The weak anti localization effect due to topological surface states of a Bi1.46Sb0. 54Te1.7Se1.3 nano ake

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states of a Bi

_1.46

Sb

_0.54

Te

_1.7

Se

_1.3

nanoflake

Bachelor project Physics at the University of Amsterdam

Joran Angevaare

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i

Report of Bachelorproject Natuur- en Sterrenkunde (15 EC) carried out between

01-04-2014 and 27-06-2014

Author:

Joran Angevaare

Student number:

10219617

Supervisor and corrector:

Dr. Anne de Visser

Second supervisor:

Yu Pan

Second corrector:

Dr. Emmanouil Frantzeskakis

University:

Universiteit van Amsterdam

Faculty:

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Institute:

Van der Waals-Zeeman Instituut

Group:

Hard condensed matter group

Address:

Science Park 904, 1098 XH Amsterdam

Date of submission:

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The bulk insulating topological insulator Bi1.46Sb0.54Te1.7Se1.3 (BSTS) is investigated via

mag-netotransport. The weak anti localization (WAL) effect is observed measuring a 130 nm thick BSTS nanoflake. The change of conductance (∆G) due to an applied magnetic field is meas-ured at different orientations of the sample surface with respect to magnetic field. The ∆G data collapses onto a single curve when plotted as a function of the perpendicular component of the magnetic field (B⊥). This provides clear evidence of the 2D nature of ∆G. The

Hikami-Larkin-Nagaoka formula for WAL is used to fit the ∆G data resulting in α ∼ −1.14 for T = 2 K and α ∼ −0.93 for T = 40 K, close to α = −1 as predicted by theory for two topological surface channels (bottom and top surface). A modified 2D Dirac model for two surface- and two bulk channels is also used to fit ∆G. This results in clear WAL behavior of the surface channels with cos ΘS ∼ 0 and unitary behavior of the bulk channels with cos ΘB∼ 0.26 − 0.36.

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Introduction

Topological insulators are exotic materials. Topological insulators (TIs) have an insulating bulk while having topological protected surface states. These surface states are due to the discontinuity of topological class at the surface of a TI and are conducting. Since TIs have been predicted by Fu et al. [7] and Moore & Balents [19] in 2007 there has been a surge of research about these materials. Hsieh et al. [11] were the first to make a 3D topological insulator (TI).

The special properties of TIs make them an interesting field of research. It has been predicted that topological insulators with induced superconductivity might harbor Majorana zero modes [6] (Majorana particles are their own antiparticles). Furthermore topological insulators are potentially interesting for applications in spintronics, thermo electrics and quantum computing.

Between theoretically predicted TIs and crystals grown at laboratories there is a difference. Even though these TIs grown at laboratories are expected to have an insulating bulk this is not observed. In these crystals the bulk channels contribute significantly to the conductivity.

After Hsieh et al. [11] reporting Bi1−xSbx to be a TI, Zhang et al. [32] predicted Bi2Te3,

Bi2Se3 and Sb2Te3 to be TIs. A singe Dirac-cone surface state was confirmed for Bi2Te3 [5] [12]

and Bi2Se3 [30]. However existence of topological surface states were left unconfirmed. Ren et al.

[24] further improved the bulk-insulating properties of Bi, Sb, Te and Se based materials resulting in Bi2−xSbxTe3−ySey (BSTS) for x = 0.5 and y = 1.3. Further optimization by [20] for the

stoichiometry with the highest bulk resistivity resulted in the composition Bi1.46Sb0.54Te1.7Se1.3.

In order to confirm the topological nature of Bi1.46Sb0.54Te1.7Se1.3 this bachelor thesis will focus

on measuring the topological surface states by the weak anti localization effect.

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Theory

Topological insulators are theoretically and experimentally challenging materials. To understand these materials some key concepts that are essential to understanding this report will be introduced in this section.

2.1 Topology

Figure 2.1: Examples of shapes with different to-pological classes. By tearing the shape as in fig-ure (a) one obtains figfig-ure (b) and gluing the ends together one can obtain figure (c). Figure taken from [18].

Topology is a term known from mathematics. The term is also used for TIs.

A topological class describes a set of objects that can be changed onto each other via a con-tinuous deformation. One example commonly used are the orange, the donut and the tea-cup. In a teacup there is one hole in the shape and by continuous deformation (without tear-ing, gluing or making holes) one can deform this teacup into a donut. However starting from an orange it is impossible to make a donut like shape by continuous deformation, since one would need to make a hole in the orange in or-der to obtain the hole which is present in the donut. Making holes is a discontinuous deform-ation and hence a donut and a teacup have a different topological class compared to an or-ange. A teacup and a donut are of the same

topological class. From figure 2.1 we can see the left shape (a) cannot be transformed into the right shape (c).

Topology for electronic materials however is due to the way the wave function of electrons is “folded” i.e. the topology of the Hilbert space. Ordinary insulating materials have a wave function with a topological class equivalent to the shape as in figure 2.1c. However for a topological insulator (TI) the topological class is equivalent to figure 2.1a. So if a TI is placed in vacuum (an insulator) the vacuum and the TI are of a different topological class. To connect these two topological classes a discontinuous deformation (tearing and gluing) has to be made on the surface as shown in figure 2.1b. These surface states have different properties than the bulk. How to calculate this difference in topological class is shown in section 2.6, but before the derivation of topological invariants can be shown, some important phenomena need to be explained.

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CHAPTER 2. THEORY 3

2.2 Spin-orbit coupling

Spin-orbit coupling (SOC) is an interaction between the momentum and the spin of an electron. If we picture in the semi-classical Bohr model an electron orbiting a nucleus this will have an effect on the energy. This effect is given by ∆H = −~µ · ~B. Where ∆H is the correction to the ordinary Hamiltonian, ~µ the magnetic moment and ~B the magnetic field. Looking from the rest frame of the electron there is a magnetic field induced by the orbiting nucleus. We can express this magnetic field using ordinary electrodynamics resulting in ~B = −_m~r×~p

ec2|

~ E ~

r| where ~r × ~p = ~L.

Furthermore we can express ~µ = −gSµB ~ S

~, with ~r the vector between the nucleus and the electron,

~

p the momentum of the electron, methe electron mass, c the speed of light, ~E the electric field, ~L

the orbital momentum, gS the electron spin g-factor and µB the Bohr magneton. This gives the

correction to the Hamiltonian:

∆H = gsµB ~mec2 ~ E ~ r ~ L · ~S

After adding a relativistic correction of ∆H = − µB

~mec2 ~ E ~ r ~ L · ~S and substituting ~ E ~ r = 1 |~r| ∂ ~V (~r) ∂|~r| , we obtain: ∆H = (gs− 1) µB ~mec2 1 |~r| ∂ ~V (~r) ∂|~r| ! ~ L · ~S

Spin-orbit coupling is very important and will be one of the conditions for topological insulators (TIs) that change the topological invariant of the bulk (section 2.6). Furthermore weak anti localization (WAL) requires strong spin-orbit coupling as will be explained in section 2.7.

2.3 Berry phase

The derivation of the Berry phase is shown in appendix A (page 25). In context of topological insu-lators the closed loop integral is along the edge of the 1st _{Brillouin zone. Due to the discontinuous}

deformation the topological surface states are expected to acquire a π Berry phase.

2.4 Time reversal symmetry

Time reversal symmetry (TRS) tells one whether a system is invariant under the time reversal operator Θ : t → − T. An example of a time reversal symmetric quantity is acceleration (~a =

∂2~x

∂t2). Examples of a broken symmetry are the velocity (~v) but also the magnetic field ( ~B), since

F = q(~v × ~B), then ΘF = Θ(m~a) = m~a = F = Θ(q(~v × ~B) = q(−~v × Θ ~B) and hence Θ ~B = − ~B.

2.4.1 Kramers pair

One important property of materials conserving time reversal symmetry (TRS) is the formation of a Kramers pair at a time reversal invariant momentum (TRIM). A derivation of Kramers pair is shown here. Starting from the Hamiltonian of a periodic system [2]

H|ψnki = Enk|ψnki

with |unki the cell periodic eigenstate of the Bloch Hamiltonian. Because of Bloch’s theorem we

can write:

|ψnki = eik·r|unki

with the Bloch Hamiltonian:

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Figure 2.2: Example of Kramers pairs with dif-ferent spin up (red) and down (blue) bands. Note the two TRIMs, ~k = 0, and ~k = π. Figure taken from [2].

The cell periodic eigenstate satisfies the re-duced Schr¨odinger equation. When H con-serves TRS ([H, Θ] = 0) then H(k) satisfies:

H(−k) = ΘH(k)Θ−1

The equation shown above implies that the en-ergy bands of a time reversal symmetric system comes in pairs since +k and −k have the same energy. These pairs are called Kramers pairs. These pairs are degenerate at the edges and the center of a 1D Brillouin zone (k = ±π or k= 0). At these points +k and −k are equi-valent since the Brillouin zone is periodic.

A momentum where +k = −k is also called a time reversal invariant momentum (TRIM). See figure 2.2. Kramers pairs are degenerate at TRIMs (also for 2D and 3D).

2.5 Parity

Parity is another important symmetry. The operator Π has the following effect on the spatial coordinates: Π :   x y x  →   −x −y −z   (2.1)

Since Π2= 1 the eigenvalue of Π i.e. ξ equals ±1. Some quantities with even parity (ξ = 1) are time, mass but also spin and angular momentum. Position, velocity and linear momentum on the other hand have an odd parity (ξ = −1).

2.6 _Z

2

invariant

A calculation of the topological class of a 3D material for a relatively simple case will be shown in this section. The result of the derivation of the topological invariant as done by Ando [2] is presented below. When inversion symmetry is assumed, and one also assumes 2N occupied bands forming N Kramer pairs, the Z2invariant (which indicates the topological class), ν, can be

calculated using (−1)ν = 8 Y i=1 N Y n=1 ξ2n(Λi) (2.2)

Where ξ2n(Λi) is the parity eigenvalue of the nth Kramer pair at a time reversal invariant

mo-mentum (TRIM), Λi. Since there are eight TRIMs in the 1st (3D) Brillouin zone there are eight

Λi’s in this formula . As can be seen from equation (2.2) ν can be either 1 or 0.

For 3D topological insulators where inversion symmetry is not assumed there are actually four Z2invariants, (ν0; ν1ν2ν3). These invariants can be calculated using a more elaborate formula than

in equation 2.2. If ν0= 1 the topological insulator is called a strong 3D topological insulator while

it is called a weak 3D topological insulator for νi= 1 for i = 1, 2, 3. Bi2−xSbxTe3−ySey (BSTS) is

shown to be a (1; 000) strong 3D topological insulator [2].

A change of parity can be induced by strong spin-orbit coupling (SOC) when due to SOC the conduction band and valence band are inverted. Band inversion takes place when the conduction

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CHAPTER 2. THEORY 5

band obtains a lower energy compared to the valence band. The energy correction ∆H as in section 2.2 can cause band inversion.

A band inversion can change the parity at a certain TRIM. This leads to a sign change in equation (2.2), which changes the Z2invariant ν. This is shown to be the case for Sb2Te3, Bi2Se3

and Bi2Te3but not for Sb2Se3[32]. It is important that an odd number parity sign changing band

inversions take place, otherwise there is no sign change in equation 2.2 and ν will not change.

2.7 Theory of weak (anti) localization

In this section one of the most important concept will be introduced i.e. weak anti localization (WAL). WAL is so important since the main measurements presented here involve the WAL effect. For didactic reasons weak localization (WL) will be explained first, followed by weak anti localization (WAL).

The weak localization (WL) effect is due to constructive interference between two wave functions with the same probability amplitudes. Assuming a path as shown in figure 2.3 i.e. a backscattering path due to adiabatic random scattering, there is a similar trajectory possible which is followed in exactly the opposite order and hence is the time reversal partner of the first path. The probability of a particle starting at ~r1 that encloses this loop and ends at ~r1= ~r2 can

be calculated using the probability amplitude A+ _{(or A}− _{for its time reversal partner).}

Figure 2.3: For this trajectory the starting point is equal to the endpoint ( ~r1= ~r2). The arrows indicate

two ways to complete this trajectory. From quantum mechanics we know the

scattering probability of a particle start-ing at ~r1 and ending at ~r2 is given by

P ( ~r1, ~r2, t) = |PiAi| 2

. If there are two trajectories as in figure 2.3, with probabil-ity amplitudes A+and A−which are time reversal partners, we need to sum over these two probability amplitudes.

Something special happens when time reversal symmetry (TRS) is preserved. If this is the case the time reversal partners A+ _{and A}− _{are equal, A}+ _{= A}−_{= A. So}

classically one could calculate the probab-ility of this backscattering and one would find:

P = P++P− = A+2+A−2= A2+A2= 2A2

But quantum mechanically we would first have to sum and then have to square so:

P = X i Ai 2 = A++ A−2 = 4A2

This is twice the result we obtained classically due to the constructive interference of the probability amplitude of two time reversal adiabatic backscattered paths. This is one of the few measurable consequences of quantum mechanics. Due to this larger chance of backscattering electrons tend to localize. This localization decreases the conductivity and increase the resistivity.

The weak anti localization (WAL) effect is the opposite effect of WL. If a material has strong spin-orbit coupling (SOC) spin and momentum are locked. In figure 2.4, two time reversal partners are shown and if they backscatter adiabatically they will rotate backwards. However as is shown in figure 2.4 one trajectory A+ _{will be rotated by π while its time reversal partner, A}−

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Figure 2.4: For two time reversal partners with strong SOC present backscattering causes a 2π phase shift between the different paths. Figure taken from [21].

Since electrons are fermions and fermions are known to have an antisymmetric wave func-tion:

ψ(θ) = −ψ(θ + 2π)

this implies that the probability amplitude of the time reversal partner of A+ _{differs a minus}

sign, A− = −A+_. _{If one now calculates}

the probability for these time reversal backs-cattered paths with a relative phase difference of 2π one finds: P = X i Ai 2 = A++ A−2 = A+− A+2 = 0 (2.3) This means that since for each adiabatic cattering path there is a time reversal backs-cattered path, and as long as TRS is pre-served, these two paths will destructively in-terfere. This decrease of backscattering

prob-ability will manifest itself in an increase of the conductivity and hence a decrease in the resistivity. Measuring weak (anti) localization can be done by magnetotransport. The enhanced con-ductivity or decreased concon-ductivity for WAL and WL respectively can be found by applying a magnetic field. If one applies a magnetic field the phase between two time reversal partners will change. Hence for the maximal destructive interference of backscattering paths (WAL) there will be less destructive interference and thus more backscattering when a magnetic field is applied. Therefore one expects when WAL is present and one applies a magnetic field the conductivity will decrease with increasing magnetic field.

For WL we have the opposite effect and hence the conductivity will increase if a magnetic field is applied, since there is no maximal constructive interference anymore between the time reversal partners. Furthermore, applying a magnetic field breaks time reversal symmetry (see section 2.4). The minimum magnitude of the magnetic field to reasonably reduce the WAL is Bc ≡ _e`~2

φ

[3] where e is the elementary charge and `φis the phase coherence length i.e. the average length over

which an electron retains its phase.

2.8 Hikami-Larkin-Nagaoka (HLN) formula

For the WAL discussed in previous section there is a 2D expression found by Hikami et al. [10] which describes the change in conductance as a function of an applied magnetic field. The Hikami-Larkin-Nagaoka formula is shown in equation (2.4) where Ψ is the digamma function, α is a dimensionless parameter and `φ is the phase coherence length i.e. the average length over which

a wave function retains its phase. In equation (2.4) e is the elementary charge and ~ is Planck’s constant divided by 2π. The parameter α tells one about the strength of the WAL.

∆σxx= − αe2 2π2 ~ ln ~ 4e`2 φB ! − Ψ 1 2 + ~ 4e`2 φB !! (2.4)

2.9 Modified 2D Dirac model

The HLN formula (equation 2.4) describes the suppression of the 2D WAL effect as a function of the applied magnetic field. Though we interpret a fit of α = −1 as two 2D WAL channels with

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CHAPTER 2. THEORY 7

each contributing α = −1₂ a more careful analysis can be made.

Figure 2.5: This schematic figure of a band struc-ture shows the energy gap between the valence-and conduction bvalence-ands (M ), the hybridization gap (∆) and the Fermi energy (EF). Figure taken

from [16]. Lu & Shen [16] proposed a model where they

used the 2D modified Dirac model as in equa-tions (2.5) and (2.6) for the topological surface bands and 2D bulk band, respectively. In equa-tions (2.5) and (2.6) τz= ±1 is the block index,

k2 _{= k}2

x+ k2y, σi are the Pauli spin matrices,

furthermore hkzi = 0 and hk2zi = πt

2

were as-sumed with T the thickness of the sample. In equations (2.5) and (2.6) A, eA, B, eB, D and

e

D are model parameters originating from the modified Dirac model using k · p theory. As shown in figure 2.5, M is the band gap between the valence and conduction band and ∆ is the hybridization gap opened up in the topological surface state. ∆ and eB are assumed to be very small for samples bigger than tens of nanomet-ers. HB = Dk2+τz M 2 − Bk 2 σz+A(σxkx+σyky) (2.5) HS = eDk2+τz ∆ 2 − eBk 2 σz+ eA(σxkx+σyky) (2.6) From the above equations Lu & Shen [16]

derived a new HLN formula as in equation (2.7). The authors explain that the derivation is similar to the derivation of that for topological surface states with a magnetically doped gap which is done by Lu et al. [17]. A new HLN formula is introduced in equation (2.7) which describes the effect of the magnetic field on the conductivity per conducting channel. For two channels one should multiply equation (2.7) with a factor of 2.

∆σ(B) = 1 X i=0 αie2 2π~2 Ψ ~ 4eB`2 φi +1 2 ! − ln ~ 4eB`2 φi !! (2.7)

In this formula there are a few differences with eq. (2.4). There are several variables in this equation which will be explained in the following paragraph. Equation (2.8) shows how to calculate `φi and αi. Even though there are many parameters the key concept is that there are two α’s, α0

and α1. Between these α’s there is a competition. These α’s can take different values, 0 ≤ α0≤ 1₂

and −1₂ ≤ α1≤ 0.

The most important variable is cos Θ (equation 2.9). If cos Θ→ 0 i.e. the WAL regime α1→ −1₂

and α0→ 0. But contrary for cos Θ→ 1 then α1→ 0 and α0→ 1₂ and we are dealing with the WL

regime. Furthermore figure 2.6 shows dependency of the W(A)L behavior due to the parameter cos Θ.

The parameters of equation (2.7) , equation (2.7) is a complicated formula with several parameters. With this formula it is possible to describe the WAL and WL with one formula [16]. The derivation of this expression is made in Lu et al. [17] and is beyond the scope of this report.

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We give the expressions of αi and `i in equation (2.8). `−2_φi ≡ 1 `2 φ + 1 `2 i , α0= η2v(1 + 2ηH) 2(1 + 1/g1) , α1= − ηv2(1 + 2ηH) 2(1/g0+ 1/g2) , ηH = − 1 2 1 − 1 ηv −` 2 `2 x , ηv=  1 − a2b2`_`22 e − `2 `2 z a4_{+ b}4   −1 , `0= _g 0 2`2_{(1 + 1/g} 1) −12 , `1= _g 1 2`2_{(1 + 1/g} 0+ 1/g2) −12 , g0= 2 a4_{+ b}4 a4 1`2 1 `2 e + 1 `2 z − 1 ! , g1= 2   1`2 1 `2 e − 1 `2 z 2a2_b2 a4_+b4 − 2 `2 x − 1  , g2= 2 a4_{+ b}4 b4 1`2 1 `2 e + 1 `2 z − 1 ! , a ≡ cos Θ 2 , b ≡ sin Θ 2 , ` = 1 `2 e + 1 `2 m −12 , `m= 1 `2 x + 1 `2 y + 1 `2 z −12 (2.8) cos(Θ) ≡ M/2 − Bk 2 f EF− Dk2_F (2.9)

Figure 2.6: Crossover from the WAL to WL as a function of cos Θ. For cos Θ = 0, 0.09 a clear neg-ative magneto conductivity indicates WAL. For cos Θ = 0.22, 0.334, 0.55 there is weak magneto conductivity, this is called the unitary or sym-plectic regime. For cos Θ = 0.82, 0.999 a clear positive magnetoconductance indicates WL. After filling in all the parameters the ∆σ still

depends on the parameters `e, `x, `y, `z, `φ,

B and Θ. The most important variable is Θ. From now on we will discuss cos(Θ) rather than Θ since cos Θ gives a quick feeling for whether we are talking about WAL or WL. The para-meter `e, the mean free path due to electric

scattering is assumed to be small while `x,y,z

are assumed to be large i.e. electric scatter-ing is assumed to be strong and magnetic scat-tering (in directions x, y and z) is assumed to be weak. Typical values assumed by Lu & Shen [16] in their paper are shown in table 2.1. One can expect a crossover from WAL to WL if cos Θ goes from 0 → 1. As discussed in section 2.7 that for WAL a negative magneto conduct-ivity is expected while for WL a positive con-ductivity is expected. A plot of this is shown in figure 2.6. For this plot equation (2.7) was used with the parameters as defined in table 2.1. One can see a clear transition from WAL to WL due to increase of cos Θ. The unitary or symplectic regime is the region of cos Θ∼ 0.33.

Symbol Value Explanation Other properties/notes `e 10nm Electric scattering length Assumed to be small i.e. `e `B

`B Magnetic length `B≡

q

~ 4eB

`φ Phase coherence length

`m 17 µm Magnetic scattering length _`12 m = _`12 x +_`12 y +_`12 z

`x, `y, `z 10 µm Components of `malong x, y and z Assumed isotropy i.e. `x= `y = `z

cos Θ Parameter as defined in eq. (2.9) 0 ≤ cos(Θ) < 1

Table 2.1: Parameters of equation (2.7). The second column, value, gives the values as used in section 4.4.

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Chapter 3

Measurements/Method

3.1 The sample

Figure 3.1: The resistivity of Bi2−xSbxTe3−ySey

as a function of temperature for different compos-itions. This figure shows the highest resistivity for Bi1.46Sb0.54Te1.7Se1.3. Panel (a) shows the x

variation while panel (b) shows the y variation. Figure taken from [20].

One of the setbacks of topological insulators is that even though their bulk is predicted to be insulating, topological insulator crystals grown in laboratories are semiconductors at best. Therefore the conductivity of these ma-terials is dominated by the bulk states instead of the surface states. In order to investigate to-pological properties one would like an isolating bulk and that the conductivity is dominated by the topological surface states.

Following Ren et al. [24], Pan et al. [20] investigated how the bulk resistivity of Bi2−xSbxTe3−ySey can be further optimized.

As is shown in figure 3.1 Bi1.46Sb0.54Te1.7Se1.3

has the highest resistivity (ρ).

The highest ρ provides the motivation for using Bi1.46Sb0.54Te1.7Se1.3as the sample

com-position used in this research. Pan et al. [20] calculated for a slightly different composition i.e. Bi1.46Sb0.54Te1.8Se1.2 for a 130 µm thick

sample a surface contribution of 27% to the total conductivity, at T = 8 K. For a 1 µm thick sample this would already be 97%. For this research the sample is made even thin-ner to obtain surface dominated conductivity: ∼ 130 nm thick would correspond to a surface contribution of 99.7% to the total conductivity. The single crystal (Bi1.46Sb0.54Te1.7Se1.3)

was grown at the University of Amsterdami _by

Dr. Y. Huang. At the University of Twenteii

the crystal was prepared into a Hall bar by M. Snelder. The BSTS nanoflake was placed on SiO2 and the residual BSTS was etched away

using E-beam lithography to obtain the Hall

i_{The van der Waals-Zeeman Institute of the University of Amsterdam (UvA).} ii_{The Faculty of Science and Technology at the University of Twente.}

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(a) Schematic figure of the Hall bar configura-tion. The direction of the magnetic field (B) is indicated. The axis of rotation is indicated by the black arrow, θ = 0◦ corresponds to the orientation with B perpendicular to the sample surface. The resistivity along the current direc-tion, ρxx, is obtained through a 4-point

meas-urement with the current applied as indicated. The Hall resistance due to the deflection of charge carriers by the Lorentz force is meas-ured over the contacts indicated by ρxy. L

and W represent the channel length and width respectively: L = 6.75 ± 0.25 µm and W = 2.00 ± 0.02 µm. The thickness of the sample is t ≈ 130 nm.

(b) Image of the Hall bar made with an optical microscope. The white region in the middle is the Bi1.46Sb0.54Te1.7Se1.3 nanoflake. The blue

background is SiO2, while the gray areas are

the etched away BSTS. The electrodes are also shown.

Figure 3.2: The sample shown schematically and an image made obtained with an optical microscope.

bar configuration as schematically shown in figure 3.2a. An image obtained with an optical mi-croscope is shown in figure 3.2b. Gold is sputtered onto the sample to make the electrodes. The thickness of the sample is ∼ 130 nm (ranging between 127 and 135 nm).

As shown in figures 3.2b and 3.2a there is a substantial error in the channel length L = 6.75 ± 0.25 µm. This is due to the thickness of the voltage contacts of the BSTS.

3.2 The DynaCool Physical Property Measurement System

(PPMS)

To measure the Bi1.46Sb0.54Te1.7Se1.3 sample we used the DynaCool Physical Property

Measure-ment System (PPMS) made by Quantum Design. This system allows one to easily obtain results from magnetotransport measurements compared to other devices available at the University of Amsterdam. For example, there is no need to manually fill cryogens like liquid helium or nitrogen, instead a pulse-tube refrigerator is used. The PPMS can measure samples with temperatures as low as T = 1.9 K and offers a stable environment for measuring samples for all temperatures between 400 K and 1.9 K. The superconducting magnet can produce magnetic fields up to 9 T.

In figure 3.3 we show a schematic picture of the PPMS. The important components shown are the cooling mechanism, the sample chamber and the superconducting magnet (cooled by the 4 K plate).

An option which is used for this research is the horizontal rotator. This offers the possibility to rotate a sample in a magnetic field. Since the magnetic field is in a fixed direction, one can examine phenomena like WAL as a function of the direction of a magnetic field with respect to the sample surface.

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CHAPTER 3. MEASUREMENTS/METHOD 11

3.3 Magnetoresistance

It has been shown by various articles that Bi2Te3 [26] and Bi2Se3 [27], [25], [23], [31] have a

π Berry phase. If Bi1.46Sb0.54Te1.7Se1.3 is a topological insulator weak anti localization will be

present. The WAL decreases the resistivity due to the anti localization of the electrons. Weak anti localization of electrons can be suppressed by applying a magnetic field because of the change of the phase shift of two time reversal paths as explained in section 2.7. By applying a magnetic field the weak anti localization will decrease and this leads to an increase of resistivity.

Figure 3.3: Schematic picture of the DynaCool Physical Property Measurement System (PPMS). Different com-ponents are indicated. Image taken from [22].

We can verify this with the Dyn-aCool Physical Property Measure-ment System (PPMS) (see section 3.2) by measuring the resistance. The resistance measured has a Hall component (Rxy) and a component

measured along the contact direction (Rxx). The Rxx component is

sym-metric, while the Rxyis

antisymmet-ric as a function of the magnetic field polarity. This (anti)symmetry im-plies R+

xx= R−xx and R+xy = − R−xy

where R+_{and R}− _{are the resistance}

for respectively the positive and neg-ative magnetic field. An antisym-metric Rxy is due to opposite

direc-tion of the induced Lorenz force on the charge carriers for an opposite magnetic field direction. Exploiting this (anti)symmetry we can separate these contributions.

The value displayed on the PPMS console will slightly differ from the value at the sample position since we sweep the magnetic field. This causes an induction current in the sample and an induction field opposite to the applied field. To account for this we first average the value of the resist-ance for an in- and decreasing mag-netic field (sweep up and sweep down

of the magnetic field). So R± = 1₂R±_↓ + R±_↑ with R↑ and R↓ sweep up and sweep down

re-spectively. One can now calculate:

Rxx= R+_{+ R}− 2 Rxy= R+_{− R}− 2

To calculate the conductance (G) using the resistance one should use the tensor relation, equation (3.1). In equation (3.1) ρ is the resistivity and σ the conductivity. Since ρxx ρxyiii the

formula σxx=_ρ1 xx is a good approximation. σxx σxy σyx σxx = 1 ρ2 xx+ ρ2xy ρxx −ρxy ρxy ρxx (3.1) iii_{At T = 2 K and |B| < 2 T, 11 > R} xx> 9.4 kΩ and 11 > Rxy> −1.5 × 102Ω.

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3.4 Measuring weak anti localization (WAL)

From Rxx we can acquire the conductance G via Rxx = _G1_xx. Instead of Gxx, G is often used

to represent the conductance along the current direction. As discussed in section 2.7 the WAL will decrease if a magnetic field is applied. A decrease of the WAL effect can be measured as an increasing resistance. An increasing resistance would imply a decreasing conductance. So if we start with the conductance at zero field G(0) and one applies a magnetic field the conductance changes with ∆G.

∆G(B) = G(B) − G(0) (3.2)

In equation (3.2) ∆G is the change in conductance due to an applied magnetic field. As is explained in appendix B on page 27 there are several ways to evaluate ∆G in the following sections we will use:

∆G(B) = G(B) − G90◦(B)

with G90◦(B) the conductance of the Hall bar when the sample surface is parallel to the magnetic

field.

3.4.1 Hikami-Larkin-Nagaoka (HLN) formula

If the ∆G for our BSTS Hall bar is due to 2D WAL the Hikami-Larkin-Nagaoka (HLN) formula [10] would describe the data. The HLN formula is given in equation (2.4) with ∆σ = _WL∆G where L is the distance between the voltage contacts (6.75 ± 0.25 µm) and W is the width of the sample (2.00 ± 0.02 µm). In equation (2.4), Ψ is the digamma function, α is a fit parameter and `φ is the phase coherence length i.e. the average length over which a wave function retains its

phase. The parameter α measures the strength of the WAL. For each topological surface channel a contribution of −1

2 to α is expected. Since our sample is thin two topological surface channels

(the top and bottom surface) contribute and hence an α of −1 [14]. Even though σ and hence ∆σ and ∆G have units of 1

Ω from now on these quantities will be

expressed in terms of the unit of conductance he2 h

i

with e the elementary charge and h Planck’s constant.

3.5 Modified 2D Dirac model.

As introduced in section 2.9 a 2D modified Dirac model was proposed by Lu & Shen [16]. The HLN formula changed to the expression given by (2.7) with the parameters defined in equation (2.8).

In section 4.4 we analyze the data with the modified 2D Dirac model. The modified HLN formula depends on the variables cos Θ, `φand B. Furthermore like in [16] we make the assumption

that two bulk channels and two surface channels are contributing. A full analysis would require to determine the band structure near the Fermi energy by angle-resolved photo emission spectroscopy (ARPES).

3.5.1 Four channel model

A four channel model is made in order to describe the data. For this model we assume two bulk and two surface channels. The modified HLN formula is used (equation (2.7)) and we assume similar a `φ for bulk and surface channels. The difference of bulk- and surface channels lies in

their dependence on the variable cos Θ.

We redefine cos Θ in equations (3.3) and (3.4) to be different for surface and bulk channels. In these equations M is the energy gap between the valence and conduction band and ∆ the energy gap opened at the Dirac point for the surface bands due to the hybridization of the bottom and top surface. These quantities are also shown in figure 2.5. EF and eEF are the Fermi energy

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CHAPTER 3. MEASUREMENTS/METHOD 13

measured from the Dirac point for the bulk-bands and surface-bands respectively. Furthermore B, eB, D and eD are model parameters of the Hamiltonians in equations (2.5) and (2.6).

cos ΘS ≡ ∆/2 − eBk2_f e EF− eDk2F (3.3) cos ΘB ≡ M/2 − Bk2 f EF− DkF2 (3.4) As discussed in section 2.9, since our sample is ∼ 130 nm thick i.e. greater than tens of nanometers, ∆ and eB are expected to be very small [15] [33]. Therefore cos ΘS is close to 0. On the other hand

cos ΘB for Bi2Se3 and Bi2Te3 is expected to range between 1₂ and 1 according to Lu & Shen [16].

Fitting the four channel model was done using Mathematica. A few more assumptions were made about Bi1.46Sb0.54Te1.7Se1.3. Firstly we assume one value of `φ for both surface and bulk

channels. Secondly we assume identical behavior for the two bulk channels. Also the surface channels are assumed to have identical behavior. In this model the change of conductance due to a magnetic field is given by:

∆σtot(B, ΘS, ΘB, `φ, `e, `x, `y, `z) = ∆σ_S(1)(B, ΘS, ` (1) φ ) + ∆σ (2) S (B, ΘS, ` (2) φ ) + ∆σ (3) B (B, ΘB, ` (3) φ ) + ∆σ (4) B (B, ΘB, ` (4) φ ) = 2∆σS(B, ΘS, `φ) + 2∆σB(B, ΘB, `φ) (3.5)

In equation (3.5) the superscripts 1,2,3 and 4 indicate the 1st_{, 2}nd_{, 3}rd_{and 4}th_{conducting channel.}

The subscripts B and S indicate the bulk- and surface channels. For each ∆σ equation (2.7) is substituted.

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Results

4.1 The sample at low temperatures

Figure 4.1: The temperature dependence of ρxx

revealing the an extra conduction channel arise at low temperature (T < 50 K).

In this section a quick overview concerning the analysis leading to the results will be presented.

4.1.1 Cooling the sample

As shown in figure 4.1 if the sample is cooled down, it obeys conventional semiconductor be-havior down to T ≈ 50 K. At ∼ 50 K the res-istance saturates. This is an indication of the emergence of an other conduction channel [14]. This cutoff of the resistivity indicates surface dominated transport below ∼ 50 K.

4.1.2 Measuring at 2 K

Figure 4.2: Rxx measured for different

orient-ations of the magnetic field with respect to the sample surface.

Once the sample has been cooled one can cal-culate the resistance along the current direc-tion Rxx(B) by sweeping the magnetic field

B : 0 → 2 → −2 → 0 T and symmetrizing as explained in section 3.3. Rxx was measured

for different orientations of the sample with re-spect to the magnetic field. The results are shown in figure 4.2. As shown in the figure there is a small term R90◦

xx. This term cannot

be due to WAL since there is no component of magnetic field perpendicular to the sample surface. How we can correct for this term is discussed in appendix B on page 27.

The change in conductance can now be calculated using ∆G = G(B) − G90◦(B) =

1 R(B)−

1

R90◦(B) resulting in figure 4.3. A

negat-ive change of conductance is observed as expec-ted (see section 2.7). One can see that for dif-ferent orientations of the sample with respect to the magnetic field ∆G is different.

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CHAPTER 4. RESULTS 15

Figure 4.3: The change of conductance (∆G) as a result of the applied magnetic field plotted as a function of the magnetic field for different orientations.

If ∆G would be due to the surface states this is a 2D effect and hence ∆G would de-pend on the component of the magnetic field perpendicular to the sample surface. Project-ing the magnetic field and then plottProject-ing the change of conductance ∆G(B⊥) results in

fig-ure 4.4. Since ∆G collapses onto one single curve when plotted as a function of B⊥ it is

clear that this data collapse is due to the sur-face states and we are dealing with 2D WAL.

4.2 Fitting the change of

the WAL effect

4.2.1 Fitting for fields up to 0.5

T

Figure 4.4: ∆G as a function of B⊥ for different

orientations of the sample. For the data collapse of ∆G in figure 4.4 one

can fit the Hikami-Larkin-Nagaoka (HLN) for-mula (2.4). One should pay attention to fit up to high enough fields i.e. B Bc = ~/e`2φ.

Fitted values of the phase coherence length (`φ)

for 2 K are 115±15 nm for different field ranges corresponding to Bc ∼ 0.05 ± 0.02 T, for T =

40 K one finds Bc ∼ 0.24 T for `φ∼ 52 nm.

Fitting should be done for B ≥ 0.25 T. How-ever increasing the fit range too much affects the quality of the fit as shown in figure 4.6.

Though it might be difficult to see on the scale of figure 4.6 the fit quality decreases for B > 0.5 T. At higher temperatures the same observation is made. Hence the choice is made to fit up to field strengths of 0.5 T.

4.2.2 WAL at different

temperat-ures

Figure 4.5: Fit parameter α obtained by fitting the HLN formula to ∆G for measurements at dif-ferent temperatures.

The weak anti localization effect was studied at different temperatures. The fit parameters α and `φ are expected to react differently to

a change of temperature. The strength, α, of the WAL effect is expected not to change as a function of temperature. On the other hand the phase coherence length (`φ) is expected to

change since the probability of inelastic scat-tering due to electron-electron interaction in-creases if the temperature is increased. Fur-thermore `φ is expected to obey a power law

(`φ∝ T−x), see section 4.2.3.

Fits at different temperatures were made, the results are shown in figures 4.7a and 4.7b. One can see that the WAL indeed mainly

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Figure 4.6: Fitting the HLN formula for different field ranges with fit parameters as indicated. Figures a, b, c and d present the ∆G fitted up to 0.25, 0.5, 1 and 2 T respectively. This figure shows that for a fit range of the magnetic field up to 0.5 T one obtains the best fits.

decreases with increasing temperature due to shortening of the phase coherence length.

Figure 4.8: The phase coherence length (`φ)

ob-tained using the HLN formula to fit ∆G. From figures 4.7a and 4.7b the fit parameters α

and `φcan be extracted and these are displayed

in figures 4.5 and 4.8. Note that the large error bar on α is mainly due to the substantial error in L as defined in figure 3.2a. The errors in L and α are closely related since ∆G = W_L∆σ. One can see that the obtained α is close to the theoretical value predicted for two topological surface states i.e. α = −1.

4.2.3 Temperature dependence of

the phase coherence length

From the temperature dependence of the phase

coherence length (`φ) we could infer the dimensionality of the system. By increasing the

temper-ature one expects more inelastic electron-electron interaction and hence shortening of `φ [4], [1].

The phase coherence length is expected to obey the relation: `φ ∝ T−

d

4 (4.1)

Here d is the dimensionality of the system. This formula is only valid for 2D and 3D systems hence d = 2 or d = 3.

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(a) Fitting the HLN formula for different tem-peratures with fit parameters as indicated. Fig-ures a, b and c show ∆G at T = 2, 4 and 10 K respectively.

(b) Fitting the HLN formula for different tem-peratures with fit parameters as indicated. Fig-ures a, b and c show ∆G at T = 15, 25 and 40 K respectively.

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Figure 4.9: The phase coherence length (`φ) for

different temperature ranges. The data points are shown in blue. Fitting a power law as in equation (4.1) for T = 2, 4, 10, 15, 25 and 40 K is shown in purple, for T = 2, 4, 10 and 15 K is shown in red and for T = 10, 15, 25 and 40 K is shown in green.

As shown in figure 4.9i _{one can see that}

fitting a power law as in equation (4.1) for all temperatures a relation of `φ ∝ T−0.18 is

found. One can see that the data and the fit in figure 4.9 do not coincide. The data points in the temperature range 10 K≤ T ≤ 40 K fall on a straight line (on this log log scale). For 2 K≤ T ≤ 15 K the same can be said. Fitting now for two ranges i.e. T ≤ 15 K and T ≥ 10 K results in two fits which sep-arately coincide with the data well resulting in `φ ∝ T−0.05 and `φ ∝ T−0.48 respectively.

This implies that for T ≥ 10 K the power law is close to `φ ∝ T−1/2 for a 2D system and

for low temperatures `φ levels off. One should

be careful however not to draw too strong con-clusions based on the limited number of data points.

4.3 Comparison to the literature

In this section the results obtained are compared to the results reported in the literature. Table 4.1 shows the values found in the literature for α and `φ. Each reference will be discussed in a

different section.

Ref. sec. Sample T (K) α `φ Bmax

Hsiung et al. [13] 4.3.1 x = 0.5, y = 1.3, t = 200 nm 2 K −0.99 121 nm 0.5 T Xia et al. [29] 4.3.2 x = 0.5, y = 1.2, t = 596 nm 2 K −0.74 180 nm 0.5 T Lee et al. [14] 4.3.3 x = 0.5, y = 1.3, t = 85 nm* 4.2 K −1.3 160 nm 6 T Table 4.1: Literature comparison of fitted values of the HLN formula for Bi2−xSbxTe3−ySey

samples with x and y as indicated. *Note that Lee et al. [14] use a back gated sample.

4.3.1 Comparison with Hsiung et al. [13]

Hsiung et al. [13] measured a Bi1.5Sb0.5Te1.7Se1.3 200 nm thick sample finding α = −0.99 and

`φ= 121 nm at T = 2 K. The results of ∆G are shown in figure 4.10.

The value found is similar to the α reported by us. The slightly smaller α might be due to the thicker (200 nm) sample since it is more likely to measure one surface if the sample is thicker.

However it should be noted that even though the same definition of ∆G = G(B) − G90◦(B) is

used as in this report (see appendix B) their data in figure 4.10 do not collapse onto one single curve. Furthermore the HLN formula is only fitted for low angle data i.e. for small values of θ.

4.3.2 Comparison with Xia et al. [29]

Xia et al. [29] measured a Bi1.5Sb0.5Te1.8Se1.2 sample. In figure 4.11 the obtained fit parameters

α and `φ for different temperatures are shown.

The (absolute value of) α found is smaller than the α measured in this report. This might be due to the thicker sample (596 nm) Xia et al. [29] used. If the sample is thicker the chance that only one surface is measured is bigger.

i_{Where a log log scale is used since if `}

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Figure 4.10: Results of Hsiung et al. [13] measuring a Bi1.5Sb0.5Te1.7Se1.3 200 nm thick

sample at T = 2 K. In this figure ∆G(B⊥) is shown and the HLN formula is fitted for low

angle data.

Figure 4.11: Results of Xia et al. [29] measuring a Bi1.5Sb0.5Te1.8Se1.2 596 nm thick sample.

In this figure the fit results for α and `φ are shown for different temperatures.

A small term for the orientation of the sample with respect to the magnetic field at θ = 90◦ (in Xia et al. [29] this orientation is defined as 0◦) is also observed. The authors attribute this term to a small unavoidable misalignment of the sample.

If one compares figures 4.11 and 4.9 it is interesting to see that Xia et al. [29] found similar behavior of the phase coherence length (`φ). Their fits find `φ ∝ T−0.19 and `φ ∝ T−0.66 for 2

K≤ T ≤ 7 K and 10 K≤ T ≤ 45 K respectively. The authors attribute this change of behavior of `φ(T ) to the effect of phonon scattering at high temperatures and paramagnetic impurities at low

temperatures.

4.3.3 Comparison with Lee et al. [14]

Lee et al. [14] found somewhat different results from the above mentioned papers. The Bi1.5Sb0.5Te1.7Se1.3

85 nm thick sample that is measured is engineered with a back gate. With a back gate it is possible to separate bottom and top surface contributions. For a 0V back gate voltage values for α ≈ −1.3 and `φ≈ 160 nm were obtained at T = 4.2 K.

The large absolute value of α is attributed to topological trivial Rashba split bands of a 2D electron gas (2DEG). The 2DEG is due to band banding at the surface. A 2D electron gas is essentially a quantum well where electrons are strictly confined in one dimension but weakly in

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the other two. The confined dimension is the z-direction along the thickness of the sample t in which direction the band bending takes place.

The dependence of `φ with respect to the temperature is found to be exactly the theoretically

expected `φ∝ T−0.5for 2D systems where `φ was obtained for temperatures T = 4.2, 5, 7, 10, 15,

30 and 50 K.

Even though the authors of [14] also observed a linear term for R90◦ they could not explain

its origin. They propose to use the definition ∆G(B) = G(B) − G90◦(B) if the linear term for

R90◦ would be due to a bulk channel. However R₉₀◦ is small compared to R₀◦. According to the

authors using ∆G = G(B) − G(0) or ∆G(B) = G(B) − G90◦(B) yields similar results for α and

`φ. The authors decided to use ∆G = G(B) − G(0) and refer to an appendix for this R90◦ for

motivating their choice of definition.

4.3.4 Comparison with Ando [2]

According to Ando [2] each topological surface channel contributes −1₂ to α in the HLN formula (equation 2.4) but if the thickness of the sample is smaller than the phase coherence length (`φ)

diffusive transport between bottom and top surface channels is possible for Bi2Se3. However if the

channels are separated by a gate voltage one could measure two independent transport channels. If diffusive transport between bottom and top surface channel is possible one would expect α ∼ −0.5. For the sample measured in this report `φ ∼ t (for T ≤ 10 K). According to Ando [2]

diffusive transport results in effectively one surface contributing and hence an α ∼ −1₂ would be expected. However an α of ∼ −1 is measured, indicating two channels contributing. Furthermore for T = 40 K, `φ ∼ 52 nm < t so diffusive transport would not be possible for T = 40 K and

hence α would increase to ∼ −1. However looking at figure 4.5 one can see this is not the case. For these reasons we expect above reasoning is not applicable to this Bi1.46Sb0.54Te1.7Se1.3 sample

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4.4 The 2D Dirac model for four conducting channels

Figure 4.12: Parameters cos ΘS and cos ΘB

ob-tained at different temperatures. This figure shows that the surface channel is in the WAL re-gime (since cos Θ ∼ 0). The bulk channels are in the unitary regime (since cos ΘB ∼ 0.26 − 0.36).

The four band model as described in section 3.5 is fitted for T = 2, 4, 10, 15, 25 and 40 K. The results are shown in figures 4.14a and 4.14b. Furthermore the fit parameters obtained are shown in figures 4.13 and 4.12. As can be seen from figure 4.12 by fitting this model we ob-tain a very small cos ΘS as expected.

How-ever cos ΘB increases for increasing

temperat-ure. One can also see from figures 4.14a and 4.14b that the bulk contributions to the con-ductance are in the unitary regime. The expec-ted value of cos ΘB for Bi2Te3 and Bi2Se3lies

between 1₂ and 1. The value of cos ΘB found

here is considerably smaller ∼ 0.2 − 0.4. The fits were made using the Nonlinear-ModelFit function of Mathematica [28]. The distinction between cos ΘS and cos ΘB was

made by the conditions that cos ΘS ∼ 0 and

cos ΘB > 0.

4.4.1 Temperature dependence of

the phase coherence length

Figure 4.13: The phase coherence length (`φ) for

different temperatures fitted for different temper-ature ranges. The data points are shown in blue. Fitting a power law as in equation (4.1) for T = 2, 4, 10, 15, 25 and 40 K is shown in purple, for T = 2, 4, 10 and 15 K is shown in red and for T = 10, 15, 25 and 40 K is shown in green with the exponents as indicated.

As shown in figures 4.12 and 4.13 cos ΘS is ∼ 0

for all temperatures, cos ΘB increases slightly

with rising temperature and `φis strongly

tem-perature dependent. Furthermore equation (3.5) is very sensitive to changing `φ.

One can see from figure 4.12 that cos ΘB

increases for increasing temperatures. The bulk contribution to ∆G becomes smaller if cos ΘB is closer to 0.33. Once can even see

that at T = 40 K the bulk contribution has a small positive magnetoconductance i.e. negat-ive magnetoresistance.

As explained before in section 4.2.3 `φ is

expected to obey a power law. The temperat-ure dependence of `φ is shown in figure 4.13.

Fitting a power law for all temperatures res-ults in `φ∝ T−0.2 and the data and the fit do

not coincide. But if we again fit for two tem-perature ranges i.e. 10 K≤ T ≤ 40 K and 2 K≤ T ≤ 15 K one obtains fits which coincide with the data. Fitting for higher temperat-ures T ≥ 10 K a power law of `φ ∝ T−0.54

is obtained while for T ≤ 15 K we obtain `φ ∝ T−0.06. One can see that this is

sim-ilar to what is observed before in section 4.2.3 with the ordinary HLN model.

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(a) Measured WAL for temperatures 2, 4 and 10 K in figures a, b and c respectively. The data is fitted using the model of equation (3.5). The fit is displayed by the curve labeled Total while 2∆σSand 2∆σBare shown as Surface and

Bulk. Fitted values as indicated.

(b) Measured WAL for temperatures 15, 25 and 40 K in figures a, b and c respectively. The data is fitted using the model of equation (3.5). The fit is displayed by the curve labeled Total while 2∆σSand 2∆σBare shown as Surface and Bulk.

Fitted values as indicated.

Figure 4.14: The 2D modified Dirac model used to fit the ∆G data for (a) T ≤ 10 K and (b) T ≥ 15 K.

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Chapter 5

Discussion and conclusion

5.1 Discussion

5.1.1 Aging of the sample

Since the sample was already made in spring 2013 and is measured and analyzed in May 2014 it is very likely that there are adsorbates present on the sample surface. Even though the sample was kept at low pressure, the vacuum chamber was opened now and then. Since we are not using surface sensitive techniques like angle-resolved photo emission spectroscopy (ARPES) or scanning tunneling microscopy (STM), but magnetotransport to investigate the sample it is expected that absorbates have less effects.

If the sample is exposed to air and adsobates are present on the surface band bending may occur, which can lead to creation of a 2D electron gas (2DEG) on the surface [14]. One could say that such a 2DEG can also be recognized as the unitary behavior of the bulk channels as fitted in section 4.4.

5.1.2 2D Dirac model for multiple channels

As mentioned in section 3.5 there are a couple of difficulties with the used modified 2D Dirac model. Since we did not measure this specific sample using ARPES we do not have access to important quantities in order to obtain cos Θ as in equation (2.9). Furthermore, in section 3.5 we assume a contribution of two surface channels and two bulk channels. Two surface channels seem to make sense due to the two surfaces. The two bulk channels however are more of an educated guess. More channels are also fitted and result in weaker unitary behavior per channel and approximately the same behavior of all bulk channels combined. The model of two bulk channels (and two surface channels) is used throughout this report.

Another note is that the parameters of table 2.1 are a bit heuristic. Even though the modified HLN formula stays quite stable (with a relative change < 0.1%) for an increase or decrease of the parameters by a factor 10 there is no experimental confirmation of these values. The values were assumed to be as such by the authors of [16] who derived equation (2.7). One can also check that for cos Θ→ 0, `e→ 0 and `x= `y= `z→ ∞ one obtains the ordinary HLN formula (2.4).

5.1.3 Error on the contact length

As shown in figure 4.5 there is a large error in α. The substantial error in α is mainly due to the error in the channel length L as introduced in figure 3.2a. Since the HLN formula is expressed in terms of ∆σ and ∆G = W_L∆σ the error in L and the error in α are closely related. The error of L is due to the thickness of the BSTS connecting the sample to the gold electrodes.

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5.2 Conclusion

Topological surface states are examined by the weak anti localization (WAL) effect on a BSTS nanoflake with composition Bi1.46Sb0.54Te1.7Se1.3. Magnetotransport measurements were made

with the DynaCool Physical Property Measurement System. The calculated difference in conduct-ance ∆G due to an applied magnetic field shows a 2D origin since it collapses onto one single curve when plotted as a function of the perpendicular component of the magnetic field with respect to the sample surface (B⊥).

Fitting the ∆G data is done using two models. First the Hikami-Larkin-Nagaoka (HLN) formula was used. This showed that α ranges between −1.14 for T = 2 K and −0.93 at T = 40 K, close to the theoretically predicted value of α = −1 for two topological surface channels.

The phase coherence length (`φ) was expected to obey a power law T−1/2 if the system is

2D and this was obtained for T ≥ 10 K. However fitting a power law for T ≤ 15 K resulted in `φ∝ T−0.05.

A 2D modified Dirac model was fitted for the ∆G data. This results in separate contributions due to bulk and surface channels. The fit results of `φare similar to the results obtained with the

ordinary HLN formula. The surface contributions are indeed found to be in the WAL regime as expected with cos ΘS ∼ 0 while the bulk channels are in the unitary behavior regime (cos ΘB ∼

0.26 − 0.36). Furthermore the temperature dependence of `φis almost identical to the dependence

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Appendix A

The Berry phase

The derivation of the Berry phase will be presented here following the derivation in reference [8]. We start of with a Hamiltonian that is independent of time and a particle that is in the nth

eigenstate |ψni

H|ψni = En|ψni (A.1)

We assume that |ψni retains the nth eigenstate only picking up a phase factor if we deform the

Hamiltonian adiabatically. We turn to the time dependent Schr¨odinger equation (A.2). |ψn(t)i = e−iEnt/~|ψni

i~_∂t∂|ψ(t)i = H(t)|ψ(t)i (A.2) We express |ψ(t)i as a linear combination of the eigenstates of the Hamiltonian

ψ(t) =X

n

cn(t)|ψn(t)ieiθn(t) (A.3)

with θn(t) = − 1 ~ Z t 0 En(τ ) dτ

After the substitution of (A.3) into (A.2), and notingi

−~cn|ψni ˙θneiθn= cnH|ψnieiθn and taking

the inner product with hψm| we obtain equation (A.4).

X n ˙cnhψm|ψnieiθn= − X n cnhψm| ˙ψnieiθn (A.4)

After invoking orthogonality of the eigenstates we get: ˙cm(t) = −

X

n

cnhψm| ˙ψnieiθn−θm

By differentiating (A.1) with respect to time and taking the inner product with hψm| for n 6= m

we obtain:

˙

H|ψni + H| ˙ψni = ˙En|ψni + En| ˙ψni

hψm| ˙H|ψni + hψm|H| ˙ψni = ˙Enδmn+ Enhψm| ˙ψni

hψm| ˙H|ψni = (En− Em)hψm| ˙ψni (A.5)

i_{With the notation ˙}_{φ =} ∂φ ∂t.

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If we substitute equation (A.5) into equation (A.4) we obtain equation (A.6) which is exact. The approximation comes in equation (A.7) leaving out the second term if we assume that ˙H is very small. The solution of equation (A.7) is given by equation (A.8).

˙cm(t) = − cmhψm| ˙ψmi − X m6=n cn hψm| ˙H|ψni En− Em e− i ~ t R 0 En(τ )−Em(τ ) dτ (A.6) ≈ − cmhψm| ˙ψmi (A.7)

The solution of equation (A.7) is given by:

cm(t) = cm(0)eiγm(t) (A.8)

with γm(t) = i t Z 0 ψm(τ )| ∂ ∂τ|ψm(τ ) dτ

Recall equation (A.3) from which we obtain the wave function:

|ψn(t)i = eiθn(t)eiγn(t)|ψn(t)i (A.9)

In equation (A.9) θn(t) is called the dynamic phase and γn(t) is the geometric phase. Now for

some parameters Rj(t), with H dependent of Rj(t):

∂|ψni ∂t = X j |∂ψni ∂Rj ∂Rj ∂t = (∇R|ψni) dR dt With R =P jRj and similarly ∇R=Pj ∂

∂Rj one can write:

γn(t) = i Rt

Z

Ri

hψn|∇Rψni dR (A.10)

If Ri = Rf (with the subscripts indicating initial and final) after some time T hence the phase

change is:

γn(T ) = i

I

hψn|∇Rψni dR (A.11)

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Appendix B

Different ways to obtain ∆G

As is shown in figure 4.2 there is a term for the resistance with the 90◦ orientation of the sample (R90◦

xx). This term cannot be due to the WAL effect since the magnetic field is parallel to the

surface. How we can correct for this term is discussed in this section. In this section we drop the subscript Rxx= R for readability. Three ways of dealing with R90

◦

by altering the definition of ∆G (the change of conductance due to an applied magnetic field) are proposed:

• ∆G(B) =R(B)−1_{− R(0)}−1 _{as used in refs. [14], [29] and [3]. This definition is discussed in}

appendix B.1.

• ∆G(B) =R(B)−1_{− R}90◦_(B)−1

as used in refs. [9] and [13]. This definition is discussed in appendix B.2.

• ∆G(B) = R(B) − ∆R90◦_(B)−1

− R(0)−1_{with ∆R}90◦_{(B) ≡ R}90◦_{(B) − R}90◦_{(0). This}

defin-ition is discussed in appendix B.3.

In the following sections we present each method separately and discuss its physical interpretation.

B.1 Ordinary definition of ∆G

Figure B.1: This figure shows ∆G(B) = R(B)−1− R(0)−1 _{at T = 2 K showing that ∆G}

is dependent on other components than the 2D WAL effect.

The usual definition as in equation (B.1) is widely used for the analysis of WAL [14], [29], [3].

∆G(B) = R(B)−1− R(0)−1 (B.1) Applying this definition to our data would im-pose a problem. As can be seen from figure 4.2 there is a dependency of B in the R90◦ _term

whereas for this analysis R90◦ _{is assumed to}

be a constant. This field dependence of R90◦

cannot be due to the 2D WAL effect since the magnetic field is perpendicular to the sample’s surface. By using this definition one obtains ∆G as displayed in figure B.1. From this figure it is clear that there is some other contribution to the ∆G than the 2D WAL effect since ∆G is changing for different angles (when plotted as a function of B⊥). ∆G would be independent

of the angle if ∆G is only due to WAL.

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B.2 Subtracting 1/R

90◦

As shown in figure B.1 in the previous section there is a term contributing which is independent of the angle of the magnetic field with respect to the sample. By altering the definition of ∆G as done in [9] and [13], we compensate for the increase of R90◦ (see figure 4.2) by using the definition: ∆G(B) = R(B)−1−R90◦(B)−1 (B.2) In [9] the authors argue that this behavior is due to the classical Lorentz deflection of carriers obeying Kohler’s rule R(B) ≈ R(B = 0)1 + µB2_{with µ the mobility.}

In the article by Lee et al. [14] a comparison of their data is made with respect to He et al. [9]. The authors of Lee et al. [14] argue that since the B2_{dependency of R}90◦ _{is not observed one}

may not use ∆G = G(B) − G(θ = 90◦). Instead the ∆G = G(B) − G(B = 0) is used. Looking at figure 4.2 for θ = 90◦ one can see that this curve is not ∝ B2 _{in fact it looks like a linear}

magnetoresistance as in [14] (figure 5). The authors of Lee et al. [14] propose to still use the definition as in equation (B.1) instead of the definition in equation (B.2) since similar results are obtained for the two definitions. However for our data these different definitions do not result in similar results as shown in table B.1.

A last note of interest could be that the classical quadratic term could be suppressed by weak (anti) localization of the bulk bands as mentioned in [16] or a misalignment of the sample. In Lee et al. [14] it is also pointed out that it is not yet clear what causes this linear-like magnetoresistance.

B.3 Subtracting 1/∆R

Figure B.2: ∆G at T = 2 K as defined in equa-tion (B.3) clearly showing the expected collapse of curves for different angles when plotted as a function of B⊥.

The last way of correcting for this linear mag-netoresistance is the following:

∆G(B) = 1 R(B) − ∆R90◦ (B)− 1 R(0) (B.3) with ∆R90◦_{(B) ≡ R(B) − R(0) at θ = 90}◦_.

This approach is motivated by the assump-tion that there is an other scattering channel on which electrons can scatter. This scatter-ing is enhanced by applyscatter-ing a magnetic field. The effect on the resistance with the perpen-dicular orientation (θ = 90◦) to the magnetic field we call ∆R90◦_{(B) ≡ R(B) − R(0). This}

would alter the definition of ∆G as shown in equation B.3. The result of implementing this definition is shown in figure B.2. In compar-ison to figure B.1 one can clearly see that the ∆G is independent of the angle (when plotted as a function of B⊥) which is expected for the

2D WAL effect.

B.4 Choice of definition of ∆G

As mentioned in appendix B.2 it is not clear where the linear magnetoresistance (shown in figure 4.2) of Rθ=90◦ originates from. Three ways of dealing with this linear magnetoresistance were proposed in sections B.1, B.2 and B.3. From figures B.1, 4.4 and B.2 it is clear that the method of appendix B.1 is not accounting for this linear magnetoresistance while the methods in sections B.2 and B.3 can account for the linear magnetoresistance since ∆G is independent of the orientation

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APPENDIX B. DIFFERENT WAYS TO OBTAIN ∆G 29

Method 1 Method 2 Method 3

Sec. B.1, ∆G = Sec. B.2, ∆G = Sec. B.3, ∆G = Magnetic R(B)−1− R(0)−1 _R(θ)−1_{− R(90}◦₎−1 _{(R(B) − ∆R)}−1

− R(0)−1

Field range with ∆R ≡ R90◦_{(B) − R}90◦₍₀₎

Range of fit α = α = α = Brange = 0.5 T −1.28 −1.14 −1.17 Brange = 1.0 T −1.18 −1.04 −1.08 Brange = 2.0 T −1.12 −0.97 −1.02 Range of fit `φ= `φ= `φ= Brange = 0.5 T 118 nm 116 nm 114 nm Brange = 1.0 T 125 nm 124 nm 122 nm Brange = 2.0 T 131 nm 133 nm 129 nm

Table B.1: Different fitvalues for the different definitions of ∆G as described in sections B.1, B.2 and B.3. Even though the definitions of method 2 and method 3 are quite different they both seem to give similar results within a few percent. All values are at T = 2 K.

of the sample’s surface with respect to the magnetic field. For T = 2 K the HLN formula (2.4) is fitted for different field ranges and these different methods of calculating ∆G. The result of these fits are displayed in table B.1.

The definition of ∆G in appendix B.2 provides us with a physical argument why there is an additional R90◦_{(B) term. Therefore we choose to use the definition ∆G(B) = R(B)}−1 ₋

R90◦_(B)−1

with the note that the origin of R90◦_{(B) is not yet clear.}

Since the fit results using the definitions as in sections B.2 and B.3 differ < 3% for the field range to 0.5 T one could also argue that both methods are usable since their results do not differ much.

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Acknowledgments

A few persons really helped me during this bachelor thesis. First of all dr. Anne de Visser helped me through the sometimes tough material. If papers were not clear he really took the time to explain their results to me which helped me a great deal. He also gave me feedback on numerous occasions and helped me to work hard and look critical at my own work.

Secondly Yu Pan has been great in explaining me all I needed to know about measuring samples, processing data and analyze the data once it was processed. Also helping me with installing the samples and writing sequences for measurements.

Also Marieke Snelder from Twente University was very helpful. She helped me measuring the sample, prepared the Hall bar and made new contacts to the sample holder.

Dr. Yingkai Huang made the single crystal that was used to make the Hall bar and therefore I would like to thank him as well.

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Appendix D

Populair wetenschappelijke

samenvatting

Figure D.1: De DynaCool (Physical Property Measurement System) waarmee de verwachte to-pologische isolator Bi1.46Sb0.54Te1.7Se1.3 is

ge-meten. Dit figuur is overgenomen van Quantum Design [22].

Topologische isolatoren zijn een bijzonder soort materialen vanwege hun uitzonderlijke geleidende eigenschappen. Een normaal ma-teriaal is of een geleider (zoals koper) of een isolator (zoals rubber). Topologische isolatoren zijn echter beide: aan het oppervlak is het een geleider terwijl het binnenste van de stof een isolator is. Dit is een materiaaleigenschap die niet kan worden veranderd. Als iemand het pervlak er af zou schrapen, zal er een nieuw op-pervlak ontstaan met dezelfde eigenschap. Dit komt doordat de stof een andere “ topologische fase” heeft dan een isolator zoals bijvoorbeeld het vacu¨um. Om de fase van de binnenkant van de stof te verbinden met de fase van het vacu¨um ontstaan deze bijzondere oppervlakte-toestanden. Dankzij deze oppervlakte toest-anden zijn bijzondere kwantumeffecten meet-baar in dit soort materialen.

Hoewel het theoretisch voorspeld is dat to-pologische isolatoren een isolerende binnenkant

hebben blijkt dit in de realiteit niet altijd te kloppen. Uit ander onderzoek is gebleken dat de stof Bi1.46Sb0.54Te1.7Se1.3 een relatief groot isolerend vermogen heeft, terwijl het nog steeds deze

bijzondere oppervlakte toestanden heeft. Deze toestanden zijn onderzocht met een heel dun mon-ster. Het monster (130 nanometer) is maar liefst duizend maal zo dun als een vel papier en honderd maal zo dun als een rode bloedcel.

Aangezien het monster zo dun is verwachten we een grote oppervlak-volume verhouding. Om-dat deze verhouding zo groot is zullen de eigenschappen van dit monster vooral worden bepaald door de oppervlakte toestanden.

Er is een bijzonder effect op de weerstand dat wordt veroorzaakt door de oppervlakte toest-anden. De beschrijving van dit effect en de resultaten van de metingen staan in dit verslag. Wat dit effect inhoudt wordt hieronder (versimpeld) uitgelegd.

Stroom is het verplaatsen van geladen deeltjes: elektronen. Deze elektronen kunnen echter ook botsen tegen onzuiverheden, atomen en anderen elektronen. Als elektronen veel terug zouden botsen (verstrooien) dan is er effectief een kleinere stroom (minder elektronen in de juiste richting). Maar door de bijzondere eigenschappen van de oppervlakte toestanden gebeurt dit verstrooien

The weak anti localization effect due to topological surface states of a Bi1.46Sb0. 54Te1.7Se1.3 nano ake

states of a Bi

1.46

Sb

0.54

Te

1.7

Se

1.3

nanoflake

Bachelor project Physics at the University of Amsterdam

Joran Angevaare

Report of Bachelorproject Natuur- en Sterrenkunde (15 EC) carried out between

01-04-2014 and 27-06-2014

Author:

Joran Angevaare

Student number:

10219617

Supervisor and corrector:

Dr. Anne de Visser

Second supervisor:

Yu Pan

Second corrector:

Dr. Emmanouil Frantzeskakis

University:

Universiteit van Amsterdam

Faculty:

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Institute:

Van der Waals-Zeeman Instituut

Group:

Hard condensed matter group

Address:

Science Park 904, 1098 XH Amsterdam

Date of submission:

Contents

Chapter 1

Introduction

Theory

2.1

Topology

2.2

Spin-orbit coupling

2.3

Berry phase

2.4

Time reversal symmetry

2.4.1

Kramers pair

2.5

Parity

2.6

Z

invariant

2.7

Theory of weak (anti) localization

2.8

Hikami-Larkin-Nagaoka (HLN) formula

2.9

Modified 2D Dirac model

Chapter 3

Measurements/Method

3.1

The sample

3.2

The DynaCool Physical Property Measurement System

(PPMS)

3.3

Magnetoresistance

3.4

Measuring weak anti localization (WAL)

3.4.1

Hikami-Larkin-Nagaoka (HLN) formula

3.5

Modified 2D Dirac model.

3.5.1

Four channel model

Results

4.1

The sample at low temperatures

_1.46

_0.54

_1.7

_1.3

_Z