Anomalous skew-scattering nonlinear Hall effect and chiral photocurrents in PT-symmetric antiferromagnets Da Ma1Arpit Arora1Giovanni Vignale2and Justin C.W. Song1
2025-04-30
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Anomalous skew-scattering nonlinear Hall effect and chiral photocurrents in
PT-symmetric antiferromagnets
Da Ma,1, ∗Arpit Arora,1, ∗Giovanni Vignale,2and Justin C.W. Song1, †
1Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore 637371
2The Institute for Functional Intelligent Materials (I-FIM),
National University of Singapore, 4 Science Drive 2, Singapore 117544
Berry curvature and skew-scattering play central roles in determining both the linear and nonlin-
ear anomalous Hall effects. Yet in PT-symmetric antiferromagnetic metals, Hall effects from either
intrinsic Berry curvature mediated anomalous velocity or the conventional skew-scattering process
individually vanish. Here we reveal an unexpected nonlinear Hall effect that relies on both Berry cur-
vature and skew-scattering working in cooperation. This anomalous skew-scattering nonlinear Hall
effect (ASN) is PT-even and dominates the low-frequency nonlinear Hall effect for PT-symmetric
antiferromagnetic metals. Surprisingly, we find that in addition to its Hall response, ASN produces
helicity dependent photocurrents, in contrast to other known PT-even nonlinearities in metals which
are helicity blind. This characteristic enables to isolate ASN and establishes new photocurrent tools
to interrogate the antiferromagnetic order of PT-symmetric metals.
Nonlinear response can be a powerful diagnostic of
a material’s intrinsic symmetries. A prime example is
the nonlinear Hall effect that manifests in time-reversal
invariant but inversion broken metals [1–13]. Arising
at second-order in an applied electric field, the nonlin-
ear Hall effect is often attributed to quantum geometric
properties of Bloch electrons such as the Berry curva-
ture dipole (BCD) [3,5,6] or skew-scattering processes
[7,8,11,13]. Such nonlinearities can persist even in
antiferromagnets (e.g., BCD nonlinear Hall effect [14])
when both inversion (P) and time-reversal (T) symme-
tries are broken. However, an unusual situation occurs
in antiferromagnets that respect the combination of P
and Tsymmetries, i.e. PT symmetry [15–17]. Even
though antiferromagnetism breaks Pand Tsymmetries
simultaneously, PT symmetry zeroes out net Berry flux
and ensures that the BCD [17] and conventional skew-
scattering nonlinearities vanish [18]. Can Berry curva-
ture or skew-scattering play any role in Hall responses of
PT-symmetric materials?
Here we reveal a new paradigm for nonlinear transport
where skew-scattering (extrinsic scattering) and Berry
curvature (quantum geometric) cooperate to produce a
second-order nonlinear Hall effect that persists in PT-
symmetric materials. This anomalous skew-scattering
nonlinear Hall effect (ASN) arises from combining a
spin-dependent anomalous velocity and a skew-scattering
spin-dependent distribution, Fig. 1b. ASN is T-odd, van-
ishing in T-symmetric materials; as such, it has been
neglected. However, as we argue, ASN is PT-even, ren-
dering PT-symmetric antiferromagnets a prime venue for
its realization.
Surprisingly, ASN also mediates a helicity dependent
chiral photocurrent in the metallic limit peaking in the
THz. This is striking since all other known intraband
chiral photocurrents active in metals [3,19] vanish in
PT-symmetric materials and are insensitive to magnetic
FIG. 1. Anomalous skew-scattering nonlinear Hall effect in
PT-symmetric materials. (a) Scattering k→k′for up spin
(green) shares the same rate as k′→kfor down spin (red) in a
PT-symmetric system. In particular, this PT-symmetry pro-
duces opposite skew-scattering contributions to the scattering
rate for up and down spins respectively [see Eq. (2)]. (b) Due
to PT symmetry, the first-order skew-scattering driven devi-
ations of the electronic distribution (from equilibrium, solid)
have opposite signs for up and down spins; blue/yellow indi-
cate sign of deviation. P T symmetry also enforces opposite
anomalous velocity for up (green horizontal arrows) and down
(red horizontal arrows) spins. When this spin dependent
anomalous velocity is combined with skew-scattering driven
deviations of the electronic distribution, a non-vanishing non-
linear Hall effect manifests even in a P T -symmetric metal.
ordering [19]. ASN, as we will see below, not only sur-
vives in PT-symmetric materials but is T-odd making it
a useful new tool for accessing helicity dependent THz
chiral photocurrents locked to magnetism.
Our work lies in the context of a recent surge of interest
in second-order nonlinearities [16–22] in PT-symmetric
antiferromagnets (e.g., CuMnAs [23,24], MnBi2Te4[25,
26]); such nonlinearities can be used to detect antifer-
arXiv:2210.14932v2 [cond-mat.mes-hall] 24 Aug 2023
2
romagnetic order, see e.g., Ref. [15]. In the metal-
lic/intraband limit, these have largely focussed on an in-
trinsic nonlinear Hall (INH) effect that arises from the
Berry connection polarizability tensor [16,17]. INH pro-
duces nonlinear Hall currents that are independent of
the scattering time, τ. In contrast, ASN is extrinsic and
depends on τat low frequencies. As a result, ASN is
expected to dominate the nonlinear Hall effect in PT-
symmetric antiferromagnets in the clean limit providing
a much needed engineering strategy for boosting nonlin-
ear Hall signals in PT antiferromagnets [15].
PT partners and spin-dependent skew-scattering: We
begin by examining the effect PT-symmetry can have on
the motion of electrons. As a simple illustration, consider
the minimal Bloch hamiltonian H(0)(k) = H(0)
↑(k) +
H(0)
↓(k) where s={↑,↓} are spins and kis the electron
wavevector. PT-symmetry enforces double degeneracy
and (PT)H(0)
↑(k)(PT)−1=H(0)
↓(k) [23] yielding
ϵ↑(k) = ϵ↓(k) = ϵ(k),⟨u↑(k)|u↑(k′)⟩=⟨u↓(k′)|u↓(k)⟩,
(1)
where |us(k)⟩is a Bloch state of H(0)(k) with a spin label
s. For brevity of notation, we have omitted the band
index. Eq. (1) conveniently relates the properties of the
PT partners ↑and ↓. For example, ↑and ↓share the same
group velocity v(k) = ∂kϵ(k)/¯h, but possess opposite
Berry curvature Ωs(k) = i⟨∇kus(k)| × |∇kus(k)⟩signs.
Eq. (1) also constrains electronic scattering. In the
presence of a scalar impurity potential V, the scat-
tering rate in a single band is given by Ws
k→k′=
(2π/¯h)|⟨us(k′)|V|ψs(k)⟩|2δ[ϵs(k)−ϵs(k′)] [27] that cap-
tures skew-scattering processes that occur beyond the
Born approximation. Here |ψs(k)⟩is an eigenstate of the
full hamiltonian H(0)(k)+Vand can be expanded order-
by-order using the self-consistency relation: |ψs(k)⟩=
|us(k)⟩+ [ϵs(k)−H0(k) + iη]−1V|ψs(k)⟩[27]. For scalar
impurities and elastic scattering, we find (see SI [28]),
W↑
k→k′=W↓
k′→k, w(S,A)
↑,k,k′=±w(S,A)
↓,k,k′,(2)
where w(S,A)
s,k,k′= [Ws
k′→k±Ws
k→k′]/2 are the symmetric
and skew (antisymmetric) scattering contributions to the
total scattering rate respectively. Crucially, the scatter-
ing process ↑,k→↑,k′is the PT partner of ↓,k′→↓,k
and have the same rate (Fig. 1a). As a result, the ↑,↓
have opposite skew-scattering contributions. This conclu-
sion persists for any PT symmetric scattering potential.
Eq. (2) applies order-by-order in V, and can be ob-
tained by employing Eq. (1) to the scattering rate. For
an intuitive illustration of the origins of Eq. (2), we ex-
amine the familiar third order in Vexpression for the
skew-scattering rate [27,29,30]
wA
s,k,k′=4π2niV3
0
¯hX
k′′
δ(ε)
k,k′δ(ε)
k,k′′ Im{Ls(k,k′′,k′)},(3)
Nonlinear Hall effects T P T references
Berry curvature dipole (BCD) + −[3,5,6,14]
Intrinsic (INH) −+ [16,17]
Conventional skew-scattering + −[7,8,11]
Anomalous skew-scattering (ASN) * −+ this work
TABLE I. Symmetry of intraband nonlinear Hall responses.
+ indicates the response is even (i.e. allowed by symmetry),
−means it is odd (i.e. forbidden by symmetry). Starred
nonlinear Hall susceptibility is the new PT-even response dis-
cussed in this work in Eq. (8) for the ASN, see also SI [28].
where V0is the impurity strength, nithe impurity
density, δ(ε)
k,k′=δ[ϵ(k)−ϵ(k′)], and Ls(k,k′′,k′) =
⟨us(k)|us(k′′)⟩⟨us(k′′)|us(k′)⟩⟨us(k′)|us(k)⟩is the Wil-
son loop associated with the Pancharatnam-Berry phase
of the skew-scattering process [30]. Directly applying
Eq. (1) to Eq. (3) yields a sign changing wAin Eq. (2).
The inclusion of both PT partners (in our case, spin) is
essential since applying the same reasoning to a spinless
system produces a vanishing wA(e.g., Ref. [18] computed
a vanishing wAto V4in a spinless system).
Eq. (1) and (2) have a profound impact on transport
behavior of PT-symmetric materials (e.g., PT-symmetric
antiferromagnets). Because Ω↑(k) = −Ω↓(k), the net
Berry flux and the net Berry curvature dipole (BCD)
vanish thereby zeroing out the intrinsic linear anoma-
lous Hall as well as the BCD nonlinear Hall effect. Sim-
ilarly, the changes to the distribution function due to
skew-scattering in Eq. (2) are opposite for ↑and ↓(see
Fig. 1b and detailed discussion below); when combined
with v(k) = ∂kϵ(k)/¯h, the conventional skew-scattering
anomalous Hall effect at both linear and second order
vanishes under PT-symmetry.
Anomalous skew-scattering nonlinear Hall effect: How-
ever, when both Berry curvature mediated anomalous ve-
locity (PT-odd) as well as the changes to the distribution
function driven by skew-scattering (PT-odd) combine, a
non-vanishing second-order ASN Hall effect (PT-even)
can be produced (Fig. 1b). To see this in a systematic
fashion, we analyze the net charge current
j(t) = −eX
k,s v(k) + eE(t)/¯hׯ
Ωs(k)fs(k, t),(4)
where −e < 0 is the carrier charge, E(t) is a time-varying
uniform electric field, fs(k, t) is the distribution function,
and ¯
Ωs(k) is the modified Berry curvature that includes
both intrinsic Bloch band Berry curvature [Ωs(k)] as well
as field-induced corrections [16,17,31]
¯
Ωs(k) = Ωs(k) + ∇k×G(k)E(t).(5)
Here G(k) is the Berry-connection polarizability tensor
in the metallic band of interest [16,17,31]. For band
3
n, [G]ab(k) = 2eRe{Pn′̸=nAnn′
a(k)An′n
b(k)/[ϵn(k)−
ϵn′(k)]}, with Ann′
b=⟨un(k)|i∂kbun′(k)⟩. Here, a, b de-
notes Cartesian coordinates and G(k) is even under PT.
For clarity, we concentrate on the intraband limit and
focus on scalar impurities that preserve PT symmetry
and conserve spin. The distribution function in Eq. (4)
can be directly computed via a spatially uniform kinetic
equation, ∂tfs(k, t)−eE(t)·∂kfs(k, t)/¯h=I{fs(k, t)}
where [27,32,33]
I{fs(k, t)}=X
k′Ws
k′→kfs(k′, t)−Ws
k→k′fs(k, t)(6)
describes the spin-dependent collision integral.
The distribution function can be solved in the standard
perturbative fashion: in powers of Eand for weak skew-
scattering by using the relaxation time approximation,
see SI for a detailed derivation [28]. As such, we expand
fs(k, t) as
fs(k, t) = f0(k) + X
ℓ,m
f(m)
ℓ,s (k, t),(7)
where the second term captures the deviation of the dis-
tribution function from the equilibrium distribution func-
tion, f0(k). Here subscript ℓ= 1,2,· · · denote order in E
and the superscript m= 0,1,· · · denote its dependence
on skew-scattering rate; m= 0 captures the purely sym-
metric part independent of skew-scattering.
Note f0(k) is the same for both ↑and ↓due to PT
symmetry. Similarly, even as m= 0 contributions
depend on the (transport) relaxation time: (τs)−1=
⟨Pk′wS
s,k′,k(1 −cos θvv′)⟩,PT symmetry in Eq. (2) en-
sure f(0)
ℓ,s (k, t) are the same for ↑and ↓since τ↑=τ↓=τ.
Here θvv′is the angle between v(k) and v(k′), and ⟨· · · ⟩
indicates an average over an energy contour. In contrast,
skew-scattering (m= 1) contributions to the distribution
function f(1)
ℓ,s (k, t) have opposite signs for opposite spins,
see Fig. 1b: a property key to ASN.
Writing E(t) = Eeiωt +c.c. and substituting the dis-
tribution functions into Eq. (4) enables to directly dis-
cern the nonlinear Hall responses. Amongst the possi-
ble second-order nonlinear Hall responses obtained (see
Table I), two are PT-even; the rest are odd. The first
PT-even response is the intrinsic nonlinear Hall (INH)
effect [16,17,31] obtained by combining the second
term in Eq. (5) with f0(k). This yields an INH current
[jINH]a(t) = Re(j0
a+j2ω
aei2ωt) with j0
a=χINH
abc [Eb]∗Ec
and j2ω
a=χINH
abc EbEc, where χINH
abc [16,17,31] depends
only on band geometric quantities. χINH
abc is independent
of τand insensitive to ωin the semiclassical limit.
The second PT-even nonlinear Hall response, ASN, is
the main result of our work. This nonlinear Hall effect
arises from combining Ωs(k)×E(t) with the skew dis-
tribution function f(1)
1,s (k, t). This produces a nonlin-
ear Hall response: [jASN]a(t) = Re(j0
a+j2ω
aei2ωt) with
j0
a=χASN
abc [Eb]∗Ecand j2ω
a=χASN
abc EbEcwith
χASN
abc = 2e3εadb
¯h2X
k,k′,s
Ωs
d(k)˜τ2
ωwA
s,k,k′∂f0(k′)
∂k′c
,(8)
where Ωs
d(k) denotes the dcomponent of Ωs(k), ˜τω=
τ/(1 + iωτ) and εadb is the Levi-Civita symbol. Since
Ωs
dand wA
sare both odd under PT , their product is
even producing a finite extrinsic nonlinear Hall effect.
Importantly, χASN
abc scales as τ2wAfor ωτ ≪1. As a
result, χASN
abc is expected to dominate the nonlinear Hall
response in clean systems. At finite ω,χASN
abc displays a
characteristic ωdependence varying rapidly on the scale
1/τ (see below); this ωdependence distinguishes it from
both the ωinsensitive χINH as well as interband effects
that have characteristic ωdependence on the scale of
interband transition energy ϵn−ϵm.
Symmetry, scattering, and chiral photocurrents: ASN
has several striking attributes. Due to its Berry curva-
ture roots, χASN
abc is antisymmetric in its first two indices
yielding a nonlinear Hall effect [34] always transverse to
the applied electric field. This antisymmetric nature
imposes additional point-group symmetry constraints
as compared to conventional skew-scattering nonlineari-
ties [7,11,13]. For example, in 2D, antisymmetric non-
linear χabc requires broken rotational symmetry [3,17].
ASN’s antisymmetric behavior contrasts with that of
another PT-even nonlinear response that arises from
combining v(k) with f(0)
2,s (k, t) [16,18] to produce a clas-
sical nonlinearity, χDrude
abc . Importantly, χDrude
abc has a sus-
ceptibilty that is completely symmetric when its indices
are permuted yielding a response that need not always be
transverse as required of Hall type responses [34]. Exper-
imentally, this fully symmetric nonlinear Drude response
can be weeded out via interchanging the directions of
driving field and response: symmetric χabc is even under
exchange, whereas nonlinear Hall responses are odd.
Perhaps most striking is how ASN produces a
helicity-dependent chiral photocurrent: [j⟲]a=
i
2Im[χabc](E∗
bEc−EbE∗
c).ASN chiral photocurrent arises
from its part quantum geometric and part skew-
scattering origins. First, since ASN depends on skew
scattering χASN
abc possesses both real and imaginary com-
ponents arising from the complex valued ˜τ2
ωin Eq. (8).
Second, because ASN proceeds from the anomalous ve-
locity Ω×E, its susceptibility is asymmetric allowing for
a non-zero j⟲after both band cindices are summed.
Importantly, ASN’s combination of geometric nature
and scattering processes is essential. For instance, even
as symmetric scattering alone enables a nonlinear Drude
conductivity χDrude
abc [16,18] that has an imaginary com-
ponent, it nevertheless is completely symmetric under
any interchange of indices yielding a zero j⟲. Similarly,
while χINH
abc is also asymmetric, it nevertheless is purely
real producing helicity blind photocurrents. As a result,
to our knowledge, χASN
abc is the only intraband nonlinearity
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Anomalousskew-scatteringnonlinearHalleffectandchiralphotocurrentsinPT-symmetricantiferromagnetsDaMa,1,∗ArpitArora,1,∗GiovanniVignale,2andJustinC.W.Song1,†1DivisionofPhysicsandAppliedPhysics,SchoolofPhysicalandMathematicalSciences,NanyangTechnologicalUniversity,Singapore6373712TheInstituteforFunction...
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