Physical Properties of Topological Insulators
『三次元ってのはな、もっと面倒くさいんだよ。』
Helical Spin Polarization
Electron spins of topological surface states lies in the plane \((k_x, k_y)\).
Electron spins are perpendicular to \(\vb{k}\).
Electrons hold helicity.
In particular, a Kramers pair consists of \((\vb{k},\vb*{\sigma})\) and \((-\vb{k},-\vb*{\sigma})\), i.e. they hold the same helicity.
Degeneracy at Dirac point protected by TRS.
TIs known as of 2014 are left-handed for Fermi levels above the Dirac point and right-handed below.
Warning
It seems that the definition of helicity here is a little bit different than that in particle physics.
Note
Dirac cone of topological surface state is non-degenerate.
Quasiparticle Interference
Quasiparticle interference: If \(\vb{k}_1\) is scattered to \(\vb{k}_2\), then these two states may interfere with each other and change the charge density.
Spin is preserved during scattering by non-magnetic centers. Therefore, in scattering of topological surface states, scattering of the form \(\vb{k} \rightarrow -\vb{k}\) are forbidden due to spin polarization.
In \(\ce{Bi2Te3}\) where the Dirac cone is replaced a complex \(C_6\) shape, scattering is not limited to \(\vb{k} \rightarrow -\vb{k}\) and therefore quasiparticle interference is observable.
Resistivity
Warning
Electrons being zero-mass Dirac particles does not implies that the effective masses are zero. Instead, the effective masses are divergent for Dirac particles.
Dirac Particles
Massless Dirac particles carries a Berry phase of \(\pi\):
\[\gamma = \oint_C \dd{\vb{k}} \cdot i \bra{u_\pm(\vb{k})} \nabla_k \ket{u_\pm(\vb{k})} = \pi.\]In ordinary metals the electrons have a higher back scattering probability due to interference, which leads to weak localization.
Negative magnetoresistance since \(\vb{A}\) breaks such interference.
For Dirac particles, with the \(\pi\) Berry phase such scattering is reduced, leading to weak antilocalization.
Positive cusp-shape magnetoresistance since \(\vb{A}\) breaks such interference.
Landau quantization depending on \(N\):
\[E_\pm(N) = \pm \sqrt{\frac{2e\hbar v_{\mathrm{F}}^2}{c} \cdot N}.\]Hall conductivity quantized: \(N\) denoting the highest filled Landau level,
\[\sigma_{xy} = -\frac{e^2}{h} \qty(N + \frac{1}{2}).\]
SdH Oscillation of Surface State
Lattice vibration only has finite effect on Landau levels.
Electronic properties mostly depend on \(\rho(E_{\mathrm{F}})\).
DoS to relaxation time:
\[\rho(E_{\mathrm{F}}) \sim \frac{1}{\tau},\]i.e. higher the DoS, higher the scattering probability.
Under strong magnetic field,
\[\begin{split}\sigma_{xx} &\approx \frac{nm^* c^2}{B^2 \tau}, \\ \sigma_{xy} &\approx \frac{nec}{B}.\end{split}\]Minima of \(\sigma_{xx}\): \(E_{\mathrm{F}}\) between two consecutive Landau levels, and therefore \(\rho(E_{\mathrm{F}})\) is small.
Maxima of \(\sigma_{xx}\): \(E_{\mathrm{F}}\) lies at a certain Landau level, and therefore \(\rho(E_{\mathrm{F}})\) is large.
Different from ordinary 2D electron systems where \(\sigma_{xy}\) is proportional to the filling factor \(\nu\), in a Dirac system \(\sigma_{xy}\) is proportional to (N+1/2) due to the Landau level at the Dirac point.
Fan Diagram
Oscillation of \(\sigma_{xx}\):
\[\Delta \sigma_{xx} \propto \cos \qty[ 2\pi\qty(\frac{F}{B} - \frac{1}{2} + \beta)].\]\(B_N\) denotes the \(N\)th \(\operatorname{argmin}\) of \(\sigma_{xx}\).
\(1/B_N\) versus \(N\) is a straight line:
\[2\pi \qty(\frac{F}{B_N} - \frac{1}{2} + \beta) = (2N - 1)\pi.\]\(\beta\) obtained by extrapolation of \(1/B_N\)-\(N\) to \(1/B_N = 0\), which hits the \(N\)-axis at \(N=\beta\).
Dirac system confirmed if \(\beta = 1/2\) by such extrapolation.
Warning
\(\sigma_{xx} \ll \abs{\sigma_{xy}}\) may not hold in experiment condition. It’s more reliable to find minima based on \(\sigma_{xy}\) instead of \(\rho_{xy}\).
Temperature Dependency of Oscillation Frequencies
Fan diagrams may not be able to tell apart the \(\pi\) phase shift due to SOC and Dirac cone.
The slop of the \(T^2\)-temperature dependency of oscillation frequency may be able to tell these cases apart [Guo2021].
More Information from SdH Oscillation
Carrier density: for 2D Dirac system,
\[n_{\mathrm{s}} = \frac{1}{(2\pi)^2} \pi k_{\mathrm{F}}^2 = \frac{e}{2\pi \hbar c}F,\]where we have assumed cylindrical sections and therefore
\[F = \frac{\hbar c}{2\pi e} \pi k_{\mathrm{F}}^2.\]Testing if SdH oscillation comes from 2D surface states:
\[F \sim \frac{1}{\cos\theta}\]for 2D electron systems where \(\theta\) is the angle between \(\vb{B}\) and the normal vector to the electron system.
Cyclotron frequency: with Lifshitz-Kosevich theory,
\[\Delta \sigma_{xx} = A_0 R_{\mathrm{T}} R_{\mathrm{D}} R_{\mathrm{S}} \cos \qty[2\pi \qty(\frac{F}{B} - \frac{1}{2} + \beta)],\]where
\[\begin{split}R_{\mathrm{T}} &= 2\pi^2 \frac{k_{\mathrm{B}} T / \hbar\omega_{\mathrm{c}}}{\sinh \qty[2\pi^2 (k_{\mathrm{B}} T / \hbar \omega_{\mathrm{c}} )]}, \\ R_{\mathrm{D}} &= \exp [-2\pi^2 (k_{\mathrm{B}} T_{\mathrm{D}} / \hbar \omega_{\mathrm{c}})], \\ R_{\mathrm{S}} &= \cos \qty(\frac{1}{2} \pi g m_{\mathrm{e}} / m_{\mathrm{c}}),\end{split}\]\(g\) is the \(g\)-factor of electron, and \(T_{\mathrm{D}}\) is the Dingle temperature given by
\[T_{\mathrm{D}} = \frac{\hbar}{2\pi k_{\mathrm{B}} \tau}.\]Obtain \(m_{\mathrm{c}}\) with fixed \(B\) and varying \(T\).
Fermi velocity: with
\[m_{\mathrm{c}} = \frac{\hbar^2}{2\pi} \qty(\pdv{A(E)}{E})_{E=E_{\mathrm{F}}}\]and (for 2D electron systems)
\[A(E_{\mathrm{F}}) = \pi k_{\mathrm{F}}^2 = \frac{ \pi E_{\mathrm{F}}^2 }{ (\hbar v_{\mathrm{F}})^2 },\]we find
\[m_{\mathrm{c}} = \frac{\hbar k_{\mathrm{F}}}{v_{\mathrm{F}}}.\]Relaxation time and mobility: \(T_{\mathrm{D}}\) may be obtained by data fitting and hence \(\tau\). Electron mobility is given by
\[\mu_{\mathrm{s}}^{\mathrm{SdH}} = \frac{e\tau}{m_{\mathrm{c}}} = \frac{e\ell^{\mathrm{SdH}}}{\hbar k_{\mathrm{F}}}\]where
\[\ell^{\mathrm{SdH}} = v_{\mathrm{F}} \tau.\]
Example: Two-Band Model
Taking bulk and surface conductivity into account, we find
\[\begin{split}\rho_{yx} &= \frac{(R_{\mathrm{s}} \rho_{\mathrm{b}}^2 + R_{\mathrm{b}}\rho_{\mathrm{s}}^2)B + R_{\mathrm{s}} R_{\mathrm{s}} (R_{\mathrm{s}} + R_{\mathrm{b}})B^3}{(\rho_{\mathrm{s}} + \rho_{\mathrm{b}})^2 + (R_{\mathrm{s}} + R_{\mathrm{b}})^2 B^2}, \\ \rho_{xx} &= \frac{\rho_{\mathrm{s}} \rho_{\mathrm{b}} (\rho_{\mathrm{s}} + \rho_{\mathrm{b}}) + (\rho_{\mathrm{s}} R_{\mathrm{b}}^2 + \rho_{\mathrm{b}} \rho_{\mathrm{s}}^2)B^2}{(\rho_{\mathrm{s}} + \rho_{\mathrm{b}})^2 + (R_{\mathrm{s}} + R_{\mathrm{b}})^2 B^2},\end{split}\]where \(\rho_{\mathrm{b}}\) and \(R_{\mathrm{b}}\) are resistivity and Hall coefficient of the bulk, respectively, while \(\rho_{\mathrm{s}}\) and \(R_{\mathrm{s}}\) are those of the surface, where
\[\begin{split}\rho_{\mathrm{s}} &= \rho_{\mathrm{2D}} t, \\ R_{\mathrm{s}} &= \frac{t}{e n_{\mathrm{s}}},\end{split}\]and \(t\) is the thickness of the sample.
Parameters \(n_{\mathrm{3D}}\), \(\rho_{\mathrm{b}}\), \(n_{\mathrm{s}}\), and \(\rho_{\mathrm{2D}}\) may be obtained by fitting the data, and hence the electron mobility.
\(n_{\mathrm{s}}\) may also be obtained from SdH data.
Additional constraint that the fitting be exact for \(\rho_{xx}(B=0)\) be imposed to reduce DoF.
Warning
\(\mu^{\mathrm{tr}}_{\mathrm{s}} > \mu^{\mathrm{SdH}}_{\mathrm{s}}\) due to difference in scattering mechanisms.
Weak Antilocalization
Weak localization: back scattering \(\vb{k}\rightarrow -\vb{k}\) has the largest amplitude due to interference.
Negative magnetoresistance because magnetic field breaks TRS and hence the interference.
Weak antilocalization: back scattering reduced due to the Berry phase of \(\pi\) for each cycle. Electron mobility is therefore higher.
Positive magnetoresistance of cusp shape because magnetic field breaks TRS and hence the interference.
Warning
Weak antilocalization may also be caused by strong spin-orbit coupling. Therefore, this phenomenon is not limited to Dirac systems.
Magnetoresistance from antilocalization: Hikami-Larkin-Nagaoka equation
\[\Delta \sigma_{xx}(B) = \alpha \frac{e^2}{\pi h} \qty[ \Psi\qty(\frac{\hbar c}{4eL_\phi^2 B} + \frac{1}{2}) - \ln\qty(\frac{\hbar c}{4e L_\phi^2 B}) ],\]where \(L_\phi\) is the phase coherence length of electron, and \(\alpha\) is \(-1/2\) for each conducting channel.
For thin film samples where the thinkness is less than \(L_\phi\) (around 100nm to 1000nm), the channels on the two surfaces are combined as one and therefore \(\alpha = -1/2\).
Topological Protection of Surface States
Aspects of Protection
Non-trivial \(\mathbb{Z}_2\) index implies the existence of surface states.
Anti-parallel spin at \(\vb{k}\) and \(-\vb{k}\) reduces back scattering and therefore protects surface transport.
Weak antilocalization due to Berry phase of \(\pi\) of massless Dirac particles antilocalizes electrons and therefore protects surface transport.
Breaking of Protection
Thickness reduced \(\Downarrow\)
States on top and bottom surfaces get mixed \(\Downarrow\)
Energy gap opens \(\Downarrow\)
Berry phase
\[\gamma \approx \pi \qty(1 - \frac{\Delta}{E_{\mathrm{F}}}) \Downarrow\]Weak antilocalization reduced \(\Downarrow\)
Electron mobility reduced.
Aharonov-Bohm Effect in Nano-Ribbons
Magnetic field applied along longitudinal direction.
Oscillation (of period \(\Phi = h/e\)) of \(\sigma\) occurs for ribbon width below 570nm.
Oscillation disappears for width above.
Coherence length inferred to be around 500nm.
Altshuler-Aronov-Spivak effect not observed.
Magnetism
Magnetism (bulk or surface) found upon doing with magnetic atoms.
Bulk magnetism found in doped \(\ce{Bi2Te3}\) with \(\ce{Mn}\).
Surface magnetism found in doped \(\ce{Bi2Se3}\) and \(\ce{Bi2(Se,Te)3}\) thin film with \(\ce{Mn}\).
Possibly due to RKKY interaction.
Quantum Anomalous Hall Effect
QAHE found in \(\ce{Cr}\)-doped \(\ce{(Bi,Sb)_2Te3}\) thin film and \(\ce{Mn}\)-doped \(\ce{Bi2(Se,Te)3}\) thin film [Chang2013].
Magnetism of doping ions breaks TRS and therefore creates gap at Dirac point.
Hence one-dimensional edge states created.
Topological Magnetoelectric Effect
Lagrangian
\[\mathcal{L} = \frac{1}{8\pi} \qty(\epsilon \vb{E}^2 - \frac{1}{\mu} \vb{B}^2) + \qty(\frac{\alpha}{4\pi^2}) \theta \vb{E}\cdot \vb{B}.\]\(\theta = \pi\) for \(\mathbb{Z}_2\) topological insulators.
Electromagnetic response:
\[\begin{split}\vb{D} &= \vb{E} + 4\pi \vb{P} - \frac{\alpha\theta}{\pi}\vb{B}, \\ \vb{H} &= \vb{B} - 4\pi \vb{M} + \frac{\alpha \theta}{\pi} \vb{E}.\end{split}\]
Topological magnetoelectric effect: an electric field induces a magnetic field in the same direction.
Dirac gap must be opened and Fermi level must be tuned therein.
Image magnetic monopole: electric charge close to the surface may induce magnetic field of the form of an magnetic monopole in the TI.
Spintronics
Spin pumping may be achieved by connecting the surface of TI to magnetic materials.
Spin Transport in the Ballistic Transport Regime
The injected spins are converted into a charge current along the Hall direction due to the spin-momentum locking on the surface state.
Direction of spin-induced voltage difference depends on the injected spin direction.
Spin Transport in the Diffusive Transport Regime
Spin transport is too small at the diffusive transport regime, i.e. where the mean free path of electrons are smaller than the sample size.
Hamiltonian of surface states:
\[H = \hbar v_{\mathrm{F}} (k_x \sigma_x + k_y \sigma_y) - \mu_{\mathrm{B}}(H_x \sigma_x + H_y \sigma_y).\]\(\vb{H}_\parallel\) shift the Dirac point.
\(H_\perp\) opens the Dirac gap.
Under zero magnetic field, the average \(k\) is shifted by
\[\Delta k = \frac{mJ}{ne\hbar}\]since
\[J = nev,\quad mv = e\tau E,\quad \Delta k = \frac{e\tau E}{\hbar}.\]Spin average: with electric field applied along \(y\)-direction,
\[\langle \sigma_x \rangle = \frac{J_y}{2ev_{\mathrm{F}}},\]too small to be detectable.
Under nonzero magnetic field,
\[\Delta k \approx \frac{\mu_{\mathrm{B}} {H}_{\parallel}}{\hbar v_{\mathrm{F}}}\]and
\[\langle \sigma_x \rangle \sim \frac{n\Delta k_y}{k_{\mathrm{F}}},\]also too small.
Optics
Low energy (THz, i.e. meV) regime is preferable.
Interval of Landau levels measurable.
AC topological magnetoelectric effect: topological magnetoelectric effect from the AC field of the light beam.
Faraday effect: rotation of the polarization plane of light transmitted, where the incident light is parallel to the magnetic field.
\[\theta_{\mathrm{F}} \approx (\nu_{\mathrm{T}} + \nu_{\mathrm{B}}) \alpha,\]where \(\nu_{\mathrm{T}}\) and \(\nu_{\mathrm{B}}\) are Landau filling factors of the top surface and the bottom surface, respectively.
Magnetic Kerr effect: rotation of the polarization plane of light reflected, where the incident light is parallel to the magnetic field.
\[\theta_{\mathrm{K}} \approx \pm \frac{\pi}{2}.\]Floquet-Bloch state: periodic pattern of \(E\)-axis of the Dirac cone with quasi-equilibrium eigenstates.
Observable using pump-probe ARPES.