Orbital Edelstein effect as a condensed-matter analog of solenoid

Subscriber access provided by MT ROYAL COLLEGE

Communication

Orbital Edelstein effect as a condensed-matter analog of solenoid Taiki Yoda, Takehito Yokoyama, and Shuichi Murakami Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b04300 • Publication Date (Web): 26 Jan 2018 Downloaded from http://pubs.acs.org on February 1, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Orbital Edelstein eect as a condensed-matter analog of solenoid Taiki Yoda,

†

Takehito Yokoyama,

†

∗,†,‡

and Shuichi Murakami

Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8551, Japan, and TIES, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8551, Japan

E-mail: [email protected] Phone: +81-3-5734-2747. Fax: +81-3-5734-2739

∗ † ‡

To whom correspondence should be addressed Department of Physics, Tokyo Institute of Technology TIES, Tokyo Institute of Technology

1

ACS Paragon Plus Environment

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 17

Abstract We theoretically study current-induced orbital magnetization in a chiral crystal. This phenomenon is an orbital version of the Edelstein eect. We propose an analogy between the current-induced orbital magnetization and an Ampère eld in a solenoid in classical electrodynamics. In order to quantify this eect, we dene a dimensionless parameter from the response coecients relating a current density with an orbital magnetization.

This dimensionless parameter can be regarded as a number of turns

within a unit cell when the crystal is regarded as a solenoid, and it represents how chiral the crystal is.

By focusing on the dimensionless parameter, one can design

band structure which realizes induction of large orbital magnetization. In particular, a Weyl semimetal with all the Weyl nodes close to the Fermi energy can have a large value of this dimensionless parameter, which can exceed that of a classical solenoid.

Keywords Edelstein eect, orbital magnetization, chiral crystal, Weyl semimetal

Coupling between charge and spin degrees of freedom leads to various conversion phenomena between charge current and spin. Typical examples of the conversion are the spin-Hall eect

14

and the Edelstein eect.

37

These eects make it possible to control magnetization

by the charge current. However, their magnitudes are limited by the size of the spin-orbit interaction since they are driven by the spin-orbit interaction. Recently, a dierent mechanism of conversion between a charge current and a magnetization has been proposed. We proposed current-induced orbital magnetization et al. proposed gyrotropic magnetic eect.

9

8

and Zhong

These eects are described by a similar response

coecient. In particular, the current-induced orbital magnetization is an orbital analog of the Edelstein eect, and we can call this eect an orbital Edelstein eect.

2


In the orbital

Page 3 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Edelstein eect, we focus on the orbital magnetic moment of the Bloch states given by

mnk =

where

Hk

is the Bloch Hamiltonian with eigenvalues

Bloch state in the state

e Im⟨∂k unk | × [Hk − εnk ]|∂k unk ⟩, 2¯h

|unk ⟩.

nth band.

εnk ,

and

This orbital magnetic moment

(1)

|unk ⟩

is the periodic part of a

mnk is associated with each Bloch

It is an expectation value of the operator of the orbital magnetic moment

taken for the Wannier orbital corresponding to the Bloch state, where charge,

−e

r is the position of an electron and v is the velocity of an electron. 1012

assume that the time-reversal symmetry is preserved, which yields inversion symmetry is also preserved, it yields

1012

mnk = mn,−k ,

− 2e r×v

is an electron Hereafter, we

mnk = −mn,−k .

leading to

mnk ≡ 0

If the

for all the

bands. Instead, we here assume that the inversion symmetry is broken; it hereby leads to nonzero

mnk

in general. In particular, in chiral crystals as we show later, this nonzero

mnk

naturally follows from the chirality of the crystals. Thus, in equilibrium, the total orbital magnetization for the whole system is zero because of cancellations between the contributions from

k

and

−k.

The distribution of the orbital magnetization in

k

space is similar to the

distribution of the spin polarization in spin-split bands in systems with spin-orbit coupling, such as Rashba systems and surfaces of topological insulators.

In such systems with the

spin-split bands, a charge current induces an unbalance between the populations at at

−k,

k

and

and the total spin polarization becomes nonzero, which is called the spin Edelstein

eect. Likewise, from Eq. (1), the orbital Edelstein eect is expected in a similar way, if we consider the distribution of the orbital magnetization

mnk

spin polarization.

3


in

k

space instead of that of the

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 17

At zero temperature, the orbital Edelstein eect is formulated as a Fermi-surface integral of the orbital magnetization,

8,9

ME Mi = αij Ej , ∑∫ ME αij = eτ n

where

(1/¯ h)∂εnk /∂k τ

dk df mnk,i vnk,j , (2π)3 dε ε=εnk

M is the orbital magnetization, E is the electric eld, τ

Fermi distribution function,

that

BZ

df /dε|ε=εnk = −δ(εnk − εF ), εF

is the electron velocity.

is constant. The tensor

αME

(2) (3)

is the relaxation time,

f

is the Fermi energy, and

is the

vnk =

Here, we adopted a relaxation-time approximation

describing this response is an axial tensor with rank 2.

The class of crystals whose symmetry allows nonzero rank-2 axial tensor is called gyrotropic. In particular, breaking of inversion symmetry is required for gyrotropic crystals, and among 21 point groups lacking inversion symmetry, only 18 point groups are gyrotropic, having nonzero orbital Edelstein eect and nonzero (spin) Edelstein eect. However, the physical origin of the orbital Edelstein eect is dierent from the Edelstein eect. While the spinorbit interaction is essential in the Edelstein eect, the orbital Edelstein eect does not need the spin-orbit interaction.

9

The size of orbital Edelstein eect is determined by the lattice

structure and hopping amplitudes between the sites in chiral crystals.

In chiral crystals,

several interesting phenomena have been revealed: an electric response in a magnetic eld such as magnetochiral anisotropy current-induced optical activity.

1316

and a magnetic response in an electric eld such as

17,18

In this letter, to quantify this eect, we introduce a dimensionless parameter

ξ,

which

corresponds to the number of turns within a unit cell when the crystal is regarded as a solenoid, and we then show that in some cases this dimensionless constant is much enhanced compared to its classical value. To dene the dimensionless constant

4


ξ,

we rst introduce a

Page 5 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

tensor

β Mj

describing the ratio between a magnetization and an electric current density

Mi = βijMj jj .

instead of

αME . τ

relaxation time that

β Mj

Since both

M

and

j

j,

(4)

are proportional to

τ E, β Mj

and can be determined by band structure only.

is independent of the We theoretically show

is largely enhanced when systems are in the Weyl semimetal phase and all the Weyl

points are close to the Fermi energy, through our calculations in a tight-binding model and in an eective Weyl Hamiltonian. We then dene the dimensionless parameter the tensor

β Mj

as a product between

ξ

ξ , by expressing

and a scale factor given by the lattice constants.

ξ

indicates a ratio between a longitudinal and a circulating components of the electric current, and it represents an eciency of the orbital Edelstein eect as compared with a classical solenoid. These results are useful for designing band structure with large orbital Edelstein eect.

(b)

(a)

z

B b2

b3

t3

y A

b1

t2

t1

t2

x

z

Figure 1: (a) One layer of the model forming a honeycomb-lattice. Dashed arrows denote vectors

b1 , b2 ,

and

b3 .

(b) Chiral hopping (t2 term in Eq. (5)) in the right-handed helix.

Red (blue) lines denote hoppings between A (B) sites.

As an example, we here introduce a simple tight-binding model with chiral crystal structure proposed in Ref. 8. The tight-binding model is composed of honeycomb-lattice layers with one orbital per site, as shown in Fig. 1(a), where

b3 = a/2(−ˆ x−

√

stacked along the

3ˆ y)

and

a

z -direction

b1 = aˆ x, b2 = a/2(−ˆ x+

√ 3ˆ y),

is a lattice constant in the honeycomb lattice. The layers are with an interlayer lattice constant

5


c.

The Hamiltonian of the

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 17

tight-binding model is written as

H = t1 [ + t2

∑

c†i,l cj,l + t3

∑∑

⟨ij⟩,l

∑

i,l s=±1

t3

ci,l

∑

c†i+bj ,l+1 ci,l +

i∈A,j,l

where

c†i,l ci,l+s ] c†i−bj ,l+1 ci,l + H.c. ,

(5)

i∈B,j,l

is an annihilation operator of an electron at the ith site in the l th layer,

are real constants, and we set

t1 > 0

t1 , t 2 ,

and

for simplicity. The Hamiltonian (5) does not include

spin-orbit interaction and spin indices are omitted. The t1 term is a nearest-neighbor hopping within the same honeycomb layers.

The

t2

term represents right-handed chiral hoppings

between sites in the same sublattice in the neighboring layers as shown in Fig. 1(b). This term breaks inversion and mirror symmetries. The The space group of the model is

P 622.

t3

term is a vertical interlayer hopping.

The Bloch Hamiltonian of Eq. (5) takes the following

form

Hk = d0 I + dk · σ, ∑ d0 = 2t2 cos(kz c) cos(k · bi )

(6)

i

dx

+2t3 cos(kz c), ∑ = t1 cos(k · ai ),

(7) (8)

i

dy = t1

∑

sin(k · ai ),

i

dz = −2t2 sin(kz c)

∑

(9)

sin(k · bi ),

(10)

i

where

ai

dk = (dx , dy , dz ),

the Pauli matrices

σi

act on the sublattice degree of freedom, and

are vectors pointing from an A site to three neighboring B sites.

We note that the

tight-binding model (6) can be mapped to the Haldane model on a honeycomb lattice replacing

kz

with a ux

ϕ.

A similar model has been proposed in acoustic systems.

6


19

20

by

The

Page 7 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

orbital magnetic moment for Bloch eigenstates in this model is given by

mnk,i

where

dk = |dk |

and

εijl

(

e 1 = − εijl 2 dk · h ¯ 2dk

t2 = 0,

(11)

is the Levi-Civitá antisymmetric tensor. Equation (11) shows that and

dz

the orbital magnetic moment vanishes at arbitrary

k.

the orbital magnetic moment is zero if any one of or

) ∂dk ∂dk × , ∂kj ∂kl

dx , dy ,

The Brillouin zone and the band structure of Eq. (6) is shown in Fig. 2. exhibits four Weyl points, whose energies are

−ε0 = −(3t2 −2t3 ) at the K and K' points.

ε0 ≡ (3t2 − 2t3 )

t1 = 0

is zero. Therefore, if

Our model

at the H and H' points and

For Weyl semimetals with time-reversal symmetry

but without inversion symmetry, the minimal number of Weyl points is four;

21

therefore, this

model is a minimal model for a Weyl semimetal without inversion symmetry. The numerical results of

t2 = 0,

αME

are shown in Fig. 3(a)(b) for

εF = 0

and

the orbital magnetization is zero due to inversion symmetry. For t2

Weyl points are at the same energy The numerical results of (Fig. 3(c)),

β Mj

β Mj

diverges at

ε = 0,

= 2t3 /3,

For

the four

and the orbital magnetization almost vanishes.

are shown in Fig. 3(c)(d) for

t2 = 2t3 /3.

εF = 0.2t1 .

εF = 0 and εF = 0.2t1 .

This divergence of

of the magnetization; instead, the divergence of

β Mj

β Mj

For

εF = 0

does not mean divergence

occurs because toward

t2 = 2t3 /3

the

current and the orbital magnetization in Eq. (4) simultaneously converge to zero, while the current density converges to zero faster than the orbital magnetization. Meanwhile, as long as the current behavior of

j

β Mj

is nonzero,

β Mj

never diverges and the magnetization stays nite.

is physically reasonable.

On the other hand,

(Fig. 3(d)) since the current density is nite except for To analyze properties of

β Mj

β Mj

is nite for

This

εF = 0.2t1

t2 = t3 = 0.

in detail, we expand the Hamiltonian around one of the

Weyl points up to the linear order of

k:

Hqµν = −νε0 I − h ¯ (µv1 qx σx + v2 qy σy ) + µν¯hv3 qz σz ,

7


(12)

Nano Letters

(b) H’=(-,-) A

4

L H=(+,-)

K’=(-,+) M Γ

K=(+,+)

Energy/t1

(a)

2 0 -2 -4 Γ

KM

Γ

K H

Γ

A

H L

A

Figure 2: (a) Brillouin zone of our model with high-symmetry points. The high-symmetry

ν ), where µ, ν = ±. t3 = 0.2t1 (red) and −0.2t1

points K, K', H, H' are specied by (µ, Hamiltonian (6) with

t2 = 0.2t1

for

(b) Energy bands of the (blue). The energy bands

are shown.

ME

α zz /(e2τt1/ħ2)

0.12 0.1

(a)

0.08 0.06

0.12

εF=0

t3=0.4t1 t3=0.2t1 t3=0 t3=-0.2t1 t3=-0.4t1

0.1

0.04 0.02

ME

0 ≤ kz ≤ π/c

α zz /(e2τt1/ħ2)

within

0

0.06

t3=0.4t1 t3=0.2t1 t3=0 t3=-0.2t1 t3=-0.4t1

εF=0.4t1

0.04 0.02 0

(c)

(d) 2 ξ=βMj zz /( 3a /2c)

15

(b)

0.08

-0.02

-0.02

2 ξ=βMj zz /( 3a /2c)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 17

10 5 0 -5 -10

t3=0.4t1 t3=0.2t1 t3=0 t3=-0.2t1 t3=-0.4t1

12 8 4 0

-15 -0.4 -0.3 -0.2 -0.1 0

-0.4 -0.3 -0.2 -0.1 0

0.1 0.2 0.3 0.4

t2/t1

(e)

z S xy

0.1 0.2 0.3 0.4

t2/t1

(f)

Iz

c ξ turns Icirc Mj from the tight-binding model (6) as a function βzz Mj Mj ME ME of t2 . (a) αzz at εF = 0. (b) αzz at εF = 0.2t1 . (c) βzz at εF = 0. (d) βzz at εF = 0.2t1 . In √ 2 Mj (c) and (d), the vertical axes show the dimensionless parameter ξ ≡ (βzz )/( 3a /(2c)). (e) is a schematic picture of the chiral circulating current Icirc within a unit cell, inducing the Figure 3: Numerical results of

ME αzz

and

orbital angular momentum. (f ) shows a solenoid, which corresponds to a unit cell of a chiral crystal. We show the case with unit cell in the

z

ξ = 8,

which means the solenoid with eight turns along the

direction.

8


Page 9 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

where

q = k − k0

√ 3 t a, and 2 1

is a displacement from the Weyl point at

k0 , ε0 ≡ 3t2 − 2t3 , h ¯ v1 = h ¯ v2 =

√ h ¯ v3 = 3 3t2 c. µ(= ±1) and ν(= ±1) denote a valley degree of freedom as shown

in Fig. 2(a), specifying one of the Weyl points. We can then calculate the orbital magnetic moment around the Weyl points and it is given by

mqµν = −ν

where

q = (qx , qy , qz ).

e v1 v2 v3 q, 2 2 2 v1 qx + v22 qy2 + v32 qz2

In an isotropic case (v1

= v2 = v3 ),

(13)

this formula reduces to the result

in Ref. 9. This result indicates that the orbital magnetic moment is enhanced around the

αME ,

Weyl points. Keeping only the contributions from the four Dirac cones to

αiiME =

∑

ME αii,µν = sgn(v1 v2 v3 )

µν

where

ME αii,µν =

sgn(v1 v2 v3 )(e

cone specied by

(µ, ν).

2

τ /3h2 )(νεF + ε0 )

ε0 = 0,

4e2 τ ε0 , 3h2

Notably, the result for

αME

in an anisotropic case, Eq. (14), is Equation (14) shows that

namely the four Weyl points are located at

magnetic moment diverges at the Weyl point, This behavior results from

q -dependence

Fermi surface is proportional to

q −1 ,

of

(14)

is the contribution from the single Dirac

the same with the isotropic case obtained in Ref. 9. zero when

we obtain

αME

ε = 0.

is

While the orbital

does not diverge even when

mqµν .

αME

εF = ±ε0 .

The orbital magnetic moment on the

and the area of the Fermi surface is proportional to

q2.

Consequently, the Fermi surface integral of Eq. (13) per Dirac cone is roughly proportional to

q.

We compare the numerical results from the tight-binding model and those from the

eective Weyl Hamiltonian, Eq. (12), as a function of

ε0 = 3t2 − 2t3

in Fig. 4(a)(b). Around

ε0 = 0, the results t well with the linear behavior, expected from Eq. (14).

Far from

ε0 = 0,

the numerical result from the tight-binding model Eq. (6) deviates from Eq. (14) because of higher-order terms in

q.

Next, we evaluate the tensor

β Mj

describing the ratio between the current and the orbital

magnetization, dened in Eq. 4. By using

ji = σij Ej , 9

where


σij

is the conductivity,

αME

can

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

be written as

Mj ME αij = βik σkj .

Page 10 of 17

The conductivity is given by the Fermi-surface integral,

σij = −eτ

∑∫ BZ

n

dk df vnk,i vnk,j , (2π)3 dε ε=εnk

(15)

where the Boltzmann transport theory with relaxation-time approximation is adopted. The conductivity for the eective Hamiltonian Eq. (12) is calculated as

With Eqs. (14), (16), (17), and

Mj

βxx

Mj βyy Mj βzz

When

Mj εF ̸= 0, βzz

εF = 0,

both

ME αzz

(16)

σzz

(17)

Mj ME αij = βik σkj ,

we obtain

√ h ¯ v2 h ¯ v3 ε0 3 3 t2 (3t2 − 2t3 ) = c, = 2¯hv1 ε2F + ε20 2 ε2F + (3t2 − 2t3 )2 √ h ¯ v3 h ¯ v1 ε0 3 3 t2 (3t2 − 2t3 ) = = c, 2¯hv2 ε2F + ε20 2 ε2F + (3t2 − 2t3 )2 √ 2 2 h ¯ v1 h ¯ v2 ε0 3 t1 a 3t2 − 2t3 = . = 2 2 2 2¯hv3 εF + ε0 24 t2 c εF + (3t2 − 2t3 )2

is zero at

and

8 e2 τ ε2F + ε20 , 3 h2 h ¯ |v3 | 8 e2 τ |v3 |(ε2F + ε20 ) . = 3 h2 h ¯ |v1 ||v2 |

σxx = σyy =

σzz

3t2 = 2t3

because of

ME αzz =0

and

(18)

(19)

(20)

σzz ̸= 0.

Meanwhile, when

are zero, which leads to a divergent behavior of

This divergence arises because

σzz

converges to zero faster than

ME αzz ,

as

ε0

Mj βzz

j

and the magnetization

zero faster than

M,

M

go to zero as

leading to the divergence of

Mj βzz

ε0

3t2 = 2t3 .

approaches zero.

In other words, as previously mentioned, for the xed electric eld and at the current

at

εF = 0,

approaches zero, but

j

both

approaches

from Eq. (4). We compare the results

from the tight-binding model with those from the Weyl Hamiltonian, Eq. (12), in Fig. 4(c) and (d).

They agree well with each other around

ε0 = 0.

When

ε0

is far from zero, the

approximation as a Weyl Hamiltonian is no longer valid due to higher-order terms in

q.

Current-induced orbital magnetization, i.e. the orbital Edelstein eect, was previously studied in tellurium

17,18,22

and in zeolite-templated carbon.

10


23

However, the physical mecha-

Page 11 of 17

(a)

0.08 0.06 0.04

0.12

εF=0

t3=0.4t1 t3=0.2t1 t3=0 t3=-0.2t1 t3=-0.4t1 Eq.(14)

0.1

0.02

ME

ME

α zz /(e2τt1/ħ2)

0.1

α zz /(e2τt1/ħ2)

0.12

0 -0.02

(b)

0.08 0.06 0.04

εF=0.4t1

t3=0.4t1 t3=0.2t1 t3=0 t3=-0.2t1 t3=-0.4t1 Eq.(14)

0.02 0 -0.02

t3=0.4t1 t3=0.2t1 t3=0 t3=-0.2t1 t3=-0.4t1 Eq.(20)

(c)

2 1 0

2 βMj zz /(t1a /t2c)

3 2 βMj zz /(t1a /t2c)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

-1 -2 -3 -4 -2 -1.5 -1 -0.5

0

0.5

1

1.5

2

0.2 (d) 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -2 -1.5 -1 -0.5

ME αzz

0

0.5

1

1.5

2

ε0 / t1

ε0 / t1

Figure 4: Numerical results of

t3=0.4t1 t3=0.2t1 t3=0 t3=-0.2t1 t3=-0.4t1 Eq.(20)

and

Mj βzz

as a function of

ε0 = 3t2 − 2t3 .

Comparison

between the numerical result from the tight-binding model (6) and Eqs. (14)(20) is shown. (a) and (b) show

ME αzz , the ratio between the orbital magnetization and the electric eld.

The

broken straight lines show the result of Eq. (14), which is the sum of the contributions from four Weyl nodes.

εF

is taken as (a)

εF = 0

and (b)

εF = 0.2t1 .

(c) and (d) show

Mj βzz ,

the

ratio between the orbital magnetization and the current. The broken lines show the result of Eq. (20).

εF

is taken as (c)

εF = 0

and (d)

εF = 0.2t1 .

nism of our study totally diers from that of the previous studies. The orbital magnetization studied previously

17,18,22,23

originates from an intracelluler or intrasite circulation current

from atomic orbitals of each atom. In our theory,

12,2428

the orbital magnetic moment

originates from intercellular or intersite circulation current.

29,30

mnk

In our model Eq. (6), no

intrasite orbital magnetization is induced because we assume that the localized basis of the model Eq. (6) has no orbital angular momentum.

The intrasite and intersite orbital

Edelstein eects do not need the spin-orbit interaction. Furthermore, the intersite orbital Edelstein eect does not need the orbital angular momentum of the parent atomic orbital. The intersite orbital Edelstein eect requires gyrotropic lattice structure, in the same way as the intrasite orbital Edelstein eect and the spin Edelstein eect. Meanwhile, geometrical structure of crystals is expected to be signicant in the intersite orbital one, compared with the other two eects. In this sense, this orbital magnetization might have some similarity with valley physics,

31,32

since they both come from lattice properties. The orbital Edelstein

eect can be measured experimentally, but in the measurement of magnetization, one should separate the spin and orbital parts. Here, intrasite and intersite orbital Edelstein eects are

11


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 17

always mixed, because the distinction between them lies in the description of eigenstates in terms of atomic orbitals. Such a description becomes suitable in the limit of large interatomic separations, but is not accurate in crystals, leading to mixing between the intersite and intrasite orbital Edelstein eects. Recently, an NMR shift in tellurium with a current has been measured, and the resulting shift is proportional to the current.

33

Both the spin

and the orbital parts may contribute to the shift, and their separation requires comparison with numerical calculations. The Bohr magneton, which is a fundamental unit of the magnetic moment of an electron, does not appear in Eq. (1). In our model, the orbital magnetic moment Eq. (1) is measured as a unit of

eta2 /2¯h instead of the Bohr magneton, where t is an energy scale of the Hamiltonian

such as the nearest-neighbor hopping energy and

a

is the lattice constant. At the bottom

of the conduction band or at the top of the valence band, the inverse eective mass is of the order of

ta2 /¯h2 .

Therefore the unit

eta2 /2¯h = (e¯h/2(¯ h2 /ta2 ))

as a magnetic moment generated by an electron with a mass

1/m∗

can be interpreted

h ¯ 2 /ta2 ∼ m∗ .

At the K

point of graphene with a staggered potential, the orbital magnetic moment agrees with the eective Bohr magneton

eta2 /2¯h = IS , ta/¯ h,

and

where

µ∗B = e¯h/2m∗ . 12,30,31

I = e(ta/¯h)/2πa

The unit

eta2 /2¯h

can be also expressed as

is a circulating current with a radius

a

and velocity

S = πa2 .

Here, we compare the size of the orbital Edelstein eect in our model with that of a solenoid in classical electrodynamics. First, we evaluate

β Mj ,

by modelling a chiral crystal

in the following way. We set the axis of the crystal be along the has a length

c

along the

z

axis and an area

Sxy

along the

current owing in the unit cell into two; one is a current circulating current

Icirc

around the unit cell within the

I

xy

xy

z -direction.

The unit cell

plane. We shall decompose the

along the

z

axis, and the other is a

plane, as shown in Fig. 3(e). From

classical electrodynamics, the circulating current induces an orbital magnetic moment

Icirc Sxy ,

and the resulting orbital magnetization

the other hand, the electric current density

jz

12

Mz

is given by

is evaluated as


mz ≃

Mz = mz /Sxy c ≃ Icirc /c. jz ≃ I/Sxy .

On

Therefore, the

Page 13 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

tensor

Mj βzz

is represented as

Mj βzz ≃ ξ(Sxy /c),

where

ξ = Icirc /I .

This dimensionless factor

ξ

indicates the ratio between two components of the current, shown in Fig. 3(e). One can draw an analogy with a classical solenoid, and evaluate eciency of the orbital Edelstein eect in a chiral crystal. In a solenoid with the number of turns per unit length

n,

an electric current

volume is given by of

Mz

I

induces a magnetic eld

M = nI .

H = nI ;

therefore, a magnetic moment per

By comparing this formula with the corresponding formula

for the orbital Edelstein eect, we obtain

ξ = nc;

namely, the dimensionless factor

ξ

indicates the number of turns within the unit cell when the crystal is regarded as a classical solenoid (see Fig. 3(f )). Thus, this dimensionless factor

ξ

represents how chiral" the crystal

is, and classically it is of the order of unity. Remarkably, as seen in Fig. 3(c), the value of

ξ can

be much larger than that expected from a geometrical structure of the crystal. For example, in the crystal structure shown in Fig. 1(b), one may naively expect this dimensionless factor to be maximally

1/3, but it is not true.

zero (Fig. 3(c)). Namely,

ξ

In our model with

ξ

εF = 0, ξ diverges as ε0 approaches

becomes large for a Weyl semimetal, with all the Weyl nodes

close to the Fermi energy. Thus, counterintuitively, the crystal structure in Fig. 1(b) works as a solenoid with many turns within the unit cell (see Fig. 3(f ) for

ξ=8

as an example) .

How a chiral crystal compares with a classical solenoid from the viewpoint of currentinduced magnetization depends on the size of the system.

In nanoscale, the size of the

current-induced magnetization for a chiral crystal can exceed that of a classical solenoid, as

ξ

can exceed unity. As the scale becomes larger, a chiral crystal becomes less favorable for

inducing large magnetization, compared with a classical solenoid; it is because for a xed amount of the current, the current density inversely scales with the area of the system within the

xy

plane, and the magnetization scales similarly, while for a classical solenoid, the area

within the

xy

plane does not aect the strength of the generated magnetic eld. Thus to

make a nanoscale solenoid, the orbital Edelstein eect in a chiral crystal may be useful compared with a classical solenoid. For this purpose, a Weyl semimetal with all the Weyl nodes located near the Fermi energy is ideal, and such a Weyl semimetal can be found close

13


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 17

to a phase transition to an insulator phase, because the phase transition is accompanied by creations of Weyl nodes. In conclusion, we have dened the tensor claried the analogy between dimensionless factor

ξ

β Mj

β Mj

to quantify the orbital Edelstein eect, and

and a solenoid.

The tensor

and lattice constants. We show that

β Mj

β Mj

is characterized by the

and

ξ

can be enhanced by

designing band structure, in particular, for a Weyl semimetal with all the Weyl nodes close to the Fermi energy.

Author Information Corresponding Authors ∗

E-mail: [email protected].

Notes The authors declare no competing nancial interest.

Acknowledgement T. Yoda is a JSPS Research Fellow. This work was supported by JSPS KAKENHI Grant No. JP16J07354 and by Grants-in-Aid for Scientic Research on Innovative Areas Topological Materials Science (KAKENHI Grant No. JP16H00988) and Nano Spin Conversion Science (Grant No.

JP17H05179 and 26103006), by MEXT Elements Strategy Initiative to Form

Core Research Center (TIES), and by JSPS KAKENHI Grant Number JP16K13834.

References (1) Murakami, S.; Nagaosa, N.; Zhang, S.-C.

Science (Washington, DC, U.S.)

2003,

301,

13481351.

(2) Sinova, J.; Culcer, D.; Niu, Q.; Sinitsyn, N. A.; Jungwirth, T.; MacDonald, A. H.

Rev. Lett.

2004, 92, 126603. 14


Phys.

Page 15 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

(3) Sinova, J.; Valenzuela, S. O.; Wunderlich, J.; Back, C. H.; Jungwirth, T.

Phys.

Rev. Mod.

2015, 87, 12131260.

(4) Manchon, A.; Koo, H.; Nitta, J.; Frolov, S.; Duine, R.

(5) Edelstein, V. M.

Solid State Commun.

(7) Inoue, J.; Bauer, G. E. W.; Molenkamp, L. W.

(9) Zhong, S.; Moore, J. E.; Souza, I.

Sov. Phys. JETP Phys. Rev. B

Sci. Rep.

Phys. Rev. Lett.

1996, 53, 70107023.

(11) Sundaram, G.; Niu, Q.

Phys. Rev. B

1999, 59, 1491514925.

(13) Rikken, G. L. J. A.; Fölling, J.; Wyder, P.

2003, 67, 033104.

2016, 116, 077201.

Phys. Rev. B

Rev. Mod. Phys.

1991, 73, 537.

2015, 5, 12024.

(10) Chang, M.-C.; Niu, Q.

(12) Xiao, D.; Chang, M.-C.; Niu, Q.

2015, 14, 871882.

1990, 73, 233.

(6) Aronov, A. G.; Lyanda-Geller, Y.; Pikus, F. G.

(8) Yoda, T.; Yokoyama, T.; Murakami, S.

Nat. Mater.

2010, 82, 19592007.

Phys. Rev. Lett.

2001, 87, 236602.

(14) Pop, F.; Auban-Senzier, P.; Canadell, E.; Rikken, G. L.; Avarvari, N.

Nat. Commun.

2014, 5, 3757. (15) Krsti¢, V.; Roth, S.; Burghard, M.; Kern, K.; Rikken, G. L. J. A.

J. Chem. Phys.

2002,

117, 1131511319. (16) Morimoto, T.; Nagaosa, N.

(17) Vorob'ev, L. E.; Shturbin, A. V.

Phys. Rev. Lett.

Ivchenko, E. L.;

JETP Lett

2016, 117, 146603.

Pikus, G. E.;

Farbshtein, I. I.;

Shalygin, V.;

1979, 29, 441.

(18) Shalygin, V. A.; Sofronov, A. N.; Vorob'ev, L. E.; Farbshtein, I. I.

2012, 54, 23622373. 15


Phys. Solid State

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(19) Haldane, F. D. M.

Phys. Rev. Lett.

1988, 61, 20152018.

(20) Xiao, M.; Chen, W.-J.; He, W.-Y.; Chan, C. T.

(21) Murakami, S.

New J. Phys.

Nat. Phys.

2015, 11, 920924.

2007, 9, 356.

(22) Ivchenko, E. L.; Pikus, G. E.

1978, 27, 604.

JETP Lett.

(23) Koretsune, T.; Arita, R.; Aoki, H.

(24) Xiao, D.; Shi, J.; Niu, Q.

Page 16 of 17

Phys. Rev. B

Phys. Rev. Lett.

2012, 86, 125207.

2005, 95, 137204.

(25) Thonhauser, T.; Ceresoli, D.; Vanderbilt, D.; Resta, R.

Phys. Rev. Lett.

2005,

95,

137205.

(26) Ceresoli, D.; Thonhauser, T.; Vanderbilt, D.; Resta, R.

(27) Shi, J.; Vignale, G.; Xiao, D.; Niu, Q.

Phys. Rev. Lett.

Phys. Rev. B

2006, 74, 024408.

2007, 99, 197202.

2010, 22, 123201.

(28) Resta, R.

J. Phys.: Condens. Matter

(29) Yafet, Y.

Solid State Physics: Advance in Research and Applications ; Academic, New

York, 1963; Vol. 14; p 1.

2008, 77, 235406.

(30) Yao, W.; Xiao, D.; Niu, Q.

Phys. Rev. B

(31) Xiao, D.; Yao, W.; Niu, Q.

Phys. Rev. Lett.

2007, 99, 236809.

(32) Xiao, D.; Liu, G.-B.; Feng, W.; Xu, X.; Yao, W.

Phys. Rev. Lett.

(33) Furukawa, T.; Shimokawa, Y.; Kobayashi, K.; Itou, T.

16


2012, 108, 196802.

Nat. Commun.

2017, 8, 954.

Page 17 of 17

Graphical TOC Entry 15 2 ξ=βMj zz /( 3a /2c)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

10 5 0 -5 -10 -15 -0.4 -0.3 -0.2 -0.1 0

0.1 0.2 0.3 0.4

t2/t1

For TOC only

17


Orbital Edelstein effect as a condensed-matter analog of solenoid

Recommend Documents