2 Nov 2017 @ NQS2017 in YITP Estimation of Pseudo Magnetic Field for Isotropic/Anisotropic Dirac Cones T oshikaze Kariyado MANA, NIMS 2 Nov 2017 arXiv:1707.08601
2 Nov 2017 @ NQS2017 in YITP Motivation & Background Landau levels without an external magnetic field. Essence Dirac cones shift as gauge field Any system with Dirac cones! Even for a system inert � k + � A to magnetic field: charge � k neutral particles, photons, phonons...
2 Nov 2017 @ NQS2017 in YITP Example: Graphene under Strain † Theory: F. Guinea et al. , Nat. Phys. 6 , 30 (2010). Exp.: N. Levy et al. , Science 329 , 544 (2010). b A B 3 a 2 D ( E ) 1 0 –0.2 –0.1 0 0.1 0.2 E (eV)
2 Nov 2017 @ NQS2017 in YITP Example: Artifitial System K. K. Gomes et al. , Nature 483 , 306 (2012). 2D electrons on Cu surface with arranged molecule deposition 0 0.15 60 Å 60 T 15 Å 0 T 60 T d ˜ ˜ B B B = 60 T g (theory) 45 T ˜ ˜ B = 60 T ˜ B = 60 T 2 nm E D c 1.0 30 T K 60 T 60 T g (experiment) M E D 45 T g (eV –1 nm –2 ) 45 T 0.5 E D ˜ 30 T 30 T 15 T ˜ 15 T 15 T E M ˜ ˜ 0.0 B = 0 T B = 0 T 0 T 5 nm –200 0 200 –200 0 200 –200 0 200 ˜ B V (mV) V (mV) V (mV)
2 Nov 2017 @ NQS2017 in YITP Quantum Oscillation Strained 3D Weyl semimetal 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 DOS(10meV) 0.02 u max 0.01 a b b 0.00 strain magnetic field σ yy (b)/ σ yy (0) 0.8 continuum B 0.6 x x 0.4 z z y 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -1 ] 1/B, 1/b [T T. Liu, D. I. Pikulin, and M. Franz, Phys. Rev. B 95 , 041201 (2017).
2 Nov 2017 @ NQS2017 in YITP Valley Imbalance a l r e d o e u p s e a l r
2 Nov 2017 @ NQS2017 in YITP Valley Imbalance Valley dependent Lorentz force in strained graphene (a) (b) y W R L 0 x A. Chaves et al. , Phys. Rev. B 82 , 205430 (2010).
2 Nov 2017 @ NQS2017 in YITP Valley Imbalance Landau level splitting in strained graphene (a) 2 b B 2 b B b B b B 0 0 b B b B (b) 2 b B 2 b B 2 D 2 D B. Roy, Z.-X. Hu, and K. Yang, Phys. Rev. B 87 , 121408 (2013).
2 Nov 2017 @ NQS2017 in YITP T opic 1. Simple setup for pseudo magnetic field generation ◮ not necessary strain 2. Concise formula to estimate pseudo magnetic field ◮ Counting number of “observable” Landau levels ◮ Effects of anisotropy of Dirac cones 3. Application to an exsisting material ◮ 3D Dirac cones in an antiperovskite family
2 Nov 2017 @ NQS2017 in YITP Setup “Simplest” configuration bulk 1 buffer L “ B ” Δ E layer Δ k y bulk 2 bulk 1 bulk 2 k z Important Parameters ◮ L : thickness of the buffer layer ◮ Δ k : size of the Dirac cone shift See also, A. G. Grushin et al. , Phys. Rev. X 6 , 041046 (2016). C. Brendel et al. , Proc. Natl. Acad. Sci. USA 114 , 3390 (2017). H. Abbaszadeh et al. , arXiv:1610.06406.
2 Nov 2017 @ NQS2017 in YITP Formulation H ( ± ) σ ←→ H ( ± ) = ℏ ( − � = ℏ ( � k ± � ∇ ± � k 0 ) · � k 0 ( y )) · � σ � k ℏ ℏ Δ k h R A ( ± ) = ∓ � � | � B | = | � ∇ × � k 0 ( y ) , A | ∼ = e 2 e e L N 2 πR Δ k = , L = N bulk 1 buffer L “ B ” Δ E layer Δ k y bulk 2 bulk 1 bulk 2 z k
2 Nov 2017 @ NQS2017 in YITP Formulation H ( ± ) σ ←→ H ( ± ) = ℏ ( − � = ℏ ( � k ± � ∇ ± � k 0 ) · � k 0 ( y )) · � σ � k ℏ ℏ Δ k h R A ( ± ) = ∓ � � | � B | = | � ∇ × � k 0 ( y ) , A | ∼ = e 2 e e L N 2 πR Δ k = , L = N R : Dirac cone shift, N : buffer thickness bulk 1 buffer L “ B ” Δ E layer Δ k y bulk 2 bulk 1 bulk 2 z k
2 Nov 2017 @ NQS2017 in YITP Formulation R : Dirac cone shift, N : buffer thickness ◮ typical case R B | ∼ 1 . 6 × 10 4 × | � [T] ∼ 5 Å → N ◮ observable Landau levels � π � 4 π 2 ℏ 2 R | n | � ℏ Δ k → | n | < NR E n = < 4 N 2 2 bulk 1 buffer L “ B ” Δ E layer Δ k y bulk 2 bulk 1 bulk 2 z k ;
2 Nov 2017 @ NQS2017 in YITP T oy Model H � k = [ 1 + δ + 2 ( cos k + cos k y )] σ z + 2 α sin k y σ y � 3 [( ˜ k − ˜ H � k ∼ − δ ) σ z + ˜ αk y σ y ] 2 π δ 2 α ˜ ˜ k = k − , δ = , α = ˜ � � 3 3 3 ˜ & ˜ δ ↔ A α ↔ y /
2 Nov 2017 @ NQS2017 in YITP Results R : Dirac cone shift, N : buffer thickness 1.0 0.5 Energy N = 0 N = 10 0.0 π π 4 NR = 0 4 NR ∼ 0 . 8 − 0.5 − 1.0 1.0 0.5 Energy N = 30 N = 50 0.0 π π 4 NR ∼ 2 . 4 4 NR ∼ 3 . 9 − 0.5 − 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 k x [2 π / a ] k x [2 π / a ] ;
wave function tail hits the boundary no longer Landau level extension of the wave function extension thickness 2 Nov 2017 @ NQS2017 in YITP Discussion bulk2 1.0 0.5 Energy 0.0 buffer − 0.5 − 1.0 0.0 0.1 0.2 0.3 0.4 0.5 k x [2 π / a ] bulk1 k
wave function tail hits the boundary no longer Landau level extension of the wave function extension thickness 2 Nov 2017 @ NQS2017 in YITP Discussion bulk2 1.0 0.5 Energy 0.0 buffer − 0.5 − 1.0 0.0 0.1 0.2 0.3 0.4 0.5 k x [2 π / a ] bulk1 k
wave function tail hits the boundary no longer Landau level extension of the wave function extension thickness 2 Nov 2017 @ NQS2017 in YITP Discussion bulk2 1.0 0.5 Energy 0.0 buffer − 0.5 − 1.0 0.0 0.1 0.2 0.3 0.4 0.5 k x [2 π / a ] bulk1 k
2 Nov 2017 @ NQS2017 in YITP Discussion bulk2 1.0 0.5 Energy 0.0 buffer − 0.5 − 1.0 0.0 0.1 0.2 0.3 0.4 0.5 k x [2 π / a ] bulk1 k wave function tail hits the boundary → no longer Landau level ◮ extension of the wave function ∼ � n B ∝ � nN/R ◮ extension < thickness → n < NR
2 Nov 2017 @ NQS2017 in YITP Anisotropy R : Dirac cone shift, N : buffer thickness Δ E = ℏ Δ k π � | n | < NR � 4 π y ℏ 2 R | n | � 4 y E n = N 2 1.0 0.5 Energy N = 50 N = 50 0.0 / y ∼ 2 / y ∼ 0 . 5 − 0.5 − 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 k x [2 π / a ] k x [2 π / a ] Anisotropy is advantageous for observing the LL structure! ;
Dirac Point 2 Nov 2017 @ NQS2017 in YITP Materials TK and M. Ogata, J. Phys. Soc. Jpn. 80 , 083704 (2011). ◮ Antiperovskite A 3 EO (A=Ca,Sr,Ba and E=Sn,Pb) family O Ca, Sr, Ba Ca 3d Sn, Pb overlap 3D Dirac cones (with tiny Pb 6p mass) by d-p overlap! O 2p 3D linear dispersion Pb 6s ;
2 Nov 2017 @ NQS2017 in YITP Materials TK and M. Ogata, arXiv:1705.08934, to appear in PRMaterials. Band overlap Ca Sr Ba overlap dominant Sn SOC Pb J = 3 SOC 2 dominant J = 1 2 (Potentially) tunable!
2 Nov 2017 @ NQS2017 in YITP Materials ◮ Ba 3 SnO (band inversion dominant) vs Ca 3 PbO (SOC dominant) Ba 3 SnO Ca 3 PbO 1.0 7 6 7 7 + 7 − + 8 0.5 6 8 − 6 6 7 Energy (eV) 8 7 − 6 + 7 8 0.0 + + 7 8 + overlap 8 7 7 6 SOC − 0.5 6 6 − 6 6 − 1.0 X � � X
2 Nov 2017 @ NQS2017 in YITP Strategy ◮ Inducing Dirac cone shift by modulating chemical composition ◮ Ca 3 SnO ↔ Sr 3 SnO ◮ Estimating R instead of | B pseudo | , to avoid computational burden
2 Nov 2017 @ NQS2017 in YITP (Quasi) Ab-Initio Estimation: Wannier Interpolation 1. Derive effective models for the two end materials Ca 3 SnO and Sr 3 SnO 2. Interpolate the parameters to obtain a model for Ca 3 ( 1 − ) Sr 3 SnO x = 4/9 0.4 x = 5/9 Energy (eV) 0.2 0 − 0.2 − 0.4 0 0.05 0.1 0.15 0.2 0.25 k x [2 π / a ]
2 Nov 2017 @ NQS2017 in YITP (Quasi) Ab-Initio Estimation: Wannier Interpolation 1. Derive effective models for the two end materials Ca 3 SnO and Sr 3 SnO 2. Interpolate the parameters to obtain a model for Ca 3 ( 1 − ) Sr 3 SnO x = 4/9 0.4 x = 5/9 Energy (eV) 0.2 0 R ∼ 0 . 006 − 0.2 − 0.4 0 0.05 0.1 0.15 0.2 0.25 k x [2 π / a ]
2 Nov 2017 @ NQS2017 in YITP (Quasi) Ab-Initio Estimation ◮ heterostructure Ca 3 ( 1 − ) Sr 3 SnO, = ( = 0 + = 1 ) / 2 1.0 O Energy (eV) Ca 0.5 k x = 0.092 0.0 x = 4/9 Sr − 0.5 Sn − 1.0 1.0 Energy (eV) 0.5 Sr k x = 0.114 0.0 x = 5/9 Ca − 0.5 − 1.0 0 0.05 0.1 0.15 0.2 0.25 k x [2 π / a ]
2 Nov 2017 @ NQS2017 in YITP (Quasi) Ab-Initio Estimation ◮ heterostructure Ca 3 ( 1 − ) Sr 3 SnO, = ( = 0 + = 1 ) / 2 1.0 O Energy (eV) Ca 0.5 k x = 0.092 0.0 x = 4/9 Sr − 0.5 Sn − 1.0 R ∼ 0 . 022 1.0 Energy (eV) 0.5 Sr k x = 0.114 0.0 x = 5/9 Ca − 0.5 − 1.0 0 0.05 0.1 0.15 0.2 0.25 k x [2 π / a ]
2 Nov 2017 @ NQS2017 in YITP Fabrication of Films ◮ Sr 3 PbO, molecular beam epitaxy, thickness 200nm-300nm D. Samal, H. Nakamura, and H. Takagi, APL Mater. 4 , 076101 (2016). ◮ Ca 3 SnO, pulsed laser deposition M. Minohara et al ., arXiv:1710.03406.
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