Insulators are known as materials with a large band gap between occupied and empty bands (Fig. 1.1a). 3D Topological insulators are similar to trivial insulators in the sense that their Fermi surfaces are both within bulk band gap. But in the case of topological in-sulator, there are additional gapless surface states inside the bulk band gap (Fig. 1.1b) that intersect at the ”Dirac point”. These additional states support 2D conducting channels at the surface, and are protected from non-magnetic impurity scatterings due to time reversal symmetry. The energy dispersion of the surface states is linear with the wave-vector k, E = vF¯hk, where vF is Fermi velocity. Due to it’s linear dispersion, the effective mass m∗ ∝ d2E/dk2 of Dirac Fermion near Dirac point is zero. Their wavefunction must be described by Dirac equation rather than Schr¨odinger equation. The charge carriers in the surface states are also called ”Dirac fermions”. One direct consequence of zero effective mass is the well defined chirality
h≡ σ·p = ±1, (1.1)
where σ is Pauli matrix and p is momentum unit vector. For typical particles with non-zero mass, p depends on the observer’s reference frame and may even change sign if the
1
observer is moving faster than the particle. Therefore, chirality is not well-defined for massive particles. But the eigenvalue of chirality operator is irrelevant to the observer’s reference frame when acting on wavefunctions described by Dirac equation. Therefore, the chirality of Dirac fermion is well-defined and the surface states are referred to as
“chiral states”. A consequence of well-defined chirality is the suppression of the 2kF -scatterings (backscattering). Since chirality is a conserved physical quantity in topologi-cal insulators, electrons moving in opposite direction are required to have opposite spins (from Eq 1.1). Therefore, backscattering is suppressed since it requires higher energy to flip electron spins. Another consequence of the linear dispersion is the π-shift in Berry’s phase, γ [6]. We know that electrons in crystals moving in a closed trajectory at con-stant energy in momentum space gains an additional phase called Berry’s phase which is zero in typical conductors and π in massless Dirac materials (without considering spin-orbit coupling). Further consideration of spin-spin-orbit interaction (which is non-negligible in topological insulators) will modify this π shift in Berry’s phase [7]. The exact value of this phase shift can be experimentally determined through Shubnikov-de Hass oscillations (to be discussed below).
Bi1−xSbx, Bi2Se3 and Bi2Te3 were first theoretically predicted to be 3D topological insulators [8, 1]. Later on, the 2D surface state of Bi1−xSbxwas experimentally confirmed using Angle Resolved Photoemission Spectroscopy (ARPES) [9]. Soon after, Bi2Se3[2]
and Bi2Te3 [10] were also experimentally confirmed by ARPES. Calculated band struc-ture of Bi2Se3 is shown in Fig. 1.2a, the warmer color denotes a higher density of states (D.O.S.). The gapless states with linear dispersion can be seen crossing bulk band-gap.
The center white line denote the position of Fermi surface. Experimental result of Bi2Se3 from ARPES is shown in Fig. 1.2b. The color code denotes ARPES spectra height, which is proportional to electron D.O.S. The yellow areas are bands with high D.O.S., indicating the position of bulk bands, the lower one and upper one are valence band and conduction band, respectively. The white dotted line marks the position of the Fermi level. A red
Energy
Figure 1.1: (a)Typical band structure of a trivial insulator. Colored area denote bands filled by electrons. Valence band is totally filled while the conduction band is empty.
They are separated by a band gap Eg, which is much larger than the thermal energy kBT . The dash line depicts the position of Fermi level EF. (b)The band structure of an ideal topological insulator. The gapless surface states are denoted by the red lines. The band gap size is typically about 300 meV, which is similar to semiconductors.
colored, gapless band with almost linear dispersion can be seen inside the bulk band gap.
The electron cyclotron mass can be estimated from the energy dispersion in Fig. 1.2b.
The cyclotron effective mass is usually defined by [11]
mc = ¯h2 2π
∂A
∂E, (1.2)
where A is k-space area enclosed by the orbit and E is the energy. Combine the above equation with the linear dispersion relation of Dirac fermions, E = vF¯hk, it can be shown that mc= ¯hk/vF. Therefore, the cyclotron mass is zero at Dirac point. The Fermi velocity estimated from Fig. 1.2b is∼ 4.6 × 105m/s, and the cyclotron mass in surface states near EF is ∼ 0.25m0 (kF ∼ 0.1 ˚A−1 and EF ∼ 300meV ), where m0 is the free electron mass. For the electrons in the bottom of conduction band in Fig. 1.2b (assume parabolic dispersion, E = ¯h2k2/2mc), the estimated cyclotron mass mc ∼ 0.12m0 for electrons near EF (kF ∼ 0.04 ˚A−1and EF ∼ 50meV ).
The crystal structure of Bi2Se3 is shown in Fig. 1.3 (figure from Ref. [1]). The prim-itive lattice vectors are denoted as t1, t2, and t3. It has layered structure with a triangular lattice within one layer. Each unit cell consists of five-atom layers along the c-axis, which
)b* )c*
Figure 1.2: Bi2Se3 electronic band structure. (a)Theoretical work based on first princi-ple calculations from Ref. [1]. The warmer color indicate higher electron D.O.S., the blue regions are bulk band-gap. The upper and lower irregular red region are the bulk conduc-tion and valence band, respectively. A linear dispersed gapless surface state can be seen inside the bulk band-gap. (b)The ARPES measurement of surface electronic band struc-ture from Ref. [2]. The warmer color denotes higher electron D.O.S., the black region is bulk band-gap. The upper and lower yellow regions are bulk conduction and valence band, respectively. The white dash line marks the position of Fermi level. A red colored, gapless band with nearly linear dispersion can be seen inside the bulk band gap.
)b* )c*
)d*
Figure 1.3: The crystal structure of Bi2Se3 (from Ref. [1]) (a)Crystal structure with three primitive lattice vectors denoted as t1, t2, and t3. A quintuple layer is indicated by the red square. (b)Top view along the z-direction. The lattice atoms in a quintuple layer have three different positions, denoted as A, B, and C. (c)Side view of the quintuple layer, which is about 1nm thick.
is known as a quintuple layer (indicated by the red square). The lattice atoms in a quin-tuple layer have three different positions, denoted as A, B, and C. The thickness of each quintuple layer is about 1nm, and the coupling force between quintuple layers is weak (van-der-Waals type).
One of the most important feature of the surface state is it’s spin texture. The electron spins of the surface states are predicted to be in-plane and perpendicular to the momen-tum (see Fig. 1.4a). The red arrows denote spin direction of each point on the intersection of Fermi surface and Dirac cone. Those spin textures have been verified by ARPES (Fig. 1.4b-d). Figure 1.4b is the top view of the Dirac cone intersection with Fermi sur-face, the yellow arrows are the expected spin orientation. The inner white area is the bottom of bulk conduction band, but a darker ring with radius∼0.1 ˚A−1is the intersection of Dirac cone with Fermi level. Figure 1.4(c)and (d) show the measured spin
polariza-tions along the kx direction (z direction is defined as out of plane). No clear signal can be seen in x and z components. But in the y component, a clear peak of opposite sign and equal value at kx ∼ ±0.1 ˚A−1 can be seen, which is in accord with the position of Dirac cone edge. This observation agrees well with the theoretical prediction [12]. We can understate this spin texture in a simple way. We know that, due to relativistic ef-fect, moving charge carriers in their rest frame will experience an effective magnetic field Bef f∝vF×E, where vF is Fermi velocity of the charge carriers, and E is the electric field exerted on the charge carriers in crystal’s rest frame. This effect is more pronounced in systems with strong spin-orbit coupling. When considering the carriers confined on the surface of a crystal, those carriers will only experience an effective electric field Esalong out-of-plane direction from symmetry argument. Therefore, the effective magnetic field will be pointing along in-plane direction and perpendicular to vF. Electrons moving on the surface will then tend to line up their spins with Bef f in order to minimize Zeeman energy, which is proportional to σ·Bef f. From this point of view, charge carriers moving in opposite direction will have their spin polarization inverted. Because the effective field is pointing along opposite direction on the opposite surface, spin polarization of charge carriers moving in same direction but on opposite surface will also be inverted.
One might ask why are there so few topological insulators. Kane and Mele developed an easy way to distinguish non-trivial topological insulators (such as Bi2Se3) from trivial ones (band insulator) using a special topological invariant, Z2 [14, 15, 8]. They found that for a material with Fermi level inside band-gap, Z2 can only be 0 (trivial insulator) or 1 (non-trivial topological insulator). Z2 can be easily determined by counting number of Dirac pairs inside the band-gap and then take modulo 2. For materials with Z2=1, the gapless surface states are protected from weak disorder and non-magnetic impurities by time reversal symmetry (TRS). For materials with Z2=0, the surface states can be easily destroyed by a small perturbation and become trivial insulators.
)b*
E
)c*k
yk
x)d* )e*
Figure 1.4: (a) Schematic drawing of Bi2Se3 surface state spin texture near Fermi energy (denoted by the blue hexagon). The red arrows denote spin direction of each point on the intersection of Fermi surface and Dirac cone. (b)ARPES result from Ref. [2]. The inner white area is the bottom of bulk conduction band. A darker ring with radius ∼0.1 ˚A−1 is the intersection of Dirac cone with Fermi surface. The yellow arrows denote the spin directions. (c)-(d) are the spin polarizations along kx (z direction being defined as out of plane) ( results from Ref. [13]). No clear signal can be seen in x and z component. But in the y component, a clear peak of opposite sign and equal value at kx ∼ ±0.1 ˚A−1 can be seen, which is in accord with the position of Dirac cone edge.