Transmission in pseudospin-1 and pseudospin-3/2 semimetals with linear dispersion through scalar and vector potential barriers

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Physics Letters A

www.elsevier.com/locate/pla

Transmission in pseudospin-1 and pseudospin-3/2 semimetals with linear dispersion through scalar and vector potential barriers

Ipsita Mandal

^a,b,∗

aFacultyofScienceandTechnology,UniversityofStavanger,4036Stavanger,Norway bNordita,Roslagstullsbacken23,SE-10691Stockholm,Sweden

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received28April2020

Receivedinrevisedform3June2020 Accepted9June2020

Availableonline23June2020 CommunicatedbyL.Ghivelder

Keywords:

Semimetals Tunneling Klein Super-Klein

We investigate the tunneling of pseudospin-1 and pseudospin-3/2 quasiparticles through a barrier consisting of both electrostatic and vector potentials, existing uniformly in a finite regionalong the transmissionaxis.First,wefindthetunnelingcoefficients,conductivitiesandFanofactorsintheabsence ofthevectorpotential.Thenwerepeatthecalculationsbyswitchingontherelevantmagneticfields.The featuresshowcleardistinctions,whichcanbeusedtoidentifythetypeofsemimetals,althoughbothof themexhibitlinearbandcrossingpoints.

©^{2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).

Contents

1. Introduction . . . 1

2. Formalism . . . 2

3. Pseudospin-1fermions . . . 3

3.1. Mode-matching . . . 4

3.2. Transmissioncoefficient,conductivityandFanofactor . . . 4

4. Pseudospin-3/2fermions . . . 5

4.1. Mode-matching . . . 6

4.2. Transmissioncoefficients,conductivityandFanofactor . . . 7

5. Summaryanddiscussions . . . 7

Acknowledgements . . . 9

References . . . 10

1. Introduction

Recently,therehasbeenasurgeofinterestincondensedmattersystemsthatcanhostmultiband(bothlinearandquadratic)crossings inthe Brillouinzone (BZ)[1], manyof whichdo nothave a high-energycounterpart. Inparticular, forthreefold aswell asfor aclass of fourfold degeneracies, the low energy Hamiltonian is of the form k·S^{, where} S ^{represents the vector consisting of three spin-1} orspin-3/2 matrices.Hence, weget three-dimensional(3d)semimetalswithpseudospin-1andpseudospin-3/2 quasiparticleexcitations, whicharenothingbutnaturalgeneralizationsoftheWeylsemimetalHamiltoniank·

σ

(

σ

representingthevectorofthePaulimatrices) featuringpseudospin-1/2quasiparticles.Allthesefermionshavealineardispersion,justlikeDiracfermions,andthebandstructureshave nonzeroChernnumbers.The pseudospin-1quasiparticlesare sometimesreferred toasMaxwell fermions[2],whilethepseudospin-3/2

*

Correspondenceto:FacultyofScienceandTechnology,UniversityofStavanger,4036Stavanger,Norway.

E-mailaddress:[email protected].

https://doi.org/10.1016/j.physleta.2020.126666

0375-9601/©^{2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).

Fig. 1.Tunnelingthroughapotentialbarrierinalinearband-crossingsemimetal.Theupperpanelshowstheschematicdiagramsofthespectrumofapairofparticle-hole symmetricbands,withrespecttoascalar(orelectric)potentialbarrierofstrengthV0inthex-direction.ThemiddlepanelshowsaconstantvectorpotentialAsuperposed inthesameregion.Theoretically,thisvectorpotentialcanbecreatedbyapplyingequalandoppositedeltafunctionmagneticfields(Band−^{B)attheedgesofthebarrier} region,orientedperpendiculartothex-axis.Thelowerpanelrepresentstheschematicdiagramofthetransportacrossthepotentialbarrier.TheFermilevelisdepictedby dottedlines,andliesintheconductionbandoutsidethebarrier,andinthevalencebandinsideit.Thebluefillingsindicateoccupiedstates.Forsimplicity,onlyonepairof particle-holesymmetricbandshasbeenshown.Generically,therecanbemorethanonesuchpair.(Forinterpretationofthecoloursinthefigure(s),thereaderisreferredto thewebversionofthisarticle.)

quasiparticles are well-known as Rarita-Schwinger-Weyl fermions [3]. By using DFT calculations and bulk-sensitive soft x-ray ARPES, B.Q. Lvetal. [4] havepredictedthecoexistenceofallthesethreetypesoftopologicalfermionsintheelectronicstructureofPdBiSe.There isa crucialdifference inthe dynamicalpropertiesof theDiracparticles (withspin-1/2) andthe spin-1quasiparticles that we consider here–thelatterexhibit super-Kleintunneling [5–7], whichmeansthat thebarrieriscompletely transparentforallincident anglesfor certainincidentenergies.NotethatbothDiracandspin-3/2particles[8] exhibitKleintunneling.

Inthispaper,we studythe behaviourofthetransmission coefficientsofthe pseudospin-1andpseudospin-3/2fermionsinpresence offinite barriers madeofscalar andvector potentials.We try to identifythe distinct features peculiarto thepseudospin value. These mightprovetobe atool toidentify/distinguishthesematerialsinexperiments.Tunneling in2dopticallattice versionsofpseudospin-1 andpseudospin-3/2fermionshavebeenstudiedearlierinRef. [9] and[10].

Thepaperis organized asfollows.In Sec.2,we explain thegeneralset-up forcarryingout thetunneling experiment.In Sec.3and 4, we apply the Landau-Büttiker formalism to compute the tunneling coefficients forthe pseudospin-1 andpseudospin-3/2 fermions, respectively.Finally,weendwithasummaryandoutlookinSec.5.

2. Formalism

Inordertostudytransport,the3dsystemismodulatedbyascalarpotentialbarrier(givingrisetoanelectricfield)ofstrengthV0and widthL,resultinginanx-dependentpotentialenergyfunction:

V

(

x

) =

V0 for 0

<

x

<

L

0 otherwise

.

^(2.1)

Inthenextstep,wesubjectthesampletoequalandoppositemagneticfieldslocalizedattheedgesoftherectangularelectricpotential, anddirectedperpendiculartothex-axis[11,12].ThiscanbetheoreticallymodeledasDiracdeltafunctionsofoppositesignsatx=^{0 and} x=^Lrespectively,andgivesrisetoavectorpotentialwiththecomponents:

A

(

x

) ≡ {

⁰

,

A_y

,

A_z

} =

{

0

,

Bz

, −

By

}

for 0

<

x

<

L

0 otherwise

.

^(2.2)

Notethatthisarisesfromthemagneticfield B= ¹₂

B_yˆj+^Bzkˆ

[δ (x=⁰)−δ (x=^L)]. Theentireset-upisdepictedpictoriallyinFig.1.

Some possiblemethodsto achieve thisset-upinreal experiments(forinstance,by placingferromagneticstripes atbarrierboundaries) havebeendiscussedinRef. [11].

We will followthe usual Landau-Büttiker procedure (see, for exampleRefs. [13–15])to compute the transport coefficients. For the sakeofcompleteness,we reviewtheimportantstepshere. Weconsiderthetunneling ofquasiparticlesinaslabofsquare cross-section (withoutanylossofgenerality),withthetransversewidthbeingW.WeassumethatW islargeenoughsuchthatthespecificboundary conditionsbeingusedinthecalculationsareirrelevantforthebulkresponse.Here,weimposetheperiodicboundaryconditions:

^tot

(

x

,

0

,

z

) =

^tot

(

x

,

W

,

z

) ,

^tot

(

x

,

y

,

0

) =

^tot

(

x

,

y

,

W

) .

(2.3) Thetransversemomentumk_⊥=(ky,kz)isconservedasnopotentialisappliedalongthosedirections,anditscomponentsarequantized as:

k_y

=

²

π

ny

W

≡

^qny

,

k_z

=

²

π

nz

W

≡

^qnz

,

(2.4)

where(nx,ny)∈Z. Thelongitudinal directioncorresponds to transport alongthe x-axis, and forthiswe need toconsider plane wave solutionsoftheforme^ikxx.Thenthefullwavefunctionisgivenby:

^tot

(

x

,

y

,

z

,

n

) =

^const.

×

n

(

x

)

eⁱ

q_{n y}y+^qnzz

,

(2.5)

with

n

= (

ny

,

nz

) .

(2.6)

Sincewe considertransmissioninsemimetalswithatleastone pairofvalence(

ε

⁻) andconduction(

ε

⁺)bandscrossinglinearlyata point,withdispersionrelationoftheform

ε

^±_{= ±¯}h vg

k²x+^k2y+^k2z (vg istheeffectivespeedofthequasiparticles), wewill dealwith thecasewhentheincidentparticlesareelectron-likeexcitations.Inotherwords,theFermienergy(E)isadjustedtolieintheconduction band outsidethe potential barrier.¹ Hence, given an arbitrary mode oftransverse momentum k_⊥, we candetermine the x-component ofthe wavevectors ofthe incoming, reflected, and transmitted waves (denoted by k), by solving

ε

⁺(k_x,n)=^{E.In the regions x}<0 andx>L,we haveonly propagatingmodes (k is real),while the x-componentsinthe scatteringregion(denoted by k),˜ are givenby k˜²=

E−V0

¯ h vg

2

−

k_⊥+^e_h¯^A2

,andmaybepropagating(k˜ isreal)orevanescent(k˜ isimaginary).

Nowwe need tousethe piece-wisesolutions forthewavefunction(), applicable intheregions inquestion (insideoroutsidethe potentialbarrier).Hence,eventhoughtheincidentwavefunctionrepresentsanelectron-likeexcitation,forV₀>E,theFermilevelwithin thepotentialbarrierlieswithin thevalenceband, andwe mustusethevalenceband wavefunctions(representinghole-likeexcitations) inthatregion. Inthenext step,we needto usetheboundaryconditions todeterminethe reflectionandtransmission coefficients.The boundaryconditionsaredetermined byintegratingtheequation H=^E (H^istheHamiltonianwritteninthepositionspace)overa smallintervalinthex-directionaroundthepointsx=^{0 andx}=^{L,andtheyensurethecontinuityofthecurrentfluxalongthe}x-direction.

3. Pseudospin-1fermions

IthasbeenshowninRef. [1] thatthespacegroup199 mayhosta3drepresentationattheP point(anditstime-reversedpartner−^P) intheBZ,whichistimereversalnon-invariant.Thelinearizedk·^pHamiltonianaboutPhostspseudospin-1fermionsandtakestheform:

H1

(

k

) = ¯

^{h v}gk

·

^S

,

(3.1)

whereSrepresentthevectorspin-1operatorwiththethree-components

S_x

= √

¹ 2

⎛

⎝

^{0 1 0}1 0 1 0 1 0

⎞

⎠ ,

S_y

= √

¹ 2

⎛

⎝

⁰

−

^{i 0} i 0

−

ⁱ

0 i 0

⎞

⎠ ,

S_z

=

⎛

⎝

^{1 0}0 0 ⁰0 0 0

−

1

⎞

⎠ ,

(3.2)

andv_g denotesthemagnitudeofthegroupvelocityassociatedwiththeDiraccone.Theenergyeigenvaluesaregivenby:

ε

₁^±

(

k

) = ±¯

^{h v}gk

, ε

₁⁰

(

k

) =

0

,

(3.3)

wherek=

k²_x+^k2y+^k2z,anddemonstratetwolinearlydispersingbandsandaflatbandcrossingatapoint.Herethe“+^”and“−^”signs refertothelinearlydispersingconductionandvalencebands,respectively.Thecorrespondingnormalizedeigenvectorsaregivenby:

s

=

¹ Ns

2kz

(

kz

+

s k

) +

k²_x

+

k²_y

(

k_x

+

^iky

)

²

,

√

2

(

k_z

+

^{s k}

)

k_x

+

^iky

,

1

T

(

wheres

= ±) ,

0

=

¹ N0

−

^kx

+

^iky

k_x

+

^iky

,

√

2k_z k_x

+

^iky

,

1

,

(3.4)

respectively,wherethe _N¹

s and _N¹

0 denotethecorrespondingnormalizationfactors.

Thecurrentoperatorforthisissystemiscapturedbyˆj= ∇kH1(k)=^vgS,whichimpliesthatthe localcurrentforaflatband plane waveisgivenby:

j₀

=

^vg

^†₀S

₀

=

⁰

.

(3.5)

Hence,itdoesnotcontributetothecurrentdensity[6],andweneedonlyconsider_±fortransportproperties.

Inpresenceofthescalarandvectorpotentials,weneedtoconsiderthetotalHamiltonian:

H^tot₁

=

H1

( −

ⁱ

∇ +

^eA

(

x

)

¯

h

) +

^V

(

x

)

(3.6)

inpositionspace,andfindtheappropriatewavefunctions.

1 TheFermienergyEcaningeneralbetunedbychemicaldopingoragatevoltage.

3.1. Mode-matching

Ascatteringstaten,inthemodelabeledbyn,isconstructedfromthestates:

n

(

x

) =

⎧ ⎪

⎨

⎪ ⎩

φ

L forx

<

0

φ

M for 0

<

x

<

L

φ

R forx

>

L

,

φ

L

=

₊

(

k

,

k_⊥

)

e^ikx

+

^rn

₊

(−

^k

,

k_⊥

)

e^−ikx

√

V

(

k

,

n

) , φ

M

=

α

n

₊

(

k

˜ ,

k

˜

_⊥

)

e^{ik x˜}

+ β

n

₊

(−˜

^k

,

k

˜

_⊥

)

e^{−ik x˜}

(

E

−

^V0

) +

α

n

₋

(

_k

˜ ,

_k

˜

_⊥

)

e^{ik x˜}

+ β

n

₋

(−˜

^k

,

_k

˜

_⊥

)

e^{−ik x˜}

(

V₀

−

^E

) , φ

R

=

^tn

₊

(

k

,

k_⊥

)

√

V

(

k

,

n

)

^e

ik(x−L)

,

V

(

k

,

n

) ≡ | ∂

_k

ε

⁺₁

(

k

,

n

) | =

^{h v}

¯

gk k

,

k

=

E²

¯

h²v²_g

−

^k2_⊥

,

_k

˜ =

(

E

−

^V0

)

²

¯

h²v²_g

− ˜

^k2_⊥

,

_k

˜

_⊥

=

^k⊥

+

^eA

(

x

)

¯

h

,

(3.7)

wherewe haveusedthevelocity V(^k,n)tonormalizetheincident,reflectedandtransmittedplane waves.Thesymbol(u)represents theHeaviside step function,asusual. Themode-matching procedureatthe edges x=^{0 and x}=^{L givesustheexplicit}expressions for tn(E,V0,B),which are toolong to write down within the manuscript. In anycase, we have to compute the transmission probability numerically,whichatanenergyE isgivenby:

T

(

E

,

V₀

, θ, φ,

B

) = |

^tn

(

E

,

V₀

,

B

) |

²

,

where

θ =

^cos−1

^{h v}

¯

gq_nz E

and

φ =

^tan−1

_q

n_y

k_,3/2

(3.8)

definetheincidentangle(solid)oftheincomingwavein3d.

At normal incidence, the analytical expression simplifies to t₀(E,V₀,0)=^eiL

E−V0

¯

h v g , which results in perfect transmission (T =^1),

also referred to as Klein tunneling. Again, tn(V0/2,V0,0)=^{eiL V0 sin2h v g¯ θcosφ,which implies the occurrenceof perfect} transmission forany incidentanglewhen E=^V0/2.Thisisthewell-knownsuper-Kleintunneling[6,7] forpseudospin-1Diracconesystems.Wealsonotethat tn=⁰(V0,V0,0)=^0.

3.2. Transmissioncoefficient,conductivityandFanofactor

WeassumeW tobelargeenoughsuch thatk_⊥ caneffectivelybetreatedasacontinuousvariable,andperformtheintegrationsover theangularvariablestoobtainconductivityandFanofactor.WeexpressE andV0inunitsof ^{h v}_L^g.

Usingk=_{h v¯}^E_g^sinθcosφ ,n_y=^{W E}_{h vg}^sinθsinφ ,n_z=_{h v}^{W E}_g^cosθ , dn_ydn_z= ^W_h2²v^E2_g²cosφsin²θdφ,inthezero-temperaturelimitandfora smallappliedvoltage,theconductanceisgivenby[16]:

G

(

E

,

V0

) =

^e2 h

n

|

^tn

|

²

→

^e2 h

|

^tn

|

^2dnxdny

=

^e2W2E2 h³v²_g

π θ=⁰

π

2

φ=−^π₂

T

(

E

,

V0

, θ, φ,

B

)

cos

φ

sin²

θ

d

φ ,

(3.9)

leadingtotheconductivityexpression:

σ (

E

,

V₀

,

B

) =

L

W

₂

G

(

E

,

V₀

)

e²

/

h

=

E hv_g

/

L

₂

^π

θ=0 π

2

φ=−^π₂

T

(

E

,

V₀

, θ, φ,

B

)

cos

φ

sin²

θ

d

φ .

(3.10)

TheFanofactorcanbeexpressedas:

F

(

E

,

V₀

,

B

) =

_π

θ=⁰

^π₂

φ=−^π2 T

(

E

,

V0

, θ, φ,

B

)

cos

φ

sin²

θ

d

φ

_π

θ=0

^π₂

φ=−^π₂ T

(

E

,

V₀

, θ, φ,

B

)

[1

−

^T

(

E

,

V₀

, θ, φ,

B

)

] cos

φ

sin²

θ

d

φ

.

(3.11)

First let usstudy the characteristics oftransmission coefficientsinthe absence ofthe magnetic fields.Fig. 2 showsthe polarplots of T(E,V0,

π

/2,φ,0) asa functionofthe incident angleφ (at θ=

π

/2), which correspondsto kz=^{0.InFig. 3,we show theangular} dependenceofT(E,V₀,θ,φ,0)incontourplots.AsE approachesthevalue V₀/2,itreachestheconditionofsuper-Kleintunnelingwhere thereisperfecttransmissionforallangles.Thesuper-Kleincontourplotisnotshownhereasthiswouldhavebeenaredundantplot.AsE goesaboveV0/2,thetransmissionregionsgetconfinedtonarrowerandnarrowerangularregions,centredaround(θ=⁰,φ=⁰).InFig.4, weillustratetheconductivity

σ

(E,V₀,0)andtheFanofactorF(E,V₀,0),asfunctionsofE/V₀,forsomevaluesofV₀.Duetosuper-Klein tunneling, F=^{0 for E}=^V0/2.

Fig. 2.Pseudospin-1semimetal: ThepolarplotsshowthetransmissioncoefficientT(E,V0,θ,^π₂,0)asafunctionoftheincidentangleφ(in thexy-planewithnokz- component)fortheparametersE=⁰.3V0(red),E=⁰.5V0(green),E=⁰.8V0(magenta),E=^V0(blue),E=¹.001V0(orange),E=¹.2V0(cyan),E=¹.5V0(pink),and E=2.0V0(purple).Super-KleintunnelingmanifestsitselfatE=V0/2,forwhichT=1.

Fig. 3.Pseudospin-1semimetal:Contourplotsofthetransmissioncoefficient(T)intheabsenceofthevectorpotential,asafunctionof(θ,φ),forvariousvaluesofV0andE.

Fig. 4.Pseudospin-1semimetal:Plotsofthe(a)conductivity,and(b)Fanofactor(F),asfunctionsofE/V0,forvariousvaluesofV0,inabsenceofthevectorpotential.Fis zeroatE=^V0/2 duetosuper-Kleintunneling.

Thepresenceofthevector potentialmodifiesthecontourplotsofT,asshowninFig.5.AlthoughT=1 forE=V0/2 (forallangles) inabsenceofmagneticfields,thisfeatureisdestroyedbytheconstantvectorpotential,asseeninFig.5(b).Forvaluesof E above V0/2, theT^{1 regionsgetrestrictedtodiscs(justlikeintheB}=^{0 case),whosecentresarenowshiftedawayfromthe}(θ=

π

/2,φ=⁰)point duetotheeffectofB=^0.

4. Pseudospin-3/2fermions

Theeight spacegroups 207-214 canhostfourfold topological degeneraciesabout the,R and/or Hpoints [1].The linearized k·^p Hamiltonianaboutsuchapointhostspseudospin-3/2fermionsandtakestheform:

H3/2

(

k

) = ¯

^{h v}gk

·

^J

,

(4.1)

Fig. 5.Pseudospin-1semimetal:Contourplotsofthetransmissioncoefficient(T)inthepresenceofthevectorpotential,asafunctionof(θ,φ),forvariousvaluesofV0and E.Thevaluesforthevectorpotentialcomponents

Ay,Az

areequalto:(a){⁰.5,0.5}^V0¯he^2vg,(b){⁰.2,−⁰.2}^V0¯he^2vg,(c){⁰.1,0.1}^V0h¯e^2vg,(d){−⁰.2,0.2}^V0h¯e^2vg. wherethethreecomponentsofJformthespin-3/2representationoftheSO(3)group,andtheirstandardrepresentationisgivenby:

Jx

=

⎛

⎜ ⎜

⎜ ⎝

0

√3

2 0 0

√3

2 0 1 0

0 1 0

√3 2

0 0

√3

2 0

⎞

⎟ ⎟

⎟ ⎠ ,

Jy

=

⎛

⎜ ⎜

⎜ ⎝

0 ⁻

√3 i

2 0 0

√3 i

2 0

−

^{i 0}

0 i 0 ⁻

√3 i 2

0 0

√3 i

2 0

⎞

⎟ ⎟

⎟ ⎠ ,

Jz

=

¹ 2

⎛

⎜ ⎜

⎝

3 0 0 0

0 1 0 0

0 0

−

^{1 0}

0 0 0

−

3

⎞

⎟ ⎟

⎠ .

(4.2)

Herevg denotesthemagnitudeofthegroupvelocityofthequasiparticles.Theenergyeigenvaluestaketheform:

ε

^±_3/2

(

k

) = ±

^{3h v}

¯

gk

2

, ε

^±_1/2

(

k

) = ± ¯

^{h v}gk

2

,

(4.3)

demonstratingfourlinearlydispersingbandscrossingatapoint.Herethe“+^”and“−^{”signs,asusual,refertotheconductionandvalence} bands,respectively.Thecorrespondingnormalizedeigenvectorsaregivenby:

^s_3/2

=

_N¹_s

3/2

s k

k²_x

+

^k2_y

+

^4k2_z

+

^kz

3k²_x

+

^3k2_y

+

^4k2_z

kx

+

^iky

3

,

√

3

2k_z

(

s k

+

^kz

) +

^k2_x

+

^k2_y

kx

+

^iky

2

,

√

3

(

s k

+

^kz

)

k_x

+

^iky

,

1

T

,

^s_1/2

=

¹ N₁^s_/2

− (

s k

+

kz

)

kx

−

iky

(

k_x

+

^iky

)

²

,− −

2kz

(

s k

+

kz

) +

k²_x

+

k²_y

√

3

k_x

+

^iky

2

,

^{s k}

+

^3kz

√

3

k_x

+

^iky

,

1

T

,

(4.4)

respectively,wheres= ±^,and _N^1s

3/2

and _N¹s 1/2

denotethecorrespondingnormalizationfactors.

Inpresenceofthescalarandvectorpotentials,weneedtoconsiderthetotalHamiltonian:

H^tot_3/2

=

H3/2

(−

ⁱ

∇ +

^eA

(

x

)

¯

h

) +

^V

(

x

)

(4.5)

inpositionspace,andfindtheappropriatewavefunctions.

4.1. Mode-matching

Wewillfollowthesameprocedureasdescribed forthepseudospin-1semimetals.Again,withoutanylossofgenerality,weconsider thetransportofoneofthepositiveenergystates,namely⁺_3/2,correspondingtoelectron-likeparticles,withtheFermileveloutsidethe potentialbarrierbeingadjustedtothevalue E=^{3h v¯}₂^{gk.Inthiscase,ascatteringstate}˜n,inthemodelabeledbyn,isconstructedfrom thestates:

˜

n

(

x

) =

⎧ ⎪

⎨

⎪ ⎩

φ ˜

L forx

<

0

φ ˜

M for 0

<

x

<

L

φ ˜

R forx

>

L

,

(4.6)

where

φ ˜

_L

=

⁺_3/2

(

k,3/2

,

_k

˜

_⊥

)

e^ik,3/2x

_V

˜ (

k_,3/2

,

n

)

+

σ=¹₂,³₂

r_n,σ

⁺_σ

(−

^k,σ

,

_k

˜

_⊥

)

e^−ik,σx

_V

˜ (

k_,σ

,

n

)

, φ ˜

M

=

σ=¹₂,³₂

α

n,σ

⁺_σ

(

k

˜

_σ

,

k

˜

_⊥

)

e^{ikσ˜ x}

+

σ=¹₂,³₂

β

n,σ

⁺_σ

(−˜

^kσ

,

k

˜

_⊥

)

e^{−ikσ˜ x}

(

E

−

^V0

)

+

σ=¹₂,³₂

α

n,σ

⁻_σ

(

_k

˜

_σ

,

_k

˜

_⊥

)

e^i˜kσx

+

σ=¹₂,³₂

β

n,σ

⁻_σ

(−˜

k_σ

,

_k

˜

_⊥

)

e^{−ikσ˜ x}

(

V0

−

E

) ,

φ ˜

R

=

σ=¹₂,³₂

tn,σ

⁺_σ

(

k,σ

,

_k

˜

_⊥

)

e^ik,σx

_V

˜ (

k_,σ

,

n

)

,

V

˜ (

k_,σ

,

n

) ≡ | ∂

_k

ε

⁺_σ

(

k

,

n

) | ,

k_,3/2

=

4E² 9^h

¯

^2v2g

−

^k2_⊥

,

_k

˜

_3/2

=

4

(

E

−

^V0

)

² 9^h

¯

^2v2g

− ˜

^k2_⊥

,

_k

˜

_⊥

=

^k⊥

+

^eA

(

x

)

¯

h

,

k_,1/2

=

4E²

¯

h²v²_g

−

^k2_⊥

,

_k

˜

_1/2

=

4

(

E

−

^V0

)

²

¯

h²v²_g

− ˜

^k2_⊥

.

(4.7)

WehaveusedthevelocityV˜(k_,σ,n)tonormalizetheincident,reflectedandtransmittedplanewaves.

Theusualmode-matchingprocedureatx=^{0 andx}=^{Lallowsustosolvefort}n,σ(E,V0,B)numerically.Thetransmissionprobabilities atanenergyE aregivenby:

T_σ

(

E

,

V₀

, θ, φ,

B

) = |

^tn,σ

(

E

,

V₀

,

B

)|

²

,

where

θ =

^cos−1

3^{h v}

¯

gq_nz 2E

and

φ =

^tan−1

_q

ny

k_,3/2

(4.8)

definetheincidentangle(solid)oftheincomingwavein3d.Fornormalincidence,wegetthesimpleanalyticalexpressiont0,σ(E,V0,0)= e

iL E−V0

3 δ_σ,3/2,whichimpliestheoccurrenceofKleintunneling withperfecttransmission(T3/2=^{1 and T}1/2=^{0).We notethat super-} Kleintunneling[6,7] isabsentforthepseudospin-3/2quasiparticles,unlikethepseudospin-1Diracconesystems.

4.2.Transmissioncoefficients,conductivityandFanofactor

Inthecontinuum limit forthetransverse momenta,usingk,3/2=₃²_{h v¯}^E_g^sinθcosφ ,ny= ²_{3h v}^{W E}_g^sinθsinφ ,nz= ²_{3h v}^{W E}_g^cosθ , dnydnz=

4W²E²

9h²v²g cosφsin²θdφ,theconductanceisgivenby[16]:

G

(

E

,

V₀

) =

^4e2W2E2 9h³v²_g

π θ=0

π

2

φ=−^π₂

σ

T_σ

(

E

,

V₀

, θ, φ,

B

)

cos

φ

sin²

θ

d

φ ,

(4.9)

leadingtotheconductivityexpression:

σ (

E

,

V₀

,

B

) =

⁴ 9

E hv_g

/

L

2

π θ=⁰

π

2

φ=−^π₂

σ

T_σ

(

E

,

V₀

, θ, φ,

B

)

cos

φ

sin²

θ

d

φ .

(4.10)

TheFanofactorisgivenby:

F

(

E

,

V₀

,

B

) =

_π

θ=⁰

^π₂

φ=−^π₂

σ

T_σ

(

E

,

V0

, θ, φ,

B

)

cos

φ

sin²

θ

d

φ

_π

θ=⁰

^π₂

φ=−^π₂

σ

T_σ

(

E

,

V₀

, θ, φ,

B

)

[1

−

^Tσ

(

E

,

V₀

, θ, φ,

B

)

] cos

φ

sin²

θ

d

φ

.

(4.11)

Inthe absenceofthe magneticfields, Fig.6 showsthepolarplots ofthe two transmissioncoefficientsasfunctionsof theincident angleφ,forthecasewhenkz=^{0 (hence} θ=

π

/2).Klein tunnelingisobservedfor T3/2 fora rangeofanglesaround normalincidence (independentoftheincidentenergy).Additionally, thereareresonanceconditionsforcertainvaluesofk˜ andL underwhichthebarrier becomescompletelytransparentfor T3/2 or T1/2.Fig.7showstheangulardependenceofT(E,V0,θ,φ,0)incontourplots.The patterns forE=^V0/2 clearly distinguishthepseudospin-3/2semimetalfromthepseudospin-1case,asthelatterexhibitsperfecttransmissionat allanglesfor E=^V0/2.Kleintunneling isobservedforT_3/2 forarangeofanglesaround(θ=⁰,φ=⁰),andinthoseregions, T_1/2=⁰ (sinceT3/2+^T1/2≤^{1).InFig.8,weillustratethe}conductivity

σ

(E,V0,0)andtheFanofactorF(E,V0,0),asfunctionsofE/V0,forsome valuesofV0.Unlikethepseudospin-1case, F doesnotgotozeroatE=^V0/2,duetotheabsenceofsuper-Kleintunneling.Both F and

σ

showmuchmoreoscillatorybehaviourcomparedtothepseudospin-1quasiparticles.

The contourplots in Fig. 9 capture how the presence of the vector potential modifies the transmission coefficients. The angles for perfecttransmissionisnowshiftedawayfromnormalincidence.Thetransmissionpatternsarealsomarkedlydifferentfromthoseforthe pseudospin-1semimetals,asseenbycomparingwithFig.5.

5. Summaryanddiscussions

Inthispaper,wehavecomputedthetransmissioncoefficientsofthepseudospin-1andpseudospin-3/2semimetalswithlineardisper- sionandnonzeroChernnumbers.Thesearethehigher-pseudospingeneralizationsofthewell-studiedWeylsemimetals.Thetransmission coefficientshavebeencalculatedinthepresenceofbothscalarandvectorpotentials,existinguniformlyinaboundedregion.Thepatterns foundclearly serve asfingerprintsof thecorresponding semimetal, althoughall of them havelinear dispersions. Similar computations weredoneforthecaseofWeylfermionsinRef. [11]. Comparingwiththoseresults,one caneasilyseethatthecharacteristicsforthese higher-pseudospincasesdifferconsiderably. Inparticular,thepseudospin-1casedemonstrates super-Kleintunneling,whichisabsent in

Fig. 6.Pseudospin-3/2semimetal:ThepolarplotsshowthetransmissioncoefficientsTσ(E,V0,θ,^π₂,0)asfunctionsoftheincidentangleφ(inthexy-planewithnokz- component)fortheparametersE=0.3V0(red),E=0.5V0(green),E=0.8V0(magenta),E=0.95V0 (blue),E=1.001V0(orange),E=1.2V0(cyan),E=1.5V0(pink), andE=².0V0(purple).Kleintunnelingisobservedfornormalincidence.

Fig. 7.Pseudospin-3/2semimetal:Contourplotsofthetransmissioncoefficient(Tσ)intheabsenceofthevectorpotential,asfunctionsof(θ,φ),forvariousvaluesofV0and E.Kleintunnelingisobservedforarangeofanglesaroundnormalincidence.

Fig. 8.Pseudospin-3/2semimetal:Plotsofthe(a)conductivity(σinunitsof4/9),and(b)Fanofactor(F),asfunctionsofE/V0,forvariousvaluesofV0,inabsenceofthe vectorpotential.

Fig. 9.Pseudospin-3/2semimetal:Contourplotsofthetransmissioncoefficients(Tσ)inthepresenceofthevectorpotential,asfunctionsof(θ,φ),forvariousvaluesofV0 andE.Thevaluesforthevectorpotentialcomponents

Ay,Az

areequalto:(a){0.3,0.3}^V0¯he^2vg,(b){0.2,−⁰.2}^V0¯he^2vg,(c){0.01,0.01}^V0h¯e^2vg,(d){−0.01,0.01}^V0h¯e^2vg.

theWeyl andpseudospin-3/2cases.The conductivities andFanofactors obtainedherealso serveasanother setof measurablequanti- tiestoidentifythedifferenttypesofsemimetals.Anotherimportantpoint isthatthiskindofcalculationswillhelp usfindthe perfect transmissionregions by tuning theFermi level and/or the magnetic fields,which hasthe potential to be used in generating localized transmissioninthebulkofthesemimetals,forexampleinelectro-opticapplications.

Thebehaviourofthequantities calculatedherecan alsobecontrastedagainst thatinquadraticband-crossingsemimetalsstudiedin Ref. [15].Infutureworks,thesetransportpropertieswillbestudiedinthepresenceofdisorder,ashasbeendoneinthecaseofWeyl[17]

anddouble-Weyl [18] nodes. Theeffectofmagnetic fieldsonthetunneling behaviourofthe 3ddouble-Weyl nodesand2d anisotropic Weylfermions [19] will be another interesting avenue to explore. Furthermore,it will be worthwhile to examine theeffects of terms whichreduce thesymmetry. Forexample,addition ofa termproportional tok_i J³_i toH_3/2 reducesthefull rotationalsymmetry to the rotationalcubicgroup.Lastly,thisexerciseneedstobecarriedoutinthepresenceofinteractions,whichcandestroythequantizationof variousphysicalquantitiesinthetopologicalphases[20,21].

Declarationofcompetinginterest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influencetheworkreportedinthispaper.

Acknowledgements

WethankAtriBhattacharyaforhelpwiththefigures.