Although the surface states of topological insulators had been identified by ARPES mea-surements as mentioned in the previous section, verification from transport meamea-surements are relatively few. The main reason is that Fermi surface is usually not in the bulk band gap, as it should be in an ideal case. Due to crystal growth difficulties, Fermi surface of Bi2Se3is typically in the bulk conduction band. Since the surface states’ D.O.S. is much smaller than the bulk band, the signal contribution from surface states are easily over-whelmed by the bulk bands. Therefore, the identification of surface states becomes tricky.
However, several methods had been used successfully to identify surface states from trans-port measurement. One is by Aharonov-Bohm interference (AB effect) [3, 16, 17]. Based on quantum mechanics, two electrons moving in a path enclosing nonzero magnetic flux ϕ (see Fig. 1.5a) will acquire a relative phase. There are two possible interference paths.
One is from the interference between two split electron beams as shown in the left picture of Fig. 1.5. Enclosed magnetic flux will cause alternating destructive and constructive in-terference with respect to the increasing magnetic field. The period of inin-terference equals h/e, where h is Plank’s constant and e is the electron charge. The other one is the
inter-ference with it’s own time-reversed path. The conductance will oscillate with period of h/2e [18]. Since backscattering is suppressed in Dirac surface states, h/2e period is also
suppressed. As a result, the resistivity along the magnetic field direction should be oscil-lating with period h/e. This effect cannot be seen in normal metal wires due to the arbitrary in electron paths. However, the only conducting channel in topological insulators is the gapless surface state. Therefore, we expect to see the resistivity oscillation with respect to field strength at a constant period h/eA, where A is the sample’s cross-section area per-pendicular to field direction. This method was first demonstrated on Bi2Se3 nano-ribbon to reveal the Dirac surface states [3] (Fig 1.5b). The normalized magnetoresistance is plotted against magnetic field in the ribbon’s longitudinal direction at 2K. The resistivity
)b* )c*
Figure 1.5: (a)Schematic of the interference paths in AB effect experiment and their ex-pected interference periods. (b)Experimental result of AB interference in Bi2Se3 nano-ribbon from Ref. [3]. The resistivity is oscillating with a period of 0.62T, which corre-sponds to h/e period.
is oscillating with a period of 0.62T, which corresponds to h/e period. The drawback of probing surface states through AB effect is that it can only work in nanowire systems, and it requires large surface-to-volume ratio. According to Ref. [3], the cross-section needs to be smaller than 104 nm2 in order to see the effect. The geometrical limitations make this method less favorable.
Another way to identify surface states is through Shubnikov-de Haas (SdH) oscil-lations. Under magnetic field, electrons will undergo cyclotron motion, and their energy band will transform into discrete Landau levels. The separation between each level is ¯hωc, where ωcis electron’s cyclotron frequency. The Landau filling factor ν ∼ EF/¯hωc, where EF is Fermi energy. Since ωc is proportional to magnetic field B, ν will decrease with increasing field. Most transport properties in materials are dominated by the electronic D.O.S. near Fermi surface. Therefore, they are expected to show oscillatory behavior as Fermi surface passes through each Landau level. The oscillations in magneto-resistance due to this effect is called SdH oscillations. It can be shown that the amplitude of SdH oscillations are described by [19]
damping factors, respectively. TD is Dingle temperature, which is a measure of disorder level with definition of kBTD = ¯h/2πτ , and τ is the scattering time. C is a proportional constant, ρ0(B) is the resistance without considering SdH oscillation, fSdH is the fre-quency of SdH oscillation, and γ is Berry phase. Onsager further showed [20] that the oscillation period 1/fSdH is related to the Fermi surface by
1
fSdH ≡ △( 1
Bm) = 4π2
ϕ0Ae, (1.4)
where Bm are the positions of SdH oscillation minima, ϕ0 = h/e is the magnetic flux quanta, and Aeis the extremal cross-sectional area of the Fermi surface in the plane per-pendicular to applied magnetic field. Therefore, SdH oscillation can map out the geometry of Fermi surface. For example, if the Fermi surface is spherical, then the period of SdH oscillation should not change with magnetic field orientation since Ae remains the same.
But for a 2D Fermi surface, Ae is proportional to 1/ cos θ and will diverge at θ = 90◦ (Fig. 4.3), where θ is the angle between field and the 2D Fermi surface’s normal direc-tion. This angular dependence of SdH oscillation period can then provide information on the dimensionality of Fermi surface. This method was first used to identify the 2D sur-face states in Bi1−xSbx [21]. In addition to the periods corresponding to 3D bulk Fermi surface, a period with 2D nature was seen. A lot of effort had been focused on Bi2Se3
[22, 23, 24] that has a larger band-gap and a single Dirac cone. However, despite the high mobility (∼104 cm2/Vs) and low carrier concentration (∼1017cm−3) achieved, SdH os-cillations are still dominated by 3D bulk band in the pristine Bi2Se3. Although Ca doping can shift Fermi level into bulk band-gap [13, 25], SdH oscillations turn out to disappear also [26]. Only in SbxBi2−xSe3 with carrier concentration as low as 2.9×1016 cm−3and magnetic field up to 60T, SdH oscillations originating from 2D surface states was finally observed [27]. This technique was latter used on pristine, non-metallic Bi2Te3 crystals with very low carrier concentration (∼1015cm−3) [4]. In Fig. 1.6a, the resistivity deriva-tive dρxx/dH of a non-metallic Bi2Te3 was plotted against the inverse perpendicular field 1/H⊥at different angles. Where H⊥ ≡ H cos θ is the magnetic field component
perpen-)b*
)c*
)d*
)e*
Figure 1.6: Identification of surface states in non-metallic Bi2Te3, data from Ref. [4].
(a)The resistivity derivative dρxx/dH versus inverse perpendicular field. Where H⊥ ≡ H cos θ, is the magnetic field component perpendicular to the sample cleavage plane. The minima fall on the same vertical lines, indicating a 2D Fermi surface. (b)Field position of the resistivity minimum (which corresponds to the ν = 3 Landau level) plots against field angle (red dots), which follows the 1/ cos θ trend (black solid curve) up to 30T. (c)Hall conductivity versus magnetic field (red circles), and its best fit to Eq. 1.6 in solid black line. The non-linearity at low field cannot be explained by one band model. (d)Surface term (red circles) and bulk term (solid thin line) of Hall conductivity extracted from (c).
dicular to the sample cleavage plane. We can see that the minima fall on the same vertical lines, indicating a 2D Fermi surface. In Fig. 1.6b, field position of a particular resistivity minimum (which corresponds to ν=3) was plotted against the field angle θ in red dots, which follows the 1/ cos θ trend (black solid curve) up to 30T as expected for a 2D Fermi surface (Fig. 4.3). The recent studies of the angular dependence of fSdH in the topological insulator Bi2Te2Se also showed similar results [28, 29], making this method probably the most promising way of identifying the surface states by transport measurements.
Another evidence of a 2D surface state in non-metallic Bi2Te3comes from an anomaly of Hall conductivity observed at low field. From semiclassical expression, the Hall
con-ductivity σxy of electrons in a single band, closed orbit under uniform magnetic field is
σxy = neµ µB
[1 + (µB)2], (1.5)
where n is carrier concentration, e is electron charge, and µ is the mobility. At small field, σxycan be approximated by neµ2B. That is, if there are more than one parallel conducting bands each described by Eq. 1.5, the total Hall conductivity will be dominated by the band with large µ. This was first observed in non-metallic Bi2Te3crystals [4]. Figure 1.6c plots the Hall conductivity against magnetic field (red circles). A clear non-linearity can be seen at low field, which cannot be explained by the single band models, such as Eq. 1.5. Now, consider a two band model composed of one 3D band and one 2D band. The total Hall conductivity
σtotalxy = σbxy+ Gxy/t, (1.6)
where t is the sample’s thickness, σxyb and Gxy are Hall conductivity from the bulk band and sheet Hall conductance from the surface states, respectively. Both σbxyand Gxy can be described by Eq. 1.5 (note that for Gxy, n in Eq. 1.5 is the sheet carrier concentration). The fitted curve is plotted in Fig. 1.6c in solid black line, which fits the data points very well.
Surface term (red circles) and bulk term (solid thin line) of Hall conductivity extracted from (c) are plotted in Fig. 1.6d. The obtained surface carrier density and mobility are in good agreement with the results extracted from SdH oscillations, suggesting that the observed anomaly truly comes from the 2D surface states. Similar results were also found in recent studies on the topological insulator Bi2Te2Se [28, 30]. Since the mobility in the surface states is typically larger than bulk bands due to backscattering suppression, Hall conductivity at low field can provide another way to identify the surface states.