Photovoltaic Chiral Magnetic Effect

0.29 MB·

by  Katsuhisa Taguchi

Photovoltaic Chiral Magnetic Effect
Katsuhisa Taguchi1,3, Tatsushi Imaeda1,3, Masatoshi Sato2, and Yukio Tanaka1,3
1Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan
2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan
3CREST, Japan Science and Technology Corporation (JST), Nagoya 464-8603, Japan
(Dated: January 5, 2016)
We theoretically predict a generation of a current in Weyl semimetals by applying circularly
polarized light. The electric field of the light can drive an effective magnetic field of order of ten
6
Tesla. Forlowerfrequencylight,anon-equilibriumspindistributionisformedneartheFermisurface.
1
Due to the spin-momentum locking, a giant electric current proportional to the effective magnetic
0
fieldisinduced. Ontheotherhand,higherfrequencylightrealizesaquasi-staticFloquetstatewith
2
no induced electric current. We discuss relevant materials and estimate order of magnitude of the

n inducedcurrent.
a
J

4 Introduction— Recently, Dirac and Weyl semimet- obtain a net current of the chiral magnetic effect.
als, which host bulk gapless excitations obeying quasi- Recent studies using femtosecondlaser pulses have es-

]
l relativisticfermionequations,haveattractedmuchatten- tablishedamethodtogeneratenon-equilibriummagnetic
l
a tion in condensed matter physics [1–14]. Dirac semimet- fieldsbycircularlypolarizedlightinferrimagnets[32–34].
h als have been theoretically predicted[1–3] and experi- Thelight-inducedeffectivemagneticfieldBeffisgivenby
- mentally demonstratedin (Bi In ) Se [4, 5], Na Bi[6,
s 1−x x 2 3 3
e 7] and Cd As [8, 9]. There are also several exper- Beff ∝iE×E∗. (1)
3 2
m
iments supporting the realization of Weyl semimetals
. in TaAs[10–13]. Moreover, Dirac and Weyl semimetals with the circularly polarized complex electric field E[35,
t
a have been theoretically predicted in a superlattice het- 36]. The generation of the effective magnetic field is due
m erostructure of topologicalinsulator (TI)/normalinsula- to the conversion of spin-angular momentum from light
- tor (NI)[3], and a Dirac semimetal has been realized in to electrons via the spin-orbit coupling[36–38]. The di-
d the GeTe/Sb Te superlattice[14]. rection of Beff depends on the chirality of the circularly
n 2 3
polarized light. Its magnitude is proportional to the in-
o Low energy bulk excitations in Dirac and Weyl
c tensity of laser and can be 20 T for a sufficient strong
semimetals come in pairs of left and right-handed
[ laser pulse[32–34].
Weyl fermions, because of the Nielsen-Ninomiya’s no go
In this Letter, we theoretically predict a current j in-
1 theorem[15]. In the low energy limit, eachchargeflow of
v duced by the effective magnetic field (Fig.1). The pho-
leftandright-handedWeylfermionspreservesclassically,
9 tovoltaic current is due to a non-equilibrium spin distri-
buttheirdifference,theaxialcurrent,isnotconservedin
7 butionnearthe Fermisurface. Forlowerfrequencylight,
3 the quantum theory, due to the chiral anomaly. In an
the conversion of spin-angular momentum between light
0 analogy of relativistic high energy physics [16–21], the
0 anomalyrelatedeffectshavebeendiscussedincondensed
.
1 matter physics[22–31]. The anomaly induced currents
0 are dissipationless and thus they have potential applica-
6 tions to unique electronics.
1
: Amongtheanomalyrelatedeffects,oneofthemostin-
v
terestingphenomena is the chiralmagneticeffect. Inthe
i
X presence of a time-dependent θ term in the Dirac-Weyl
r theory, a current proportional to an applied static mag-
a
neticfieldhasbeenpredictedtheoretically[16–20,23–26].
Theflowduetothestaticmagneticfield,however,might FIG. 1: (Color online) Schematic illustration of photovoltaic
be problematic in condensed matter physics. First, in chiralmagneticeffect: (a)Foralowerfrequencylightregime,
Weyl semimetals, the time-dependent θ termis obtained electrons near the Fermi surface are excited by the incident
in the ground state, in the presence of the energy differ- lightthroughtheRamanprocessillustrated. Asaresult,afi-
nitespindistributionisgeneratedneartheFermisurface,and
ence ofleft andright-handedWeyl points [25]. However,
thespinofWeylfermionsisalignedinthedirectionoftheef-
thesystemstaysthegroundstateunderastaticmagnetic fective magnetic field Beff, on average. (b) For the reason
field, so no actual current should flow eventually[23].
above, the circularly polarized light aligns the spin of Weyl
Moreover, the detection can be difficult because there fermions. Because of (pseudo)spin-momentum locking, Weyl
isno drivingforcetogetoutthe currentinsuchanequi- fermions with helicity σ=1 (σ=−1) movein thesame (op-
librium state. Hence, instead of a static magnetic field, posite)directionasthespin,whichresultsinnonzerocurrent
one should consider a non-equilibrium magnetic field to jσ.
2
and electrons occurs only near the Fermi surface. Thus, Current induced by circularly polarized light—
liketherelativistictheory,thelowenergydescriptionus- We calculate the current induced by light, using the
ingWeylfermionsgivesagoodapproximationtoevaluate Keldysh Green’s function technique[40]. Below, we as-
the chiral magnetic effect. On the basis of the Keldysh sume that ~Ω is muchlowerthan the band width, so the
Green’s function, we show that a net current is obtained lowenergyeffectiveHamiltonian(3)givesagoodapprox-
byapplyingcircularlypolarizedlight. Thecurrentispro- imation. For Eq.(3), the current is defined by
portional to the effective magnetic field in an analogous
form of the chiral magnetic effect. On the other hand, hji≡evFhψ†(x,t)σzsψ(x,t)i, (8)
when light has a frequency higher than the energy scale
which is decomposed as
of the band width, a quasi-static Floquet state is real-
ized, where the chiral magnetic effect is cancelled due to
hji≡hj i+hj i, (9)
+ −
the occupied band electrons. In the latter case, Weyl
points are shifted in the momentum space, resulting in with hj i ≡ σev hψ†sψ i. Here ψ† = (ψ† ,ψ† )
the change of the anomalous Hall effect, instead. σ=± F σ σ σ σ,↑ σ,↓
is the creation operator of Weyl fermions with helicity
Model—WeconsiderthefollowingHamiltoniantode- σ = ±. There is no mixing term between ψ† and ψ in
scribe Weyl/Dirac semimetals in the presence of circu- + −
H, and thus hj i and hj i can be calculated separately.
+ −
larly polarized light
For a while, we consider the b=0 case.
In terms of the Keldysh Green’s function,
H =H +H +V . (2)
Weyl em imp
the chiral current hj i is represented as hj i =
σ σ
ThefirsttermistheHamiltonianofWeyl/Diracsemimet- −σi~ev tr[sG<(x,t : x,t)] with the 2×2 matrix lesser
F σ
als. In low energy, it takes the form Green function G<(x,t : x,t) = −i~hψ†(x,t)ψ (x,t)i.
σ σ σ
The contribution from Beff ∝ iE ×E∗ is given by the
HWeyl = ψk†HWeylψk, (3) diagrams in Fig.2. It is written as
Xk
HWeyl =~vFσz(k−σzb)·s−µσ0s0−µ5σzs0, (4) hjσii=−i~evF[Iσijk(Ω)+Iσijk(−Ω)]EjE∗k (10)
where ψ = t(ψ ψ ψ ψ ) is the annihilation with
k ↑,+ ↓,+ ↑,− ↓,−
operatorofelectronwith (pseudo)spin (↑,↓)andhelicity
e2v2
(+,−). sµ andσµ arethePaulimatricesof(pseudo)spin Iijk(Ω)= F C(I),ijk. (11)
σ 4Ω2 σ
and helicity, vF is the Fermi velocity, and µ is the chem- I=Xa,b,c,d
ical potential. The parameters 2b and 2µ denote the
5
difference of the position of left and right-handed Weyl Each diagram in Fig.2 gives the following C(I=a,b,c,d),ijk
σ
pointsinthemomentumandenergyspaces,respectively.
For Dirac semimetals, b = 0 and µ5 = 0. The sec- Cσ(a),ijk = tr sigk,ω,σsjgk,ω+Ω,σskgk,ω,σ <,
ond term in Eq. (3) is the gauge coupling between Xk,ω (cid:2) (cid:3)
Weyl/Dirac semimetals and light <
C(b),ijk = tr sig Sj g skg ,
σ k,ω,σ ω,ω+Ω k,ω+Ω,σ k,ω,σ
H =− j·Aem, (5) Xk,ω h i
em
Xk Cσ(c),ijk = tr sigk,ω,σsjgk,ω+Ω,σSωk+Ω,ωgk,ω,σ <,
where j denotes the charge current, and Aem is the vec- Xk,ω (cid:2) (cid:3)
tor potential of light. For circlarily polarized light, the C(d),ijk = tr Si g sjg skg <,(12)
σ ω,ω k,ω,σ k,ω+Ω,σ k,ω,σ
electric field Eem =−∂tAem is given by Xk,ω (cid:2) (cid:3)
Eem =Re EeiΩt , (6) where g< is given by
k,ω,σ
(cid:2) (cid:3)
where E is a complex vector and Ω is the angular fre-
g< =f ga −gr (13)
quency of light. The third term in Eq. (2) expresses the k,ω,σ ω k,ω,σ k,ω,σ
(cid:2) (cid:3)
impurity scattering in Weyl/Dirac semimetals[39, 40], withtheFermidistributionfunctionf andtheretarded
ω
and advanced Green’s functions
V = ψ† σ0s0u (q)ψ . (7)
imp k+q imp k
Xk,q gr = ~ω−σ~v k·s+µ+σµ + i~ −1,
k,ω,σ (cid:20) F 5 2τ (cid:21)
The impurity scattering potential u is assumed to e,σ
imp
be short-ranged and triggers a finite relaxation time, ga = gr †. (14)
k,ω,σ k,ω,σ
which is given within the Born approximation as τe,σ = (cid:2) (cid:3)
~/(πν n u2 )withaconcentrationofnonmagneticim- Si is the vertex correction due to the nonmagnetic
e,σ c imp ω,ω′
purities n . impurity scattering V [41].
c imp
3
EE ·· (a)! Ej (b)! (c)! (d)! magnetic field near the Fermi surface
!!jjσσii""=!si!kk,,ωω,,σσ!!ssjk!!k,ω+EΩE∗,kσ!+! SSSSωωωωjjjj,,,,ωωωω!!!!++++ΩΩΩΩ"""" +! SSSSSSSSωωωωkkkk++++ΩΩΩΩ,,,,ωωωω +! SSSS SSSSωωωωiiii,,,,ωωωω Bσeff ≡χσΩiE×E∗ =σLχσΩ|E|2qˆ, (19)
with χ ≡ 2e2vF2τe4,σ. Here qˆis the unit vector of the di-
FIG.2: Diagrammaticrepresentationofachargecurrenthjσii rectionσoflig3htgµpBro~pagation,σL =±1specifiesthechiral-
via photovoltaic chiral magnetic effect (a) without and (b-d)
ity (clockwise or counter-clockwisepolarization)oflight,
with a vertex correction of nonmagnetic impurity scattering.
g is the Land´e factor, and µ is the Bohr magneton.
A wavy line denotesan electric field of thepolarized light. B
Itisnotedthatthelight-inducedcurrenthasasimilar-
ity to the chiral magnetic effect. In both cases, the cur-
(I=a,b,c,d),ijk rentflowsinthedirectionofanappliedmagneticoreffec-
Using Eq.(13), one canrewriteC in terms
σ
tive magnetic field, and its magnitude is proportional to
oftheretardedandadvancedGreen’sfunctions. For|µ+
σµ |≫~/τ , we find that the difference of the chemical potential between left and
5 e
right-handed fermions. Indeed, like our case, the spin-
C(I=a,b,c,d),ijk ∝ (f −f ). (15) polarization and the spin-momentum locking are essen-
σ ω+Ω ω
Xω tialtoobtainthecurrentinthechiralmagneticeffect[17].
Under astatic magneticfield, electronsformthe Landau
This means that only fermions near the Fermi surface levels. For Weyl fermions, the zeroth Landau level is
contribute the light-induced current, which justifies our fully spin-polarized in the direction of the applied mag-
Weyl fermion approximation. We also find that Cσ(I),ijk neticfield,andthusthegroundstateofthesystemisalso
containsbothoftheretardedandadvancedGreen’sfunc- spin-polarized. As a result, the current flows due to the
tions,anditisexpressedbytheirproduct. Thisisasignal spin-momentum locking[17]. We dub our light-induced
of a non-equilibrium process[40]. current effect as photovoltaic chiral magnetic effect.
After some calculation [41], we obtain Here we would like to mention that there is an impor-
tant difference between our photovoltaic chiral magnetic
2ν e3v3τ4
hj i=σ e,σ F e,σΩi(E×E∗), (16) effect and the original one. In the original case, the chi-
σ 3~ ral magnetic effect is caused by a static magnetic field,
and thus the resultant current is equilibrium (and dis-
where νe,σ = (2µπ+2σ~µ3v5F)32 is the density of state of the Weyl sipationless). In condensed matter physics, however, an
cone with helicity σ. From Eq. (16), the total current analogouscurrentofWeylfermions,evenifexists,iscom-
hji is pletelycancelledbyothercurrentintheconductionband
[23]. Ontheotherhand,thephotovoltaicchiralmagnetic
hji= 2(νe,+τe4,+−νe,−τe4,−)e3vF3Ωi(E ×E∗), (17) effect is due to the time-dependent electric field, so the
3~ current is non-equilibrium and dissipative. The current
comes only from Weyl fermions near the Fermi surface,
whichisnon-zerowhenν τ4 6=ν τ4 ,namelywhen
e,+ e,+ e,− e,− so no cancellation occurs.
µ 6=0.
5 The effective magnetic field also generates the ax-
The obtained current originates from a non-
ial current, which is the difference between charge cur-
equilibrium distribution of spin: When one exposes the
rents with different helicity: hj i ≡ hj i − hj i =
axial + −
system to circularly polarized light, the conversion of
ev [hψ†sψ i+hψ†sψ i]. Asmentionedabove,forlower
spin-angular momentum between light and electrons oc- F + + − −
Ω, the system is well described by Weyl fermions, and
curs due to the spin-orbit interaction. As a result, there
thus the axial current can be well-defined as well. The
arises a non-equilibrium distribution of spin near the
axial current is nonzero even for Dirac semimetals with
Fermi surface [Fig.1(a)]. For Weyl fermions, because
b = µ = 0. The axial current can be detected as total
of the spin-momentum locking,the non-equilibrium spin 5
spin polarization, by using pump-probe techniques[32].
distribution gives rise to the current flow [Fig.1(b)]. In-
Wecaneasilygeneralizetheaboveresultforhjiinthe
deed, for Eq.(3), the current operator is essentially the
case with b 6= 0. Since b behaves like a static Zeeman
same as the spin operator, and thus, from the same cal-
field in H , it can shift hψ†sψ i by the Pauli param-
culation,one canshowthat the circularlypolarizedlight Weyl σ σ
agnetism. However,b cannot drive a net currentsince it
induces a non-zero spin polarization of electrons
is static. Moreover, the circularly polarized light affects
2ν e2v2τ4 only electrons near the Fermi surface, which structure
hψσ†sψσi= e,σ3~F e,σΩi(E×E∗), (18) does not depend on b. Therefore, we have the same cur-
rent hji in Eq.(17) even when b6=0.
near the Fermi surface. We estimate the magnitude of Beff and hji by using
σ
Since the circularly polarized light induces the spin- materialparametersforTaAs[42],v =3×105m/s,τ =
F e
polarization of electrons, it effectively acts as a Zeeman 4.5×10−11 s, and µ=11.5 meV. If the difference of the
4
chemicalpotentialisµ =1meV,|Beff|canbeestimated in Eq.(19) originates from a dissipative process so it de-
5 σ
as |Bσeff=±| = (4.3∓2.6)×10−16([s−Ω1])([V|2E/|m22]) T. For pends on τe,σ, BFeffloquet in Eq.(21) is independent of the
|E| = 4 kV/m and Ω = 2.2×109 s−1, |Beff | can reach impurity scattering. Furthermore, the former magnetic
σ=± fieldonlyaffectsonelectronsneartheFermisurface,but
upto15∓9T.Then,theinducedchargecurrentbecomes
the latter acts on the whole of the band. Consequently,
|hji|≃2×106A/m2,whosecurrentdensityismuchlarger
the resultant phenomena can be different.
thantheanomalousHallcurrentdensityduetothechiral
WefindthatnonetcurrenthjiisobtainedbyBeff :
anomaly[43]. This giant current density is caused from Floquet
the giant magnetic field Beff. We would like to point According to Eq.(20), BFeffloquet just provides a uniform
σ
Zeeman splitting (or shift) in the whole band spectrum
out that hji is distinguished from the longitudinal [39]
oftheWeylsemimetal,likeastaticZeemanfield. There-
and the transverse charge current [26–28, 43], since hji
fore, in a steady state, electrons fill the band up to the
is parallel to the light traveling direction and it flows in
Fermi energy. In this situation, one can use the same
an opposite direction when the chirality of the light is
argument in Ref.[23], and prove that hji = 0. Whereas
reversed.
Weyl fermions may have a nonzero spin hψ†sψ i due
Floquet state— So far, we have assumed that the σ σ
to the Pauli magnetism of Beff , the current due to
frequencyΩoflightismuchlowerthanascaleoftheband Floquet
the spin-momentum locking is totally cancelled by the
width. Now,weconsidertheoppositecase. Incontrastto
current from the rest of the band. In other words, no
the lower Ω case, in which only electrons near the Fermi
photovoltaic chiral magnetic effect occurs for the higher
surfaceareinfluencedbylight,thehigherfrequencylight
frequency light.
can affect the whole electrons in valence bands.
It is helpful to regard the frequency Ω as an energy
To consider this situation, we adapt the Floquet
cutoffforthechiralmagneticeffect. ForlowerΩ,thelight
method: Because H in Eq.(2) is periodic in t, i.e.
can excite only Weyl fermions near the Fermi surface,
H(t) = H(t + 2π/Ω), the wave function of the
Scho¨dinger equation i~∂ ψ(t) = H(t)ψ(t) has the form and thus the quasi relativistic phenomena like the chiral
t
of ψ(t) = φ e−i(ε+m~Ω)t/~, where the summa- magneticeffectmayoccur. AsΩincreases,electronsina
m m lowerpositionofthebandcanparticipateinthecurrent,
tion is takenPfor all integers m. Substituting this
then eventually, when Ω is large enough to affect the
form into the Scho¨dinger equation, we have the Flo-
quet equation, H φ = (ε + m~Ω)φ , with wholespectrumofthe band,the chiralmagneticeffectis
n m,n n m completely cancelled.
H = (Ω/2π) P2π/ΩdtH(t)ei(m−n)Ωt +m~Ωδ . For
m,n 0 m,n Instead, for higher Ω, one can expect the light in-
the HamiltonianRin Eq.(2), the diagonalterm of the Flo-
duced anomalous Hall effect. Substituting Eq.(3) for
quet Hamiltonian is given by H = H +V +
m,m Weyl imp H in Eq.(20), one finds that Beff shifts b by
m~Ω,andtheoff-diagonalonesareH =H∗ = Weyl Floquet
m,m+1 m+1,m δb = −(gµ /2~v )Beff . The change of b induces
(Ω/2π) 02π/ΩdtHeme−iΩt =−iev2FΩ|E|σz(sx−iσLsy)when the change oBf θ-teFrm iFnlotqhueetWeyl semimetals [25], which
lightis aRlongz axis. Other off-diagonalterms areidenti- results in
cally zero. Eachsolution of the Floquet equation gives a
2αcǫ 2αcǫ
periodic steady state. hδρi= 0δb·B, hδji=− 0δb×E, (22)
π π
For large Ω, the diagonal terms are dominant, so one
can treat the off-diagonal ones as a perturbation. In in the presence of external magnetic and electric fields,
the zeroth order, our system is described by H0,0 = B and E. Here α is the fine structure constant, c is
HWeyl +Vimp, then the first non-zero correction in the the speed of light, and ǫ0 is the vacuum permittivity.
perturbation theory appears in the second order as The lightinduced chargepumphδρiandanomalousHall
~1Ω[H0,−1,H0,1]. Thus, we obtain the following effective currenthδjihavebeendiscussedrecentlyinRefs.[44–46].
Hamiltonian Conclusion— We theoretically predict photovoltaic
chiral magnetic effect, which is induced by the effective
e2v2
H =H +V +iσ0 F(E ×E∗)·s, (20) magneticfieldduetocircularlypolarizedlight. Inthelow
eff Weyl imp ~Ω3
lightfrequencyregime,theeffectivemagneticfieldaffects
onlyfermionsneartheFermisurface. Asaresult,theef-
by which a periodic steady state of our system is de-
fectivemagneticfieldplaystheroletotriggerafinitespin
scribed.
polarizationof Weyl fermions anddrive the finite charge
From Eq. (20), it is found that the higher frequency
current in Eq.(17). On the other hand, in the high fre-
light induces a different effective Zeeman magnetic field,
quencyregime,theFloquetquasisteadystateisrealized.
e2v2 The circularly polarized light induces the effective mag-
Beff ≡ F iσ0(E ×E∗). (21)
Floquet gµ ~Ω3 netic field in Eq.(21), which is completely different from
B
thatinthelowerfrequencyregime. Themagneticfieldin
Here we note that the physical origin is completely dif- the high frequency regime behaves like the Zeeman field
ferent from that in the lower frequency case. While Beff andshiftsthewholebandstructure. ThecurrentofWeyl
σ
5
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