new$perspectives$in$spintronic and$mesoscopic physics ...at issp university of tokyo 6/12/2015...

Coupled charge and magnetizationin a Weyl semimetal

New Perspectives in Spintronic and Mesoscopic Physicsat ISSP University of Tokyo 6/12/2015

K. NomuraInstitute for Materials Research, Tohoku University

in collaboration withDaichi Kurebayashi

Introduction: What is a Weyl semimetal1. Weyl semimetal in a magnetic Topological insulator2. Magnetization-dynamics-induced charge pumping3. Charge-induced spin torque

outline



What is a Weyl semimetal?

A Weyl semimetal is three-‐dimensional analogue of graphene

H =vFm0 px − ipy

px + ipy −m0

"

#

$$

%

&

'' H =vF

pz px − ipypx + ipy −pz

"

#

$$

%

&

''

Dirac cone is stable in 3D

kykx

E

K

K’

K’

K’

K

K

2D 3D

m0= 0 (for massless)

Spin-orbit coupling

Topological states of matters

Topologically nontrivial states

E

_

+ _

+

Dirac semimetalNI TI

m0 (Band gap)

• Topological insulators (gapped)

Kane & Mele (2005)

• Topological semimetals (gapless)

strong SOCweak SOC

Weyl semimetals

Dirac semimetals degenerate

non-‐degenerate

2D (graphene) Wallace (1947), …

3D (accidental) Herring (1937), …

(symmetry protected)Wang et al., Young et al.(2012),..

3D (I-‐symmetry broken) Murakami (2007)Halasz&Balents (2012), ..

3D (T-‐symmetry broken) Wan et al. (2011)Burkov&Balents (2012), ..

Dirac-Weyl semimetals

Weyl semimetals

Dirac semimetals

Dirac-Weyl semimetals

HDirac = kxα1+kyα2 +kzα3 +m0α4

m0=0 (massless) ai: 4x4 Dirac matrix

{αi ,α j} = 2δij

HWeyl = pxσ1 + pyσ 2 + pzσ 3

si: 2x2 Pauli matrix

k = p+K0

degenerate

non-‐degenerate

Dirac hamiltonian

Symmetry breaking

degenerate

b=0HDirac = kxα1+kyα2 +kzα3 +m0α4

Θ−1HDirac (k)Θ =HDirac (−k)Π−1HDirac (k)Π =HDirac (−k)

αi =0 σ i

σ i 0

!

"##

$

%&&, α4 =

I 00 −I

!

"#

$

%&

T-‐symmetry

I -‐symmetry

Symmetry breaking

αi =0 σ i

σ i 0

!

"##

$

%&&, α4 =

I 00 −I

!

"#

$

%&

spin of electrons

Σi =σ i 00 σ i

"

#$$

%

&''

+ bSz

Dirac hamiltonian


degenerate

b=0


T-‐symmetry

I -‐symmetry

Symmetry breaking

αi =0 σ i

σ i 0

!

"##

$

%&&, α4 =

I 00 −I

!

"#

$

%&

spin of electrons

Σi =σ i 00 σ i

"

#$$

%

&''

+ bSz

Dirac hamiltonian


degenerate

b=0

non-‐degenerate

2b


T-‐symmetry

I -‐symmetry

Symmetry breaking

+ JMSz

local spins

Dirac hamiltonian


2JM

degenerate

M=0

non-‐degenerate

Hsd = J S(Ri ) ⋅ Σi=1

Nimp∑

M = -‐ <S>

Symmetry breaking

local spins

F = 12χs

M 2 +12χe

m2 − JMm

=12χs

1−J2χeχs( )M 2

+12χe

m− χeJM( )2

local spin electron spin

+ JMSz

Dirac hamiltonian


2JM

degenerate

M=0

non-‐degenerate

Hsd = J S(Ri ) ⋅ Σi=1

Nimp∑

M = -‐ <S>

Symmetry breaking

local spins

F = 12χs

M 2 +12χe

m2 − JMm

=12χs

1−J2χeχs( )M 2

+12χe

m− χeJM( )2

local spin electron spin

2JM

degenerate

M=0

non-‐degenerate1−J2χsχe < 0

1−J2χsχe > 0

“Van Vleck ferromagnetism”

M ≠ 0

M = 0

Introduction: What is a Weyl semimetal1. Weyl semimetal in a magnetic TI2. Magnetization-dynamics-induced charge pumping3. Charge-induced spin torque

outline



Self-consistent theory

HeMF =

k∑ ck

+ H0 (k )+ xJMΣz[ ]ck

H0 (k ) = Ri (k )αi +m0 (k )α4 +ε(k )Ii=1

3

∑

HSMF = Jm

I=1

Nimp

∑ Sz (rI )

Kurebayashi, KN JPSJ 83 (2014) 063709.

local spins

_

+

Htotal = HeMF + Hs

MF – Nimp J Mm

Electrons local spins

Virtual crystal approximation


HeMF =

k∑ ck

+ H0 (k )+ xJMΣz[ ]ck HSMF = Jm

I=1

Nimp

∑ Sz (rI )


m =1N

〈ci+Σzci 〉

i=1

N

∑ , M =1Ni

〈Sz (R!"I )〉

I=1

Nimp

∑

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 20 40 60 80 100 120 140

x=2%x=4%x=6%x=8%

x=10%

M

T [K]

Htotal = HeMF + Hs

MF – Nimp J Mm


Virtual crystal approximation

Magnetic transition

kzkzkz

M = 0 M ≠ 0

0

-1 -0.5 0 0.5 1

0

-1 -0.5 0 0.5 1

0

-1 -0.5 0 0.5 1


HeMF =

k∑ ck

+ H0 (k )+ xJMΣz[ ]ck HSMF = Jm

I=1

Nimp

∑ Sz (rI )


Htotal = HeMF + Hs

MF – Nimp J Mm


0

20

40

60

80

100

120

140

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Tem

pera

ture

[K]

Concentration [%]

Magnetic transition

kzkzkz

M = 0 M ≠ 0

0

-1 -0.5 0 0.5 1

0

-1 -0.5 0 0.5 1

0

-1 -0.5 0 0.5 1


Weyl semimetal in a magnetic TI

0

10

20

30

40

50

60

70

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Tem

pera

rture

[K]

M0 [eV]

Phase diagram

strong SOCweak SOC

0

-1 -0.5 0 0.5 1

0

-1 -0.5 0 0.5 1

Weyl semimetals

Dirac semimetals


Experiments of magnetic TIs

Science 339, 1582 (2013)

Bi2-‐yCry(Sex Te1-‐x)3

Anomalous Hall effect

ρxy= R0B + RsM

The Hall effect in materials with-‐ ferromagnetic order-‐ spin-‐orbit coupling

Bi2-‐yCry(Sex Te1-‐x)3

ρxy∝M (B→ 0)

J

VH

M

Ji= sij Ej


-0.5 0 0.5kz

E

3d Weyl SM

Hweyl(kx,ky,kz) = kxs1 + kys2 + Dm(kz)s3

ky=0kx=0ky=0


kx

ky

kz

Dm(kz) <0

Dm(kz) >0

Dm(kz) >0

-0.5 0 0.5kz

E


kx=0ky=0


kx

ky

kz

Dm(kz) <0

Dm(kz) >0

Dm(kz) >0

σ 2Dxy (kz ) =

e2

2h[1− sgn(Δm (kz ))]

sxy=e2/h

sxy= 0

sxy= 0


σxy3D =

dkz

2π∫ σ 2Dxy(kz)

=e2

2πh2KWeyl


σ 2Dxy (kz ) =

e2

2h[1− sgn(Δm (kz ))]

Kubo formula by D. Kurebayashi


!2Dxy ðkzÞ ¼

e2

h!

X

n

Zd2k?ð2"Þ2

fð#nk?ðkzÞÞ"znk?

ðkzÞ; ð18Þ

"nk ¼ rk $ Ank; ð19ÞAnk ¼ ihnkjrkjnki; ð20Þ

where "nk is the Berry curvature, and Ank is the Berryconnection. The anomalous Hall conductivity is given byintegrating the two-dimensional Chern number over kz,

!xy ¼Z "=az

%"=az

dkz2"

!2Dxy ðkzÞ; ð21Þ

where az is a lattice constant. Figure 5 shows the numericallyobtained anomalous Hall conductivity !xy at T ¼ 0K as afunction of the concentration of doped magnetic impurities.We see that the anomalous Hall conductivity is proportionalto the concentration of magnetic impurities when theconcentration is small. This behavior is due to the fact thatthe two-dimensional Chern number is quantized in units ofe2=h between two Weyl points, and vanishes at othermomenta.6) Namely the anomalous Hall conductivity isproportional to the momentum of the Weyl point kD which iseasily evaluated kD as kD ¼ &JMzx=A2az. This analyticalresult also shows the anomalous Hall conductivity !xy ¼JMz

A2azx which is consistent with our numerical result. At the

concentration x ' 30%, anomalous Hall conductivity reachesquantized value !xy ¼ e2=ðhazÞ. In this region, a conductionband and a valence band are completely inverted by thestrong exchange interaction and no longer possess three-dimensional massless Weyl fermions. This phase is under-stood as the quantum anomalous Hall insulator phase.

In conclusion, we have estimated possible conditions forrealizing the Weyl semimetal phase by doping magneticimpurities in Bi2Se3 family within the mean-field theory.We found that ferromagnetic ordering in these materials isseen below Tc ¼ 40K with magnetic impurity concentrationx ¼ 5% and m0 ¼ 0 eV which corresponds to the Diracsemimetal in the undoped limit. This magnetic ordering isintroduced by the Van-Vleck paramagnetism, and thus thecritical temperature depends on the strength of the spin–orbitinteraction. We confirmed that the critical temperaturedecreases with weakening the strength of the spin–orbit

interaction. As the exchange field increases, the conductionband and the valence band invert. The Weyl semimetal phaseis realized as the gapless phase, as shown in Fig. 3.

Acknowledgment This work was supported by Grants-in-Aid for ScientificResearch (Nos. 24740211 and 25103703) from the Ministry of Education,Culture, Sports, Science and Technology, Japan (MEXT).

1) S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S.von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger,Science 294, 1488 (2001).

2) T. Jungwirth, J. Mašek, J. Kučera, and A. H. MacDonald, Rev. Mod.Phys. 78, 809 (2006).

3) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev.B 83, 205101 (2011).

4) X. Wan, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. Lett. 108,146601 (2012).

5) G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Phys. Rev. Lett. 107,186806 (2011).

6) A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011).7) D. Bulmash, C. Liu, and X. Qi, Phys. Rev. B 89, 081106 (2014).8) H.-J. Kim, K.-S. Kim, J.-F. Wang, V. Kulbachinskii, K. Ogawa, M.

Sasaki, A. Ohnishi, M. Kitaura, Y.-Y. Wu, L. Li, I. Yamamoto, J.Azuma, M. Kamada, and V. Dobrosavljevi, Phys. Rev. Lett. 110,136601 (2013).

9) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010); X.-L.Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).

10) Y. L. Chen, J.-H. Chu, J. G. Analytis, Z. K. Liu, K. Igarashi, H.-H.Kuo, X. L. Qi, S. K. Mo, R. G. Moore, D. H. Lu, M. Hashimoto, T.Sasagawa, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen,Science 329, 659 (2010).

11) J. Zhang, C.-Z. Chang, P. Tang, Z. Zhang, X. Feng, K. Li, L.-L. Wang,X. Chen, C. Liu, W. Duan, K. He, Q.-K. Xue, X. Ma, and Y. Wang,Science 339, 1582 (2013).

12) R. Yu, W. Zhang, H.-J. Zhang, S. Zhang, X. Dai, and Z. Fang, Science329, 61 (2010).

13) T. Sato, K. Segawa, K. Kosaka, S. Souma, K. Nakayama, K. Eto, T.Minami, Y. Ando, and T. Takahashi, Nat. Phys. 7, 840 (2011).

14) H. Jin, J. Im, and A. J. Freeman, Phys. Rev. B 84, 134408 (2011).15) H.-J. Kim, K.-S. Kim, J.-F. Wang, M. Sasaki, N. Satoh, A. Ohnishi, M.

Kitaura, M. Yang, and L. Li, Phys. Rev. Lett. 111, 246603 (2013).16) H. Zhang, C. Liu, X. Qi, X. Dai, Z. Fang, and S. Zhang, Nat. Phys. 5,

438 (2009).17) C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S. C. Zhang, Phys.

Rev. B 82, 045122 (2010).18) G. Rosenberg and M. Franz, Phys. Rev. B 85, 195119 (2012).19) C. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou,

P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z.Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue,Science 340, 167 (2013).

20) S. Ryu and K. Nomura, Phys. Rev. B 85, 155138 (2012).

Fig. 5. (Color online) Anomalous Hall conductivity !xy as a function ofimpurity concentration with a parameter m0 ¼ 0 eVat T ¼ 0K. Red trianglesshow the simulation result and the black solid line is the analytical result,!xy ¼ 2JMzx=A2az.

Fig. 4. (Color online) (Upper figure) Chemical potential dependence ofCurie temperature with m0 ¼ 0 eVand x ¼ 5%, 10%. (Lower figure) Densityof states with m0 ¼ 0 eV and x ¼ 5%.

J. Phys. Soc. Jpn. 83, 063709 (2014) Letters D. Kurebayashi and K. Nomura

063709-4 ©2014 The Physical Society of Japan

σxy3D =

dkz

2π∫ σ 2Dxy(kz)

=e2

2πh2KWeyl

Lattice model

2JM

Remarks

0

10

20

30

40

50

60

70

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Tem

pera

rture

[K]

M0 [eV]

Magnetic order in TI as “Van Vleck ferromagnetism”.

Weyl semimetal can be realized in magnetic TIs. (Cr-‐doped Bi2(SexTe1-‐x)3)

Anomalous Hall effect is a signature of the Weyl semimetal.

Kubo formula by D. Kurebayashi

!2Dxy ðkzÞ ¼

e2

h!

X

n

Zd2k?ð2"Þ2

fð#nk?ðkzÞÞ"znk?

ðkzÞ; ð18Þ

"nk ¼ rk $ Ank; ð19ÞAnk ¼ ihnkjrkjnki; ð20Þ

where "nk is the Berry curvature, and Ank is the Berryconnection. The anomalous Hall conductivity is given byintegrating the two-dimensional Chern number over kz,

!xy ¼Z "=az

%"=az

dkz2"

!2Dxy ðkzÞ; ð21Þ

where az is a lattice constant. Figure 5 shows the numericallyobtained anomalous Hall conductivity !xy at T ¼ 0K as afunction of the concentration of doped magnetic impurities.We see that the anomalous Hall conductivity is proportionalto the concentration of magnetic impurities when theconcentration is small. This behavior is due to the fact thatthe two-dimensional Chern number is quantized in units ofe2=h between two Weyl points, and vanishes at othermomenta.6) Namely the anomalous Hall conductivity isproportional to the momentum of the Weyl point kD which iseasily evaluated kD as kD ¼ &JMzx=A2az. This analyticalresult also shows the anomalous Hall conductivity !xy ¼JMz

A2azx which is consistent with our numerical result. At the

concentration x ' 30%, anomalous Hall conductivity reachesquantized value !xy ¼ e2=ðhazÞ. In this region, a conductionband and a valence band are completely inverted by thestrong exchange interaction and no longer possess three-dimensional massless Weyl fermions. This phase is under-stood as the quantum anomalous Hall insulator phase.

In conclusion, we have estimated possible conditions forrealizing the Weyl semimetal phase by doping magneticimpurities in Bi2Se3 family within the mean-field theory.We found that ferromagnetic ordering in these materials isseen below Tc ¼ 40K with magnetic impurity concentrationx ¼ 5% and m0 ¼ 0 eV which corresponds to the Diracsemimetal in the undoped limit. This magnetic ordering isintroduced by the Van-Vleck paramagnetism, and thus thecritical temperature depends on the strength of the spin–orbitinteraction. We confirmed that the critical temperaturedecreases with weakening the strength of the spin–orbit

interaction. As the exchange field increases, the conductionband and the valence band invert. The Weyl semimetal phaseis realized as the gapless phase, as shown in Fig. 3.

Acknowledgment This work was supported by Grants-in-Aid for ScientificResearch (Nos. 24740211 and 25103703) from the Ministry of Education,Culture, Sports, Science and Technology, Japan (MEXT).

1) S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S.von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger,Science 294, 1488 (2001).

2) T. Jungwirth, J. Mašek, J. Kučera, and A. H. MacDonald, Rev. Mod.Phys. 78, 809 (2006).

3) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev.B 83, 205101 (2011).

4) X. Wan, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. Lett. 108,146601 (2012).

5) G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Phys. Rev. Lett. 107,186806 (2011).

6) A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011).7) D. Bulmash, C. Liu, and X. Qi, Phys. Rev. B 89, 081106 (2014).8) H.-J. Kim, K.-S. Kim, J.-F. Wang, V. Kulbachinskii, K. Ogawa, M.

Sasaki, A. Ohnishi, M. Kitaura, Y.-Y. Wu, L. Li, I. Yamamoto, J.Azuma, M. Kamada, and V. Dobrosavljevi, Phys. Rev. Lett. 110,136601 (2013).

9) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010); X.-L.Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).

10) Y. L. Chen, J.-H. Chu, J. G. Analytis, Z. K. Liu, K. Igarashi, H.-H.Kuo, X. L. Qi, S. K. Mo, R. G. Moore, D. H. Lu, M. Hashimoto, T.Sasagawa, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen,Science 329, 659 (2010).

11) J. Zhang, C.-Z. Chang, P. Tang, Z. Zhang, X. Feng, K. Li, L.-L. Wang,X. Chen, C. Liu, W. Duan, K. He, Q.-K. Xue, X. Ma, and Y. Wang,Science 339, 1582 (2013).

12) R. Yu, W. Zhang, H.-J. Zhang, S. Zhang, X. Dai, and Z. Fang, Science329, 61 (2010).

13) T. Sato, K. Segawa, K. Kosaka, S. Souma, K. Nakayama, K. Eto, T.Minami, Y. Ando, and T. Takahashi, Nat. Phys. 7, 840 (2011).

14) H. Jin, J. Im, and A. J. Freeman, Phys. Rev. B 84, 134408 (2011).15) H.-J. Kim, K.-S. Kim, J.-F. Wang, M. Sasaki, N. Satoh, A. Ohnishi, M.

Kitaura, M. Yang, and L. Li, Phys. Rev. Lett. 111, 246603 (2013).16) H. Zhang, C. Liu, X. Qi, X. Dai, Z. Fang, and S. Zhang, Nat. Phys. 5,

438 (2009).17) C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S. C. Zhang, Phys.

Rev. B 82, 045122 (2010).18) G. Rosenberg and M. Franz, Phys. Rev. B 85, 195119 (2012).19) C. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou,

P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z.Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue,Science 340, 167 (2013).

20) S. Ryu and K. Nomura, Phys. Rev. B 85, 155138 (2012).

Fig. 5. (Color online) Anomalous Hall conductivity !xy as a function ofimpurity concentration with a parameter m0 ¼ 0 eVat T ¼ 0K. Red trianglesshow the simulation result and the black solid line is the analytical result,!xy ¼ 2JMzx=A2az.

Fig. 4. (Color online) (Upper figure) Chemical potential dependence ofCurie temperature with m0 ¼ 0 eVand x ¼ 5%, 10%. (Lower figure) Densityof states with m0 ¼ 0 eV and x ¼ 5%.

J. Phys. Soc. Jpn. 83, 063709 (2014) Letters D. Kurebayashi and K. Nomura

063709-4 ©2014 The Physical Society of Japan

Lattice model

new$perspectives$in$spintronic and$mesoscopic physics ...at issp university of tokyo 6/12/2015...

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