Download 1. Jisoon IHM_jihm_axion at TI_APCTP_20160325

Axion electrodynamics on the surface
of topological insulators
March 25, 2016
Jisoon Ihm
Department of Physics
POSTECH
Collaborators
Yea-Lee Lee (Postech)
Hee Chul Park (IBS)
Young-Woo Son (KIAS)
Y.-L.Lee et al, PNAS 112, 11514 (2015)
1. Motivations
– Work function
W
F, W
F’, W’
constant potential
inside the metal
e (f – f’) = -(W-W’)
: potential gradient
-W’
• In general, the work function (W) depends on surface orientations.
• Thus, there should be a potential gradient across the facets.
1. Motivations
– Surface dependent work function on TI
W
W’
Metallic states exist on
all surfaces of TI;
work function(vs.
ionization potential) is
well-defined.
Surface dependent
work function
Insulating
bulk
e (f – f’) = -(W-W’)
: potential gradient
Nontrivial topology of Bi2Se3
Topological magnetoelectric effect
described by axion electrodynamics
2 . E l e c t r o n i c s t r u c t u r e s o f B i 2S e 3
– Crystal structures
top surface (111)
side surface (110)
2 . E l e c t r o n i c s t r u c t u r e s o f B i 2S e 3
– Band structures from ab-initio study
top surface (111)
side surface (110)
Well defined single Dirac cone on each surface (distorted on [110])
2 . E l e c t r o n i c s t r u c t u r e s o f B i 2S e 3
– Work functions depending on surfaces
top surface (111)
side surface (110)
• Work function of (111) = 5.84 eV, Work function of (110) = 5.04 eV
• 0.80 eV difference in work functions between (111) and (110) facets
2 . E l e c t r o n i c s t r u c t u r e s o f B i 2S e 3
– Work function around nanorod
Bi2Se3 (insulator with surface states)
Al (metal)
C.J.Fall et al., PRL 88, 156802 (2002)
cf: Characteristics of TI compared with NI
L.Fu and C.L.Kane PRB 74, 195312 (2006) and 76, 045302 (2007)
3 . To p o l o g i c a l m a g n e t o e l e c t r i c e f f e c t
– Modified Maxwell equations by axion field
(
, and q is axion field determined by topology.)
is a total derivative and doesn’t contribute to dynamics. However, if 𝜃

depends on r or t, it contributes to dynamics.
In strong CP problem, 𝜃(pseudo-scalar) is promoted to an axion field.
𝜃 is the relative phase between topologically distinct vacuum structures
(instanton); Analogy to TI holds.
In QCD, the measured 𝜃 becomes (adjusts itself to) zero by axion(through
promotion to a dynamic variable) and TRS is restored; Analogy to TI is
unclear and to establish it is a challenge for the future.
3 . To p o l o g i c a l m a g n e t o e l e c t r i c e f f e c t
– Modified Maxwell equations by axion field
Modified Maxwell equations
constitutive relations
F. Wilczek PRL 58, 1799 (1987)
𝜃 = 0 for NI and 𝜋 for TI
X.-L.Qi et al., PRB 78, 195424 (2008):
S.-C.Zhang group
: topological charge (𝜌𝑡 )
: topological current ( 𝑡 )
Topological magnetoelectric effect can be described phenomenologically
in terms of axion electrodynamics.
3 . To p o l o g i c a l m a g n e t o e l e c t r i c e f f e c t
– TME in TI with broken TRS
Is there ambiguity if we choose −𝜋 ? (No, sign is chosen by TRS breaking.)
Is there ambiguity if we choose 3𝜋, 5𝜋, etc.? (No, it corresponds to excitation.)
TRS-breaking gap for surfaces states by FM;
one sign of current (
) is chosen.
Fermi level should lie inside the gap.
Apply external electric fields E
(In our case, E already exists in TI.)
Circulating Hall current flows
𝑛: excitation
X.-L.Qi et al., PRB 78, 195424 (2008)
𝑗𝑡
1 𝑒2
(Magnetization: 𝑀𝑡 = − = −(𝑛 + ) 𝐸 )
𝑐
2 ℎ𝑐
Why is independent of the TRS breaking B?
DOS ∝ B, velocity ∝ 𝐸/𝐵,
∝ 𝐵 ∙ 𝐸/𝐵 = 𝐸
𝑡
: dissipationless (bound current)
4. Axion electrodynamics in TI
– The model
Assume that
1) T-breaking gap for all surfaces
2) Fermi level is inside the gap
4. Axion electrodynamics in TI
– A new numerical approach
Variational problem of ‘axion electrodynamics’
Minimization of F with Dirichlet boundary conditions.
(boundary conditions)
Numerically solve it using finite element method
4. Axion electrodynamics in TI
– Po t e n t i a l s
Electric potential (V)
Magnetic scalar potential (10-6C/s)
4. Axion electrodynamics in TI
– Fields
Electric field (107V/m)
Magnetic field (Gauss)
At 5 nm above the corner, E ~ 4x107 V/m and B ~ 140mGauss
4. Axion electrodynamics in TI
– Smoothing boundary conditions
Electric field (107V/m)
Magnetic field (Gauss)
At 5 nm above the corner, E ~ 2.6x107 V/m and B ~ 130mGauss
4. Axion electrodynamics in TI
– Near the edges
Electric potential (V)
Electric field (107V/m)
Magnetic scalar potential (10-6C/s)
Magnetic field (Gauss)
4. Axion electrodynamics in TI
– Comparison with the previous result
potential gradient
F, W
F’, W’
TI
X.-L.Qi et al., Science 323, 1184 (2009)
Work function difference of 0.8eV
B ~ 140 mGauss
at 5 nm above the corner
Electron gas of n=1011/cm2, R=1mm
B ~ 1.7 mGauss
Conclusions
1.
There is a large work function difference between surfaces of
different orientations of TI.
2.
Large electric fields inside the TI give rise to the magnetic
ordering along the edges through the topological
magnetoelectric effect.
3.
Our demonstration can be a useful basis to realize the axion
electrodynamics in real solids.
1. Motivations
– Work functions
e (f – f’) : potential gradient
2
F, W
F’, W’
1
metal
3
Zero total work is done in taking an
electron from an interior level at the
Fermi energy over the path, returning
it at the end to an interior level at the
Fermi energy (Ashcroft & Mermin)
12 : W
23 : e (f - f’) = -(W-W’)
31 : -W’
• In all crystals, a work function of a surface depends on its orientation.
• Thus there should be a potential gradient across the facets.
2 . E l e c t r o n i c s t r u c t u r e s o f B i 2S e 3
– Work function naïve approach
From the bulk Hamiltonian of Bi2Se3, surface energy bands can
be obtained by appropriate projections [PRB 86, 075302 (2012)]
where
Then,
and
2 . E l e c t r o n i c s t r u c t u r e s o f B i 2S e 3
– Work function surface dipoles
z
ρ(z)
 ( z) 

2d
e
V ( z) 
4
 0 sin

d
z, z  [ d , d ] otherwise 0.
d
PR
 R3 d A where P   d z  ( z )dz  e 0d
Vvac
e2 0 d

where  =(1+ TI ) / 2
2
Assuming rombohedral structure with a = 4.08 Å and c = 29.8 for (111) and
[101] facets and using a fact from graphene,  TI  1/ vx v y
2
c
,
d

 V(111)  0.64 eV
2
12
3a
3
a
[101] facet:  0  , d 
 V[101]  0.13 eV
ac
2 3
(111) facet:  0 
ΔW = 0.5 eV
Phys. Rev. 49, 653 (1936)
3 . To p o l o g i c a l m a g n e t o e l e c t r i c e f f e c t
– TME in TI with broken TRS
PRB 78, 195424 (2008)
Hall conductance
Circulating Hall current
Magnetization generated by Hall current
Topological contribution
to bulk magnetization
Topological contribution
to bulk polarization
3 . To p o l o g i c a l m a g n e t o e l e c t r i c e f f e c t
– Definitions
Linear magnetoelectric polarizability
, where
(
)
The last pseudoscalar term is not originated from the motion of ions, instead
This term is a total derivative not affecting electrodynamics
PRL 102, 146805 (2009)
4. Axion electrodynamics in TI
– A new numerical approach
Variational problem of ‘axion electrodynamics’
4. Axion electrodynamics in TI
– A new numerical approach
Solving a variational problem of ‘axion electrodynamics’ by the Ritz
method using the triangularization of the whole domain.
W on Γ
Supplementary Information
– Charge density
Electric charge (109C/m2)
Magnetic charge (1015C/ms)
Collaborators
Yea-Lee Lee (Postech)
Hee Chul Park (IBS)
Young-Woo Son (KIAS)
Y.-L.Lee et al, PNAS 112, 11514 (2015)