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 F, W F’, W’ e (f – f’) = -(W-W’) : potential gradient constant potential inside the metal W -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 e (f – f’) = -(W-W ’) : potential gradient Metallic states exist on all surfaces of TI; work function(vs. ionization potential) is well-defined. Insulating bulk Nontrivial topology of Bi2Se3 Surface dependent work function Topological magnetoelectric effect described by axion electrodynamics

2. Electronic structures of Bi2Se3 – Crystal structures top surface (111) side surface (110)

2. Electronic structures of Bi2Se3 – Band structures from ab-initio study top surface (111) side surface (110) Well defined single Dirac cone on each surface (distorted on [110])

2. Electronic structures of Bi2Se3 – 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. Electronic structures of Bi2Se3 – 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. Topological magnetoelectric effect – 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 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. Topological magnetoelectric effect – Modified Maxwell equations by axion field 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 ( 𝜌 𝑡 ) : topological current ( 𝑡 ) Topological magnetoelectric effect can be described phenomenologically in terms of axion electrodynamics.

3. Topological magnetoelectric effect – 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 𝑀 𝑡 =− 𝑗 𝑡 𝑐 =−(𝑛+ 1 2 ) 𝑒 2 ℎ𝑐 𝐸 (Magnetization: ) X.-L.Qi et al., PRB 78, 195424 (2008) 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 T-breaking gap for all surfaces 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 – Potentials 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) Magnetic scalar potential (10-6C/s) Electric field (107V/m) Magnetic field (Gauss)

4. Axion electrodynamics in TI – Comparison with the previous result F, W F’, W’ potential gradient 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 There is a large work function difference between surfaces of different orientations of TI. Large electric fields inside the TI give rise to the magnetic ordering along the edges through the topological magnetoelectric effect. Our demonstration can be a useful basis to realize the axion electrodynamics in real solids.

1. Motivations – Work functions F, W F’, W’ e (f – f’) : potential gradient 2 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’ 1 metal 3 In all crystals, a work function of a surface depends on its orientation. Thus there should be a potential gradient across the facets.

2. Electronic structures of Bi2Se3 – Work function naïve approach From the bulk Hamiltonian of Bi2Se3, surface energy bands can be obtained by appropriate projections [PRB 86, 075302 (2012)] where and Then,

2. Electronic structures of Bi2Se3 – Work function surface dipoles z ρ(z) Assuming rombohedral structure with a = 4.08 Å and c = 29.8 for (111) and [101] facets and using a fact from graphene, ΔW = 0.5 eV Phys. Rev. 49, 653 (1936)

3. Topological magnetoelectric effect – 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. Topological magnetoelectric effect – 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)