2}\end{array}\right). $$

(1.7)

Electronic states near the surface are in-plane Bloch waves while in the z-direction they can be described by a z-dependent spinor:

$$ {\varPsi}_{\uparrow }=A\left(\begin{array}{c}\left|1,\Big\rangle \right.\\ {}\left|2,\Big\rangle \right.\end{array}\right)\exp \left(\lambda\ z\right). $$

From the Schr $$ \ddot{o} $$ dinger equation H0Ψ↑ = EΨ↑, we get the characteristic equation for λ:

$$ \mathrm{Det}\left(\begin{array}{cc}-\varDelta +{B}_1{\lambda}^2-E&amp; -{iA}_1\lambda \\ {}-{iA}_1\lambda &amp; \varDelta -{B}_1{\lambda}^2-E\end{array}\right)=0. $$

(1.8)

The equation has four solutions: :±λ1, ± λ2:

$$ {\lambda}_{1,2}=\frac{1}{B_1\sqrt{2}}{\left[F\mp \sqrt{F^2+4{B}_1^2\left({E}^2-{\varDelta}^2\right)}\right]}^{1/2}, $$$$ F={A}_1^2+2{B}_1\varDelta . $$

(1.9)

Real λ1, 2 guarantee the exponential decay of the wave function away from the surface and exist only in the energy interval inside the energy gap |E| < |Δ|. Outside the gap, the imaginery part of λ makes the wave function oscillatory and merges with the bulk states. For an electron energy that falls within the gap, additional constrains following from (1.9) ensure the existence of well-defined surface states:

$$ {B}_1\varDelta &lt;0;\kern1em {A}_1^2&gt;4\left|{B}_1\varDelta \right|. $$

(1.10)

So, surface states exist if the bulk energy spectrum in the kz-direction is inverted. In order to find the energy spectrum of surface electrons, we present the wave function

$$ \left(\begin{array}{c}\left|1,\Big\rangle \right.\\ {}\left|2,\Big\rangle \right.\end{array}\right) $$

as the superposition of two linearly independent eigenfunctions of the Hamiltonian (1.7) corresponding to real λ1, λ2. As we consider a semi-infinite sample with a single surface at z = 0, the wave function is expressed as:

$$ {\varPsi}_{\uparrow }(z)={C}_1\left(\begin{array}{c}{B}_1{\lambda}_1^2+E-\varDelta \\ {}{iA}_1{\lambda}_1\end{array}\right)\;\exp\;\left(-{\lambda}_1z\right)+{C}_2\left(\begin{array}{c}{B}_1{\lambda}_1^2+E-\varDelta \\ {}{iA}_1{\lambda}_2\end{array}\right)\;\exp\;\left(-{\lambda}_2z\right). $$

(1.11)

The zero boundary condition Ψ↑(z) = 0 becomes a system of homogeneous equations for coefficients C1, 2:

$$ X\left(\begin{array}{c}{C}_1\\ {}{C}_2\end{array}\right)=0, $$$$ X=\left(\begin{array}{cc}{B}_1{\lambda}_1^2+E-\varDelta &amp; {B}_1{\lambda}_2^2+E-\varDelta \\ {}{iA}_1{\lambda}_1&amp; {iA}_1{\lambda}_2\end{array}\right). $$

(1.12)

The energy of surface state E = 0 compatible with real nonzero λ1, 2 follows as a solution to Det(X) = 0:

$$ \left({B}_1{\lambda}_1^2+E-\varDelta \right){\lambda}_2-\left({B}_1{\lambda}_2^2+E-\varDelta \right){\lambda}_1=0. $$

(1.13)

One more solution E = 0 corresponds to Ψ↓(z) and follows from the lower block of the Hamiltonian H0. The exact E = 0 position of the surface level at k = 0 is the consequence of the electron-hole symmetry assumed in the Hamiltonian (1.3). If this condition were relaxed, the level would be shifted from the middle of the gap still located somewhere inside the gap of the bulk spectrum. It should be noted that the surface level pinned inside the bandgap discerns TI from conventional solids, where Tamm–Shockley surface states may appear due to the termination of the periodic potential at the surface. However, the existence and positions of Tamm–Shockley levels depend on the termination potential which may place them (if any) inside the conduction or the valence band. The level exists due to inverted bands, and its topological nature is manifested in the fact that the level is pinned to the bandgap of a bulk crystal irrespective of the surface potential and surface defects. The fact is related to the band topology and will be discussed in the next chapters.

Topological surface states are not unique to the TI solids specified above. The zero mode was first predicted in a heterostructure, where two IV–VI semiconductors with opposite signs of the gap parameter are brought into contact [3]. As will become clear later, the IV–VI single crystal also has surface states in the middle of the bulk gap energy region. This material class is called a crystalline topological insulator and includes the Pb1 − xSnxTe(Se) system. The topological state is protected by mirror symmetry and occurs when Sn content exceeds the critical value at which conduction and valence bands invert their positions on an energy scale [4–7].

1.4 Thin Film

Below we consider the full TI model with two surfaces and look at surface modes at finite in-plane wave vector k ≠ 0.

1.4.1 Wave Functions

We start with the Hamiltonian H = H0 + H1, where H0 is given in (1.6) and

$$ {H}_1=\left[{V}_S(z)+{V}_{AS}(z)\right]I+\left(\begin{array}{cccc}-{B}_2{k}^2&amp; 0&amp; 0&amp; {A}_2{k}_{-}\\ {}0&amp; {B}_2{k}^2&amp; {A}_2{k}_{-}&amp; 0\\ {}0&amp; {A}_2{k}_{+}&amp; -{B}_2{k}^2&amp; 0\\ {}{A}_2{k}_{+}&amp; 0&amp; 0&amp; {B}_2{k}^2\end{array}\right), $$

(1.14)

where I = diag (1, 1, 1, 1) and the external field V(z) may include band bending near surface and is written as the sum of VS(z) = [V(z) + V(−z)]/2 and VAS(z) = [V(z) − V(−z)]/2, even and odd parts, respectively. A nonzero VAS(z) implies spatial inversion asymmetry.

First, we consider the Hamiltonian (1.7) in a film with two surfaces at z =  ± L/2, so the wave function includes terms with ±λ1, 2:

$$ {\displaystyle \begin{array}{c}{\varPsi}_{\uparrow }(z)={C}_{11}\left(\begin{array}{c}{B}_1{\lambda}_1^2+E-\varDelta \\ {}-{iA}_1{\lambda}_1\end{array}\right)\;\exp\;\left({\lambda}_1z\right)+{C}_{12}\left(\begin{array}{c}{B}_1{\lambda}_1^2+E-\varDelta \\ {}{iA}_1{\lambda}_1\end{array}\right)\;\exp\;\left(-{\lambda}_1z\right)\\ {}\kern4.5em +{C}_{21}\left(\begin{array}{c}{B}_1{\lambda}_2^2+E-\varDelta \\ {}-{iA}_1{\lambda}_2\end{array}\right)\;\exp\;\left({\lambda}_2z\right)+{C}_{22}\left(\begin{array}{c}{B}_1{\lambda}_2^2+E-\varDelta \\ {}{iA}_1{\lambda}_2\end{array}\right)\;\exp\;\left(-{\lambda}_2z\right).\end{array}} $$

(1.15)

The problem has the spatial inversion symmetry: the surfaces at z =  ± L/2 are identical, so we can reduce the number of coefficients by assuming symmetric (C11 = C12, C21 = C22) and antisymmetric (C11 =  − C12, C21 =  − C22) superpositions in (1.15). Next we consider the two cases separately.

1. C11 = C12 = A, C21 = C22 = B. The wave function takes the form

$$ {\chi}_{\uparrow }(z)=A\left(\begin{array}{c}\left({B}_1{\lambda}_1^2+E-\varDelta \right)\mathit{\cos}\kern-0.12em h\left[{\lambda}_1z\right]\\ {}-{iA}_1{\lambda}_1\mathit{\sin}\kern-0.12em h\left[{\lambda}_1z\right]\end{array}\right)+B\left(\begin{array}{c}\left({B}_1{\lambda}_2^2+E-\varDelta \right)\mathit{\cos}\kern-0.12em h\left[{\lambda}_2z\right]\\ {}{iA}_1{\lambda}_2\mathit{\sin}\kern-0.12em h\left[{\lambda}_2z\right]\end{array}\right). $$

(1.16)

Zero boundary conditions χ↑(±L/2) = 0 become a system of homogeneous equations for coefficients A, B:

$$ M\left(\begin{array}{c}A\\ {}B\end{array}\right)=0, $$$$ M=A\left(\begin{array}{c}\left({B}_1{\lambda}_1^2+E-\varDelta \right)\;\mathit{\cos}\kern-0.12em h\;\left[{\lambda}_1L/2\right]\kern1em \\ {}{iA}_1{\lambda}_1\;\mathit{\sin}\kern-0.12em h\;\left[{\lambda}_1L/2\right]\end{array}\begin{array}{c}\left({B}_1{\lambda}_2^2+E-\varDelta \right)\;\mathit{\cos}\kern-0.12em h\;\left[{\lambda}_2L/2\right]\\ {}{iA}_1{\lambda}_2\;\mathit{\sin}\kern-0.12em h\;\left[{\lambda}_2L/2\right]\end{array}\right), $$

(1.17)

and the equation Det(M) = 0 determines the surface energy level E−:

$$ \frac{\left({B}_1{\lambda}_1^2+E-\varDelta \right){\lambda}_2}{\left({B}_1{\lambda}_2^2+E-\varDelta \right){\lambda}_1}=\frac{\mathit{\tan}\kern-0.12em h\left[{\lambda}_1L/2\right]}{\mathit{\tan}\kern-0.12em h\left[{\lambda}_2L/2\right]}. $$

(1.18)

The wave function corresponding to this energy level can be found from (1.16) providing coefficients A, B are known. Using (1.17) and (1.18), one finds

$$ B=-A\frac{B_1{\lambda}_1^2+E-\varDelta }{B_1{\lambda}_2^2+E-\varDelta}\cdot \frac{\mathit{\cos}\kern-0.12em h\left[{\lambda}_1L/2\right]}{\mathit{\cos}\kern-0.12em h\left[{\lambda}_2L/2\right]}. $$

(1.19)

Solving (1.18) with respect to E − Δ, we get

$$ E-\varDelta +{B}_1{\lambda}_1^2=\frac{B_1{\lambda}_1\left({\lambda}_1^2-{\lambda}_2^2\right)\mathit{\tan}\kern-0.12em h\left[{\lambda}_1L/2\right]}{\lambda_1\mathit{\tan}\kern-0.12em h\left[{\lambda}_1L/2\right]-{\lambda}_2\mathit{\tan}\kern-0.12em h\left[{\lambda}_2L/2\right]}, $$$$ E-\varDelta +{B}_1{\lambda}_2^2=\frac{B_1{\lambda}_2\left({\lambda}_2^2-{\lambda}_1^2\right)\mathit{\tan}\kern-0.12em h\left[{\lambda}_2L/2\right]}{\lambda_2\mathit{\tan}\kern-0.12em h\left[{\lambda}_2L/2\right]-{\lambda}_1\mathit{\tan}\kern-0.12em h\left[{\lambda}_1L/2\right]}. $$

(1.20)

Now substituting