Chapter 1 Introduction
1.1 Topological Insulator
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Chapter 1 Introduction
1.1 Topological Insulator
Topological Insulator has become a hot topic recent. In 2007, the existence of 3D topological insulator were published. After one year later, 3D topological insulator was successfully found by experiment in Bi-Te bases system. This discovery is the-state-of-the-art field in solid state physics.
Figure 1.0 3D TI Band structure and Dirac cone (Green line for spin down and orange line for spin down)
3D topological insulators are special materials which insulating inside the bulk but carring the current on its surface. The bulk’s band structure is familiar to a typical insulator which has a gap between valence and conduction band. The carriers would go from valence band to conduction band. This kind of material has particular band structure which exist the special state connecting valence band and conduction band of
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electron as Figure 1.0. This unusual states are named the surface state which allowing electron ballistic transport and called Dirac cone. This states are the direct characteristics of topological insulators and the topological phase is determined by non-trivial topological invariant.
This material has topological non-trivial phase and resulting from the spin-orbit coupling. It could lead a topological trivial state to topological non-trivial state. The carriers with spin up moved to opposite direction to the spin down carrier and leading to spin polarized conduction. The spin-orbit coupling combines with time reversal symmetry causing to the topological protected surface states.
Topological insulators have unique physical properties which not only for the academia research but also to the application area. Just like the strong spin-orbit interaction could make it possible to pump the spin current without any joule heat. The spin-polarized surface feature could directly contributes to spintronic science and quantum computation device. In this thesis, Bi-Te base materials with different Selenium dopants have been done and physical properties have been studied..
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1.1.1 2D Topological insulator & Quantum hall state
In this case, we consider a two-dimensional world. The electrons are restricted in the xy-plane and couldn’t have any motion to the z-direction. When it applied a magnetic field B perpendicular to this 2D plane, the electrons would exhibit orbital motions by the Lorentz force. In the microcosmic world, the physic quantities are not continuous but quantized so that the electrons could only occupied the specific energy value. The frequency for the orbit motion is called Cyclotron frequency and was defined by
ω𝑐 = 𝑒𝐵
𝑚∗ (Equation 1-1)
Where e is electron charge, B is magnetic field, m* is the effective mass. In the ideal case, the Fermi energy is a fixed value and we could project it to the 2D system and it could be sketched an imagination picture (Figure 1.1).
Figure 1.1 Landau level for 2D TI
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All of this orbit motion corresponds to their particular energy and the energy are also got quantized. This energy levels are called Landau Levels and the energy could be determined by
ϵ𝑛 = ℏω𝑐(𝑛 +1
2) (Equation 1-2)
where n is any positive integer value. Each one of Landau levels are separated with energy ℏωc. When the quantum number n is large enough, the quantum mechanical behavior would be closed to the classical behavior and this principle only suitable in ℏωc.
To discuss the insulator case, we could imagine that the Landau Levels below the Fermi level are occupied. When the Landau Levels are higher than Fermi level, they are unoccupied. And it could produce a band gap and the band structure is similar to a typical insulator. This kind system was called integer Quantum Hall state but there is a difference between a typical insulator and integer Quantum Hall state. The electrons in typical insulator are bound to atoms but the electrons are allowed to drift while applying an electrical field. This electrons caused to a Hall current with Hall conductivity which was determined by
σ𝑥𝑦 =𝑁𝑒2
ℎ (Equation 1-3)
where e is electron charge, ℎ is Planck’s constant and N is the filling factor. From this conductivity, we start linking the Integer Quantum Hall state and 2D topological insulator system. Now let’s think about a real world of the x-y plane. At the edge of the xy plane, the electrons at this area couldn’t fulfill an entire cyclotron orbit. Finally, the electrons bounce from the edge and lead a skipping motion. (Figure 1.2). The cyclotron orbit direction is dependent on the external magnetic field and carriers the skipping
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motion is also fixed to only one direction along one edge. The motion is called chiral and could be presented by a band structure as a gapless state. The unidirectional conducting edges are causing to a change in topological Chern invariants which come from n=0 to n=1.
Figure 1.2 Electron’s skipping motion on the edge
The difference between an integer quantum Hall system and typical insulators are the topological. Topological insulator is named by its classification system based on topological order. An example for topological is the doughnut and coffee cup. This two objects are known to be topological equivalent because their shape could be transform to each other and still keep their surface continuous. It could also extend to band structure that Bloch Hamiltonians could be topological equivalent if they could be continuously distorted into each other. The topological invariant is the characteristic to this material.
We have introduced the fundamental principle of the Integer Quantum Hall state and the difference of the typical insulator. The IQH state and 2D topological have something similar. Both of them show an insulating inside the bulk but conducting on the edge which is topologically protected. 2D topological insulator’s spin degenerates are the difference of IQH state and 2D TI. Therefore, 2D topological insulator is an
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example of Quantum spin Hall (QSH) state which is very familiar to Integer Quantum Hall state. The QSH state had two Hall conductivity, one comes from spin up and the other from spin down electrons, respectively (Figure 1.3). This two current values are equal but directions are opposite. It shows a zero net current but has a quantized spin Hall conductivity and Hall conductivity has its independent Chern number as n↑ and n↓
with n↑ + n↓ = 0.
Figure 1.3 QSH state’s Hall conductivity
2D topological exists an additional topological invariant, ν which can take values of 1or 0. QSH state shows ν=1 which allows this system having two current along the edge which are for spin up and spin down, respectively. Dirac point would exhibit in 2D topological insulator system (Figure 1.4).
Figure 1.4 3D TI Band structure and Dirac cone
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1.1.2 3D Topological Insulator
In last section, we have introduced 2D topological insulator system and some basic principle for this materials. It is an important different between Quantum Spin Hall state and a 3D topological. Both of them could have the conduction edge or surface, but the results are different. In QSH state, it have to apple an external magnetic field.
However, 3D topological insulator needs not any field, it is causing to spin-orbit coupling rather than applied field. 3D TI isn’t simply stacking many 2D TI but is a brand new system. The 3D TI’s surface states are a non-trivial bulk topology effect.
Topological invariants are determined by calculating the parity of this system whether it is topological non-trivial or not. It could start from the energy level of Bismuth (6s26p3) and Selenium (4s24p3) of time-reversal invariant points near the Fermi level. Because the electrons near to Fermi level are almost p-orbitals so we could neglected the s-orbits electron here. First effect is that chemical bond between Bi and Se would concern to the state energy. Second effect is that the crystal field splitting are affect the spin-orbital coupling. Finally, the spin-orbit coupling Hamiltonian which is defined by
ℋ𝑆𝑂𝐶 = 𝜆𝐿 ∙ 𝑆 (Equation 1-4)
Where L is the orbit momentum and S is the spin momentum. Spin orbital coupling would lead the level reversed and closed to Fermi energy when λ is large enough. The reversion could change the parity and could transform a trivial insulator to a non-trivial insulator.