Table Of ContentDirac semimetal thin films in in-plane magnetic fields
Zhuo Bin Siu and Mansoor B. A. Jalil
Computational Nanoelectronics and Nanodevices Laboratory,
Electrical and Computer Engineering Department, National University of Singapore, Singapore
Seng Ghee Tan
Data Storage Institute, Agency for Science, Technology and Research (A*STAR), Singapore
Inthiswork,westudytheeffectsofin-planemagneticfieldsonthinfilmsoftheDiracSemimetal
(DSM)Na Biwhereoneofthein-planedirectionsisperpendiculartothek-separationbetweenthe
3
6 two Weyl nodes that exist for each spin orientation. We show numerically that the states localized
1 nearthesurfacesofthesethinfilmsarerelatedtotheFermiarcstatesinsemi-infiniteslabs. Dueto
0 the anisotropy between the two in-plane directions, the application of a magnetic field along these
2 directions have differing effects. A field parallel to the k space separation between the Weyl nodes
n leads to a broadening of the surface state band and the formation of an energy plateau, while a
u perpendicularfieldshiftstheenergywheretheholeandparticlebandsmeetupwards,andsharpens
J thetipsofthebands. Weillustratetheeffectsofthesechangestothedispersionrelationbystudying
thetransmissionfromasourcesegmentwithoutamagneticfield,toadrainsegmentwithanin-plane
8
1 magnetic field for various combinations of field and source-drain interface directions.
]
l I. INTRODUCTION minatedperpendiculartothek-spaceseparationbetween
l
a the Weyl nodes. We then consider DSM thin films with
h
finitethickness,andnextmoveontodiscusstheeffectsof
- TheDiracSemimetal(DSM)[1–4]isarecentlydiscov-
s in-plane magnetic fields on the dispersion relations. We
e ered topologically non-trivial state which has attracted finally showcase one consequence of the magnetic fields
m muchattention. Likethesurfacestatesofthemoreestab-
by considering the transmission from a source DSM thin
. lishedthree-dimensionaltopologicalinsulators(TI)[5,6] film segment without a magnetic field, to a drain DSM
t
a ,thelowenergyspectrumofDSMstaketheformofDirac segment with a magnetic field for various directions of
m cones. Unlike the Dirac cones in three-dimensional TI
magnetic fields and source-drain interfaces.
- surface states which have linear dispersion only in the
d two dimensions in the plane of a TI surface, the DSM
n
Dirac cones disperse linearly in all three k-space dimen- II. BULK EIGENSTATES
o
sions in bulk DSMs about the Weyl nodes. Moreover,
c
[ unlike in conventional TIs where there is a odd number The Hamiltonian for bulk Na Bi reads
of Dirac points, the Weyl nodes in DSMs occur in pairs 3
3
v wvaittuhreonseoumrceemabnedr tohfeeoatchherpamiremacbteinrgacatsinag aBseraryBecrurry- M((cid:126)k) Ak+ 0 0
538 criusrevtaotusruerfsaicnek.stTatheessienptahiersfoorfmWoefyFlenromdieasricnsDlinSkMinsggtihvee H =(cid:15)0((cid:126)k)+A0k− −M0((cid:126)k) M0((cid:126)k) −A0k−
5 two points in k-space when the bulk DSM is truncated 0 0 −Ak+ −M((cid:126)k)
0 perpendicular to the direction of the k-space separation
1. between the Weyl nodes. These Weyl nodes are topo- where (cid:15)0 =C0+C1kz2+C2k2 and M((cid:126)k)=M0−M1kz2−
0 logically stable against perturbations which preserve the M2k2,k2 =kx2+ky2andk± =kx±iky [1,2]. Thisconsists
6 translational symmetry. To date, two materials, Cd As oftwouncoupledblocksrepresentingthespinupandspin
3 2
1 [7–10] and Na Bi [11–13] have been experimentally con- downstates, whichwecanconsiderseparately. Wefocus
3
v: firmed to host the DSM state. on the spin up states.
i Diagonalizing the spin up block yields the eigenvalues
X The effects of electric fields applied perpendicular to
the surfaces of quasi-two dimensional thin films and (cid:112)
r (cid:15) =(cid:15) ± (Ak)2+M2 (1)
a quasi-one dimensional Na Bi nanowires have been stud- ± 0
3
ied recently [14, 15]. Whereas the effects of magnetic and the (unnormalized) eigenspinors
fields on DSMs have been studied previously (for exam-
ple, in Refs. 16–18 ), these works have tended to focus (cid:18)M ±(cid:112)(Ak)2+M2(cid:19)
. (2)
on magnetic fields perpendicular to the plane of DSM A(k −ik )
x y
slabs and thin films. In this work, we study the effects
of in-plane magnetic fields on DSM thin films. To lay ThebulkenergydispersionisshowninFig. 1. Thedis-
the foundations for our subsequent discussion, we first persionrelationconsistsoftwoparaboliccone-likestruc-
reviewtheenergydispersionofbulkNa Biandtheemer- tures above (below) (cid:15) corresponding to the + (−) sign
3 0
gence of Fermi arcs on semi-infinite slabs of Na Bi ter- of Eq. 1 with the cone tips squashed inwards to form
3
2
cusps, so that the points where the cone tip starts curv- eigenspinors Eq. 2. Within the same ± branch, we
(k2)
ing inwards correspond to the lowest (highest) energy at havetwovaluesofk =± (cid:113)k2 . Ifxk wereimag-
the two Dirac points. The two Dirac points have energy x (kx) x;±(kx2) x
inary, then one of these will blow up in the wrong di-
(cid:15) and lie along the k =0 line.
0 z
rection. If k is real, then the upper component of the
x
eigenspinor has the same value for both signs of ± ,
(kx)
but the k in the lower component has opposite signs
x
800
so it is impossible to form a linear combination of the
two such that the eigenspinor vanishes. We therefore
600
conclude that the linear superposition must consist of
different ± branches.
400 (k2)
x
Next, we investigate whether various quantities are
V
me 200 real, complex or imaginary. E, kz and ky are given to
E / be real. From E = (cid:15) ±(cid:112)(Ak)2+m2 we conclude that
0
0 (cid:15) must be real because E is real and if (cid:15) has any imag-
0 0
inarycomponenttheimaginarycomponentcannotbesi-
-200
multaneously canceled off by both signs of the square
1 root term. (cid:15) being real in turn restricts k2 to be real,
-400 0 0 x
2 1 0 k -1 -2 -1 k ssotrakixnsis(cid:112)ei(tAhekr)2re+alMor2 itmoabgeinraerayl.. kE2biseirnegal,resaolMals2omcounst-
z
be real as well. Thus the upper component of the eigen-
spinorisreal. Earlierwenotedthatforthewavefunction
FIG. 1. The dispersion relation of bulk Na Bi as a function
of k=(cid:112)k2+k2 and k . 3 to disappear at the boundary the two eigenspinors must
x y z
come from different ± branches. We now know that
(k2)
x
the upper components of both branches are real, so the
relative weights of the two eigenspinors are real. This
in turn forces the k to be imaginary, so that the lower
III. SEMI-INFINITE SLABS x
componentsofbotheigenspinorsareimaginary. (Assume
that k is real so that the only imaginary part of the
OnehallmarkofaDSMistheemergenceofFermiarcs x
lower eigenspinor component comes from the ik por-
when a bulk DSM is truncated perpendicular to the k- y
tion. Then for the imaginary part of the lower eigen-
space separation between the Weyl nodes. We therefore
spinor component to cancel off we need the coefficients
considerasemi-infiniteslabterminatedatthexdirection
of the lower eigenspinor branches to be equal and op-
so that the slab extends to infinity along the y and z
posite. This forces k to have the same value in both
directions, and k and k are good quantum numbers. x
y z eigenspinors however, which gives the trivial solution of
We look for linear combinations of the eigenstates of
both eigenspinor components being 0.)
the spin up block of Eq. 2 which vanish at x = 0. In
These insights allow us to suggest a numerical scheme
solving for k from the energy equation Eq. 1
x
to find the allowed values of (k ,k ) for a given value of
y z
(cid:112)
E±(E) =(cid:15)0±(E) (Ak)2+M2 E for a ±x terminated slab. An equation in ky and kz
can be formed in the following way – for a given (k ,k )
y z
weshiftthe(cid:15)0 totheLHSandsquaretheresultingterm, we solve for the two values of kx corresponding to the
so the ± distinction in the square root is lost. Here, the ± brancheswiththecorrectsignoftheimaginarypart
(k2)
subscript(E)indicatesthatthis±pertainstoE inorder depxendingonwhetherthesemi-infiniteslabisterminated
todistinguishitfromtheother±sthatoccurlaterinthe alongthe+xor−xdirection. Denotingthesetwovalues
text. of k as k , we back substitute them into Eq. 1 to
In detail, we have x x±(kx2)
determine which of the ± branches they correspond
(E)
(cid:112)
E =(cid:15) ± (Ak)2+M2 to, so that the correct form of the eigenspinors can be
0
⇒(E−(cid:15) )2 =(Ak)2+M2 obtained. With these two eigenspinors, which we denote
0
as |± (cid:105), we can then calculate the determinant of the
which is a quadratic equation in k2 so that we have, in linear(kexq)2uations in the unknown coefficients c and c
+ −
turn, two values for kx2 for a given ky. This quadratic in |+(k2)(cid:105)c++|−(k2)(cid:105)c−. The allowed values of (ky,kz)
equation is cumbersome and not particularly insightful are thexn those for wxhich the determinant vanishes.
and will not be presented here. We then obtain two val-
Fig. 2 shows the calculated Fermi arcs for spin up at
ues for k2, which we designate as k2 . In general,
x x;±(kx2) variousvaluesofenergyforsemi-infiniteslabsterminated
only one of these roots will actually be the solution for a along the x = 0 line and extending to x → ±∞ respec-
given +/− sign value of the ± . tively, as well as the infnite-sized, bulk Na Bi equal en-
(E) 3
Nowseekingalinearsuperpositionofeigenstateswhich ergycontours(EECs)atk =0atthosevaluesofenergy.
x
disappear at the slab surface boundary, we examine the The spin up Fermi arcs for semi-infinite slabs extending
3
1 x = W, W being the thickness of the film. Under these
-50 meV
0.8 0 meV boundary conditions, the spatial part of the eigenstate
50 meV wavefunctions can be expanded as a linear combination
0.6 100 meV
of the normalized eigenstates of the infinite potential
(cid:112)
0.4 well |φ (cid:105) where (cid:104)x|φ (cid:105) = φ (x) = 2/W sin(nπx/W).
n n n
0.2 Since the spin up and spin down states are decoupled in
-1m the Hamiltonian, we consider each spin separately. For
n 0
k / y each spin and given value of ky and kz the relevant part
-0.2 oftheHamiltonianH(k ,k ) (the±subscriptreferring
y z ±
-0.4 to the spin up / down parts of the Hamiltonian) can be
expanded in the basis of the |φ (cid:105) states. This gives a a
-0.6 n
numerical matrix H˜ with matrix elements H˜ =
± ±,[i,j]
-0.8 (cid:112) (cid:82)W
2/W dx sin(iπx/W)(H (k ,k )(sin(jπx/W)).
0 ± y z
-1 The matrix can then be diagonalized numerically to
-3 -2 -1 0 1 2 3
k / nm-1 obtain the eigenenergies and eigenstates.
z
Fig. 3showsthedispersionrelationsfora20nmthick
FIG.2. ThethicklinesindicatetheFermiarcsatthevalues film at various values of ky as a function of kz.
of energy indicated in the legend. Solid lines correspond to
thespinupFermiarcsata+xterminatedsurfaceonasemi- 300
infinite slab extending to x → −∞ while dotted lines corre- 250
spondtoarcsonxterminatedsurfacesextendingtox→+∞. 200
The thin dotted lines are the bulk, infinitely sized blocks at 150 ky = 0 / nm
kx =0 at the indicated values of energy. E \ meV 10500 kkyy == 00..015 / /n nmm
0
−50
tox→±∞arereflectionsofeachotheralongthek =0
y
−100
line. (The spin down Fermi arcs are the reversal of the
−150
spinupFermiarcs). Thearcsemergeoncetheenergyis
−200
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
increased from −∞ to the bottom of the bulk hole band kz / nm−1
cusp and continue to exist as the energy is increased to-
wards +∞. The arcs are tangential to the bulk EECs FIG. 3. The dispersion relations for a 20 nm thick film
calculated with the wavevector perpendicular to the sur- as a function of kz for various values of ky indicated by the
colors of the lines. The dotted lines show the energy of bulk
face ( k in this case ) set to 0. (The two circles, one
x
unbounded Na Bi at k =0 and the corresponding values of
on the left and the other on the right, for each of the 3 x
k and k . The thicker line in (a) denotes what we shall, for
EECs at the three lower values of energies are the cross y z
brevity,refertoasthe‘lowestenergyparticleband’(LEPB).
sections on the two ‘humps’ on either side of the cusp in
the hole bands in Fig. 1. The single circle in the middle
Similar to the semi-infinite slab, the hole bands bend
for E = 50 meV is the cross section of the ‘cone’ of the
downwards in energy at small |k | to give rise to a cusp
bulk particle band. This value of energy lies above the z
betweenthetwoDiracpoints. Thefinitethicknessofthe
top of the cusp of the bulk particle band. )
thin film here leads to the formation of subbands.
We shall later see that the arcs play a significant role
There is a subband, highlighted with slightly thicker
in the dispersion relations of thin films.
linesinpanel(a)ofthefigure,whichtracesthebulk(i.e.
infinitely sized DSM block) k = 0 particle band for k
x z
values outside the bulk cusp, and the bulk k = 0 hole
x
IV. THIN FILMS
band inside the bulk cusp. This band is special, as we
shall explain later. For brevity we shall refer to it as the
Wenextconsiderthinfilmsofinfinitedimensionsalong Lowest Energy Particle Band (LEPB), and the subband
they andz direction, andfinitethicknessalongthez di- immediately below it in energy as the Highest Energy
rection. k and k are hence good quantum numbers. In Hole Band (HEHB). Except for the LEPB, the particle
y z
contrast to the semi-infinite slab which has only a single band bottoms are bounded in energy above the k = 0
x
surface,athinfilmhasbothanupperandalowersurface. bulkparticleband. Theenergiesoftheholebandsat|k |
z
Asaresultweexpecttoseeaspectswhichwearefamiliar lyingoutsidethetwobulkDiracpointsarealsobounded
with from the previous discussion on semi-infinite slabs, below the k = 0 bulk hole band. At energies above
x
aswellasfeatureswhichnowemergebecauseofthefinite the Dirac point corresponding to |k | lying outside the
z
thickness. two bulk Weyl nodes, the energy of the LEPB follows
We solve the eigenspectrum of the thin film numer- thatofthebulkparticlestatesclosely. Atenergiesbelow
ically. We adopt the hard-wall boundary conditions the Dirac point the energy of the lowest energy particle
under which the wavefunctions vanish at x = 0 and states lie slightly above the top of the ‘cusp’ in the hole
4
energy bands for k lying between the two Weyl nodes. Fermi arcs regardless of the thickness of the film. The
z
The band bottoms (tops) of the particle (hole) bands stateswhichlieneartheFermiarcsarelocalizednearthe
increase (decrease) in energy monotonously with |k |. surface ( see the states labeled (i) to (iii) ) the arcs cor-
y
Foranotherperspectivewenowplotthek spaceEECs respond to, although the extent of localization decreases
for the 20 nm thick thin film which we have been con- with increasing |k | (compare state (iii) with states (i)
z
sidering, as well as that of a thicker 50 nm thick film at and (ii) ). States (i) and (iii) share approximately the
E =−50meV. This is a value in energy which lies above same k value.
y
the bottom of the LEPB but below the two ‘humps’ sur- Point (i) with the smaller |k | lies on the LEPB and
z
rounding the small |k | cusp. point (iii) with the bigger |k | lies on the HEHB. The
z z
states lying in the vicinity of the bulk Fermi arcs for
(a)00..45 20 nm vvi (b) ii i sinmtearllse|kctz|esamchalloetrhtehralniethoen|kthz|ewLhEicPhBt,hewthwiloeFtehremsitaartcess
0.3 50 nm
that lie on the Fermi arcs for larger values of |k | lie
0.2 z
-1m 0.1 i iiivi (c) iii iv oanlwathysesHurEfaHcBe.stTahtees.stTahteesstoantetshoenLtEhPeBHEaHreBt(hie.er.eftohree
k / ny-0.10 ii outermost rings of the onions) for small |kz| lying on the
-0.2 (d) v vi Fermi arcs are also surface states (e.g. iii), but those
-0.3 states on the HEHB with larger values of |k | (e.g. vi)
z
-0.4 are not surface states.
-0.5-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 50 An examination of the EECs and density distribution
k / nm-1 x / nm
z at other values of energies (not shown) indicates that
the LEPB states at a given energy are always located
FIG. 4. In panel (a), the thick solid lines around the k-
near the semi-infinite slab Fermi arcs and are localized
space origin represent the semi-infinite slab Fermi arcs for
slabs infinite in the y and z direction, and semi-infinite in neareithertheupperorlowersurface, whereastheother
the +x and −x direction at E =−50meV. The dotted green states present at the same value of energy located away
linesarethebulkEECsatthesamevalueofenergy,whilethe from the bulk Fermi arcs are bulk states. The k-space
dots trace out the EECs for a 20 nm, and 50 nm thick film locations of the LEPB states at a given energy are only
as indicated by the color of the dots. The relative densities very weakly dependent on the film thickness. Together,
of the states at various points in k-space and E = −50meV these indicate that the LEPB states originate from the
marked i to vi on the EECs are plotted out in the panels on
bulkFermiarcsandmaysharethetopologicalprotection
the right.
of the latter. This robustness makes the LEPB states of
particular interest in potential device applications. We
One dominant feature of EECs is the presence of two
shall consequently concentrate on the low energy regime
series of concentric rings of EECs, one series of rings on
inwhichonlytheLEPBandafewbulkstatesarepresent.
the left and the other series on the right which we shall
for simplicity refer to as the left and right onions. These
twoonionscorrespondtothetwo‘humps’inFig. 3that,
V. EFFECTS OF AN IN-PLANE MAGNETIC
at higher energies, eventually form the two Weyl nodes.
FIELD
The empty space between the two onions corresponds to
the cusp between the two humps in Fig. 3. The outer-
The effects of an out of plane electric field on a DSM
most rings of each onion correspond to the HEHB. The
thin film / quasi one dimensional nanostructure was in-
states that lie outside the two semi-infinite slab Fermi
vestigated in Refs. [14, 15]. Here we investigate how a
arcs are bounded by the bulk EECs indicated by the
magnetic field in the y and z directions affect the dis-
green dotted lines. These states, such as states (iv) to
persion relations. The modification of the EECs by an
(vi) in the diagram, are bulk states where there is signif-
in-plane magnetic field, and the anisotropy between the
icant charge density away from the two boundaries. In
k-space directions perpendicular and parallel to the k
particular, away from the Fermi arcs, the bulk states (v)
space separation between the Dirac points shall, as we
and (vi) resemble the usual infinite quantum well states
show in the next section, lead to dramatic differences in
withincreasingnumberofdensitynodesaswemoveinto
the transmission from a source DSM segment without
the interior of the onion. As one would expect from the
a magnetic field to a drain segment with a field as the
infinitequantumwell,alargerthicknessresultsinalarger
energy and magnetic field are varied.
number of states at a given energy.
We model a magnetic field in the y direction via the
We now focus on the k-space around the vicinity of canonicalsubstitutionk →(k +B˜ x)andafieldinthe
the semi-infinite slab Fermi arcs. The semi-infinite slab z z y
z direction via k →(k −B˜ x) where we have absorbed
Fermi arcs for the upper and lower surface ‘penetrate’ y y y
each other so that for fixed value of |k | < 0.15nm−1 a the various physical constants into B(cid:126)˜. In this choice of
y
line of constant k cuts through both the +x and −x gaugethespatialdependenceoftheelectromagneticvec-
z
terminated semi-infinite slab Fermi arcs. There are al- tor potential is in the x direction so k and k remain
y z
waysthinfilmstateswhichlienearthesemi-infiniteslab good quantum numbers. In our numerical calculations
5
we adopt units where e = (cid:126) = 1, energy is in meV and a magnetic field.
lengths in nm. The units for k are in nm−1, so the Fig. 6 shows the dispersion relations when a magnetic
y
quantitiesofB(cid:126)˜ whichappearsubsequentlycarryunitsof field in the z direction parallel to the k space separation
between the two Dirac nodes is applied.
nm−2.
Fig. 5 shows the dispersion relations calculated as
(a) 150 (b) 150
functionsofthey andz wavevectorswiththewavevector
0
in the other direction set to 0 at various values of mag- 100 00..015 100
50 0.2 50
neticfieldinthey directionperpendiculartothek-space
separation between the two Dirac points. (We have split E / meV 0 E / meV 0
the k dispersion relations into two separate panels for −50 −50 0
y 0.05
clarity. ) −100 −100 00..12
−150−4 −3 −2 −1 kz / n0m−1 1 2 3 4 −150−3 −2 −1 ky / n0m−1 1 2 3
(a) (b)
150 150 FIG. 6. The dispersion relations for a 20 nm thick slab for
0
100 00..015 100 (a) as a function of kz at ky =0, and (b) as a function of ky
at k =0 at the values of B˜ indicated in the figure legends.
50 50 0.1 z z
E / meV E / meV 00..23 The LEPBs are indicated by thicker lines.
0 0
−50 −50 Panel (a) shows that the dispersion relation of the
−100−4 −3 −2 −1 0 1 2 3 4 −100−4 −3 −2 −1 0 1 2 3 4 LEPB along the kz direction is not very much affected
(c) kz / nm−1 kz / nm−1 by the magnetic field. A larger field however does push
150
the other particle bands upwards in energy, and the hole
100 0.1 bands downwards in energy, as well as increases the en-
0.2
50 0.3 ergy separation between the subbands. The z magnetic
E / meV field has a larger effect on the dispersion relation along
0
the k direction. As the magnetic field is increased,
−50 y
the band bottom of the LEPB flattens out and forms
−100−1 −0.5 0 0.5 1 a plateau for an increasing range of |k |.
ky / nm−1 y
FIG. 5. The dispersion relations for a 20 nm thick slab (a)
and (b) as a function of kz at ky = 0, and (c) as a function VI. TRANSMISSION
of k at k = 0 at the values of B˜ indicated in the figure
y z y
legends. The LEPBs are indicated by thicker lines.
Here we consider the transmission of the LEPB states
from a source DSM film segment without a magnetic
Focusing first on panel (a) of the figure, we see that a field, to a drain segment with a magnetic field. The
smallBy leadstotheHEHBstateswithsmall|kz|bend- setup is shown schematically in the top panel of Fig. 7
ing upwards in energy to form a small upwards pointing where an applied bias, represented by the battery in the
kink centered at kz =0. The band bottom of the LEPB figure, drives a current flowing from the source segment
band sharpens to form a downwards pointing kink at to the drain segment. The bias supplies charge carriers
kz = 0 so that the k space vicinity around where the at the source segment. The source states incident on
LEPB and HEHB touch at (cid:126)k = 0 is now reminiscent thesource-draininterfaceareeithertransmittedintothe
of a Dirac cone. The portion of the band on the left drain segment, or reflected back to the source segment.
and right side of this downwards kink form a plateau at Thestatestransmittedthroughtheinterfacetothedrain
around 60 meV in energy. The “Dirac point” where the result to a current flowing through the circuit, which we
LEPB and HEHB touch shifts upwards in energy as B representbytheammeterinthefigure. Thetransmission
y
increases until it hits the band plateau (see panel (b) ) coefficient for a source state of wavevector(cid:126)k and energy
at around 60 meV. At this stage the downwards point- E can be calculated by matching the wavefunctions at
ing kink in the particle band bottom disappears into the the source-drain interface. The net transmission at an
plateau. As By increases even further the middle of the energy E is obtained by integrating over all the source
LEPB ‘plateau’ becomes concave, the upwards kink of states of that energy propagating in the direction from
the HEHB blunts out and an energy gap between the the source to the drain.
LEPB and the HEBB opens up. Panel (c) of the figure We consider the cases where the interface between the
shows that in contrast to the complicated dispersion re- two segments lies along either the y =0 or z =0 line, so
lation along the kz direction, the energy of the LEPB thatthesourceanddrainsegmentshavesemi-infiniteex-
increases monotonically with |ky|. We shall show the tents in the in-plane (yz) direction perpendicular to the
EECs at various values of energy and B˜ in the next interface and infinite extent parallel to it. The momen-
y
section when we discuss the transmission from a source tum parallel to the interface direction is hence a good
segment without magnetic field to a drain segment with quantum number and is conserved in the transmission.
6
Sincetheintegrationoverallthesourcestatescontribut- (a) 3 (b) 3
0
ingtothetransmissionatagivenenergyisessentiallyan 2 0.1 2.5
0.2
integrationoverthetransversemomentumrangespanned 1 0.3 2 0
0.1
bthyethoevseerlsatpatbese,twtheeentrathnesmriasnsigoensiosfvetrraynmsvuecrhseaffmeoctmedenbtya -1k / nmy0 ρ1.5 00..23
-1 1
spanned by the source and drain states.
-2 0.5
Theanisotropybetweenthek-spacedirectionsparallel
(z) and perpendicular to the k-space separation leads to -3-1 -0.8 -0.6 -0.4 -0.2kz / 0nm-10.2 0.4 0.6 0.8 1 00 2 4 6 8 x /1 n0m 12 14 16 18 20
differenttransmissionprofilesdependingonthedirection
of the interface. FIG.8. (a)TheEECsfora20nmthickDSMfilmat70meV
at various strengths of B indicated in the legend. (b) The
z
particle density profiles within the thickness of the thin film
85 0.999 at various values of Bz at ky =0.5 nm−1 as indicated by the
80 dotted line on the EEC plots on the (left).
x
A 75 0.9985
z
Bz E / meV6750 0.998 For an interface along the y direction, we integrate
y 60 0.9975 along the ky range spanned by the source state. Al-
though the k range spanned by the drain EECs com-
55 y
0.997
pletely overlaps that spanned by the source EEC, the
50
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 profile of the EEC at a given k between the source
Bz y
85 and drain segments are quite different – the source EEC
0.9
80 (where Bz = 0) curves outwards along the ky direction
B 0.8
x z z 75 0.7 but the drain EECs flatten out with increasing Bz. This
flatteningoutofthedrainEECsisaccompaniedbyashift
E / meV6750 00..56 in the position along the thickness of the film (in the x
direction) where the charge carriers with group velocity
60 0.4
from the source to the drain are concentrated. Fig. 8(b)
y 55 0.3
showsthatwhereasatlowfieldsthechargecarriersarelo-
50 0.2
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 calizedneartheboundariesofthefilm, theyincreasingly
Bz
getshiftedtowardstheinteriorofthefilmwithincreasing
FIG. 7. The transmission from a source segment of 20 nm Bz. This reduces the overlap between the wavefunctions
thickDSMthinfilmwithnomagneticfieldtoadrainsegment oftheLEPBstatesinthesourceanddrainsegmentsand
with a magnetic field in the z direction for (top) an interface leads to the trend of decreasing transmission with B at
z
parallel to the z direction and (bottom) an interface parallel a given energy.
to the y direction. We now shift our focus to the effects of applying a
magnetic field in the y direction.
Fig. 7 shows the transmission profiles from a source Fig. 9 shows that the transmission from a source seg-
segmentwithoutamagneticfieldtoadrainsegmentwith ment to a drain segment with a magnetic field in the y
a magnetic field in the z direction. (The energy range is direction for an interface parallel to the y direction de-
chosen so that the only propagating states in the source creases with increasing magnetic field.
aretheLEPBstates. ) Thetransmissionforaninterface
parallel to the z direction is almost unity and exhibits a 85
decrease with magnetic field for an interface parallel to 80 0.95
B
the y direction. x z y 75 0.9
The reason for these trends can be found in the EECs meV70 0.85
of the source and drain segments shown in Fig. 8. E / 65 0.8
The k range spanned by the EECs increases with B 0.75
y z 60
while the kz range spanned remains constant. For an y 55 0.7
interface along the z direction (top of Fig. 7 ) we es- 0.65
50
sentially integrate across the k range spanned by the 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
z By
source states in order to calculate the transmission. At
any given value of k the profile of the source and drain FIG. 9. The transmission from a source segment without a
z
EECs have a similar trend of curving outwards in the k magneticfieldtoadrainsegmentwithamagneticfieldinthe
y
direction. This, and the fact that the k range spanned y direction for an interface parallel to the y direction.
z
by the source and drain EECs are almost the same and
that the EECs have the same shape of bowing outwards This can be explained from the shapes of the EECs as
alongthek directioncontributetowardsthelargetrans- shown in Fig. 10.
y
mission. Comparing against Fig. 5, the single lobe for B˜ = 0
z
7
1 allel to the z direction.
0.8
0.6 0 Thedecreaseintransmissionatagivenvalueofenergy
0.1
0.4 0.2 withBy canalsobeexplainedbytheEECprofileinFig.
-1m 0.2 00..34 10. Here since the interface is parallel to the z direction
k / ny-0.20 kz is conserved. From the figure it is evident that as
B is increased the EECs of the drain states are shifted
-0.4 y
-0.6 towards larger |k | so that the k overlap between the
z z
-0.8 source EEC centered around k =0 and the drain EECs
z
-1
-4 -3 -2 -1 0 1 2 3 4 decreases and eventually drops to 0.
kz / nm-1
We now look at what happens when we fix B and
y
FIG. 10. The EECs at E =50 meV at various values of B˜ vary the energy. Differing from the transmission profiles
z
indicated in the legend. considered earlier, here we have at larger values of B a
y
trend where the transmission at a given value of B˜ first
y
drops to 0 as the energy is increased, and then increases
correspondstothecrosssectionoftheLEPB‘cone’. The
from 0 as the energy is increased further. The region
small, narrow ‘waist’ of the B˜ = 0.1 corresponds to
z wherethetransmissionis0essentiallycoincideswiththe
the cross section of the downwards pointing kink in the
energy band gap that opens up between the LEPB and
LEPB. The central lodes enclosing(cid:126)k =0 surrounded by
HEHB.
side lobes for B˜ =0.2,0.3 nm2 are the cross sections of
z
the HEHB. There is no central lobe in the B˜z = 0.4 be- 1
causethetopoftheHEHBhasfallenbelowE =50meV. 0.8
50 meV
Calculating the transmission across an interface par- 0.6 60 meV
0.4 70 meV
allel to the y direction involves integrating over the k
rbayngtheespdarnaninedEbEyCthsedseocurrecaeseEsEwCi.thThienckryearasinnggeBspa.nnAedys -1k / nmy-00..220 100
B˜y increases from zero, gaps in the EECs begin toy open --00..64 E / meV
apartsothattheEECwhich,isoriginallyasingleclosed -0.8 0-4 kz / nm-1 4
Bc˜urv=e a0t.1B,0y.2=nm0,2birneatkhseinfitgourteh.reTehseepgaarpasteincukrv-sepsaacet -1-3 -2 -1 kz / 0nm-1 1 2 3
y
between the EEC curves represents an absence of prop- FIG. 12. The EECs at B˜ = 0.25 (solid lines) and B˜ = 0
y y
agating states at those values of ky and lead to a drop attheenergieslabeledonthelegend. Theinsetshowsthekz
in the transmission. Once B˜y increases beyond around dispersion relation for B˜y = 0.25 at ky = 0 with the dotted
0.2 where the energy at (cid:126)k = 0 of the LEPB rises above horizontallinesindicatingtheenergylevelsatwhichtheEECs
are drawn
the energy at finite |(cid:126)k|, the band top of the HEHB drops
in energy with increasing B˜ . An energy gap opens up.
y This is also reflected in Fig. 12 which show the source
ThisismanifestedastheabsenceofthecentralEEClobe
and drain EECs at B˜ = 0.025, and E = 50,60 and
atthelargervaluesofB˜ inthefigure. Atthesametime, y
y 70meV. Theseenergiescorrespondtothoseatthevalue
theky rangesspannedbythesidelobesdecreasewithB˜y. of B˜ where the transmission decreases between 50 meV
y
All of these contribute to the drop in transmission with
to 60 meV, then increases between 60 meV to 70 meV.
B˜ .
y The 50 meV drain EEC has a relative large k overlap
x
withEECduetothemiddleEECcurveenclosing|(cid:126)k|=0
85
0.9 whichcorrespondstothecrosssectionoftheHEPB.The
80
0.8 next value of energy 60 meV corresponds to an energy
x
75 0.7
z wherethek-spacerangeenclosedbythesourceEECfalls
0.6
By E / meV6750 00..45 imnistshieongaispzbeerotw.eTehnetthoepHmEoHstBvaalnudeoLfEePneBrgayndistahbeovtreatnhse-
y 60 0.3 bottom of the LEPB where the EECs corresponding to
0.2
55 the cross section of the LEPB increase in k space area
0.1
50 0 and start to overlap with the source EEC again, giving
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
By rise to finite transmission.
FIG. 11. The transmission from a source segment without
a magnetic field to a drain segment with a magnetic field in
the y direction for an interface parallel to the z direction. VII. CONCLUSION
Finally, Fig. 11 shows the transmission from a source Inthisworkwestudiedtheeffectsofin-planemagnetic
segmentwithoutamagneticfieldtoadrainsegmentwith fields on a Na Bi DSM thin film. We showed that in the
3
amagneticfieldinthey directionwiththeinterfacepar- absence of a magnetic field, there is a special band, the
8
LEPB,whichoriginatesfromthesemi-infiniteslabFermi very recent work [19] on the effects of in-plane magnetic
arcs, and is localized near the surface of the thin films. fields on Weyl and Dirac semimetals. In that work, the
The application of a magnetic field in the y direction authors reasoned that an in-plane magnetic field applied
leads to the sharpening of the LEPB band bottom and paralleltothekspaceseparationbetweentheWeylnodes
HEHB band top at (cid:126)k = 0, and a shifting of the energy (thezdirectionhere)leadstoarangeofky perpendicular
wheretheLEPBandHEPBmeetat(cid:126)k =0upwardswhen to the field in which there is no dispersion along the kz
the field is first applied. As the field is increased further, direction. This agrees with our numerical results in Fig.
the energy of the LEPB band at (cid:126)k = 0 is shifted above 8a. Whereas Ref. 19 did not explicitly study an in-plane
field magnetic field with only components perpendicu-
that of the k-space vicinity and an energy gap opens be-
lar to the Weyl node k-space separation, its authors did
tween the LEPB and HEHB. Applying a magnetic field
studyanin-planefieldappliedatanangletothek-space
in the z direction leads to the broadening of the LEPB
separation. Ourresultssupporttheirconclusionthatthe
andtheformationofanenergyplateaualongthek space
components of the in-plane field perpendicular to the k-
y direction.
space separation (B ) will lead to a shearing of the EEC
y
These changes affect the transmission from a source
along the k direction. Our EECs for a magnetic field
z
segmentwithoutamagneticfieldtoadrainsegmentwith
in the y direction in Fig. 10 shows that B pushes the
z
a magnetic field as the overlap in the k-space direction
regions of the EEC with differing signs of k in opposite
y
parallel to the interface between the source and drain
directions along the k direction. Our results show the
z
EECs varies. In theparticular case ofan interface paral-
additionalfeaturethatatlargevaluesofB ,thisshearing
z
leltotheydirectionandmagneticfieldinthezdirection,
causes the EEC to break up into separate closed curves.
thekspaceoverlapisnotverymuchaffectedbythemag-
nitude of the field. The reduction in transmission comes
insteadfromthereductioninspatialoverlapbetweenthe ACKNOWLEDGMENT
sourceanddrainwavefunctionswhenthedrainwavefunc-
tionsgetlocalizedfurtherinsidethethicknessofthefilm
The authors acknowledge the Singapore National
awayfromthesourcewavefunctionlocalizednearthefilm
Research Foundation for support under NRF Award
surface.
Nos. NRF-CRP9-2011-01andNRF-CRP12-2013-01,and
Afterthecompletionofthismanuscript,welearntofa MOE under Grant No. R263000B10112.
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