SciPos Phys. 16, 077 (2024)
Fa aday o a ion and ansmi ance as ma ke s
o opological phase ansi ions in 2D ma e ials
Manuel Calix o1,2⋆, Albe o Mayo gas1†, Nicolás A. Co de o2,3,4‡,
El i a Rome a2,5◦and Oc a io Cas años6§
1Depa men o Applied Ma hema ics, Uni e si y o G anada,
Fuen enue a s/n, 18071 G anada, Spain
2Ins i u e Ca los I o Theo e ical and Compu a ional Physics (iC1),
Fuen enue a s/n, 18071 G anada, Spain
3Depa amen o de Física, Uni e sidad de Bu gos,
Plaza Misael Bañuelos s/n, 09001 Bu gos, Spain
4In e na ional Resea ch Cen e in C i ical Raw Ma e ials o Ad anced Indus ial
Technologies (ICCRAM), Plaza Misael Bañuelos s/n, 09001 Bu gos, Spain
5Depa men o A omic, Molecula and Nuclea Physics, Uni e si y o G anada,
Fuen enue a s/n, 18071 G anada, Spain
6Ins i u o de Ciencias Nuclea es, Uni e sidad Nacional Au onoma de Mexico,
Apdo. Pos al 70-543, 04510, CDMX, Mexico
⋆calix o@ug .es , † albmay ey97@ug .es , ‡ [email p o ec ed] ,
◦e ome a@ug .es , § [email p o ec ed]
Abs ac
We analyze he magne o-op ical conduc i i y (and ela ed magni udes like ansmi -
ance and Fa aday o a ion o he i adia ed pola ized ligh ) o some elemen al wo-
dimensional Di ac ma e ials o g oup IV (g aphene analogues, buckled honeycomb la -
ices, like silicene, ge manene, s annane, e c.), g oup V (phospho ene), and zincblende
he e os uc u es (like HgTe/CdTe quan um wells) nea he Di ac and gamma poin s, un-
de ou -o -plane magne ic and elec ic ields, o cha ac e ize opological-band insula o
phase ansi ions and hei c i ical poin s. We p o ide plo s o he Fa aday angle and
ansmi ance as a unc ion o he pola ized ligh equency, o di e en ex e nal elec ic
and magne ic ields, chemical po en ial, HgTe laye hickness and empe a u e, o une
he ma e ial magne o-op ical p ope ies. We ha e shown ha abso ance/ ansmi ance
acqui es ex emal alues a he c i ical poin , whe e he Fa aday angle changes sign, hus
p o iding ine ma ke s o he opological phase ansi ion. In he case o non- opological
ma e ials as phospho ene, a minimum o he ansmi ance is also obse ed due o he
ene gy gap closing by an ex e nal elec ic ield.
Copy igh M. Calix o e al.
This wo k is licensed unde he C ea i e Commons
A ibu ion 4.0 In e na ional License.
Published by he SciPos Founda ion.
Recei ed 31-05-2023
Accep ed 19-12-2023
Published 18-03-2024 Check o
upda es
doi:10.21468/SciPos Phys.16.3.077
1
SciPos Phys. 16, 077 (2024)
Con en s
1 In oduc ion 2
2 Some wo-band 2D-Di ac ma e ial models 3
2.1 G aphene analogues: Silicene, ge manene, e c 4
2.2 HgTe/CdTe quan um wells 5
2.3 Phospho ene as an aniso opic ma e ial 8
3 Magne o-op ical conduc i i y 11
3.1 Magne o-op ical p ope ies o g aphene analogues 12
3.2 Magne o-op ical p ope ies o zincblende he e os uc u es 14
3.3 Magne o-op ical p ope ies o phospho ene and e ec o aniso opies 17
4 Conclusions 21
A Landau le els plo e sus ex e nal magne ic ield 22
B Silicene conduc i i y in he ci cula ly pola iza ion basis 23
C HgTe quan um well conduc i i y wi h Zeeman e ec 24
D Anima ions o he ene gy spec um and conduc i i ies 24
E Fa aday angle con ou plo s 25
Re e ences 28
1 In oduc ion
Two-dimensional (2D) ma e ials ha e been ex ensi ely s udied in ecen yea s (and a e ex-
pec ed o be one o he c ucial esea ch opics in u u e yea s) especially because o hei
ema kable elec onic and magne o-op ical p ope ies which make hem hope ul candida es
o nex gene a ion op oelec onic de ices. G aphene is he a che ype o a 2D nanoma e ial
wi h excep ional high ensile s eng h, elec ical conduc i i y, anspa ency, e c. In spi e o be-
ing he hinnes one, i exhibi s a gian Fa aday o a ion (ΘF∼6◦) on pola ized ligh in single-
and mul ilaye a angemen s [1–6]wi h expe imen al con i ma ion [7]. Fa aday o a ion is a
undamen al magne o-op ical phenomenon used in a ious op ical con ol, lase echnology
and magne ic ield sensing echniques.
Magne o-op ical p ope ies o o he buckled honeycomb la ices, like silicene [8], ha e
been s udied in [9–11], oge he wi h o he monolaye ansi ion me al dichalcogenides [12]
and aniso opic e sions like phospho ene [13]. Magne o-op ical measu emen s also p o ide
signa u es o he opological phase ansi ion (TPT; see [14–16] o s anda d ex books on he
subjec ) in in e ed HgTe/CdTe quan um wells (QW), dis inguishing quan um Hall (QH) om
quan um spin Hall (QSH) phases [17], whe e one can une he band s uc u e by ab ica ing
QWs wi h di e en hicknesses λ. A uni e sal alue o he Fa aday o a ion angle, close o
he ine s uc u e cons an , has been expe imen ally obse ed in hin HgTe QW wi h c i ical
hickness [18].
To de e mine expe imen ally he Fa aday o a ion e ec in Di ac ma e ials i is con enien
o conside : (1) A ans e se-magne ic-pola ized wa e inciden om he le on o a single
2
SciPos Phys. 16, 077 (2024)
opological insula o sandwiched by dielec ic laye s, which yields an enhancemen o he
Fa aday o a ion wi h an angle la ge han 700 m ad and wi h a ansmission highe han
90% [19]. (2) A g aphene shee sandwiched by one-dimensional opological pho onic c ys als
also an enhancemen o he Fa aday o a ion can be achie ed wi h high ansmi ance [20]. (3)
In hin ilms o 3D opological insula o s [21]o by conside ing hin ilms o Floque opological
insula o s whe e gian Fa aday and Ke o a ions ha e been obse ed unde he ac ion o a
pe pendicula magne ic ield o in a non- esonan op ical ield [22]. The in e se Fa aday e ec
(IFE) has been s udied in Di ac ma e ials in 2D and 3D, and hese s udies ha e concluded
ha IFE is s onge han in con en ional semiconduc o s. Then he Di ac ma e ials can be
po en ially use ul o he op ical con ol o magne iza ion in op oelec onic de ices [23].
In o ma ion heo e ic measu es also p o ide signa u es o he TPT in silicene [24–28]and
HgTe/CdTe QWs [29], as an al e na i e o he usual opological (Che n) numbe s. They also
accoun o semime alic beha io o phospho ene [30,31]unde pe pendicula elec ic ields.
In his pape we pe o m a compa a i e s udy o he magne o-op ical p ope ies o se e al
2D Di ac ma e ials, looking o TPT signa u es when he band s uc u e is uned by apply-
ing ex e nal ields o by changing he ma e ial cha ac e is ics. Fo his pu pose, we ocus on
ansmi ance and Fa aday o a ion nea he c i ical poin o he opological phase ansi ion
o opological ma e ials such as silicene and HgTe quan um wells. We ound ha , o hese
ma e ials, ansmi ance a ains an absolu e minimum T0a he c i ical TPT poin o a ce ain
alue Ω0o he no mal inciden pola ized ligh equency. This minimal beha io does no
depend on he chosen alues o magne ic ield, chemical po en ial and empe a u e, al hough
he loca ion o Ω0 a ies wi h hem. An in lec ion poin o he Fa aday angle is obse ed a
each peak o he ansmi ance, coinciding in equency. As a no el pe spec i e, we s udy
ha non- opological ma e ials, such as phospho ene, also exhibi an ex emal alue o he
ansmi ance when he ene gy gap is closed by an ex e nal elec ic ield.
The o ganiza ion o he a icle is as ollows. In Sec. 2we discuss he s uc u e o ime
independen Bloch Hamil onians o gene al wo-band 2D-Di ac ma e ial models, hei Che n
numbe s and hei minimal coupling o an ex e nal pe pendicula magne ic ield. We pa icu-
la ize o g aphene analogues (silicene, ge manene, e c.) in Sec. 2.1, zincblende he e os uc-
u es (HgTe/CdTe quan um wells) in Sec. 2.2 and aniso opic ma e ials like phospho ene in
2.3, calcula ing hei ene gy spec um and Hamil onian eigens a es (Landau le els) and de-
sc ibing hei opological phases (when hey exis ). In Sec. 3we ecall he Kubo-G eenwood
o mula o he magne o-op ical conduc i i y enso σo a 2D elec on sys em in a pe pen-
dicula magne ic ield Band an oscilla ing elec ic ield o equency Ω. In pa icula , we a e
in e es ed in analyzing he ansmi ance and Fa aday o a ion o linea ly pola ized ligh o
equency Ω o no mal incidence on he 2D ma e ial. Magne o-op ical p ope ies o g aphene
analogues, zincblende he e os uc u es and phospho ene a e analyzed in Sec ions 3.1,3.2 and
3.3, espec i ely. Fo opological insula o ma e ials, we ind ha he c i ical poin is gene ally
cha ac e ized by a minimum ansmi ance T0a a gi en ligh equency Ω0, whe e he Fa aday
angle changes sign. The e ec o aniso opies is also discussed in phospho ene in Sec ion 3.3.
Finally, Sec. 4is de o ed o conclusions.
2 Some wo-band 2D-Di ac ma e ial models
The ime independen Bloch Hamil onian o a wo-band 2D insula o has he gene al o m
H(k) = ε0(k)τ0+d(k)·τ, (1)
whe e τ= (τx,τy,τz)is he Pauli ma ix ec o , τ0deno es he 2 ×2 iden i y ma ix and
d(k)pa ame e izes an e ec i e spin-o bi coupling nea he cen e Γo he Di ac alleys K
3
SciPos Phys. 16, 077 (2024)
and K′o he i s B illouin zone (FBZ), wi h k= (kx,ky) he wo-dimensional wa e ec o .
The ene gy o he wo bands is ε±(k) = ε0(k)±|d(k)|.
To dis inguish be ween band insula o and opological insula o phases, one can use
he TKNN (Thouless-Kohmo o-Nigh ingale-Nijs) o mula [32]p o iding he Che n-Pon yagin
numbe ( ela ed o he quan um spin Hall conduc ance and he Be y phase [33])
C=1
2πZ ZFBZ
d2k∂ˆ
d(k)
∂kx×∂ˆ
d(k)
∂ky·ˆ
d(k), (2)
wi h ˆ
d=d/|d|, which coun s he numbe o imes (winding numbe ) he uni ec o ˆ
d(k)
w aps a ound he uni sphe e as kw aps a ound he en i e FBZ. The Che n numbe Cusually
depends on he sign o some ma e ial and (ex e nal) con ol pa ame e s in he Hamil onian
H(see la e o some examples), aking di e en alues in di e en phases. We shall see ha
magne o-op ical conduc i i y measu es also cap u e he opological phase ansi ion.
We shall conside he in e ac ion wi h a pe pendicula magne ic ield B= (0,0,B). P o-
mo ing he wa e ec o k o he momen um ope a o k→p/ħh=−i∇, his in e ac ion is in-
oduced h ough he usual minimal coupling, p→P=p+eAwi h A= (Ax,Ay)=(−B y,0)
he elec omagne ic po en ial (in he Landau gauge) and e he elemen a y cha ge (in absolu e
alue). A e Peie ls’ subs i u ion, which esul s in
kx→Px/ħh=a†+a
p2ℓB
,ky→Py/ħh=a†−a
ip2ℓB
, (3)
he Hamil onian (1) can be e en ually w i en in e ms o c ea ion a†and annihila ion
a=ℓB
p2ħh(Px−iPy) = −1
p2ℓB
(y−y0+iℓ2
Bpy/ħh), (4)
ope a o s, whe e ℓB=pħh/(eB)is he magne ic leng h and y0=ℓ2
Bkxis he coo dina e o he
conse ed cen e o he cyclo on o bi .
Le us e iew some ele an physical examples.
2.1 G aphene analogues: Silicene, ge manene, e c
Silicene, ge manene, and o he ansi ion me al dichalcogenides (o he Xene ype) exhibi an
in insic non-ze o spin-o bi coupling Hso =−1
2sξ∆soτz(s=±1 is he spin o he elec on
and ξ=±1 e e o he Di ac alleys Kand K′) due o second neighbo s hopping e ms in
he igh binding model [34]. Spin-o bi in e ac ion Hso combined wi h and ex e nal pe -
pendicula elec ic ield coupling H∆z=1
2∆zτz, gi es d(k)=( ħhξkx, ħhky,∆sξ), whe e
∆sξ= (∆z−sξ∆so)/2 esul s in a unable (Di ac mass) gap (see e.g. [35–38]). In Table 1
we show a compa a i e o spin-o bi coupling and Fe mi eloci y alues o se e al 2D ma e-
ials.
The Che n numbe (2) u ns ou o be
Csξ=ξsign(∆sξ), (5)
whe e we ha e in eg a ed on he whole plane, as co esponds o he FBZ in he con inuum
limi (ze o la ice cons an ). The e o e, he opological phase is de e mined by he sign o
he Di ac mass a each alley ξ. Mo e p ecisely, he e is a TPT om a opological insula o
(TI, |∆z|<∆so) o a band insula o (BI, |∆z|>∆so), a a cha ge neu ali y poin (CNP)
∆(0)
z=sξ∆so, whe e he e is a gap cancella ion be ween he pe pendicula elec ic ield and
he spin-o bi coupling.
4
SciPos Phys. 16, 077 (2024)
Table 1: App oxima e alues o model pa ame e s ∆so (spin-o bi coupling), l(in e -
la ice dis ance) and (Fe mi eloci y) o wo dimensional Si, Ge, Sn and Pb shee s.
These da a ha e been ob ained om i s -p inciples compu a ions in [38](∆so and
l) and [39,40]( ).
∆so (meV) l(Å) (105m/s)
Si 4.2 0.22 4.2
Ge 11.8 0.34 8.8
Sn 36.0 0.42 9.7
Pb 207.3 0.44 –
Using he gene al p esc ip ion (3), he minimal coupling wi h a pe pendicula magne ic
ield B hen esul s in a di e en Hamil onian Hξ o each alley ξ=±1
H1=∆s,1 ħhωa
ħhωa†−∆s,1 ,H−1=∆s,−1−ħhωa†
−ħhωa−∆s,−1, (6)
whe e ω=p2 /ℓBdeno es he cyclo on equency. The eigen alues o bo h Hamil onians
a e simply:
Esξ
n=¨sgn(n)Ç|n|ħh2ω2+∆2
sξ,n=0,
−ξ∆sξ,n=0, (7)
and he co esponding eigens a es a e w i en in e ms o Fock s a es ||n|〉, o Landau le el
(LL) index n=0,±1,±2,... [ alence (−)and conduc ion (+) s a es], as spino s
|n〉sξ=

Asξ
n|n|− ξ+1
2¶
Bsξ
n|n|+ξ−1
2¶
, (8)
wi h coe icien s (see [9,41–43] o simila esul s)
Asξ
n=¨sgn(n)
p2q1+sgn(n)cosθsξ
n,n=0,
(1−ξ)/2, n=0,
Bsξ
n=¨ξ
p2q1−sgn(n)cosθsξ
n,n=0,
(1+ξ)/2, n=0,
(9)
whe e θsξ
n=a c anħhωp|n|/∆sξ, ha is, cosθsξ
n=∆sξ/|Esξ
n|. No e ha Asξ
nand Bsξ
ncan
e en ually be w i en as cos(θsξ
n/2)o sin(θsξ
n/2), depending on sgn(n).
In Figu e 1we plo he low ene gy spec a o silicene, gi en by (7), as a unc ion o he
ex e nal elec ic ield ∆z, oge he wi h he cha ge neu ali y (c i ical) poin s ∆(0)
z=±|∆so|
(ma ked by e ical dashed lines) a which he TPT akes place.
2.2 HgTe/CdTe quan um wells
In [44–47]i was shown ha quan um spin Hall e ec can be ealized in me cu y ellu ide-
cadmium ellu ide semiconduc o quan um wells. Simila e ec s we e also p edic ed in Type-II
semiconduc o quan um wells made om InAs/GaSb/AlSb [48]. The su ace s a es in hese
3D opological insula o s can be desc ibed by a 2D modi ied e ec i e Di ac Hamil onian
H=H+0
0H−,Hs(k) = ε0(k)τ0+ds(k)·τ, (10)
5

SciPos Phys. 16, 077 (2024)
s=1, ξ=1
s=-1, ξ=1
-4-2 0 2 4
-3
-2
-1
0
1
2
3
Δz/Δso
En
sξ/Δso
E0--
E0++
E0-+
E0+-
Figu e 1: Low ene gy spec a o silicene as a unc ion o he ex e nal elec ic po en ial
∆z(in ∆so uni s) o B=0.05 T. Landau le els n=±1,±2 and ±3[ alence (−)and
conduc ion (+)], a alley ξ=1, a e ep esen ed by hin solid lines, blue o s=−1
and ed o s=1 ( o he o he alley we simply ha e Es,−ξ
n=E−s,ξ
n). The edge s a es
n=0 a e ep esen ed by hick lines a bo h alleys: solid a ξ=1 and dashed a
ξ=−1. Ve ical dashed g ay lines indica e he cha ge neu ali y poin s sepa a ing
band insula o (|∆z|>∆so) om opological insula o ( |∆z|<∆so) phases.
whe e s=±1 is he spin and H−(k) = H∗
+(−k)( empo a ily e e sed). The expansion o
Hs(k)abou he cen e Γo he i s B illouin zone gi es [45]
ε0(k) = γ−δk2,ds(k) = (αskx,αky,µ−βk2), (11)
whe e α,β,γ,δand µa e expansion pa ame e s ha depend on he he e os uc u e ( he HgTe
laye hickness λ). The mos impo an one is he mass o gap pa ame e µ, which changes
sign a a c i ical HgTe laye hickness λcwhen going om he no mal (λ < λco µ/β < 0) o
he in e ed (λ > λco µ/β > 0) egime [49]. Typical alues o hese pa ame e s o di e en
HgTe laye hickness (below and abo e λc) can be ound in [49]and in Table 2(γcan be
neglec ed).
The ene gy o he wo bands is
ε±(k) = ε0(k)±Æα2k2+ (µ−βk2)2. (12)
Table 2: Ma e ial pa ame e s o HgTe/CdTe quan um wells wi h di e en HgTe
laye hicknesses λ[49].
λ(nm) α(meV·nm) β(meV·nm2)δ(meV·nm2)µ(meV)
5.5 387 -480 -306 9
6.1 378 -553 -378 -0.15
7.0 365 -686 -512 -10
6
SciPos Phys. 16, 077 (2024)
The TKNN o mula (2) o ds(k)p o ides he Che n numbe
Cs=s[sign(µ) + sign(β)], (13)
whe e we ha e in eg a ed on he whole plane, as co esponds o he con inuum limi . Ac-
co ding o Table 2,βdoes no change sing and, he e o e, he opological phase ansi ion
occu s when µchanges sign, as al eady men ioned. In e e ence [49], he no mal and in-
e ed egimes a e equi alen ly gi en by he sign o µ/β.
Using again he gene al p esc ip ion (3), he minimal coupling wi h a pe pendicula mag-
ne ic ield Bnow esul s in
H+=

γ+µ−(δ+β)(2N+1)
ℓ2
B
p2α
ℓBa
p2α
ℓBa†γ−µ−(δ−β)(2N+1)
ℓ2
B

,
H−=

γ+µ−(δ+β)(2N+1)
ℓ2
B−p2α
ℓBa†
−p2α
ℓBaγ−µ−(δ−β)(2N+1)
ℓ2
B

,
(14)
wi h N=a†a. A Zeeman e m con ibu ion
HZ
s=−s
2BµBge
τ0+τz
2+gh
τ0−τz
2, (15)
can also be added o he Hamil onian, wi h µB≃0.058 meV/T he Boh magne on and ge,h
he e ec i e (ou -o -plane) g- ac o s o elec ons and holes (conduc ion and alence bands).
Using (Fock s a e) eigen ec o s ||n|〉 o he (Landau le el) numbe ope a o N=a†a, one
can analy ically ob ain he eigenspec um
Es
n=γ−2δ|n|−sβ
ℓ2
B−sge+gh
4BµB+sgn(n)
2α2|n|
ℓ2
B
+µ−2β|n|−sδ
ℓ2
B−sge−gh
4BµB2, (16)
o LL index n=±1,±2,±3,... [ alence (−)and conduc ion (+)], and
Es
0=γ−sµ−δ−sβ
ℓ2
B−BµBs+1
4gh+s−1
4ge, (17)
o he edge s a es n=0, s=±1. These eigen alues coincide wi h hose in [17,50,51] o he
iden i ica ion s={−1,1}={↑,↓}.
The co esponding eigen ec o s a e
|n〉s=As
n|n|− s+1
2
Bs
n|n|+s−1
2, (18)
wi h coe icien s
As
n=sgn(n)
p2Æ1+sgn(n)cosϑs
n,n=0,
(1−s)/2, n=0,
Bs
n=s
p2Æ1−sgn(n)cosϑs
n,n=0,
(1+s)/2, n=0,
(19)
7
SciPos Phys. 16, 077 (2024)
whe e
ϑs
n=a c an
p2|n|α/ℓB
µ−2β|n|−sδ
ℓ2
B−sge−gh
4BµB
. (20)
As o he g aphene analogues in (9), he coe icien s As
nand Bs
ncan e en ually be w i en as
sine and cosine o hal angle, depending on sgn(n).
Acco ding o (17), he band in e sion o edge s a es occu s when
E+
0=E−
0⇒Bin =µ
eβ/ħh−µB(ge+gh)/4, (21)
which gi es he c i ical magne ic ield Bcwhich sepa a es he QSH and QH egimes [51]. Fo
example, o he ma e ial pa ame e s in Table 2co esponding o a QW hickness λ=7.0 nm
and g- ac o s ge=22.7, gh=−1.21, one ob ains Bin ≃7.4 T. See also Figu e 2 o a g aphical
ep esen a ion o his band in e sion.
F om now on we shall disca d Zeeman coupling o he sake o con enience since ou main
conclusions emain quali a i ely equi alen . We add ess he in e es ed eade o Appendix C
whe e we ep oduce some esul s o Re e ence [17] o non-ze o Zeeman coupling and con as
wi h he ze o Zeeman coupling case.
We shall use a linea i
µ(λ) = 77.31 −12.53λ,
α(λ) = 467.49 −14.65λ,
β(λ) = 283.58 −138.16λ,
δ(λ) = 458.46 −138.25λ,
(22)
o he ma e ial pa ame e s in Table 2as a unc ion o he HgTe laye hickness λ(dimensionless
uni s and λin nm uni s). In all cases he coe icien o de e mina ion is R2>0.99. Looking a
µ(λ)in (22), we can ob ain an es ima ion o he c i ical HgTe hickness a which he opological
phase ansi ion occu s as
µ=0⇒λc=6.17 nm. (23)
In Figu e 2we plo he low ene gy spec a gi en by (16) and (17) as a unc ion o he HgTe laye
hickness λ, whe e we ha e ex apola ed he linea i (22) o he in e al [4nm, 8nm]. When
neglec ing Zeeman coupling, he band in e sion o edge s a es (21) occu s o B=ħhµ/(eβ)
which, using he linea i (22), p o ides a ela ion
λin (B) = 368.31 −2.05B
59.7 −B, (24)
be ween he applied magne ic ield B(in Tesla) and he HgTe laye hickness λin (B)(in
nanome e s) a which he band in e sion E+
0=E−
0 akes place. No e ha λin (B)≃λc=6.17 nm
o low B≪1 T, and ha E+
0=E−
0≃0 meV a his poin as shows Figu e 2.
2.3 Phospho ene as an aniso opic ma e ial
The physics o phospho ene has been ex ensi ely s udied [53–67]. The e a e se e al ap-
p oaches o he low ene gy Hamil onian o phospho ene in he li e a u e. Rudenko e al. [68]
and Ezawa [69]p opose a ou -band and i e-neighbo s igh -binding model la e simpli-
ied o wo-bands [69]. Se e al app oxima ions o his wo-band model ha e been used
in [13,70–72]. We shall choose o ou s udy he Hamil onian
H=Ec+αxk2
x+αyk2
yγkx
γkxE −βxk2
x−βyk2
y, (25)
8
SciPos Phys. 16, 077 (2024)
s=1
s=-1
45678
-40
-20
0
20
40
HgTe
laye
hickness λ(nm)
Ene gy En
s(meV)
λin
E0+
E0-
Figu e 2: Low-ene gy spec a Es
no a HgTe/CdTe quan um well as a unc ion o he
HgTe laye hickness λ o B=0.5 T. Landau le els n=±1,±2,±3[ alence (−)and
conduc ion (+)]a e ep esen ed by hin solid lines, blue o spin s=−1 and ed o
s=1. Edge s a es (n=0) a e ep esen ed by hick lines. A e ical dashed black line
indica es he HgTe hickness λin (0.5) = 6.20 nm ≃λcwhe e he band in e sion o
edge s a es occu s o B=0.5 T acco ding o (24).
p oposed by Zhou and collabo a o s [13]. This co esponds o a Bloch Hamil onian (1) wi h
ε0(k) =
Ec+E + (αx−βx)k2
x+ (αy−βy)k2
y
2, (26)
d(k) = γkx,0, Ec−E + (αx+βx)k2
x+ (αy+βy)k2
y
2.
The Hamil onian (25) p o ides a i ial Che n numbe (2), e en in he p esence o a unable
pe pendicula cons an elec ic ield (see below), which means ha monolaye phospho ene
does no ha e a opological phase pe se. I has been shown ha opological ansi ions can
be induced in phospho ene when apidly d i en by in-plane ime-pe iodic lase ields [73];
hese a e called in gene al “Floque opological insula o s” (see e.g. [74–76]), bu we shall
no conside his possibili y he e. Al hough phospho ene is no a opological ma e ial, we
will see in Sec. 3.3 ha he c i ical magne o-op ical p ope ies (e.g., minimum ansmi ance)
obse ed o silicene and HgTe QWs a e s ill alid in phospho ene when closing he ene gy
gap h ough an ex e nal elec ic ield. Ano he possibili y o modi y he ene gy gap could be
by applying s ain [60,70](see la e in Sec. 3.3).
The ma e ial pa ame e s o phospho ene can be w i en in e ms o conduc ion (c) and
alence ( ) e ec i e masses as (see [13] o mo e in o ma ion)
αx,y=ħh2
2mcx,cy
,βx,y=ħh2
2m x, y
, (27)
wi h
mcx=0.793me,mcy=0.848me,
m x=1.363me,m y=1.142me,(28)
9
SciPos Phys. 16, 077 (2024)
-10
-5
0
5
10
15 λ=5.50 nm
-10
-5
0
5
10 λ=6.17 nm
-10
-5
0
5
10 λ=7.00 nm
-15
-10
-5
0
5λ=5.50 nm
-15
-10
-5
0
5λ=6.17 nm
0 10 20 30 40 50 60
-15
-10
-5
0
5λ=7.00 nm
Conduc i i y σij(Ω)/σ0
ℏΩ (meV)
μ
F=12.5
meV
B
=0.5 T
T=1 K
η=0.5 meV
Re[σxx(Ω)]
Re[σxy(Ω)]
Im[σxx(Ω)]
Im[σxy(Ω)]
Figu e 6: Real and imagina y pa s o he longi udinal σx x and ans e se
Hall σx y magne o-op ical conduc i i ies in a bulk HgTe QW o hickness
λ=5.50,6.17,7.00 nm, as a unc ion o he pola ized ligh equency Ωand in
σ0=e2/huni s. We se he conduc i i y pa ame e s as µF=12.5 meV, B=0.5 T,
T=1 K and η=0.5 meV.
and some ep esen a i e alues o he chemical po en ial µF=12.5 meV, empe a u e T=1 K
and sca e ing a e η=0.5 meV. Fo ħhΩ∈[0,60]meV, we achie e con e gence wi h 100 LLs,
ha is, es ic ing he sum in (36) as P∞
n=−∞ →P100
n=−100. Mo e explici ly, o he pa ame e s
men ioned abo e,

n=100
X
n=−100
σi j −
n=99
X
n=−99
σi j
/σ0≤








10−5, i σi j =Re(σx x ),
10−4, i σi j =Re(σx y ),
10−3, i σi j =Im(σx x ),
10−7, i σi j =Im(σx y ).
(46)
Simila o silicene, we can see in Figu e 6 ha he e a e mul iple peaks in he abso p i e
componen s Re(σx x )and Im(σx y ), co esponding o ansi ions be ween occupied and unoc-
cupied LLs obeying he selec ion ules |n|=|m|±1. A lowe equencies ħhΩ∈[0,30]meV,
inside each cu e o Figu e 6, we ind he main peaks co esponding o he ansi ions 0 →1
o spin s=1 and s=−1. Bo h peaks me ge app oxima ely a λ≃λc=6.17 nm. This is
because he ene gy di e ences E+
1−E+
0and E−
1−E−
0a e simila when λ≃λc o low magne ic
16

SciPos Phys. 16, 077 (2024)
0 1 2 3 4 5
6.12
6.13
6.14
6.15
6.16
6.17
B
(
T
)
λ*(nm)
Es ima e S anda d E o -S a is ic
a1218.447 0.01218 000
a217.30 0.15 110
a335.4004 0.0017 20 000
a42.770 0.025110
λ i *(B)= a1-a2B
a3-a4B
R2=0.9999999951
Nume ical Non-linea i
Figu e 7: Nume ical solu ions λ∗(in nm, blue do s) o he equa ion E+
1−E+
0=E−
1−E−
0
(ene gies (16,17) o HgTe QW) o 50 di e en alues o he ex e nal magne ic ield
B. In o ange, non-linea i (47) o he nume ical alues.
ields B≪1 T, acco ding o equa ions (16,17). In o de o ex end his esul o highe alues o
he magne ic ield, we inse he pa ame e i s (22) in o he equa ion E+
1−E+
0=E−
1−E−
0, and
sol e i nume ically o λ∗=λ∗(B), ob aining he alues ep esen ed by blue do s in Figu e 7.
These alues i he equa ion
λ∗
i (B) = 218.4 −17.3B
35.4 −2.8B, (47)
which is ep esen ed as an o ange cu e in Figu e 7. Consequen ly, only o small magne ic
ields, we can in e he c i ical hickness λcwhe e he TPT in HgTe QW occu s om he con-
duc i i y Re(σx x )plo , ha is, λ∗≃λc=6.17 nm o B≪1 T.
The beha io o he Fa aday angle and he ansmi ance as a unc ion o he pola ized
ligh equency Ωa ound he c i ical HgTe laye hickness λc=6.17 nm (a which he ma e ial
pa ame e µchanges sign/Che n numbe ) is shown in Figu e 8. As o silicene, we ocus on
he lowe equencies ħhΩ∈[0,30]meV whe e he main peaks a e loca ed, and ind again a
minimum o he ansmi ance, his ime T0=0.78, a he c i ical poin λcand ħhΩ=15.0 meV.
Fo his ma e ial, he “minimal” beha io does depend on he pa icula alues o magne ic
ield, as we saw in equa ion (47). Howe e , o small magne ic ields like B=0.5 T in Figu e
8, he minimum o he ansmi ance s ill akes place a λ∗≃λc=6.17 nm. The Fa aday
angle a he c i ical poin (black cu e in Figu e 8) changes sign a he minimum ansmi ance
equency ħhΩ=15.0 meV, a beha io sha ed wi h silicene.
Fo comple eness, in Appendix Ewe show se e al con ou plo s o he Fa aday angle using
di e en c oss sec ions in he {ħhΩ,λ,B,T,µF}pa ame e space.
3.3 Magne o-op ical p ope ies o phospho ene and e ec o aniso opies
F om he phospho ene Hamil onian (25), he cu en ope a o (35) is
js
x=e
ħh(γτx+kx(τ0(αx−βx) + τz(αx+βx))) ,
js
y=e
ħhkyτ0(αy−βy) + τz(αy+βy), (48)
17
SciPos Phys. 16, 077 (2024)
λ(nm)
6.97
6.70
6.43
6.17=λc
5.90
5.63
5.37
0.80
0.85
0.90
0.95
1.00

0.78
0 10 20 30
-3
-2
-1
0
1
2
3
ΘF
15.
ℏΩ (meV)
μ
F=12.5 meV
B=0.5 T
T=1 K
η=0.5 meV
Figu e 8: T ansmi ance Tand Fa aday angle ΘF(in deg ees) in a bulk HgTe QW
as a unc ion o he pola ized ligh equency Ω, and o hickness λ < λc,λ=λc
and λ>λc, wi h λc=6.17 nm (black line). We se he conduc i i y pa ame e s as
µF=12.5 meV, B=0.5 T, T=1 K and η=0.5 meV.
which, a e minimal coupling, acco ding o p esc ip ion (29), esul s in
js
x=e
ħhγτx+a†+a
p2αy x ℓB
(τ0(αx−βx) + τz(αx+βx)),
js
y=e
ħh
αy x (a†−a)
ip2ℓBτ0(αy−βy) + τz(αy+βy).
(49)
Plugging hese ma ix elemen s in o he gene al exp ession (36) we ob ain he magne o-
op ical conduc i i y o phospho ene. No e ha , unlike silicene and HgTe QW, he e is now
a la ge asymme y be ween σx x and σy y (abou one o de o magni ude di e ence), as e i-
denced by Figu e 9. This asymme y was al eady highligh ed by [71], whe e unable op ical
p ope ies o mul ilaye black phospho us hin ilms we e s udied o B=0. In Figu e 9we
plo he eal and imagina y pa s o he conduc i i y enso componen s σi j (in σ0=e2/h
uni s) o phospho ene as a unc ion o he pola ized ligh equency Ω, o some alues o
he elec ic po en ial a ound ∆(0)
z=−Eg=−1.52 eV (closing he ene gy gap), a magne ic
ield o B=0.5 T, like in Figu e 3, and some ep esen a i e alues o he chemical po en ial
µF=−0.417 eV, empe a u e T=1 K and sca e ing a e η=0.2 meV. We a e using he same
h eshold o N=1000 Fock s a es ha we used o ind con e gence in he i s 6 Hamil o-
nian eigens a es o he nume ical diagonaliza ion in Figu e 3. This con e gence is ensu ed
o ħhΩ∈[0,20]meV. The aniso opic cha ac e o phospho ene also implies ha he cu en
18
SciPos Phys. 16, 077 (2024)
Re[σxx(Ω)]
Im[σxx(Ω)]
-20
0
20
40 Δz=-1.533 eV
-20
0
20 Δz=-1.529 eV
-20
0
20 Δz=-1.525 eV
-0.10
-0.05
0.00
0.05
0.10 Δz=-1.533 eV
Re[σyy(Ω)]
Im[σyy(Ω)]
-0.10
-0.05
0.00
0.05
0.10 Δz=-1.529 eV
-0.10
-0.05
0.00
0.05
0.10 Δz=-1.525 eV
-1.5
-1.0
-0.5
0.0
0.5 Δz=-1.533 eV
Re[σxy(Ω)]
Im[σxy(Ω)]
-1.5
-1.0
-0.5
0.0
0.5 Δz=-1.529 eV
5 10 15 20
-1.5
-1.0
-0.5
0.0
0.5 Δz=-1.525 eV
Conduc i i y σij(Ω)/σ0
ℏΩ (meV)
μF=-0.417 eV
B=0.5 T
T=1 K
η=0.2 meV
Figu e 9: Real and imagina y pa s o he longi udinal σx x ,σy y and ans e se Hall
σx y magne o-op ical conduc i i ies in a phospho ene monolaye , as a unc ion o
he pola ized ligh equency Ωand in σ0=e2/huni s. Phospho ene is unde a
pe pendicula elec ic ield po en ial ∆(0)
z=−Eg=−1.52 eV closing he ene gy gap
in Figu e 3. The y-axis icks ha e di e en alues in each subplo as he conduc i i ies
σx y and σy y a ain smalle alues han σx x (phospho ene aniso opy). We se he
conduc i i y pa ame e s as µF=−0.417 meV, B=0.5 T, T=1 K and η=0.2 meV.
js
yis signi ican ly lowe han js
x[ he Hamil onian (25) is o second o de in ky]. This makes
ans e sal componen s o he conduc i i y signi ican ly lowe han longi udinal componen s.
This is why we ha e disposed Figu e 9in a sligh ly di e en manne om Figu es 4 o silicene
and 6 o HgTe QW, which display a mo e iso opic s uc u e.
Due o he pa i y symme y o he Hamil onian (30), only he elec onic ansi ions be-
ween LLs o di e en pa i ies a e allowed [30]. The main peak (smalle equency) o he
conduc i i y Re(σx x )in Figu e 9co esponds o he elec onic ansi ions Ee en
0→Eodd
3and
Eodd
1→Ee en
2, which ha e app oxima ely he same ene gy di e ence o all ∆z<−1.53 eV
wi h a ole ance ≤10−14 eV. Tha is, Ee en
0and Eodd
1, and Ee en
2and Eodd
3, a e degene a e o all
∆z<−1.53 eV as he spec um in Figu e 3shows. When he degene a ion is b oken a ound
he elec ic po en ial ∆z≃−1.53 eV, he main conduc i i y Re(σx x )peak spli s in o wo as we
can see in Figu e 9.
The aniso opic cha ac e o phospho ene also a ec s he Fa aday angle, which a ains
much lowe alues (in absolu e alue) han o silicene o HgTe QWs. Indeed, in Figu e 10
we plo Fa aday angle and ansmi ance as a unc ion o he pola ized ligh equency Ω o
19
SciPos Phys. 16, 077 (2024)
di e en elec ic ield po en ials −1.535 ≤∆z≤ −1.519 eV. Like o silicene and HgTe QWs,
we ind a minimal beha io in he ansmi ance o phospho ene T0=0.50 o a pola ized
ligh equency ħhΩ=2.6 meV a elec ic ield po en ial ∆(0)
z=−1.523 eV, which is close o
minus he ene gy gap −Eg=−1.52 eV. No e ha his alue o he minimal ansmi ance o
phospho ene is much smalle han o silicene and HgTe QWs; ac ually, he assump ion o low
abso ance in o mula (39) is no longe alid he e and we ha e used he exac exp essions o
Tand ΘFin (39). Mo eo e , unlike o g aphene analogues and HgTe QWs, his minimum o
he ansmi ance does no seem o be ela ed o he union o wo conduc i i y peaks in o a
bigge one; a he , i is simply ela ed o he ene gy gap closu e. Ac ually, he c i ical elec ic
po en ial ∆(0)
zwhe e he ansmi ance o phospho ene eaches a minimum depends on he
magne ic ield Bchosen, as Figu e 11 shows. We pe o m a non-linea i o he nume ical
alues o ∆(0)
z(B)and ob ain he equa ion (Bin dimensionless uni s)
∆(0)
z i (B) = −77.4 −3.5B
50.9 +2.2BeV, (50)
which is ep esen ed as a o ange cu e in Figu e 11. Fo small magne ic ields, we can deduce
ha he c i ical elec ic ield po en ial is simila o minus he ene gy gap −Ego phospho ene,
ha is ∆(0)
z(B)≃ −Eg=−1.52 eV o B≪1 T. We ha e also checked nume ically ha he
c i ical elec ic po en ials ∆(0)
z(B)a e independen o he pa ame e s µFand η o a ixed
magne ic ield B. Howe e , we se di e en alues o µF o small ields B≤2 T (see cap ion
o Figu e 11), in o de o a oid blocking he elec ic ansi ion Eodd
1→Ee en
2o he main peak
o he ansmi ance. We also inc emen Nas Bdec eases in o de o achie e con e gence in
he diagonaliza ion.
Addi ionally, Figu e 10 shows how one peak o he ansmi ance spli s in o wo a ound
∆z≃ −1.53 eV (blue lines), since he LL Ee en
0b eaks i s degene a ion app oxima ely o
∆z>−1.53 eV (see Figu e 3). Fo ∆z=∆(0)
z=−1.523 eV ( hick black line), he big peak on
he le in Figu e 10 co esponds o he elec onic ansi ion Eodd
1→Ee en
2, and mo es owa d
smalle alues o ħhΩwhen inc easing ∆z. The o he small peak in he black line co esponds
o he elec onic ansi ion Ee en
0→Eodd
3, which mo es owa d bigge alues o ħhΩwhen
inc easing ∆z. The Fa aday angle also p esen s in lec ion poin s a he equencies whe e he
peaks o he ansmi ance a e loca ed.
The e o e, we see ha aniso opies a ec he alues o he Fa aday angle and ansmi -
ance. The e a e mechanical ways o in oducing aniso opies in 2D ma e ials by subjec ing
hem o s ain (like o s ained [86]o ippled [87]g aphene). This kind o aniso opies can
be ea ed by eplacing he scala Fe mi eloci y by a 2×2 symme ic enso (see e.g. [82]).
Namely, o g aphene, he Hamil onian (1) ec o dcomponen s dj=ħh kja e eplaced by
dj=ħhki i j,i=1,2,d3=0. Ac ually, o uni o mly s ained g aphene wi h s ain enso ϵ, he
Fe mi eloci y enso is (up o i s o de ) = (τ0−βϵ)(see e.g. [82,88]), whe e β∼2. The
ela ion be ween he iso opic σ0and he aniso opic σmagne o-op ical conduc i i y enso s
is simply σ(Ω,B) = σ0(Ω,B) /de ( ), wi h B=Bde ( )/ 2an e ec i e magne ic ield.
In e es ing discussions on how measu emen s o dich oism and anspa ency o wo di e en
ligh pola iza ion di ec ions can be used o de e mine he magni ude and di ec ion o s ain can
be ound in [81]. Also, pho oelas ic e ec s in g aphene [86], s ain-modula ed aniso opies in
silicene [89,90], e c. The band gap Eg=E −Eco phospho ene can be u he mo e modula ed
by s ain and by he numbe o laye s in a s ack [60,70].
20
SciPos Phys. 16, 077 (2024)
0.5
0.6
0.7
0.8
0.9
1.0

0.5
1 2 3 4 5 6
-0.4
-0.2
0.0
0.2
0.4
0.6
ΘF
ℏΩ (meV)
μ
F=-0.417 eV
B=0.5 T
T=1 K
η=0.2 meV
Δz(eV)
-1.519
-1.521
-1.523=Δz(0)
-1.525
-1.527
-1.529
-1.531
-1.533
-1.535
E0
e en →E3
odd
↖
E1
odd →E2
e en
↖
2.6
Figu e 10: T ansmi ance Tand Fa aday angle ΘF(in deg ees) in a phospho ene
monolaye as a unc ion o he pola ized ligh equency Ω, and o elec ic ields
−1.535 <∆z<−1.519 eV a ound he minus ene gy gap −Eg=−1.52 eV. The
black line co esponds o he elec ic po en ial ∆(0)
z=−1.523 eV≃ −Egwhe e he
ansmi ance a ains a minimum o T0=0.5 a ħhΩ=2.5 meV. We se he conduc i i y
pa ame e s as µF=−0.417 meV, B=0.5 T, T=1 K and η=0.2 meV.
4 Conclusions
We ha e s udied magne o-op ical p ope ies o di e en 2D ma e ials, ocusing on ansmi -
ance and Fa aday o a ion nea he c i ical poin o he opological phase ansi ion o opo-
logical insula o s like silicene and HgTe quan um wells. We ha e seen ha , in all opological
2D ma e ials analyzed, ansmi ance a ains an absolu e minimum T0a he c i ical TPT poin
o a ce ain alue Ω0o he no mal inciden pola ized ligh equency. This is a uni e sal
beha io o g aphene analogues, ha is, he minimal beha io o he ansmi ance does no
depend on he chosen alues o magne ic ield, chemical po en ial and empe a u e, al hough
he loca ion o Ω0 a ies wi h hem. In addi ion, we ha e ound ha each peak o he ansmi -
ance coincides in equency wi h an in lec ion poin o he Fa aday angle, o a ixed selec ion
o he elec ic ield, magne ic ield, chemical po en ial and empe a u e pa ame e s.
This ex emal uni e sal beha io is sha ed wi h o he opological 2D ma e ials like HgTe
quan um wells as long as he applied magne ic ield emains small enough B≪1 T. In HgTe
quan um wells we ha e e i ied ha he e is a minimum o he ansmi ance T0a he c i ical
HgTe laye hickness a a gi en equency Ω′
0( o his ma e ial his minimal beha io depends
on he magne ic ield) and he Fa aday angle a he c i ical poin changes sign a he minimum
ansmi ance equency Ω′
0.
21

SciPos Phys. 16, 077 (2024)
0 2 4 6 8 10 12 14
-1.545
-1.540
-1.535
-1.530
-1.525
-1.520
B(T)
Δz(0)(eV)
Es ima e S anda d E o -S a is ic
a1-77.4090.022-3500
a23.49 0.438.2
a350.8806 0.0095 5300
a42.21 0.288.0
(Δz(0)) i (B)= a1-a2B
a3-a4B
R2=0.9999999501
Nume ical Non-linea i
Figu e 11: Elec ic ield po en ial a which phospho ene ansmi ance eaches a min-
imum, as a unc ion o di e en magne ic ields. In o ange, non-linea i (50) o he
nume ical alues. In gene al, we se he conduc i i y pa ame e s µF=−0.41 eV,
T=1 K, η=1 meV, and use N=300 Fock s a e in he nume ical diagonal-
iza ion, o all B≥3 T. Fo smalle magne ic ields B=0.1,0.5,1,2 T, we se
µF=−0.419,−0.418,−0.416,−0.416 eV espec i ely. In he case o B=0.1 T we
also se η=1 meV and N=500 Fock s a es o achie e ene gy diagonaliza ion con-
e gence.
Fo o he non- opological aniso opic ma e ials like phospho ene, his minimal beha io o
he ansmi ance s ill emains when he ene gy gap is closed, he Fa aday angle being much
smalle (in absolu e alue) han in silicene and HgTe QWs. In his case he c i ical elec ic
po en ial whe e he ansmi ance eaches a minimum depends on he magne ic ield.
The e o e, hese ex emal p ope ies o ansmi ance/abso ance and chi ali y change o
Fa aday angle a he c i ical poin u n ou o p o ide sha p ma ke s o ei he he opological
phase ansi ion o he ene gy gap closu e.
Acknowledgmen s
Funding in o ma ion We hank he suppo o G an PID2022-138144NB-I00 unded by
Spanish MICIU and Jun a de Andalucía h ough he p ojec s FEDER/UJA-1381026 and FQM-
381. AM hanks he Spanish MIU o he FPU19/06376 p edoc o al ellowship. OC is on
sabba ical lea e a G anada Uni e si y, Spain, since he 1s o Sep embe 2023. OC hanks
suppo om he p og am PASPA om DGAPA-UNAM.
A Landau le els plo e sus ex e nal magne ic ield
We p o ide an addi ional plo o he Landau le els o he h ee di e en ma e ials as a unc-
ion o he ex e nal magne ic ield B. C i ical alues o he elec ic ield and laye hick-
ness a e selec ed, ha is, in he case o he silicene ∆z=∆so =4.2 meV, o he HgTe QW
λ=λc=6.17 nm, and o he phospho ene ∆z=−Eg=−1.52 eV.
22
SciPos Phys. 16, 077 (2024)
s=1, ξ=1
s=-1, ξ=1
0 1 2 3 4 5
-15
-10
-5
0
5
10
15
B(T)
Ene gy En
sξ/Δso
E0--
E0++
E0-+
E0+-
Δz=Δso=4.2meV
(a) Silicene.
s=1
s=-1
012345
-50
0
50
100
B
(
T
)
Ene gy En
s(meV)
λ=λc=6.17nm
E0+
E0-
(b) HgTe QW.
E en
Odd
012345
-0.46
-0.44
-0.42
-0.40
-0.38
B(T)
Ene gy En(eV)
E0e en
Δz=-Eg=-1.52 eV
(c) Phospho ene.
Figu e 12: Ene gies (Landau le els) o (a) Silicene, (b) HgTe QW, and (c) Phos-
pho ene as a unc ion o he ex e nal magne ic ield B. C i ical alues o he
elec ic ield and laye hickness a e selec ed, ha is, in he case o he silicene
∆z=∆so =4.2 meV, o he HgTe QW λ=λc=6.17 nm, and o he phospho-
ene ∆z=−Eg=−1.52 eV.
B Silicene conduc i i y in he ci cula ly pola iza ion basis
We comple e he analysis o magne o-op ical p ope ies o g aphene analogues by discussing
he case o ci cula ly pola ized ligh . In his case, he conduc i i y is σ±(Ω)=σx x (Ω)±iσx y (Ω)
o igh -handed (+) and le -handed (-) pola iza ion [91]. The e o e, he abso p i e pa is
Re(σ±) = Re(σx x )∓Im(σx y ). In Figu e 13, we p esen bo h abso p i e pa s Re(σ±) o a
silicene monolaye unde an elec ic po en ial ∆z=0.5∆so as a unc ion o he equency
o he inciden ligh Ω. The conduc i i y pa ame e s a e speci ically chosen o ep oduce he
esul s in [9], ha is, µF=3.0∆so,B/∆2
so =657 G/meV2,T=0 K and η=0.05∆so. No e
ha we ha e de ined he conduc ance quan um as σ0=e2/h=38.8µS, whe eas he au ho s
in e e ence [9] ake σ0=e2/(4ħh).
23
SciPos Phys. 16, 077 (2024)
Re[σ+(Ω)]
Re[σ-(Ω)]
0 5 10 15 20
0
10
20
30
40
50
ℏΩ/Δso
Conduc i i y σ±(Ω)/σ0
Δ
z=2.1 meV
μ
F=12.6 meV
B
=1.15895T
T
=0 K
η
=0.21 meV
Figu e 13: Conduc i i y abso p i e pa s Re(σ±) = Re(σx x )∓Im(σx y ) o igh -
handed (+) and le -handed (-) pola iza ion in a silicene monolaye unde an elec ic
po en ial ∆z=0.5∆so, as a unc ion o he pola ized ligh equency Ω(in σ0=e2/h
uni s). We se he conduc i i y pa ame e s µF=3.0∆so,B/∆2
so =657 G/meV2,
T=0 K and η=0.05∆so as in Re . [9].
C HgTe quan um well conduc i i y wi h Zeeman e ec
We ecalcula e he conduc i i y o he HgTe quan um well wi h and wi hou Zeeman coupling
o suppo he a gumen ha he esul s a e quali a i ely equi alen , he quan i a i e di e -
ences being small. A laye hickness o λ=7.0 nm is selec ed, so he ma e ial pa ame e s
a e α=365 meV·nm, β=−686 meV·nm2,δ=−512 meV·nm2, and µ=−10 meV, as aken
om Re . [49]. In Figu e 14, we plo he eal and imagina y pa s o he longi udinal σx x and
ans e se σx y conduc i i ies as a unc ion o he pola ized ligh equency Ω. The conduc-
i i y pa ame e s a e chosen o ep oduce he esul s in [17]wi h Zeeman coupling, ha is,
µF=8 meV, B=5 T, T=1 K and η=1 meV. The conduc ance quan um used he e is again
σ0=e2/h=38.8µS, whe eas he au ho s in e e ence [17] ake σ0=e2/ħh.
D Anima ions o he ene gy spec um and conduc i i ies
A ached in he Supplemen al Ma e ial [52]is a se ies o anima ions called:
-Silicene_Conduc i i y_and_Ene gy_VS_Omega.gi ,
-HgTe_Conduc i i y_and_Ene gy_VS_Omega.gi ,
-Phospho ene_Conduc i i y_and_Ene gy_VS_Omega.gi ,
whe e we plo he ene gy spec um a igh , and he eal pa Re[σx x (Ω)] and Re[σx y (Ω)]
o he conduc i i y componen s a le , o h ee di e en ma e ials s udied in he main ex :
silicene, HgTe QW, and phospho ene. The ex e nal elec ic ield ∆zin he case o he silicene
and phospho ene, and he laye hickness λo he HgTe QW, a e used as “ ime coo dina e” on
he anima ions, so each ame co esponds o one alue o hese con ol pa ame e s.
The conduc i i ies a e plo ed as a unc ion o he pola ized ligh equency Ω, and
hey change in each ame acco ding o he alues o ∆zo λ. The e o e, we can obse e
how he main peaks o he longi udinal conduc i i y Re(σx x )me ge o he c i ical alues
∆(0)
z=∆so =4.12 meV (silicene) o λ=λc=6.17 nm (HgTe QW), whe e he opological
phase ansi ion occu s in hese 2D ma e ials.
24
SciPos Phys. 16, 077 (2024)
Re[σxx(Ω)]
Im[σxx(Ω)]
-10
-5
0
5
10
15
Re[σxy(Ω)]
Im[σxy(Ω)]
0 50 100 150 200
-15
-10
-5
0
5
10
Conduc i i y σij(Ω)/σ0
ℏΩ (meV)
λ
=7 nm
μ
F=8
meV
B
=5T
T=1 K
η=1 meV
(a) Zeeman
Re[σxx(Ω)]
Im[σxx(Ω)]
-5
0
5
10
15
Re[σxy(Ω)]
Im[σxy(Ω)]
0 50 100 150 200
-15
-10
-5
0
5
10
Conduc i i y σij(Ω)/σ0
ℏΩ (meV)
λ
=7 nm
μ
F=8
meV
B
=5T
T=1 K
η=1 meV
(b) No Zeeman
Figu e 14: Real and imagina y pa s o he longi udinal σx x and ans e se Hall σx y
(magne o-)op ical conduc i i ies in a bulk HgTe QW o a hicknesses λ=7.0 nm, as
a unc ion o he pola ized ligh equency Ω(in σ0=e2/huni s) wi h and wi hou
Zeeman coupling. We se he conduc i i y pa ame e s µF=8 meV, B=5 T, T=1 K
and η=1 meV, as in Re . [17].
In he case o he phospho ene, we only obse e he degene a ion o Landau le els n=0
and n=1 in he conduc i i y a ound he elec ic po en ial ∆z≃ −1.53 eV. Tha is, he elec-
onic ansi ions Eodd
1→Ee en
2and Ee en
0→Eodd
3ha e a simila ene gy and sha e a longi udi-
nal conduc i i y peak (main peak a le in he gi ), un il he degene a ion b eaks o elec ic
ields app oxima ely highe han −1.53 eV, when bo h elec onic ansi ions will ha e di e en
ene gies so he main peak will spli in o wo.
On he o he hand, he ene gy spec um is s a ic on he anima ion, as i is plo ed as a
unc ion o all he alues ha ∆zo λ ake. Howe e , we plo a mo ing e ical dashed line
on i , ep esen ing he alue o ∆zo λin he conduc i i y ame. On op o his e ical
line, we also d aw a ows ep esen ing he elec onic ansi ions allowed be ween Landau
le els (LLs) o he speci ic alue ∆zo λ, whe e he Fe mi ene gy µFis ep esen ed by an
ho izon al dashed line. The colo o he a ows is he same as he colo o he poin s plo ed
on he op o he longi udinal conduc i i y main peaks. The leng h o he a ows ep esen s
he ene gy di e ence |En−Em|be ween he co esponding Landau le els in his pa icula
elec onic ansi ion n↔m, which also coincides wi h he equency ħhΩo he longi udinal
conduc i i y peak associa ed wi h his ansi ion. The e o e, when wo a ows ha e he same
leng h, we can obse e wo longi udinal conduc i i y peaks me ging a he c i ical poin . We
ha e only d awn he a ows o he main peaks o lowe Landau le el elec onic ansi ions o
he sake o simplici y.
E Fa aday angle con ou plo s
Fo comple eness, in Figu e 15 we show he a iabili y o he Fa aday angle o silicene ac oss
he pa ame e space: pola ized ligh equency ħhΩ, elec ic ield po en ial ∆z, magne ic ield
B, empe a u e Tand chemical po en ial µF}, using se e al con ou plo s co esponding o
di e en c oss sec ions. Also, in Figu e 16 we do he same o he Fa aday angle in HgTE
quan um wells using di e en c oss sec ions in he {ħhΩ,λ,B,T,µF}pa ame e space, whe e
25
SciPos Phys. 16, 077 (2024)
[51]B. Scha , A. Ma os-Abiague and J. Fabian, Magne ic p ope ies o HgTe quan um wells,
Phys. Re . B 86, 075418 (2012), doi:10.1103/PhysRe B.86.075418.
[52]A. Mayo gas, Supplemen a y ma e ial GIF, Zenodo (2024), h ps://zenodo.o g/ eco ds/
10807672.
[53]D. E. C. Co b idge, Phospho us: Chemis y, biochemis y and echnology, CRC P ess, Boca
Ra on, USA, ISBN 9781439840887 (2013).
[54]P. W. B idgman, Two new modi ica ions o phospho us, J. Am. Chem. Soc. 36, 1344 (1914),
doi:10.1021/ja02184a002.
[55]P. W. B idgman, Fu he no e on black phospho us, J. Am. Chem. Soc. 38, 609 (1916),
doi:10.1021/ja02260a008.
[56]Z. Zhu and D. Tománek, Semiconduc ing laye ed blue phospho us: A compu a ional s udy,
Phys. Re . Le . 112, 176802 (2014), doi:10.1103/PhysRe Le .112.176802.
[57]H. Guo, N. Lu, J. Dai, X. Wu and X. C. Zeng, Phospho ene nano ibbons, phospho-
us nano ubes, and an de Waals mul ilaye s, J. Phys. Chem. C 118, 14051 (2014),
doi:10.1021/jp505257g.
[58]J. Guan, Z. Zhu and D. Tománek, Phase coexis ence and me al-insula o ansi ion in
ew-laye phospho ene: A compu a ional s udy, Phys. Re . Le . 113, 046804 (2014),
doi:10.1103/PhysRe Le .113.046804.
[59]H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek and P. D. Ye, Phospho ene: An
unexplo ed 2D semiconduc o wi h a high hole mobili y, ACS Nano 8, 4033 (2014),
doi:10.1021/nn501226z.
[60]A. Ca alho, M. Wang, X. Zhu, A. S. Rodin, H. Su and A. H. Cas o Ne o,
Phospho ene: F om heo y o applica ions, Na . Re . Ma e . 1, 16061 (2016),
doi:10.1038/na e ma s.2016.61.
[61]R. Wan, X. Cao and J. Guo, Simula ion o phospho ene Scho ky-ba ie ansis o s, Appl.
Phys. Le . 105, 163511 (2014), doi:10.1063/1.4900410.
[62]H. Liu, Y. Du, Y. Deng and P. D. Ye, Semiconduc ing black phospho us: Syn hesis,
anspo p ope ies and elec onic applica ions, Chem. Soc. Re . 44, 2732 (2015),
doi:10.1039/C4CS00257A.
[63]M. Akh a , G. Ande son, R. Zhao, A. Al uqi, J. E. M oczkowska, G. Sumanaseke a and J.
B. Jasinski, Recen ad ances in syn hesis, p ope ies, and applica ions o phospho ene, npj
2D Ma e . Appl. 1, 5 (2017), doi:10.1038/s41699-017-0007-5.
[64]L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen and Y. Zhang, Black phospho-
us ield-e ec ansis o s, Na . Nano ech. 9, 372 (2014), doi:10.1038/nnano.2014.35.
[65]P. Chen, N. Li, X. Chen, W.-J. Ong and X. Zhao, The ising s a o 2D black phospho us
beyond g aphene: Syn hesis, p ope ies and elec onic applica ions, 2D Ma e . 5, 014002
(2017), doi:10.1088/2053-1583/aa8d37.
[66]X. Ling, H. Wang, S. Huang, F. Xia and M. S. D esselhaus, The enaissance o black phos-
pho us, P oc. Na l. Acad. Sci. 112, 4523 (2015), doi:10.1073/pnas.1416581112.
32

SciPos Phys. 16, 077 (2024)
[67]R. Xu e al., Ex ao dina ily bound quasi-one-dimensional ions in wo-
dimensional phospho ene a omic semiconduc o s, ACS Nano 10, 2046 (2016),
doi:10.1021/acsnano.5b06193.
[68]A. N. Rudenko and M. I. Ka snelson, Quasipa icle band s uc u e and igh -binding
model o single- and bilaye black phospho us, Phys. Re . B 89, 201408 (2014),
doi:10.1103/PhysRe B.89.201408.
[69]M. Ezawa, Topological o igin o quasi- la edge band in phospho ene, New J. Phys. 16,
115004 (2014), doi:10.1088/1367-2630/16/11/115004.
[70]A. S. Rodin, A. Ca alho and A. H. Cas o Ne o, S ain-induced gap modi ica ion in black
phospho us, Phys. Re . Le . 112, 176801 (2014), doi:10.1103/PhysRe Le .112.176801.
[71]T. Low, A. S. Rodin, A. Ca alho, Y. Jiang, H. Wang, F. Xia and A. H. Cas o Ne o, Tun-
able op ical p ope ies o mul ilaye black phospho us hin ilms, Phys. Re . B 90, 075434
(2014), doi:10.1103/PhysRe B.90.075434.
[72]M. Ezawa, Highly aniso opic physics in phospho ene, J. Phys.: Con . Se . 603, 012006
(2015), doi:10.1088/1742-6596/603/1/012006.
[73]C. Du eix, E. A. S epano and M. I. Ka snelson, Lase -induced opological an-
si ions in phospho ene wi h in e sion symme y, Phys. Re . B 93, 241404 (2016),
doi:10.1103/PhysRe B.93.241404.
[74]N. H. Lindne , G. Re ael and V. Gali ski, Floque opological insula o in semiconduc o
quan um wells, Na . Phys. 7, 490 (2011), doi:10.1038/nphys1926.
[75]T. Ki agawa, T. Oka, A. B a aas, L. Fu and E. Demle , T anspo p ope ies o nonequilib-
ium sys ems unde he applica ion o ligh : Pho oinduced quan um Hall insula o s wi hou
Landau le els, Phys. Re . B 84, 235108 (2011), doi:10.1103/PhysRe B.84.235108.
[76]J. Cayssol, B. Dó a, F. Simon and R. Moessne , Floque opological insula o s, Phys. Rapid
Res. Le . 7, 101 (2013), doi:10.1002/pss .201206451.
[77]G.-H. Chen, Y.-N. Chen, Y.-W. Zhou, Y.-L. Sun and E.-J. Ye, S ain and elec ic ield unable
elec onic anspo in a mchai phospho ene nanode ice wi h no mal-me al elec odes, AIP
Ad . 10, 105012 (2020), doi:10.1063/5.0021775.
[78]G. D. Mahan, Many-pa icle physics, Sp inge , Bos on, USA, ISBN 9781441933393
(2000), doi:10.1007/978-1-4757-5714-9.
[79]P. B. Allen, Elec on anspo , in Concep ual ounda ions o ma e ials - A S anda d
Model o g ound- and exci ed-s a e p ope ies, Else ie , Ams e dam, Ne he lands, ISBN
9780444509765 (2006), doi:10.1016/S1572-0934(06)02006-3.
[80]T. S aube , N. M. R. Pe es and A. K. Geim, Op ical conduc i i y o g aphene in he isible e-
gion o he spec um, Phys. Re . B 78, 085432 (2008), doi:10.1103/PhysRe B.78.085432.
[81]M. Oli a-Ley a and G. G. Naumis, Tunable dich oism and op ical abso p ion o g aphene by
s ain enginee ing, 2D Ma e . 2, 025001 (2015), doi:10.1088/2053-1583/2/2/025001.
[82]M. Oli a-Ley a and C. Wang, Magne o-op ical conduc i i y o aniso opic wo-dimensional
Di ac-Weyl ma e ials, Ann. Phys. 384, 61 (2017), doi:10.1016/j.aop.2017.06.013.
[83]K. W. Chiu, T. K. Lee and J. J. Quinn, In a ed magne o- ansmi ance o a wo-dimensional
elec on gas, Su . Sci. 58, 182 (1976), doi:10.1016/0039-6028(76)90132-1.
33
SciPos Phys. 16, 077 (2024)
[84]R. F. O’Connell and G. Wallace, Ellip ici y and Fa aday o a ion due o a wo-dimensional
elec on gas in a me al-oxide-semiconduc o sys em, Phys. Re . B 26, 2231 (1982),
doi:10.1103/PhysRe B.26.2231.
[85]A. Chak abo y, G. Bian and G. Vignale, F equency-dependen Fa aday and Ke o a-
ion in aniso opic nonsymmo phic Di ac semime als, Phys. Re . B 107, 245120 (2023),
doi:10.1103/PhysRe B.107.245120.
[86]V. M. Pe ei a, R. M. Ribei o, N. M. R. Pe es and A. H. Cas o Ne o, Op ical p op-
e ies o s ained g aphene, Eu ophys. Le . 92, 67001 (2010), doi:10.1209/0295-
5075/92/67001.
[87]J. Schie ele, L. Ma in-Mo eno and F. Guinea, Fa aday e ec in ippled g aphene:
Magne o-op ics and andom gauge ields, Phys. Re . B 94, 035401 (2016),
doi:10.1103/PhysRe B.94.035401.
[88]F. M. D. Pelleg ino, G. G. N. Angilella and R. Pucci, Linea esponse co ela ion unc ions in
s ained g aphene, Phys. Re . B 84, 195407 (2011), doi:10.1103/PhysRe B.84.195407.
[89]M. Fa okhnezhad, M. Esmaeilzadeh and K. Shakou i, S ain-modula ed aniso opy o
quan um anspo p ope ies in single-laye silicene: Spin and alley il e ing, Phys. Re .
B96, 205416 (2017), doi:10.1103/PhysRe B.96.205416.
[90]Z. B. Siu and M. B. A. Jalil, E ec i e Hamil onian o silicene unde a bi a y s ain om
mul i-o bi al basis, Sci. Rep. 11, 7575 (2021), doi:10.1038/s41598-021-86947-z.
[91]A. Pound, J. P. Ca bo e and E. J. Nicol, Magne o-op ical conduc i i y in
g aphene including elec on-phonon coupling, Phys. Re . B 85, 125422 (2012),
doi:10.1103/PhysRe B.85.125422.
34