Quantum transport in topological insulators:
Weak Anti-localization&
Electron-electron interaction
Hai-Zhou Lu (卢海舟)Department of Physics
The University of Hong Kong
Topological insulator
Reviews: Hasan & Kane, RMP (2010);
Qi & Zhang, RMP (2011); Shen, Topological insulators (Springer, 2012).
always gapless=
Super metal !
Bulk: Gapped = Insulating
Surface: Gapless = Metallic
How about their electronic transport?
Molecular Beam Epitaxy: Zhang, K.H.Wu et al, APL 95, 053114 (2009) (IOP, China); Zhang, Xue, Nature Physics 2010 (IoP&Tsinghua).
Bi2Se3 and Bi2Te3 in transport measurement
Moore, Nature, 2010;Zhang et al, Nat. Phys. 5, 438 (2009);Xia et al, Nat. Phys. 5, 398 (2009)
Mechanical Exfoliation (like for graphene): Teweldebrhan et al, Nano Lett.10 1209 (2010) (UC, Riverside)
Measuring conductivityOng group (Princeton), PRL, 2009
Bi2Se3, Bi2Te3• Thermoelectric materials• Narrow-gap semiconductors
Crystal Flake Device
What observed ?
Weak antilocalization (WAL)
Bi2Se3 (Tsinghua, IOP)Liu, Yayu Wang, et al, PRB 83, 165440 (2011); PRL 108, 036805 (2012).
Bi2Se3 (IOP)Chen, Wu, Li, Lu, et al , PRL 105, 176602 (2010)
Bi2Se3 (Princeton)Checkelsky, Ong, et al PRL 103, 246601 (2009); PRL 106, 196801 (2011).
Bi2Se3 (Stanford)H.L. Peng, Yi Cui et al , Nat. Mat. 9, 225 (2009).
Bi2Te3 (HKUST&HKU)He, JN Wang, et al, PRL 106, 166805 (2011).
Bi2Se3 (PKU, Penn State)Wang, Moses Chan, et al, PRB 83, 245438 (2010).
Resistance
Quantum diffusion
Mean free path (due to elastic scattering by static centers)
Quantum diffusive
Phase coherence length (due to inelastic scattering by phonons, e-e interaction,etc.)
Ballistic
Classical diffusive
L System size
L L
Diffusive
FFsc k
heDNe
mne 2
22
Einstein's relation
Weak (Anti-)localization
2/
1~ pT
Quantum interference
qi
scxx
Semiclassical
Conductivity
ln~
ln~ WAL
WL
Mean free path (due to elastic scattering, no temperature dependence)
Phase coherence length (due to inelastic scattering, )
Time-reversed scattering loops
Quantum interference corrects conductivity
Thouless, PRL 39, 1167 (1977)
Weak (Anti-)localization
Quantum interference
ln~
ln~ WAL
WL
Temperature
Time-reversed scattering loops
2/
1~ pT
Mean free path (due to elastic scattering, no T dependence)
Phase coherence length (due to inelastic scattering, )
qi
scxx
Semiclassical
Conductivity
Thouless, PRL 39, 1167 (1977)
XMagnetic field
Quantum interference
Magnetic field Temperature
ln~
ln~ WAL
WL
Weak (Anti-)localization
Time-reversed scattering loops
qi
scxx
Semiclassical
Conductivity
XMagnetic field
Quantum interference
Magnetic field Temperature
ln~
ln~ WAL
WL
Weak (Anti-)localization
Time-reversed scattering loops
qi
scxx
Semiclassical
Conductivity
Why do you call them Weak
(anti-)localization ?
Weak (Anti-)localization
Quantum interference
ln~
ln~ WAL
WL
Temperature
Time-reversed scattering loops
qi
scxx
Semiclassical
Conductivity
Why do you call them Weak
(anti-)localization ?
Normal metals "Anderson localization"
No conductivity
Electrons are localized
Weak (Anti-)localization
Quantum interference
ln~
ln~ WAL
WL
Temperature
Topological insulators
(1) No gap; (2) No localizationSuper metal !
Time-reversed scattering loops
qi
scxx
Semiclassical
Conductivity
Normal metals "Anderson localization"
No conductivity
WL or WAL is everywhereSi/SiO2 Rosenbaum et al, PRL 47, 1758 (1981); PRL 46, 568 (1981); Bishop, Dynes, & Tsui, PRB 26, 773 (1982).
GrapheneWu, de Heer et al, PRL 98, 136801 (2007); Tikhonenko, et al, PRL 103, 226801 (2009)
Bi2Te3 HKUST & HKUHe et al, PRL 106, 166805 (2011)
Bi2Se3 (IOP)Chen, Wu, Li, Lv, et al , PRL 105, 176602 (2010)
MgBergmann, PRL 48, 1046 (1982).
GaAsMiller et al, PRL 90, 076807 (2003);Neumaier et al, PRL 99, 116803 (2007).
MoS2, WSe2HZ Lu, Di Xiao, Wang Yao & SQ Shen, PRL, 110, 016806 (2013); Iwasa et al Science 2012; Nat. Phys. 2013 (Tokyo); Peide Ye et al ACS Nano 2013 (Purdue)
Weak localization and Weak anti-localization
Hikami, Larkin, and Nagaoka Prog. Theor. Phys. 63 7071(1980).
Symmetry classes of random ensembles [F. J. Dyson, J. Math. Phys. 1962]
Impurity scattering Symmetry Time-reversal Spin-rotational Transport
Scalar Orthogonal WL
Spin-orbit Symplectic WAL
Magnetic Unitary 2B
WAL in Topological insulators
Poor mobility Mean free path
Phase coherence length at ~ 1K
(2) Quantum diffusion regime
L
Why WAL in TIs
(1) Surface Dirac fermions
nm 10~
nm1000100~
)( xyyx kkH
Berry phase of massless surface states
)( xyyx kkH
Surface Dirac fermions
Spin-momentum
lockingxkyk
Energy
Berry phase of massless surface states
)( xyyx kkH
ik ie1
21Spinor
wave function
x
y
kk
tanMomentum angle
Surface Dirac fermions
Spin-up
Spin-down
Spin-momentum
lockingxkyk
Energy
Berry phase of massless surface states
)( xyyx kkH
ik ie1
21
2
0
2
0
2
0
2
0
21
0),1(
2
1),1(
2
d
eiedi
ieiedi
di
ii
ii
kk
Spinor wave
function
x
y
kk
tanMomentum angle
Surface Dirac fermions
Spin-up
Spin-down
Berry phase
Spin-momentum
lockingxkyk
Energy
WAL of massless surface states
Suzuura & Ando, PRL 89, 266603 (2002)
Berry phase
ln~ qi
Destructive Quantum
Interference
)( xyyx kkH
kkdi2
0
Berry phase is the reason
Suppress Back
Scattering
WAL of massless surface states
Suzuura & Ando, PRL 89, 266603 (2002)
Berry phase
ln~ qi
Destructive Quantum
Interference
)( xyyx kkH
kkdi2
0
Can we change
Berry phase?
Suppress Back
Scattering
Berry phase
02
,
12
,2
)2
1(
F
F
F
E
E
E
Berry phase WL
WAL
Large-gap
gapless
Based on the Berry phase argument, we predicted a crossover from WAL to WL
Massive Dirac fermions
(1) Magnetically doped surface states[Chen, Shen et al, Science 2010; Wray, Hasan et al, Nat. Phys. 2011]
(2) Thin film finite size effect [Lu, Shan, Yao, Niu & Shen PRB 81, 115407 (2010); Liu, Zhang et al, PRB(R) 2010; Linder et al, PRB 2009.]
(3) 2D Bulk subband[Lu & Shen, PRB 84, 125138 (2011)]
Top-bottomcoupling
Model
zxyyx kkH 2
)(
i
ii RrurU )()(
Dirac model
Disorder potential
Lu, Shi & Shen, PRL, 107 076801 (2011)
Maximally crossed diagram:Langer and Neal, PRL 16, 984 (1966)Ladder diagram correction to velocity:Shon and Ando, JPSJ 67, 2421 (1998) The dressed Hikami boxes:McCann, Kechedzhi, Fal’ko, Suzuura, Ando, and Al'tshuler, PRL 97, 146805 (2006)Reviews for non-Dirac FermionsBergmann Phys. Rep. 1984Lee and Ramakrishnan, RMP 1985
We calculate magnetoconductivity using Feynman diagrams
WAL-WL crossover
Lu, Shi & Shen, PRL, 107 076801 (2011)
Mag
neto
cond
uctiv
ity
02
,
12
,2
)2
1(
F
F
F
E
E
E
WAL-WL crossover
Liu, Yayu Wang et al, PRL 108, 036805 (2012)(Tsinghua & IOP, China)3QL Bi2-xCrxSe3
Finite size effect: Lu, Shan, Yao, Niu & Shen PRB 81, 115407 (2010); Liu, Zhang et al, PRB(R) 2010; Linder et al, PRB 2009.Magnetic doping:Chen et al, Science 2010; Wray et al Nat. Phys. 2011.
Lu, Shi & Shen, PRL, 107 076801 (2011)
WAL-WL crossover
WAL-WL crossover as a signature of ferromagnetic phase transition with Tc~5K
Zhang, Samarth et al, (Penn State)Mn-Bi2Se3WAL-WL crossover as a signature of ferromagnetic phase transition with Tc~5K
Lu, Shi & Shen, PRL, 107 076801 (2011)
WAL-WL crossover
Also:EuS/Bi2Se3Wei, Moodera, et al (MIT), APS March meeting 2013
Yang, Kapitulnik (Stanford)PRB 88, 081407(R) (2013) Editors' suggestionEuS/Bi2Se3Europium Sulfide
Lu, Shi & Shen, PRL, 107 076801 (2011)
WAL-WL crossover
Lang, K. L. Wang (UCLA)Nano Lett. (2013)4 QL Bi1.14Sb0.86Te3
intersurfacecoupling
Finite size effect: Lu, Shan, Yao, Niu & Shen PRB 81, 115407 (2010); Liu, Zhang et al, PRB(R) 2010; Linder et al, PRB 2009.
Lu, Shi & Shen, PRL, 107 076801 (2011)
WAL
Mechanism of WAL:
Berry phaseof
Surface Dirac fermions
WAL
Wang, Moses Chan (Penn state), et al, PRB 83, 245438 (2010).Bi2Se3 and Pb-Bi2Se3 (Tsinghua, IOP)
How abouttemperature
dependence in experiments?
Dilemma in topological insulators
A sad truthWang, Moses Chan (Penn state), et al, PRB 83, 245438 (2010).Bi2Se3 and Pb-Bi2Se3 (Tsinghua, IOP)
Dilemma in topological insulators
Liu, Yayu Wang (Tsinghua), et al, PRB 83, 165440 (2011)Bi2Se3 ultrathin films
Conductivity
Dilemma in topological insulators
Chen, Y.Q. Li, K. H. Wu, L. Lu, et al (IOP), PRB, 83, 241304 (R) (2011)Bi2Se3
Dilemma in topological insulators
Takagaki et al (Berlin), PRB 85, 115314 (2012)Bi2Se3
Dilemma in topological insulators
Roy et al (Austin), APL 102, 163118 (2013);Bi2Te3 ultrathin film
Dilemma in topological insulators
Roy et al (Austin), APL 102, 163118 (2013);Bi2Te3 ultrathin film
WL?
Dilemma in topological insulators
An embarrasing
situation
J. E. Moore, "The birth of topological insulators", Nature 464, 194 (2010)
Dilemma in topological insulators
What missed?
Dilemma in topological insulators
Electron-electron interaction ?
Ostrovsky, Gornyi, & Mirlin, PRL 105, 036803 (2010).
localized
metalLd
dgg,0,0
ln)(
Based on one-loop RG argument: Interaction ‘‘kills’’ the supermetal.
No interaction
With interaction
Dilemma in topological insulators
Electron-electron interaction ?
Altshuler-Aronov effect ?Solid State Communications 30, 115 (1979).
Wang, Moses Chan (Penn state), et al, PRB 83, 245438 (2010).Liu, Yayu Wang (Tsinghua), et al, PRB 83, 165440 (2011).
Fock self-energy (exchange interaction)
Electron-electron interaction
Altshuler-Aronov effect
Disorder scattering
Altshuler-Aronov effect
Weak e-e interaction +
Strong disorder scattering
1D 2D 3D
,,,
/1ln
||
/1
/12
2
T
TT
Dqqd
T m
TT /1
Thermal diffusion
length
Altshuler-Aronov effect
First-order diagrams:
Altshuler, Aronov, and Lee, Phys. Rev. Lett. 44, 1288 (1980).H. Fukuyama, J. Phys. Soc. Jpn. 48, 2169 (1980).
Altshuler-Aronov effect
Pierre et al, PRL 86, 1590 (2001).
Conventionalelectrons m
p2
2Electron-electron
interaction +
Disorder scattering
Conductivity correction from e-e interaction
Suppress the density of states at the Fermi energy
WL-like temperaturedependence
(Thermal diffusion, so conductivity decreases
with decreasing T)
In topological insulator ?
Altshuler-Aronov theory was established for conventional electrons.
However
Surface states: massless Dirac fermions
Bulk states: massive Dirac fermions
Questions:
AA effect works for Dirac fermions?
A material-dependent analysis?
mp2
2
)( xyyx kkH
Model
qkkqkqkkkqkkqkkee ccccqvV
,','''' ))()((
21
kik
ei
2sin
2cos
zxyyx kkH 2
)(
i
ii RrurU )()(
Dirac model
Disorder potential
Electron-electron interaction
,2
)(0
2
qeqv
r
Now we need to include interaction
Lu & Shen, PRL 112, 146601 (2014)
Feynman diagrams
Quantum interference(WL / WAL)
qi
Semiclassical(Drude)
Electron-electron interaction(AA effect)
ee
sc
Now we need to include interaction
Altshuler, Aronov, and Lee, Phys. Rev. Lett. 44, 1288 (1980).H. Fukuyama, J. Phys. Soc. Jpn. 48, 2169 (1980).
Maximally crossed diagramsBergmann Phys. Rep. 1984McCann, Kechedzhi, Fal’ko, Suzuura, Ando, and Al'tshuler, PRL 97, 146805 (2006)
Lu & Shen, PRL 112, 146601 (2014)
Temperature and magnetic field dependence
)21(
2ln)1(
]ln)21([
2
22
2
22
1,02
2
2
22
B
T
T
ee
i
B
i
Bi
qi
Fh
eFh
e
he
Variables:
(1) Perpendicular magnetic field B
(2) Temperature T eBB 4
TkD
BT 2
Magnetic length
Thermal diffusion length
2/
1~ pT Phase coherence length
Dirac model parameters:
(1) Mass
(2) Velocity
Sample parameters:
(1) Mean free path
(2) Phase coherence length
(3) Screening factor
),( rF
vFE2
Relative dielectric constant
Quantum interference
Electron-electron interaction
Lu & Shen, PRL 112, 146601 (2014)
Conductivity vs. Temperature
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
/2EF=0
ee+
qi [e
2 /h] B=0
B=0
B=0 Tesla
WAL
Electron-electron
interactionQuantum
interference Total
AA
WL-like
We do have the Altshuler-Aronov
effect
C
ondu
ctiv
ity [e
2 /h]
Temperature [K]
Lu & Shen, PRL 112, 146601 (2014)
Conductivity vs. Temperature
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
/2EF=0
ee+
qi [e
2 /h] B=0
B=0
B=0 Tesla
WAL
Electron-electron
interactionQuantum
interference Total
AA
WL-like
... and Weak Anti-Localization
C
ondu
ctiv
ity [e
2 /h]
Temperature [K]
Lu & Shen, PRL 112, 146601 (2014)
Conductivity vs. Temperature
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
/2EF=0
ee+
qi [e
2 /h] B=0
B=0
B=0 Tesla
WAL
Electron-electron
interactionQuantum
interference Total
AA
WL-like
Together, a Weak Localization-like
temperature dependence
C
ondu
ctiv
ity [e
2 /h]
Temperature [K]
Lu & Shen, PRL 112, 146601 (2014)
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
/2EF=0
ee+
qi [e
2 /h] B=0
B=0
B=0 Tesla
WAL
Electron-electron
interactionQuantum
interference Total
AA
WL-like
Why interaction dominates the temperature
dependence ?
Check the slope
Conductivity vs. Temperature
C
ondu
ctiv
ity [e
2 /h]
Temperature [K]Te
hln2
Lu & Shen, PRL 112, 146601 (2014)
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
/2EF=0
ee+
qi [e
2 /h] B=0
B=0
B=0 Tesla
WAL
Electron-electron
interactionQuantum
interference Total
AA
WL-like
Slope from interference is
-1/2
Conductivity vs. Temperature
C
ondu
ctiv
ity [e
2 /h]
Temperature [K]
121 pqi
A universal valueMcCann et al, PRL 97, 146805 (2006); Lu, Shi & Shen, PRL, 107 076801 (2011).
p=12/~ pT
Peng, Cui et al (Stanford), Nat. Mater. 9, 225 (2010); Checkelsky, Ong et al (Princeton), PRL 106, 196801 (2011).
Lu & Shen, PRL 112, 146601 (2014)
Phase coherence length
EEI or EM E‐ph
1D 2/3
2D 1 3
3D 3/2
2/
1~ pT
Universal conductance fluctuationCheckelsky, Ong et al (Princeton), PRL 106, 196801 (2011)
AB oscillationPeng, Yi Cui et al (Stanford), Nat. Mater. 9, 225 (2010).
p in disordered metals
p depends on dimensionality and decoherence mechanisms
In experiments, p is relaxed to be a fitting parameter
p ~ 1 in topological insulators
Altshuler et al, J. Phys. C 15, 7367 (1982); Fukuyama, JPSJ, 53, 3299, (1984); Lee & Ramakrishnan, RMP 57, 287 (1985).
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
/2EF=0
ee+
qi [e
2 /h] B=0
B=0
B=0 Tesla
WAL
Electron-electron
interactionQuantum
interference Total
AA
WL-like
Slope from interference is
-1/2
Slope from interaction depends on Dielectric constant
Conductivity vs. Temperature
C
ondu
ctiv
ity [e
2 /h]
Temperature [K]
0 20 40 60 80 100
0.4
0.6
0.8
1.0
Slop
e
Dielectric constant
Teh ee
ee
ln2
Lu & Shen, PRL 112, 146601 (2014)
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
/2EF=0
ee+
qi [e
2 /h] B=0
B=0
B=0 Tesla
WAL
Electron-electron
interactionQuantum
interference Total
AA
WL-like
0 20 40 60 80 100
0.4
0.6
0.8
1.0
Slop
e
Dielectric constant
εr =113 (Bi2Se3) 75 (Bi2Te3) Richter, Köhler, Becker, Phys. Status Solidi (b) 84, 619 (1977).
Topologicalinsulators
Slope from interaction depends on Dielectric constant
Slope from interference is
-1/2
Conductivity vs. Temperature
C
ondu
ctiv
ity [e
2 /h]
Temperature [K]
Teh ee
ee
ln2
Lu & Shen, PRL 112, 146601 (2014)
]1 ,0[)0(
)'(
VkkV
F Screening
factor
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
1
10
3
F
/2EF
r = 100
Dielectric constant: εr =113 (Bi2Se3) 75 (Bi2Te3) Richter, Köhler, Becker, Phys. Status Solidi (b) 84 (1977) 619.
Slope FTe
h43
21
ln2
Conductivity: massless
)/exp()(2
rrerV ]0 ,[ Screening length
Noscreening
Fullyscreened
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
/2EF=0
ee+
qi [e
2 /h] B=0
B=0
B=0 Tesla
WAL
Electron-electron
interactionQuantum
interference Total
AA
WL-like
C
ondu
ctiv
ity [e
2 /h]
Temperature [K]
In magnetic field(1) WAL is suppressed;
(2) Interaction part ? little change
0 2 4 60.0
0.5
1.0
0 2 4 6-1.0
-0.5
0.0
0 2 4 60.0
0.5
1.0
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
ee(B)+qi(B)
qi(B)
ee(B) /2EF=0
ee+
qi [e
2 /h]
510.1
B=0510.1
B=0
B=0~5 Tesla
WAL
Electron-electron
interactionQuantum
interference Total
AA
WL-like
Conductivity vs. Magnetic field
C
ondu
ctiv
ity [e
2 /h]
Temperature [K]
Lu & Shen, PRL 112, 146601 (2014)
MagnetoconductivityElectron-electron
interactionQuantum
interference Total
-1 0 1-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
ee
[ F
e2 /h ]
-1 0 1
qi
[e2 /h
]
B [Tesla]-1 0 1
/2EF = 0
/2EF = 0
/2EF = 0
ee
+ qi
[e2 /h
]
WAL
)0()( B
0 2 4 60.0
0.5
1.0
0 2 4 6-1.0
-0.5
0.0
0 2 4 60.0
0.5
1.0
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
ee(B)+qi(B)
qi(B)
ee(B) /2EF=0
ee+
qi [e
2 /h]
510.1
B=0510.1
B=0
B=0~5 Tesla
AA
WL-like
(1) Magnetoconductivity mainly from WAL
(2) Interaction part? little contribution
C
ondu
ctiv
ity [e
2 /h]
Temperature [K]
Magnetic field [T] Mag
neto
cond
uctiv
ity [e
2 /h]
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
1
10
3
F
/2EF
r = 100
Lu & Shen, PRL 112, 146601 (2014)
Theory & ExperimentsElectron-electron
interactionQuantum
interference Total
-1 0 1-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
ee
[ F
e2 /h ]
-1 0 1
qi
[e2 /h
]
B [Tesla]-1 0 1
/2EF = 0
/2EF = 0
/2EF = 0
ee
+ qi
[e2 /h
]
WAL
0 2 4 60.0
0.5
1.0
0 2 4 6-1.0
-0.5
0.0
0 2 4 60.0
0.5
1.0
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
ee(B)+qi(B)
qi(B)
ee(B) /2EF=0
ee+
qi [e
2 /h]
510.1
B=0510.1
B=0
B=0~5 Tesla
AA
Chen, Y.Q. Li, K. H. Wu, L. Lu, et al (IOP), PRB 83, 241304 (R) (2011)
WL-like
Theory & experiments:
Conductivity, temperatures and magnetic fields,
All of same ordersTemperature [K]
Magnetic field [T]
The higher bound of the temperature is given by
Tkv
TkD
BBT 22
eV.A3v
The theory is valid for temperatures well below T~55K when the mean free path is about 10nm, and
Lu & Shen, PRL 112, 146601 (2014)
-1 0 1-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
ee
[ F
e2 /h ]
-1 0 1
qi
[e2 /h
]
B [Tesla]-1 0 1
/2EF=0.999/2EF=0.999
/2EF = 0.999
ee
+ qi
[e2 /h
]
0 2 4 60.0
0.5
1.0
0 2 4 60.0
0.5
1.0
0 2 4 60.0
0.5
1.0
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
ee(B)+qi(B)qi(B)
B=0~5 Tesla
/2EF=0.999
ee+
qi [e
2 /h]
5
10.1
B=0
510.1
B=0
ee(B)
WL
AA
Theory & ExperimentsElectron-electron
interactionQuantum
interference Total
WAL
AA
Chen, Y.Q. Li, K. H. Wu, L. Lu, et al (IOP), PRB 83, 241304 (R) (2011)
Not consitent with large mass limit.
Massless Dirac fermions dominate
Magnetic field [T]
Large-gap
Temperature [K]
Lu & Shen, PRL 112, 146601 (2014)
Conductivity & MagnetoconductivityElectron-electron
interactionQuantum
interference Total
-1 0 1-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
ee
[ F
e2 /h ]
-1 0 1
qi
[e2 /h
]
B [Tesla]-1 0 1
/2EF = 0
/2EF = 0
/2EF = 0
ee
+ qi
[e2 /h
]
WAL
)0()( B
(1) Interaction Temperature dependence of conductivity
(2) Quantum interferenceMagnetoconductivity
0 2 4 60.0
0.5
1.0
0 2 4 6-1.0
-0.5
0.0
0 2 4 60.0
0.5
1.0
0.1 1 10-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ee [e
2 /h]
0.1 1 10
qi [e
2 /h]
Temperature [K]0.1 1 10
ee(B)+qi(B)
qi(B)
ee(B) /2EF=0
ee+
qi [e
2 /h]
510.1
B=0510.1
B=0
B=0~5 Tesla
AA
WL-like
C
ondu
ctiv
ity [e
2 /h]
Mag
neto
cond
uctiv
ity [e
2 /h] Temperature [K]
Magnetic field [T] Lu & Shen, PRL 112, 146601 (2014)
Summary
Quantum interference + electron-electron interaction explains
conductivity and magnetoconductivity of TIs [Lu & Shen, PRL 112, 146601 (2014)].
Generalization to other Dirac systems: MoS2, silicene, etc [Lu, Xiao, Yao & Shen, PRL, 110, 016806 (2013);
Iwasa group (Tokyo) Science 2012; Nat. Phys. 2013; Peide Ye group (Purdue) ACS Nano 2013]
Electron-electroninteraction
Quantuminterference
WL-like
WAL-like
C
ondu
ctiv
ity
Magnetic field Temperature
Thanks a lot!!!Collaborators: Prof. Shun-Qing Shen (HKU), Prof. Junren Shi (PKU), Dr. Hong-Chao Liu (HKUST), Dr. Hong-Tao He (HKUST&SUSTC), Prof. Jian-Nong Wang (HKUST), Prof. Wang Yao (HKU), Prof. Di Xiao (CMU), Prof. Fu-Chun Zhang Acknowledgment: Prof. Michael Ma (Cincinnati & HKU)
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