Conductance oscillations of core-shell nanowires in transversal magnetic fields

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by  Andrei Manolescu

Conductance oscillations of core-shell nanowires in transversal magnetic fields
Andrei Manolescu,1,∗ George Alexandru Nemnes,2,3 Anna Sitek,4,5
Tomas Orn Rosdahl,6 Sigurdur Ingi Erlingsson,1 and Vidar Gudmundsson4
1School of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, Iceland
2University of Bucharest, Faculty of Physics, MDEO Research Center, 077125 Magurele-Ilfov, Romania
3Horia Hulubei National Institute for Physics and Nuclear Engineering, 077126 Magurele-Ilfov, Romania
4Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
6 5Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,
1
Wroclaw University of Technology, 50-370 Wroclaw, Poland
0 6Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
2
We analyze theoretically electronic transport through a core-shell nanowire in the presence of
y
a transversal magnetic field. We calculate the conductance for a variable coupling between the
a
M nanowire and the attached leads and show how the snaking states, which are low-energy states
localized along the lines of vanishing radial component of the magnetic field, manifest their exis-
tence. In the strong coupling regime they induce flux periodic, Aharonov-Bohm-like, conductance
3
oscillations, which, by decreasing the coupling to the leads, evolve into well resolved peaks. The

flux periodic oscillations arise due to interference of the snaking states, which is a consequence of
]
l backscattering at either thecontacts with leads or magnetic/potential barriers in the wire.
l
a
h PACSnumbers: 73.63.Nm,71.70.Ej,73.22.Dj
-
s
e
m I. INTRODUCTION the wire axis. In particular, the field induces a com-
plex topology of the electronic states. Low-energy elec-
.
at Design and technological realization of quantum nan- trons may be found around two channels along the CSN
m odevices requires nanoscale systems of well defined and axis where the radial component of the field vanishes by
controllableproperties. Recently, tubular semiconductor changing sign. Carriers on both sides of the lines are
-
d structures turned out to be promising building blocks deflected towards opposite directions and thus confined
n of such appliances. Nanotubes of very narrow, but fi- into so calledsnaking states [19–22]. Higher energy elec-
o nite thickness may be achieved in few different ways. In trons start to occupy Landau states and form cyclotron
c
the case of quantum wires built of narrow-gap materi- orbits localized in the areas where the radial component
[
als surface states may induce Fermi level pinning above ofthefieldtakesmaximalvalues. Withincreasingenergy

2 the conduction band edge which results in accumulation electrons move towards the sample ends and form edge
v of electrons in the vicinity of the surface [1]. Nowadays states [22]. To the best of our knowledge experimental
7
it has become feasible to combine two (or even more) investigationofmagnetotransportin ballistic CSNs with
7
4 different materials into one vertical structure, i.e., core- magneticfield transversalto the nanowire,andofthe ef-
1 shell nanowires (CSNs). This provides a possibility to fectsofsnakingstates,haveonlyrecentlybeenattempted
0 achieve thin tubular shells surrounding a core nanowire [23].
. or other shells. One of the advantages of these systems In this paper we focus on thin cylindrical conductive
1
0 is a possibility to establish band alignment through the shellssinceinsuchsystemscarrierlocalizationorconduc-
6 thicknessesofthecomponents[2–4]andthusgrowstruc- tive channels are induced only by an external magnetic
1 turesinwhichelectronsareconfinedonlyinnarrowshell fieldandthus suchsamplesallowtoobservepurelymag-
v: areas [5, 6]. Moreover,the core part may be etched such netic effects. According to our recent calculations the
i that separated nanotubes are formed [7, 8]. existence of snaking states leads to resolved resonances
X
Most commonly CSNs have hexagonal cross-sections of the conductance when the CSN is weakly coupled
r [6–8], but triangular [9, 10] and circular [11] systems to external leads [24]. In the present paper we extend
a
have also been achieved. Electrons confined in prismatic these results and analyze signatures of snaking states in
CSNs may form conductive channels along the sharp the conductance for wide range of sample-lead coupling
edges [2, 12–17]. Rich quantum transport phenomena strength which can be controlled by variable potential
havebeenobservedinCSNs,e.g.,fluxperiodic,similarto barriers. Weshowthatinterferenceofthesnakingstates,
Aharonov-Bohm(AB), magnetoconductance oscillations due to backscattering from magnetic or potential barri-
[6, 18], single electron tunneling, or electron interference ers,mayleadto fluxperiodic magnetoconductanceoscil-
[6, 11]. Very interesting effects have been predicted in lations, detectable in transport experiments.
the presence of a strong magnetic field perpendicular to Thepaperisorganizedasfollows. Wedefinethemodel
of the system in Sec. II and describe the computational
method in Sec. III. The results and their discussion are
presentedinSec.IVwhilethefinalremarksarecontained
∗ [email protected] in Sec. V.
2
constrictions, at zero magnetic field [28–31]. The ap-
proach consists of two parts. In the first part the wave
function ofanelectronata givenenergyE is built, both
intheleadsandinthecentralscatteringregion(herethe
CSN),andmatchedbycontinuityconditionsatthejunc-
tions. In the second part one obtains the S matrix, the
transmission function, and the conductance.
We assume that in the leads, close to the junctions
with the CSN, the wave function can be written as a
combination of plane waves and orbital states
z) (b)
V( ψ (r)= ψin e−ikm(z−zl)+ψouteikm(z−zl) u (ϕ)σ ,
-Lz/2 -b 0 b Lz/2 l mσl mσl m | i
z Xmσh i (1)
wherel=S,Disalabelforthetwoleads,z = L /2de-
FIG. 1. (Color online) (a) The CSN is a cylindrical surface l z
∓
of radius ρ and length Lz nm (blue color). The wavy lines note the coordinatesofthe junctions, σ is the spinlabel,
with arrows indicate snaking orbits propagating along the um(ϕ) = eimϕ/√2π are the eigenvectors of the angular
axesofzeroradialmagneticfield(thinreddottedlines). The momentum, with m = 0, 1, 2,.... The wavevector km
± ±
contacts with the source and the drain electrodes are shown corresponds to the longitudinal motion of a particular
(goldcolor). (b)Gaussianpotentialbarriersofvariableheight circular mode m, being determined by the energy of the
placed along theCSN. incoming electron,
~2 m2 2Em m2
II. THE MODEL E = 2meff (cid:18)km2 + ρ2 (cid:19), km =s ~2eff − ρ2 , (2)
The model of a CSN used here is a simple cylindri- where m is the effective mass of the material. Note
eff
cal surface of radius ρ and length L , through which a that at a fixed energy, depending on m, k can be real
z m
currentcanflowfromoneendtoanotherduetoapoten- or imaginary. The real values describe open channels,
tial bias. We treat the electrons like a cylindrical two- propagating from one lead to another lead, whereas the
dimensional electron gas. We consider a magnetic field imaginaryvaluesdescribeclosed(orevanescent)channels
perpendicular to the axis ofthe cylinder. When the field whicharestatesboundaroundthescatteringregion[17].
issufficientlystronglow-energyelectronscantravelalong Although we are formally treating the leads as semi-
the snaking orbits createdalong the two lines of zero ra- infinite extensions of the CSN, with the same circular
dial component, at polar angles ϕ =π/2 and ϕ =3π/2, symmetry, in fact the wave functions (1) are important
as illustratedin Fig. 1(a). Two leadsare attachedto the only at (or close to) the boundaries z . Therefore, in
l
CSN, one at eachend of it, which are treated as particle principle, the shape of the leads can be arbitrary. In
reservoirs,without a specified shape. Conventionally,we experimental setups the leads are usually perpendicular
call them Source (S) and Drain (D). ”finger”electrodesattachedto thenanowiresample[18].
Two potential barriers are placed along the nanowire, Moreover the magnetic field has completely different ef-
at z = b, symmetrically around the center z = 0. fects in the leads and in the measured sample. This also
±
The barriers are defined as Gaussian functions V(z) = motivates us to neglect in our model the magnetic field
V exp[ ((z b)/c)2] with a width parameter c, and inthe leads. Inadditionto simplicity, this assumptionis
0
− −
height V , Fig. 1(b). The potential barriers are indepen- helpful to define the contact between the leads and the
0
dent on the polar angle ϕ. Their role is to either control CSNnotonlyasamathematicalboundary,butalsoasa
the contact strength, if placed at the contacts between magnetic barrier. Then, by using the potential barriers
the nanowire and the leads, and/or to contribute to the well inside the CSN we can also simulate new bound-
backscattering of the wave functions. aries of the scattering area, this time with a continuous
magnetic field.
In order to calculate the wave function in the CSN
region one has to find the eigenstates of the Wigner-
III. THE COMPUTATIONAL METHOD
Eisenbud (WE) Hamiltonian H˜, satisfying Neumann
boundary conditions at the points z (instead of the
l
Inordertocalculatetheconductanceoftheopencylin-
Dirichlet conditions familiar for hard wall boundaries),
der, i.e. the CSN in contact with the external leads,
we use the scattering formalism based on the R-matrix H˜χ =ǫ χ ,
a a a
method. This method has been used in similar trans-
port problems in quantum dots connected to the leads where a = 1,2,... is a generic quantum number label-
via quantum point contacts [25], in planar nanotransis- ing the WE energies ǫ in increasing order. The WE
a
tors [26, 27], or in cylindricalbulk nanowires with radial eigenstates χ a are expanded in a basis set q =
a
≡ | i | i
3
u (ϕ)u (z)σ , with u (z) = A cos[nπ(z/L + 1/2)], Having the S matrix one can calculate the transmission
m n n n z
n=0,1,2,..|.aind normalizationfactor A = 1/L and matrix between the open channels ν and ν′
0 z
A = 2/L for n>0.
n z p 2
The Hamiltonian H˜ is formally built as a regular Tνν′(E)= K1/2SK−1/2 ,
p νν′
Hamiltonian H, with the kinetic term containing the (cid:12)(cid:16) (cid:17) (cid:12)
modifiedmomentumoperatorpz+eAz =pz+eBρsinϕ, and finally the condu(cid:12)(cid:12)ctance G, by sum(cid:12)(cid:12)ming all con-
the vector potential being defined in the Landau gauge tributions from separate leads, i.e. ν = mσS and
A = (0,0,By), and the Zeeman term depending on the ν′ = m′σ′D , | i | i
| i | i
effective g-factor of the material,
e2 ∂
H =−2m~e2ffρ2∂∂ϕ22 + (pz +2eBmρeffsinϕ)2 − 12geffµBBσx . G= h Z dE(cid:18)−∂FE(cid:19)mXmσσ′TmσS,m′σ′D(E) , (6)
With Neumann boundary conditions, implemented via where denotes the Fermi function.
F
the cosine functions of the basis q , the resulting lin-
| i
ear term in p is not Hermitean. Therefore the ma-
z
trix elements of the WE Hamiltonian are defined as IV. RESULTS AND DISCUSSION
H˜qq′ = Hqq′ +Hq∗′q /2. This procedure is equivalent
to correctingthe momentum pz with a surfaceBlochop- Inthe numericalcalculationswe usedmaterialparam-
oertahteorraLu(cid:0)t=ho−rsi~[3[δ2(,z3−3](cid:1).zD)−δ(z−zS)]/2 as discussed by edtiuerssooffthIneACsS,Nmewffas=fix0e.0d23toaρnd=g3e0ffn=m−a1n4d.9le.ngTthhewraas-
The wave function in the CSN can be written as a L = 300 or 2000 nm. The results were convergent in
z
superposition of WE eigenstates, a basis q truncated to orbital momenta with m 10
| i | | ≤
andlongitudinalmodesn 130. Allchannels,bothopen
ψ(r,E)= αa(E)χa(r), (3) andclosed,wereusedto c≤alculatethe R andS matrices.
a The temperature was fixed to T =0.5 K.
X
and the coefficients α are determined by the continuity
a
conditionsofthewavefunctionsandtheirfirstderivatives
A. Zero magnetic field
at the z boundaries. Detailed calculations can be found
l
in Appendix A of Ref. [26]. By introducing one more
First we show in Fig. 2(a) the conductance at zero
composite label ν = mσl the amplitudes of the wave
| i | i magnetic field as a function of the chemical potential µ
function in the leads, Eq. (1), can be seen as the vectors
ψin ψin and ψout ψout , which are related via for several heights of the potential barriers. In this case
≡ { ν } ≡ { ν } L =300 nm, and the barriers are situated very close to
the continuity conditions, as z
thecontacts,atb= 147.5nm,havingawidthofc=2.5
±
ψin+ψout = iRK ψin ψout . (4) nm. The results are as expected for ballistic transport
− −
in a quantum wire. Without the barriers (V = 0) the
0
In Eq. (4) we have introduced(cid:0)two matrice(cid:1)s, the matrix conductance has the familiar steps given by the num-
of wavevectors with elements Kνν′ = kmδνν′, and the ber of open channels, which is the number of m values
so-called R matrix defined as yielding a real wavevector k for the energy E = µ in
m
Eq.(2),multipliedbythetwospinstatesσ = 1. Byin-
~2 ν a ν′ a † ±
creasingtheheightofthebarrierstheconductancedrops,
Rνν′(E)= h | ih | i . (5)
−2meff E ǫa because the energy of the electronic states within the
a −
X CSN increases,and thus a smallernumber ofopen chan-
The notation ν a stands for the scalar product of the nels remains available up to the fixed Fermi level. Also
h | i
orbitalandspinstatesincorporatedineachfactor,atthe the conductance begins to oscillate, as a results of the
two frontiers zl, i.e. (Fabry-Perot) interference between transmitted and re-
flected waves,and evolve towards resonances with peaks
2π
ν a = σ u∗ (ϕ)χ (ϕ,z )dϕ . indicating the density of states in the scattering region.
h | i h | m a l This is a consequence of the fact that the coupling be-
Z0
tween the scattering region and the leads decreases.
The scattering problem is solved by calculating the S
matrix,whichtransformsthe ”in”states in”out”states,
B. Magnetic barriers with adjustable contacts
ψout =Sψin,
and using Eq. (4) it is obtained as Another way to create backscattering of the electrons
at contacts is to consider a magnetic barrier, i.e. a
S = (1 iRK)−1(1+iRK) . magnetic field that exists only in the CSN, but not in
− −
4
tions, but the fine structure of the flux periodic oscilla-
10 V 00 (meV) (a) B = 0 tions remains visible. The resonances at high potential
8 5 barriers are now shifted to higher magnetic fields.
h) 10
2G (e/ 46 122505
30
2 C. WE Energy spectra and wave functions
0
0 2 4 6µ (meV) 8 10 12 14
WecangainmoreunderstandingoftheABoscillations
6.0 (b) µ=3 meV by looking at the WE energy spectra vs magnetic field
5.0 shown in Fig. 3. In the case of strong coupling between
2G (e/h) 234...000 ftohremCbSrNaiadnsdhtahpeelpeaadttse,ren.sg.foVr0B=>0,1thTe,lFowig.e3n(ear)g.yTlehveesles
oscillations affect the conductance, Eq. (6), through the
1.0
denominatorsofthetheR-matrix,Eq.(5),whicharesen-
0.0
0 1 2 3 4 5 sitive to such small changes in the WE energies ǫ , and
B(T) a
6.0 (c) µ=5 meV induce the AB oscillations. In fact, the R-matrix has a
5.0 form similar to the Green functions used by other au-
h) 4.0 thorsforsuchscattering-transportcalculations[34]. The
2e/ 3.0 braids are an indication of snaking states interference in
G ( 2.0 theopenCSN.Weverifiedthatsuchanenergyspectrum
1.0
is also obtained for a CSN with a finite thickness of 10
0.0
0 1 2 3 4 5 nm, by including in the basis q radial wave functions
B (T) | i
vanishing at the surfaces.
FIG.2. (Coloronline)(a)Conductancevs. chemicalpotential In the presence of high potential barriers the braids
µ without magnetic field, for contact barriers of height V0 = shrink and converge towards nearly double degenerate
0,5,...,30 meV,and thenvs. magnetic fieldperpendicularto
eigenstates, Fig. 3(b), as obtained for the isolated cylin-
the nanowire, with (b) µ = 3 meV and (c) µ = 5 meV. The
der [24]. The WE spectra can also explain the transi-
nanowire length is Lz =300 nm and thebarriers are close to
thecontacts, at b=±147.5 nm, of width c=2.5 nm. tion from flux-periodic magnetoconductance to resonant
peaks,observedwhiletheheightofthepotentialbarriers
is being increased. These peaks occur at those magnetic
fields for whichthe snakingstates arecrossingthe Fermi
the leads, as we assumed in Section III. In addition, by
energy.
adding the potential barriers we can modify the contact
Further information on the snaking states can be ob-
strength. InFig. 2(b), we showhowthe conductancede-
tained from the wave functions. In the scattering region
pends on the strengths of the transverse magnetic field
they are obtained with Eq. (3), using the coefficients α
andon the coupling to the leads for chemicalpotential 3 a
expressed with the S matrix [28],
meV. We distinguish two regimes, corresponding to low
and high potential barriers, respectively. For weak po-
tentials, i.e. V0 up to 5 meV, regular conductance oscil- αa(E)= ~2 i 1 ν a ∗km 1 Sνν′ ,
lationsareobtained,withperiods∆B slightlyincreasing 2meff √2πE−ǫa ν h | i − ν′ !
from 0.20 to 0.22 T in the interval B = 2–4 T. Each of X X
theseoscillationsnearlycorrespondstoagainofoneflux where by summing over all labels ν we consider elec-
unitΦ =h/ethroughtheareaofthecylinderprojected trons incoming from both leads, with wave functions of
0
onthe yz plane, A=2ρL . Accordingtothis estimation equalamplitudes. Nevertheless,since the wavefunctions
z
the period should be ∆B = Φ /A = 0.23 T. Therefore themselves are not directly involved in the conductance
0
these oscillations can be considered a kind of AB inter- calculations, we look at them only to correlate the be-
ference ofsnakingstates propagatingonthe lateralsides haviorofthe snakingstates inthe scatteringregionwith
of the cylinder. the WE spectrum and the conductance oscillations.
By increasing the height of the potential barriers the InFig.4weshowtwodistinctsituations,forafixeden-
AB-like oscillations smear out and the broader conduc- ergyE =3 meV, which is the chemicalpotentialused in
tance peaks emerge. These peaks are produced by the Fig.2(b), andnopotentialbarrier,V =0. ForB =2.91
0
same snaking states, but now as individual resonances T the pair of snaking states have in-phase longitudinal
occurring in the nearly isolated CSN, only weakly con- oscillations, Fig. 4(a), corresponding to a crossing of the
nected to the leads. This case was described by Rosdahl braided WE energies, Fig. 3(a), and to a conductance
et al. by modeling the contacts with a tunneling param- minimum, Fig. 2(b). For B = 3.02 T the two snaking
eter [24]. states have out-of-phase longitudinal oscillations, Fig.
InFig.2(c)weshowthemagnetoconductancewiththe 4(b). Inthiscasethebraidsaremaximallyopen,andthe
chemicalpotential increasedto 5 meV. The contribution conductancehasamaximum. Theout-of-phasestructure
of higher energy levels results in more complex fluctua- of the snaking states does not exist in the isolated CSN,
5
8.0
7.0 6.0 V0 = 0
meV) 56..00 5.0 V0 = 7 meV
Energy ( 1234....0000 (a) V0=0 2G (e/h) 34..00
0.0 2.0
0 1 2 3 4 5
8.0
1.0
7.0
V) 6.0 0.0
me 5.0 0.0 0.5 1.0 1.5 2.0 2.5
y ( 4.0 B (T)
g
er 3.0
En 2.0 FIG. 5. (Color online) Conductance vs. magnetic field for
1.0 (b) V0=20 meV a nanowire of length Lz = 2000 nm. The upper curve was
0.0 0 1 2 3 4 5 obtained without potential barriers (V0 = 0), and the lower
B(T) curvewithtwobarrierssituatedatb=±150nm,asindicated
FIG. 3. (Color online) The Wigner-Eisenbud (WE) energies in Fig. 1(b), of height V0 = 7 meV, and width parameter
ǫa (a) without contact barriers and (b) with barriers of V0 = c=20 nm. The chemical potential is µ=6.5 meV.
20 meV. The blue dotted horizontal lines show the chemical
potentials used inthemagnetoconductance calculations, µ=
3 meV and µ=5 meV. periodisinverselyproportionaltothewirelength. Inthe
absence of any potential barrier the computed magneto-
2.0 conductance has smooth steps as shown in Fig. 5. The
1.5 steps reflect the number of propagating (open) channels
ϕ / π 1.0 (a) associated to the complex subband structure of the en-
0.5 ergy spectra of an infinite hollow cylinder in transverse
magnetic field [19, 20, 22, 34]. A detailed analysis of
0.0
-100 -50 0 50 100
these steps is nevertheless beyondthe aim of the present
2.0
paper.
1.5
With potential barriers situated at 150 nm from the
ϕ / π 1.0 (b) ±
centerofthe nanowirewecanobtainagainthe flux peri-
0.5
odicoscillations,comparabletotheonesforthenanowire
0.0 of 300 nm length. The oscillations are now less regular
-100 -50 0 50 100
z (nm) than before, but with an average period of 0.20-0.22 T
FIG.4. (Color online) Probability densitiescorrespondingto withintheinterval0.4-2.4T.Inordertoreducethetrans-
the wave functions within the CSN, at energy E = 3 meV parencyofthebarriersandtoincreasethebackscattering
and V0 = 0, for (a) B = 2.91 T when the snaking states are weusedwiderbarriersthanbefore,withc=20nm. The
in phase, and (b) B = 3.02 T when they are in anti-phase.
lengthofthescatteringzoneinbetweenthepotentialbar-
Here we show the open cylindrical surface, the vertical axis
riers is less sharply defined than in the case of the pure
beingthepolarangleandthehorizontalaxisthelongitudinal
magneticbarriers,andsoistherealareaofthemagnetic
z coordinate.
flux,comparedtothereferencecrosssectionareabetween
the barriers, A = 4ρb. Nevertheless, regardless of these
imperfections,theoscillationscorrespondreasonablywell
but only the in-phase one [24]. Here, for the open CSN,
with the expected periodicity ∆B =Φ /A=0.23 T.
by imposingthe potentialbarriersthe lateralshiftofthe 0
out-of-phase maxima gradually reduces, until they align
like in Fig. 4(a). Seen from this angle the braids of the
WE spectrum and the AB oscillations are related to the
relative phase of the snaking states.
V. CONCLUSIONS
D. Potential barriers on a long wire
In conclusion we predict that the existence of the
snaking states in a CSN in a transversal magnetic field
In the next example we chose a much longer CSN, of can be experimentally observed as flux periodic oscil-
lengthL =2000nm. Themagneticfieldstillvanishesin lations of the magnetoconductance, with a short CSN
z
the leads, and so there are magnetic barriers at the con- stronglycoupledto leads. Inthis casethe snakingstates
tacts,whichcausebackscatteringandhenceinterference. behave like transmitted and reflected waves which inter-
However,the corresponding conductance oscillations are fere at the contacts with the leads. In the limit of weak
nowtoodenseandtooweaktoberesolved,becausetheir couplingthesnakingstatescanbeseenasindividualres-
6
onances of the conductance. In our model the contacts ACKNOWLEDGMENTS
are primarily simulated by matching two different types
of wave functions, in the presence of a magnetic barrier
resulting from neglecting the magnetic field in the leads.
Inordertofurthermodifythetransmissionandreflection
at the contacts we included potential barriers, which re-
duced the coupling CSN-leads, and thus the amplitude This work was financially supported by the research
of the transmitted waves. Another way to observe the funds of Reykjavik University and of the University of
flux periodic oscillations in a transverse magnetic field, Iceland, and by the Icelandic Research Fund. T. O.
although possibly less regular, may be by creating scat- Rosdahl acknowledges support from an ERC Starting
tering regions with potential barriers, as produced by Grant. We are thankful to Thomas Scha¨pers and Se-
finger gates placed over a long nanowire [18]. bastian Heedt for very interesting discussions [23].
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