Estimation of Pseudo Magnetic Field for Isotropic/Anisotropic Dirac - - PowerPoint PPT Presentation

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Estimation of Pseudo Magnetic Field for Isotropic/Anisotropic Dirac Cones

T

• shikaze 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

• k +

A

• k

Dirac cones shift as gauge field Any system with Dirac cones! Even for a system inert to magnetic field: charge 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). 1 3 2 –0.2 –0.1 E (eV) 0.1 0.2 D (E)

b

a

A B

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Example: Artifitial System

2D electrons on Cu surface with arranged molecule deposition

1.0

c

2 nm K 0.5 ED M ED EM –200 200 0.0 5 nm g (eV–1 nm–2) ˜ V (mV)

0.15 d 0 T 60 T 15 Å B = 60 T ED 45 T 60 T 45 T 60 T 15 T 30 T 15 T 30 T –200 200 –200 200 V (mV) V (mV) 60 Å 60 T 45 T 15 T 0 T B = 60 T ˜ B = 60 T ˜ B = 0 T ˜ B = 0 T ˜ B ˜ B ˜ g (theory) ˜ g (experiment) ˜ B ˜ 30 T

• K. K. Gomes et al., Nature 483, 306 (2012).

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Quantum Oscillation

Strained 3D Weyl semimetal

z x y umax a x z b b B

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.00 0.01 0.02 DOS(10meV) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1/B, 1/b [T

• 1]

0.0 0.2 0.4 0.6 0.8 σyy(b)/σyy(0)

strain magnetic field continuum

• T. Liu, D. I. Pikulin, and M. Franz, Phys. Rev. B 95, 041201 (2017).

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Valley Imbalance

p s e u d

• r

e a l r e a l

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Valley Imbalance

Valley dependent Lorentz force in strained graphene

L W R y x (a) (b)

• A. Chaves et al., Phys. Rev. B 82, 205430 (2010).

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Valley Imbalance

Landau level splitting in strained graphene

b B 2 b B b B 2 b B b B 2 b B b B 2 b B 2 D 2 D (a) (b)

• B. Roy, Z.-X. Hu, and K. Yang, Phys. Rev. B 87, 121408 (2013).

2 Nov 2017 @ NQS2017 in YITP

T

• pic
• 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

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Setup

“Simplest” configuration

“B” bulk 2 bulk 1

buffer layer

L

 y z

k Δk ΔE bulk 2 bulk 1

Important Parameters

◮ L: thickness of the buffer layer ◮ Δk: 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.

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Formulation

H(±)

• k

= ℏ( k ± k0) · σ ←→ H(±) = ℏ(− ∇ ± k0(y)) · σ

• A(±) = ∓

ℏ e

• k0(y),

| B| = | ∇ × A| ∼ ℏ e Δk L = h e2 R N Δk = 2πR  , L = N

“B” bulk 2 bulk 1

buffer layer

L

 y z

k Δk ΔE

bulk 2 bulk 1

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Formulation

H(±)

• k

= ℏ( k ± k0) · σ ←→ H(±) = ℏ(− ∇ ± k0(y)) · σ

• A(±) = ∓

ℏ e

• k0(y),

| B| = | ∇ × A| ∼ ℏ e Δk L = h e2 R N Δk = 2πR  , L = N R: Dirac cone shift, N: buffer thickness

“B” bulk 2 bulk 1

buffer layer

L

 y z

k Δk ΔE

bulk 2 bulk 1

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Formulation

◮ typical case

 ∼ 5Å → | B| ∼ 1.6 × 104 × R N [T]

◮ observable Landau levels

En =

• 4π2ℏ2R|n|

N2 < ℏΔk 2 → |n| < π 4 NR

“B” bulk 2 bulk 1

buffer layer

L

 y z

k Δk ΔE

bulk 2 bulk 1

R: Dirac cone shift, N: buffer thickness ;

2 Nov 2017 @ NQS2017 in YITP

T

• y Model

H

k = [1 + δ + 2(cos k + cos ky)]σz + 2α sin kyσy

H

k ∼ −

• 3[( ˜

k − ˜ δ)σz + ˜ αkyσy]

˜ k = k − 2π 3 , ˜ δ = δ

• 3

, ˜ α = 2α

• 3

˜ δ ↔ A & ˜ α ↔ y/

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Results

−1.0 −0.5 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 0.5 Energy kx [2π/a] −1.0 −0.5 0.0 0.5 1.0 Energy 0.0 0.1 0.2 0.3 0.4 0.5 kx [2π/a] N = 0

π 4 NR = 0

N = 10

π 4 NR ∼ 0.8

N = 30

π 4 NR ∼ 2.4

N = 50

π 4 NR ∼ 3.9

R: Dirac cone shift, N: buffer thickness ;

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Discussion

−1.0 −0.5 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 0.5 Energy kx [2π/a] k buffer bulk1 bulk2 wave function tail hits the boundary no longer Landau level extension of the wave function extension thickness

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Discussion

−1.0 −0.5 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 0.5 Energy kx [2π/a] k buffer bulk1 bulk2 wave function tail hits the boundary no longer Landau level extension of the wave function extension thickness

2 Nov 2017 @ NQS2017 in YITP

Discussion

−1.0 −0.5 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 0.5 Energy kx [2π/a] k buffer bulk1 bulk2 wave function tail hits the boundary no longer Landau level extension of the wave function extension thickness

2 Nov 2017 @ NQS2017 in YITP

Discussion

−1.0 −0.5 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 0.5 Energy kx [2π/a] k buffer bulk1 bulk2 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

ΔE = ℏΔk En =

• 4πyℏ2R|n|

N2 |n| < π 4  y NR −1.0 −0.5 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 0.5 Energy kx [2π/a] 0.0 0.1 0.2 0.3 0.4 0.5 kx [2π/a] N = 50 /y ∼ 2 N = 50 /y ∼ 0.5 Anisotropy is advantageous for observing the LL structure! R: Dirac cone shift, N: buffer thickness ;

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Materials

◮ Antiperovskite A3EO (A=Ca,Sr,Ba and E=Sn,Pb) family

               

Pb 6s O 2p Pb 6p

• verlap

Ca 3d

Dirac Point O Ca, Sr, Ba Sn, Pb

3D Dirac cones (with tiny mass) by d-p overlap! 3D linear dispersion

TK and M. Ogata, J. Phys. Soc. Jpn. 80, 083704 (2011).

;

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Materials

Band overlap SOC Ca Sr Ba Pb Sn J = 3

2

J = 1

2

• verlap

dominant SOC dominant

(Potentially) tunable!

TK and M. Ogata, arXiv:1705.08934, to appear in PRMaterials.

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Materials

◮ Ba3SnO (band inversion dominant) vs Ca3PbO (SOC dominant)

−1.0 −0.5 0.0 0.5 1.0 X

• Energy (eV)

Ba3SnO

• X

Ca3PbO

+

8

+

7

+

8

−

6

−

8

6 6 7 6 7 6 7 7 −

6

+

8

−

8

+

8

+

7

6 7 6 6 6 7 7 7

• verlap

SOC

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Strategy

◮ Inducing Dirac cone shift by modulating chemical composition

◮ Ca3SnO ↔ Sr3SnO

◮ Estimating R instead of |Bpseudo|, to avoid computational burden

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(Quasi) Ab-Initio Estimation: Wannier Interpolation

• 1. Derive effective models for the two end materials Ca3SnO and

Sr3SnO

• 2. Interpolate the parameters to obtain a model for

Ca3(1−)Sr3SnO −0.4 −0.2 0.2 0.4 0.05 0.1 0.15 0.2 0.25 Energy (eV) kx [2π/a] x = 4/9 x = 5/9

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(Quasi) Ab-Initio Estimation: Wannier Interpolation

• 1. Derive effective models for the two end materials Ca3SnO and

Sr3SnO

• 2. Interpolate the parameters to obtain a model for

Ca3(1−)Sr3SnO −0.4 −0.2 0.2 0.4 0.05 0.1 0.15 0.2 0.25 Energy (eV) kx [2π/a] x = 4/9 x = 5/9

R ∼ 0.006

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(Quasi) Ab-Initio Estimation

◮ heterostructure Ca3(1−)Sr3SnO,  = (=0 + =1)/2

−1.0 −0.5 0.0 0.5 1.0 0.05 0.1 0.15 0.2 0.25 kx = 0.114 x = 5/9 Energy (eV) kx [2π/a] −1.0 −0.5 0.0 0.5 1.0 kx = 0.092 x = 4/9 Energy (eV)

O Ca Sr Sn Sr Ca

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(Quasi) Ab-Initio Estimation

◮ heterostructure Ca3(1−)Sr3SnO,  = (=0 + =1)/2

−1.0 −0.5 0.0 0.5 1.0 0.05 0.1 0.15 0.2 0.25 kx = 0.114 x = 5/9 Energy (eV) kx [2π/a] −1.0 −0.5 0.0 0.5 1.0 kx = 0.092 x = 4/9 Energy (eV)

O Ca Sr Sn Sr Ca

R ∼ 0.022

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Fabrication of Films

◮ Sr3PbO, molecular beam epitaxy, thickness 200nm-300nm

• D. Samal, H. Nakamura, and H. Takagi, APL Mater. 4, 076101 (2016).

◮ Ca3SnO, pulsed laser deposition

• M. Minohara et al., arXiv:1710.03406.

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Summary

◮ Concise formulae for the pseudo magnetic field & pseudo

Landau levels B ∼ h e2 R N , |n| < π 4  y NR

◮ Anisotropic Dirac cones are better to observe LL structures.

◮ Estimation of R for an existing material

Perspective

◮ Interesting physical consequences!

◮ eg. coexistence with a real magnetic field

TK, arXiv:1707.08601