Transcript TIs and STMx - UC Davis Canvas

Introduction to topological
insulators and STM/S on TIs
0
Today
• Introduction to TIs – basic concepts
• Symmetry and topology.
• Analogy to QHE (the first topological phase).
• Surface states with spin-momentum locking.
• Using QPI to detect the TI surface states in Bi1-xSbx.
• Transmission through surface defects in Sb.
• Particle in a box.
• Landau Level Spectroscopy in Bi2Se3.
1
Introduction to TIs
Basic concepts
2
3D Topological Insulator
This is the prediction to be tested experimentally:
Dirac cone surface states with momentum-spin locking.
Topology and symmetry
Typically we characterize the different phases of matter by their symmetries.
Characterizing Hamiltonians in terms of their symmetries simplifies our thinking.
It allows different physical systems to be mapped onto one another.
Ferromagnetism
Broken rotational symmetry
Charge Density Wave
Broken translational symmetry
Topology and Symmetry
A new way to classify physical systems by topology has now emerged in solid state physics.
One can deform the torus to the coffee mug but not into a sphere.
Gauss-Bonnet Theorem
𝐾𝑑𝐴 = 4𝜋(1 − 𝑔)
The geometry in this space determines
the genus g.
For the sphere g = 0, for the torus g = 1.
Quantum Hall Effect
The first known topological phase
Source: wikipedia
•
•
•
Typical Hall experiment.
QHE observed in 2D electron gases .
Integer QHE: Nobel prize 1985 to Klaus von
Klitzing (MOSFETs, GaAs structures).
𝐸𝑛 = ℏ𝜔𝑐 𝑛 +
•
•
1
2
𝜔𝑐 =
𝑒𝐵
𝑚
In Quantum Hall “Insulator” a magnetic field is applied
along z, and the 2D electron gas develops Landau levels.
If the chemical potential is in between LLs one would have
expected an insulator, but instead 𝜌𝑥𝑥 = 0 !
Quantum Hall Effect
The answer is in the boundary!
Source: wikipedia
•
•
•
Typical Hall experiment.
QHE observed in 2D electron gases .
Integer QHE: Nobel prize 1985 to Klaus von
Klitzing (MOSFETs, GaAs structures).
𝐸𝑛 = ℏ𝜔𝑐 𝑛 +
•
•
•
1
2
𝜔𝑐 =
𝑒𝐵
𝑚
The boundary of a 2D system is in 1D space.
Skipping orbits lead to finite conductivity on the
boundary. EDGE STATES.
The sign of the magnetic field Bz determines the
orientation of the 1D current on the boundary.
Quantum Hall Effect
The answer is in the boundary!
What happens to the edge state if it finds a defect on the edge of the sample?
Quantum Hall Effect
The answer is in the boundary!
These edge states are immune to defects simply because there are no states
propagating in the opposite direction.
Back scattering (k to –k)
is not allowed.
So what about topology?
•
The Hall conductance is quantized.
𝜎𝑥𝑦
•
n is an integer (called the Chern number).
𝑛=
•
𝑒2
=𝑛
ℎ
1
2𝜋
𝛻𝑘 × 𝐴(𝑘𝑥 , 𝑘𝑦 ) 𝑑 2 𝑘
𝑧
And A is the Berry connection which depends on
the geometry of the bands 𝑢𝑘 in k-space.
𝐴 = −𝑖 𝑢𝑘 𝛻𝑘 𝑢𝑘
n is a topological invariant. It cannot change when the Hamiltonian varies smoothly.
•
The original idea of a topological invariant for this Quantum Hall states was found by Thouless,
Kohmoto, Nightingale and den Nijs. It was called the TKNN invariant. T = Thouless who won the
Nobel prize in 2016.
The quantization of the Hall conductivity has been measured to 1 part in 109 !
Quantum Hall “insulator” at B=0?
Look for large spin orbit coupling
•
•
In a semi-classical approach we can consider electrons orbiting the nucleus.
In the rest frame of the electron, the electric field turns into a magnetic field.
𝑩=−
•
𝒗×𝑬
∝𝒓×𝒑=𝑳
𝑐2
And the SOC term is simply 𝝁𝒔 ∙ 𝑩 . Momentum and the spin are perpendicular to one another.
This internal B-field from the spin-orbit coupling leads to the topological surface states.
Because of the strong SOC, the spin S and the momentum k are locked perpendicularly.
•
•
Originally proposed in 2D by Mele and
Kane, 2005.
The spin texture prevents back
scattering.
3D Topological Insulator
•
•
•
First generation of 3D topological insulators include: Bi1-xSbx, Bi2Se3, Bi2Te3, Sb2Te3, Sb, etc…
The “edge” of a 3D material is a 2D surface.
Due to the spin texture, back scattering is still forbidden.
Some review articles and books.
13
Great news for ARPES and STM
• Surface probes!
• ARPES can probe the surface band structure and distinguish
it from the bulk (using photon energy mapping).
• ARPES can measure the spin texture with spin detectors
and/or by taking advantage of the photon polarization.
• STM/S can also probe the surface band structure.
• STM/S can probe the directional scattering through QPI.
U-turns?.
14
First STM measurements
Bi1-xSbx – Roushan et al. Nature 2009.
15
What they knew from ARPES
• Schematic of the 3D Brillouin of Bi1-xSbx and 2D projection onto 111 surface.
• Fermi contours of the surface states, with spin texture superposed .
D. Hsieh, et al. Nature 2008.
Measure the surface states using QPI
Density of states.
Topography
Measure the surface states using QPI
• Measurements of dI/dV show QPI with many patterns.
Calculate the QPI patterns using the JDOS
• The Joint Density of States (JDOS) calculation takes into account all possible ki
to kf scattering events.
• It can be calculated from a auto-convolution of the ARPES constant energy
contours.
Is there good agreement between JDOS and the STM/S data?
Calculate the QPI patterns using the JDOS
• The Joint Density of States (JDOS) calculation takes into account all possible ki
to kf scattering events.
• It can be calculated from a auto-convolution of the ARPES constant energy
contours.
Some scattering wave vectors missing in the experimental data.
Add the spin texture.
•
The QPI can be calculated with a scattering matrix T which depends on the spin of the electrons.
Add the spin texture.
•
Focusing along a high-symmetry direction we can link the different scattering processes to the QPI
peaks. Indeed the back-scattering events are missing.
STM/S
ARPES
Note: QPI to ARPES correspondence
•
A full assignment between ARPES and STM can be accomplished
Note: QPI to ARPES correspondence
•
After correctly associating the QPI wave vectors to the ARPES band structure we can compare the
energy dispersion relations.
Transmission through
surface barriers
Sb – J. Seo et al. Nature 2010.
25
Transmission and Reflection of TI surface states
•
Step edges can also produce QPI patterns. And one can measure the transmission and reflection.
L. Burgi et al. Phys. Rev. Lett., 81 5370, 1998.
•
Unlike conventional 2D electron states, topological surface states are expected to be immune to
backscattering and should better overcome barriers from material imperfections.
Step edges in Sb
• Terraces lead to the quantization of surface electronic states – particle in a box…
• DOS averaged along the step edges show quantized states.
• What is the quantization condition from boundary conditions?
2𝜋𝑛
𝑞𝑛 =
𝐿
Quantization condition
• Terraces lead to the quantization of surface electronic states – particle in a box…
• What is the quantization condition from boundary conditions?
• Two conditions need to be satisfied.
2𝜋𝑛
𝑞𝑛 =
𝐿
𝑞𝑛 𝐸𝑛 = 𝑘𝑓 𝐸𝑛 − 𝑘𝑖 𝐸𝑛
The energy quantization agree with a linear
Dirac dispersion.
Dispersion of surface states
• Although back scattering is not allowed, other processes can happen.
• The (1D) Fourier transforms show the dispersion relation which match the ARPES data.
2𝜋𝑛
𝑞𝑛 =
𝐿
Determining the reflection
• Consider multiple scatterings in a box.
Determining the reflection
• Consider the symmetric process and obtain the density of states.
• “Fitting to the data” one obtains a reflection coefficient of 42%.
• This reflectivity is much lower than what has been reported in noble metals (~70%).
Determining the transmission.
• A lack of reflection doesn’t necessarily mean high transmission.
• Consider an electron on the surface scattering off a step edge. Where can it go?
It can transmit, reflect or go into the bulk.
• A small terrace (110Å) and a large terrace (>2500Å).
• On the large terrace the patterns are almost continuous.
The energy quantization
suppression from the
narrow terrace can be
seen in the large ONE!
Determining the transmission.
Sb Topological states.
Transmission 35%
Reflection 42%
Absorption 23%
Similar experiment on Ag 111
Transmission 0%
Reflection 50% to 80%
Absorption 20% to 50%.
L. Burgi et al. Phys. Rev. Lett., 81 5370, 1998.
Landau Level
Spectroscopy in Bi2Se3.
Sb – T. Hanaguri et al. PRB 2010.
34
Landau levels for classical and relativistic particles
There is always a n=0 LL at the Dirac point.
Landau level spectroscopy with the STM
Scaled data for multiple Bfields yields the dispersion
relation of the Dirac cone.
Summary
No back scattering
Transmission
LL Spectroscopy