Chiral Phonon Induced Spin-Polarization J. Fransson Department of Physics and Astronomy Box 516 751 20 Uppsala University Uppsala Sweden

2025-05-02 0 0 7.85MB 6 页 10玖币

侵权投诉

Chiral Phonon Induced Spin-Polarization

J. Fransson∗

Department of Physics and Astronomy, Box 516, 751 20, Uppsala University, Uppsala, Sweden

(Dated: October 25, 2022)

The current understanding of chirality suggests the existence of a connection between structure and angular

momentum, including spin. This is particularly emphasised in the chiral induced spin selectivity eﬀect, where

chiral structures act as spin ﬁlters. However, the recent discovery of chiral phonons have demonstrated that

phonons too may carry angular momentum which also can be regarded as magnetic moments which add to

the total moment. Here, it is shown that chiral phonons may induce a non-trivial spin-texture in an otherwise

non-magnetic electronic structure. By considering a set-up in which electrons and phonons are interfaced with

each other, it is shown that chiral phonons may transfer its angular momentum into the electron reservoir which,

thereby, becomes spin-polarized. It is, moreover, shown that an equivalent mechanism does not exist whenever

the electrons are interfaced with achiral phonons.

Phonons represent the collective nuclear motion within a

structure. As such, phonons are the quantum mechanically

deﬁned quantities with which the mechanical degrees of free-

dom are eﬀectively incorporated into the framework of the

general quantum ﬁeld theory. While traditionally being re-

garded as a quantity carrying linear momentum, it is only re-

cently that angular momentum of phonons have been consid-

ered. An incomplete list of important results are phonon Hall,

phonon spin Hall, and phonon angular momentum Hall ef-

fects [1–7], phonon contribution to spin-relaxation processes

[8–11] and the Einstein-de Haas eﬀect [10,12–14], phonon-

ically mediated spin-spin interactions [15,16], temperature

gradient induced phonon angular momentum [17], and opti-

cally activated chiral phono-magnetic eﬀects [18–20]. Exper-

imentally, progress has been made in observations of chiral

phonons [21–25], and phonon induced magneto-thermal prop-

erties [26–28].

The existence of phonon angular momentum opens up the

possibility to couple the electronic spin-degrees of freedom

with the mechanical. It is well-established that spin and nu-

clear motion are coupled directly through, e.g., the Elliot-

Yafet mechanism [29–34], but also indirectly via the elec-

tronic structure [15], and it has been demonstrated that such

coupling opens for a viable explanation of the chiral induced

spin selectivity eﬀect [34–37].

Hitherto, however, the angular momentum of phonons and

electrons have been considered as separate from one an-

other, where the magnetic moment associated with the chi-

ral phonons have been studied in its own right. While this

is deﬁnitely pertinent, the eﬀects angular momentum trans-

fer between phonons and electrons has been discussed in a

semi-classical model [38], in which a spin-dependent cou-

pling between phonons and electrons is assumed and it is

demonstrated that chiral phonons may give rise to the spin-

Seebck eﬀect, something that was also recently observed in

experiments [39]. Nonetheless, the mechanism that enables

the angular momentum transfer is yet to be discussed. The

purpose with this Letter is to present a coherent theory that

ties the existence of phononic angular momentum to a broken

electronic spin-degeneracy. It is shown that the mechanism is

provided through a vibronically assisted spin-orbit interaction

and while this coupling is always present, chiral phonons are

required for inducing a spin-polarization.

Chirality is a geometrical property where the structure lacks

both inversion and reﬂection symmetries. It is easy to demon-

strate that chiral phonons must carry a non-vanishing angular

momentum Jph. This can be seen directly from the deﬁnition

Jph(t)=ZQ(r,t)×˙

Q(r,t)dr,(1)

where Qis the nuclear displacement, which is connected

to phonons through the relation Q(r,t)=PqlqqQp(t)eiq·r,

where Qq=bqµ+b†

qµ, is the quantum phonon displacement

operator. Here, bqand b†

qdenote the phonon destruction and

creation operators, respectively, with q=(q,µ) (¯q=(¯

q,µ)=

(−q,µ)) comprising the wave vector qand normal mode µ,

and lq=p}/2ρvωqwhich deﬁnes a length scale in terms of

the phonon energy ωq, density ρ, and volume v, whereas qis

the displacement polarization vector.

The expectation value of the angular momentum, hence, as-

sumes the form

hJphi(t)=lim

t0→ti∂tX

lplqp×¯qD>

pq(t0,t)Zei(p−q)·rdr,(2)

where D>

pq(t0,t)=(−i)hQp(t0)Q¯q(t)ideﬁnes the correlations

between the phonons Qp(t0) and Q¯p(t). The expression in Eq.

(2) shows that a ﬁnite phonon angular momentum requires

non-collinear polarisations pand ¯q.

Non-collinear polarizations can on the one hand be

achieved when there is a mechanism that mixes the phonon

modes pand q. Such mode mixing may originate from, e.g.,

anharmonic eﬀects or scattering oﬀdefects acting upon the

otherwise orthogonal modes deﬁned in the harmonic approxi-

mation.

In the harmonic approximation, the phonons can, up to

a constant, be summarised the Hamiltonian form as Hph =

Pqωqb†

qbq. In this form, it is assumed that the phonon modes

do not mix which, therefore, leads to a vanishing angular mo-

mentum. However, the introduction of a component of the

kind Wqµνb†

qµbqνprovides a mode mixing, which can be un-

derstood as chirality.

arXiv:2210.12722v1 [cond-mat.mes-hall] 23 Oct 2022

To see this, consider the phonon spinor Φq={bqµi}N

i=1, for

Nmodes, which enables us to write the phonon model as

Hph =X

Φ†

qωqΦq.(3)

Written like this, the phonon spectrum is deﬁned through

the matrix ωq=ω0qτ0+ω1q·τ, where ω0and ω1represent

the mode conservative and mode mixing components, respec-

tively, whereas τ0and τare the N-dimensional identity and

vector of spin matrices. While in this model it is assumed that

the mixing only takes place between modes µiand µjwith

the same momentum q, it is straight forward to generalise the

model to also include mixing between diﬀerent momenta.

Considering a structure with two modes, such that τre-

duces to the Pauli matrices, the phonon Green function

Dq(z)=hhΦq|Φ¯

qii(z), can be written as

Dp(z)=2ωq

z2−ω2

0q−ω2

1q+2ω0qω1q·τ

(z2−ω2

0q−ω2

1q)2−4ω2

0qω2

(4a)

s=±1

2ωqs

z2−ω2

qsτ0+sˆω1q·τ,(4b)

where ωqs=ω0q+sω1q,ω1q=|ω1q|and ˆω1q=ω1q/ω1q. The

form of this propagator written in Eq. (4b) explicitly describes

two modes with opposite helicity, or, chirality.

On the other hand, the polarization for a chiral mode is

neither reﬂection nor inversion symmetric, where the lat-

ter condition leads to that ∗

qµ=¯

qµ,qµ, since equal-

ity in the last relation requires inversion symmetry. For,

e.g., a helical structure with transversal and longitudi-

nal lattice parameters aand c, respectively, the polariza-

tion may be expressed as q=(acosφq,asinφq,¢φq)/d(φq),

where ¢ =c/2πand d(φq)=qa2+¢2φ2

q, which displays

a variation of the mode that depends on the azimuthal

angle φq. Hence, for such a mode, the vector prod-

uct q×¯

q=(−a¢φqsinφq,a¢φqcosφq,−a2sin2φq)/d2(φq),

which suggests that a free chiral phonon mode, for which

q(t,t0)=(−i)[nB(ωq)e−iωq(t−t0)−nB(−ωq)eiωq(t−t0)], where

nB(ω) is the Bose-Einstein distribution function, carry the

non-vanishing angular momentum

hJphi=ωql2

πd2(φq)





−a¢φqsinφq

a¢φqcosφq

−a2πsin2φq







.(5)

It is important to notice that the oﬀ-diagonal components of

the phonon propagator, Eq. (4b), carry the phase factors e±iφp,

where φpdeﬁnes the azimuthal angle of the momentum vector

p. This phase dependence reoccurs also in the helical polar-

ization vector and is an essential feature of chirality. Due to

this phase dependence, the phonons necessarily carry angular

momentum.

The objective in this Letter, is to show that the phononic

angular momentum can be transferred to the electronic sub-

system and, hence, induce a spin-polarization. For this sake,

consider the coupling between electrons and phonons, which

generally can be written as [34]

He-ph =X

ψ†

p+kUkqψkQq,(6)

where the coupling matrix Ukq=Ukqσ0+Jkq·σaccounts

for a spin-conservative electron-phonon coupling, Ukq, and

an electron-phonon assisted spin-orbit interaction, Jkq.

The spin-polarisation hMkiof the electrons are calculated

using the identity hMki=(−i)spσRG<

kk(ω)dω/4π, where

kk0(ω) deﬁnes the lesser form of the general single electron

Green function Gkk0(z)=hhψk|ψ†

k0ii(z), whereas sp denotes

the trace over spin 1/2 space. To the second order (Hartree-

Fock) approximation in the electron-phonon coupling, this

Green function can be calculated from the Dyson equation

Gkk0(z)=δ(k−k0)gk(z)+gk(z)X

Σ(HF)

kκ(z)Gκk0(z),(7a)

Σ(HF)

kk0(z)=(−i)δ(k−k0)X

qq0

Dqq0UkqspZG<

kk(ω)Uk¯q0dω

2π

−1

βX

zνqq0

UkqGkk0(z+zν)Uk0¯q0Dqq0(zν),(7b)

where Dqq0=Rδ(ω)Da

qq0(ω)dω.

While this equation should be self-consistently solved, for

an analysis of the induced spin symmetries, it is suﬃcient to

replace the electronic Green functions in these expressions

with its unperturbed form gk(z)=σ0gk(z), where gk(z)=

1/(z−εk) and g<

k(ω)=2πσ0f(ω)δ(ω−εk), whereas f(ω) is

the Fermi-Dirac distribution function. These replacements

lead to the simpliﬁed self-energy

Σ(HF)

k(z)=2f(εk)X

qq0

δ(q)UkqDqq0Uk¯q0

−1

βX

νqq0

gk−q(z−zν)UkqDqq0(zν)Uk¯q0.(8)

The Hartree (ﬁrst) contribution is to lowest order linear in

the component Jkq·σwhich, hence, shows that the coupling

between the electrons and phonons may break time-reversal

symmetry. However, this contribution vanishes in this approx-

imation since there are no phonons at p=0. By contrast, the

exchange (second) contribution does not only open for break-

ing the time-reversal symmetry, since in general

UkpUkq=UkqUkq0+Jkq·Jkq0

+UkqJkq0+JkqUkq0+iJkq×Jkq0·σ,(9)

but it also provides a correlation between the electronic and

phononic degrees of freedom.

However, before showing that chiral phonon leads to a

breaking of the electronic spin-degeneracy, it is pertinent to

discuss the origin of the phonon induced spin-polarization and

spin-ﬂip processes which are enabled by Jkq. For this purpose,

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ChiralPhononInducedSpin-PolarizationJ.FranssonDepartmentofPhysicsandAstronomy,Box516,75120,UppsalaUniversity,Uppsala,Sweden(Dated:October25,2022)Thecurrentunderstandingofchiralitysuggeststheexistenceofaconnectionbetweenstructureandangularmomentum,includingspin.Thisisparticularlyemphasisedinthechira...

展开>> 收起<<

Chiral Phonon Induced Spin-Polarization J. Fransson Department of Physics and Astronomy Box 516 751 20 Uppsala University Uppsala Sweden.pdf

共6页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Chiral Phonon Induced Spin-Polarization J. Fransson Department of Physics and Astronomy Box 516 751 20 Uppsala University Uppsala Sweden

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: