Realization of Axion Electrodynamics on Topological Insulators Jisoon IhmJisoon Ihm Department of Physics POSTECH June 1, 2016

Axion Electrodynamics on Topological Insulators

Collaborators Yea-Lee Lee (Postech) Hee Chul Park (IBS) Young-Woo Son (KIAS) Y.-L.Lee et al, PNAS 112, (2015)

1. Motivations – Work function In general, the work function ( W ) depends on surface orientations. Thus, there should be a potential gradient across the facets. F, W F’, W’ e (  –  ’) = -(W-W’) : potential gradient constant potential inside the metal W -W’-W’

1. Motivations – Surface dependent work function on TI W W ’W ’ e (  –  ’) = -(W-W ’) : potential gradient Insulating bulk Metallic states exist on all surfaces of TI; work function( vs. ionization potential) is well-defined. Topological magnetoelectric effect described by axion electrodynamics Nontrivial topology of Bi 2 Se 3 Surface dependent work function

2. Electronic structures of Bi 2 Se 3 – Crystal structures top surface (111)side surface (110)

2. Electronic structures of Bi 2 Se 3 – Band structures from ab-initio study Well defined single Dirac cone on each surface (distorted on [110]) top surface (111)side surface (110)

2. Electronic structures of Bi 2 Se 3 – Work functions depending on surfaces 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 top surface (111)side surface (110)

2. Electronic structures of Bi 2 Se 3 – Work function around nanorod C.J.Fall et al., PRL 88, (2002) Al (metal) Bi 2 Se 3 (insulator with surface states)

Topology of 2D Brillouin Zone : 2-Torus

cf I. Characteristics of TI (compared with NI) s-state p-state

cf II. Characteristics of TI (compared with NI) Background knowledge

cf II. Characteristics of TI (compared with NI) L.Fu and C.L.Kane PRB 74, (2006) and 76, (2007) 0T/2 0 TI NI 00 G/2 0 TI NI Time-reversed pairs of Wannier states (switching partners or not) Correspondence to energy bands

cf III. Characteristics of TI (compared with NI) L.Fu and C.L.Kane PRB 74, (2006) Because of restriction by TRS, wavefunctions cannot be defined continuously over the grey region. (which is called “obstruction” : two patches(A and B) are required to cover the grey region.) Integral of the transformation matrix between two patches in the overlapping region becomes nonzero. This value actually agrees with the one in cf II (Product of Pfaffians, or number of zeros of Pfaffians).

cf IV. Characteristics of TI (compared with NI): Phenomenological description using field theory (, and  is axion field determined by topology.) Unusual electronic structure of TI may be represented by Axion Lagrangian.

3. Topological magnetoelectric effect – Modified Maxwell equations by axion field Modified Maxwell equationsconstitutive relations F. Wilczek PRL 58, 1799 (1987) X.-L.Qi et al., PRB 78, (2008): S.-C.Zhang group Topological magnetoelectric effect can be described phenomenologically in terms of axion electrodynamics.

3. Topological magnetoelectric effect – TME in TI with broken TRS X.-L.Qi et al., PRB 78, (2008) Fermi level should lie inside the gap. Apply external electric fields E (In our case, E already exists in TI.) Circulating Hall current flows TRS-breaking gap for surfaces states by FM; one sign of current ( ) is chosen. (Magnetization: ) : 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’ Numerically solve it using finite element method (boundary conditions) Minimization of F with Dirichlet boundary conditions.

4. Axion electrodynamics in TI – Potentials Electric potential (V)Magnetic scalar potential (10 -6 C/s)

4. Axion electrodynamics in TI – Fields At 5 nm above the corner, E ~ 4x10 7 V/m and B ~ 140mGauss Electric field (10 7 V/m)Magnetic field (Gauss)

4. Axion electrodynamics in TI – Smoothing boundary conditions Electric field (10 7 V/m)Magnetic field (Gauss) At 5 nm above the corner, E ~ 2.6x10 7 V/m and B ~ 130mGauss

4. Axion electrodynamics in TI – Near the edges Electric potential (V)Magnetic scalar potential (10 -6 C/s) Electric field (10 7 V/m)Magnetic field (Gauss)

Electron gas of n=10 11 /cm 2, R=1  m B ~ 1.7 mGauss Work function difference of 0.8eV B ~ 140 mGauss at 5 nm above the corner X.-L.Qi et al., Science 323, 1184 (2009) 4. Axion electrodynamics in TI – Comparison with the previous result F, W F’, W’ potential gradient TI

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.