Experimental Techniques
Quantum Oscillations
Quantum oscillations describes a series of related experimental techniques used to map the Fermi surface of a metal in the presence of a strong magnetic field. [Coldea2010]
Laudau Quantization
Strong magnetic field
Band structure as perturbation on Landau levels.
Landau gauge:
\[\vb{A} = (0, Bx, 0).\]Hamiltonian:
\[\begin{split}H(p_x, p_y) &= \frac{1}{2m} \qty(\vb{p} - \frac{e}{c} \vb{A})^2 \\ &= \frac{1}{2m}\qty[p_x^2 + \qty(p_y - \frac{e}{c} Bx)^2]. \\ H(p_x, \hbar k_y) &= \frac{1}{2m} p_x^2 + \frac{m\omega_c^2}{2}(x-X)^2.\end{split}\]where
\[\begin{split}p_y &= \hbar k_y, \\ \omega_c &= \frac{eB}{mc}, \\ l_B &= \sqrt{\frac{\hbar c}{eB}}, \\ X &= \frac{\hbar c k_y}{eB} = l_B^2 k_y.\end{split}\]Solution:
\[\psi_{k_y, n}(x, y) = \frac{1}{\sqrt{L_y}} e^{ik_y y} \phi_{k_y, n}(x)\]where
\[\phi_{k_y, n}(x) \propto e^{-\frac{1}{2} (x/l_B)^2} H_n\qty(\frac{x - X}{l_B}).\]Energy levels:
\[E_n = \hbar \omega_c\qty(n+\frac{1}{2}).\]Degeneracy: number of states at each level,
\[m_{\mathrm{max}} = \frac{L_x L_y}{2\pi l_B^2}.\]Note
Stronger the magnetic field, higher the degree of degeneracy.
Filling factor:
\[\nu = \frac{N_{\mathrm{e}}}{m_{\mathrm{max}}} = 2\pi l_B^2 \frac{N_{\mathrm{e}}}{L_x L_y}.\]
Weak magnetic field
Magnetic field as perturbation on band structure.
\(k_y\) still a good quantum number.
Observation of Quantum Oscillation
Classical Theory of Electron Motion
\(\vb{k}\) runs along the constant-energy surface, rotating around \(\vb{B}\).
With \(\vb{r}_\perp\) denoting the component of \(\vb{r}\) perpendicular to \(\vb{B}\), we have
\[\vb{r}_\perp(t) - \vb{r}_\perp(0) = -\frac{\hbar c}{eB} \hat{\vb{B}} \times \qty[\vb{k}(t) - \vb{k}(0)].\]Note
The orbit in the \(\vb{r}\) space is given by the orbit in the \(\vb{k}\) space rotated by \(\pi/2\) and scaled by \(\dfrac{\hbar c}{eB}\).
Period:
\[T = \frac{\hbar^2 c}{eB} \pdv{A(E,k_z)}{E} = \frac{2\pi m_{\mathrm{c}} c}{eB}.\]\(A\) is the area encicled by the orbit.
Cyclotron mass:
\[m_{\mathrm{c}} = \frac{\hbar^2}{2\pi} \pdv{A(E,k_z)}{E}.\]
With Bohr’s correspondence principle, we find that the Landau levels for large \(\nu\)’s are given by the Onsager’s semiclassical quantization condition
\[A(E_\nu, k_z) = \qty(\nu + \frac{1}{2} - \beta) \frac{2\pi eB}{\hbar c},\]where \(\beta = 0\) for free electrons.
Maximal number of states thereon when the outest Landau tube is tangent to the Fermi surface. At two consecutive extrema, one has
\[\begin{split}A(E_{\mathrm{F}}) &= A(E_{\nu}, k^0_z) = \qty(\nu + \frac{1}{2} - \beta) \frac{2\pi e B_\nu}{\hbar c}, \\ A(E_{\mathrm{F}}) &= A(E_{\nu - 1}, k^0_z) = \qty(\nu - 1 + \frac{1}{2} - \beta) \frac{2\pi e B_{\nu - 1}}{\hbar c},\end{split}\]respectively, where \(A(E_{\mathrm{F}})\) denotes the area encircled by an extremal orbit. Therefore, such exterma occurs periodically with
\[\Delta \qty(\frac{1}{B}) = \frac{2\pi e}{\hbar c A(E_{\mathrm{F}})}.\]
Note
We may therefore obtain the area encircled by the extremal orbits of the Fermi surface from the observed \(\Delta \qty(1/B)\).
Phenomena
De Haas-van Alphen oscillation: \(\chi\) versus \(1/B\).
Shubnikov-de Haas oscillation: resistivity versus \(1/B\).
Warning
As \(T\) goes up, the Fermi surface is Blurred by \(O(k_{\mathrm{B}}T)\) which may exceed \(\Delta E\), which may therefore suppress quantum oscillation.
ARPES
Measuring the energy and direction of an electron when the surface is hit by a monochrome beam, whence we may obtain \(E_n(\vb{k})\).
Conservation Laws
Conservation of Energy
where
Conservation of Momentum
Analyzing ARPES Data
Work function \(\phi\) is already known.
\(\vb{k}_{\mathrm{f}}\) of the escaping electron is measured.
\(\vb{k}_{\mathrm{i}\parallel}\) is determined by moving \(\vb{k}_{\mathrm{f}}\) to the first BZ.
Assuming 2D materials, we have
\[E_{\mathrm{i}}(\vb{k}_{\mathrm{i}\parallel}) = \frac{\hbar^2 k_{\mathrm{f}}^2}{2m} + \phi - h\nu.\]Assuming 3D materials, we have
\[E_{\mathrm{i}}(\vb{k}_{\mathrm{i}\parallel}, k_{\mathrm{i}\perp}) = \frac{\hbar^2 k_{\mathrm{f}\parallel}^2}{2m} + \frac{\hbar^2 k_{\mathrm{f}\perp}^2}{2m} + \phi - h\nu.\]\(k_{\mathrm{i}\perp}\) is unknown.
Ansatz
\[E_{\mathrm{i}}(\vb{k}_{\mathrm{i}}) + h\nu = \frac{\hbar^2 (\vb{k}_{\mathrm{i}} + \vb{G})^2}{2m} + V_0 = \frac{\hbar^2 k_{\mathrm{f}}^2}{2m} + \phi.\]
Telling apart 2D bands and 3D bands, i.e. bulk states and surface states.
Note
The energies are relative to \(E_{\mathrm{F}}\), i.e. \(E_{\mathrm{F}} = 0\).
Sensitivity to Surface Condition
ARPES probes only 1nm into the surface. Ultra-high vacuum and clear cleavage are necessary.
Determination of Topological Insulators
Determination of 2D Topological Insulators
Edge states confirmed by quantization of conductivity.
Confirmed also partially by edge charge density with STM.
Chiral spin-polarization confirmed by spin current.
Determination of 3D Topological Insulators
Confirmed by existence of Dirac cone on surface with ARPES.
Surface states also confirmed by SdH oscillation.
Dirac carriers confirmed by phase shift of \(\pi\) in oscillation due to Berry phase.
Dirac carriers also confirmed by STS.
Laudau levels for a Dirac cone takes the form \(\sqrt{BN}\) where \(N=0\) fixes the Dirac point.
The pattern may be tested by measuring local charge density versus bias voltage.
May also be confirmed by magneto-optical spectroscopy.
Warning
Since charge accumulation may occur at surface and may create a conductive layer, the dependence of resistivity versus sample size does not confirm the existence of topological surface states.
CD-ARPES
CD (Circular dichroism)-ARPES showing CR (circularly polarized right)-CL (circularly polarized left) difference spectrum may indicates non-degenerate spin configurations.
Glossary
- ARPES/角度分解電子分光/角分辨光电子能谱
Angle-resolved photoemission spectroscopy
- Work Function/仕事関数/功函数
The minimum amount of energy required to remove an electron from the crystal.
- STS/走査型トンネル分光法/扫描隧道光谱
Scanning tunneling spectroscopy.