Magnetoconductance oscillations in quasiballistic multimode nanowires

0.65 MB·

by  Yaroslav Tserkovnyak

Magnetoconductance oscillations in quasiballistic multimode nanowires
Yaroslav Tserkovnyak
Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, USA and
Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
7
Bertrand I. Halperin
0
Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
0
(Dated: February 6, 2008)
2
n Wecalculate theconductanceof quasi-one-dimensional nanowires with electronic states confined
a to a surface charge layer, in the presence of a uniform magnetic field. Two-terminal magnetocon-
J ductance (MC) between two leads deposited on the nanowire via tunnel barriers is dominated by
6 density-of-states(DOS)singularities, whentheleadsarewellapart. Thereisalsoamesoscopiccor-
1 rection due to a higher-order coherent tunneling between the leads for small lead separation. The
correspondingMCstructuredependsontheinterferencebetweenelectronpropagation viadifferent

] channelsconnectingtheleads,whichinthesimplestcase,forthemagneticfieldalongthewireaxis,
l can be crudely characterized by relative winding numbers of paths enclosing the magnetic flux. In
l
a general,theMCoscillations areaperiodic, duetotheZeemansplitting, fieldmisalignment withthe
h
wire axis, and a finite extent of electron distribution across the wire cross section, and are affected
-
s byspin-orbitcoupling. Thequantum-interferenceMCtracescontainawealth ofinformation about
e the electronic structure of multichannel wires, which would be complimentary to the DOS mea-
m surements. We propose a four-terminal configuration to enhance the relative contribution of the
higher-order tunneling processes and apply our results to realistic InAs nanowires carrying several
.
t quantumchannels in the surface charge-accumulation layer.
a
m
PACSnumbers: 73.63.Nm,73.23.Ad,73.50.Jt
-
d
n
I. INTRODUCTION a quasi-1Dwire with a finite cross section penetrated by
o
a magneticfield. While a significantvolume ofthe exist-
c
[ ing literature is devoted to the MC studies of disordered
Magnetoconductance (MC) of very small conductors,
wires (see, e.g., Refs. 3 and 4), the interference effects
2 e.g., quantumdots, wires, andrings,is a topic of intense
in magnetotransport of the quasiballistic nanowires did
v
research since the early days of mesoscopic physics. In
not attract the same attention (see, however, Refs. 5 for
8
the simple case of phase-coherent conductors topologi-
6 a discussion of weak-localization effects in narrow two-
cally identical to the ring, one can generally distinguish
4 dimensional wires, which are ballistic across the wire
8 between two classes of MC oscillations, differing in fre- width). Recent advances in nanofabrication techniques,
0 quencybythefactorof2. Firstofall,theconductanceos- however, make it possible to experimentally probe the
6 cillateswithaperiodofh/einmagneticflux(Aharonov- MC of nearly ballistic semiconducting nanowires,6 rais-
0 Bohm effect), since the magnetic field through the loop
ing questions about the conductance oscillations related
/
t canberemovedbyagaugetransformationinintegermul- to the finite separation of the leads. See also Ref. 7,
a
tiples of the flux quantum h/e. These oscillations are
m where ballistic carbon nanotubes were used to build a
sensitive to the disorder distribution, however, so that
Fabry-P´erotelectronresonator,andRefs.8,reportingan
- averaging over various disorder configurations in nom-
d observationand theory of the interference effects in tun-
n inally identical conductors, or self-averaging in larger neling between parallel ballistic quantum wires through
o samples, reduces the strength of the oscillations. The a finite-length barrier.
c quantum weak-localization (or antilocalization) correc-
In this paper, we are studying theoretically the mag-
v: tion with a period of h/2e survives the averaging in dif- netotransportinsinglequasiballisticnanowires. Suppose
i fusive samples.1 Depending on the concentration of the
X the wirecarriesmorethanonequantumchannel,sothat
disorder in a given sample, either h/e- or h/2e-type os-
the electron propagation between the injecting and de-
r cillations can be more pronounced. A given realization
a tecting leads can interfere giving rise to conductance os-
of the disordered phase-coherent mesoscopic conductor,
cillationsinthemagneticfield,whichdependonthelead
such as an open quantum dot or quasi-one-dimensional
separation length. If the electron density is confined to
(1D) wire, also exhibits universal conductance fluctua-
a narrow surface layer and the Zeeman energy is dis-
tions (UCF) with amplitude e2/h, which vanish on
sample averaging.2 The weak-∼localization correction re- regarded, the MC in the parallel magnetic field can be
decomposedintoharmonicswithfrequenciesinmultiples
lated to h/2e MC oscillations in rings is usually mani-
of h/e approximately corresponding to relative winding
fested as a smooth background with the superimposed
numbers of various electron paths around the wire axis
aperiodic UCF oscillations.
between the two leads. Generally, different transverse
Alloftheaforementionedphenomenacanplayarolein modeswilleffectivelyhavedifferentcrosssectionsforac-
2
cumulating the longitudinal magnetic flux resulting in lead2ordisappearinthe reservoirsAandB attheends
aperiodic interference patterns. The aperiodicity is fur- of the wire. If the reservoirs are floating, an electron
ther enhanced by Zeeman energy and, furthermore, the that enters the reservoirwill eventually reenter the wire,
spin degrees of freedom affect transport nontrivially due butwillhavelostallcoherencewiththeoriginalinjection
to the spin-orbit coupling. In addition, strong magnetic process. Themagneticfieldappliedeitherparallelorper-
fieldscanhaveasignificanteffectontheone-dimensional pendiculartothewireaffectsinterferencebetweendiffer-
band structure of a long wire, leading to van Hove mag- ent electron trajectories connecting the contacts, which
netofingerprints in the conductance. Varying the direc- accumulate the phase by enclosing a magnetic flux, re-
tion of the magnetic field with respect to the wire axis sulting in conductance oscillations. In addition, the field
gives an additional degree of freedom for probing elec- can modify the number of quantum channels N at the
tronicstructure. Forexample,alargeperpendicularfield Fermi level resulting in van Hove, i.e., density-of-states
eventually leads to a formation of Landau levels having (DOS), singularities.
a profound effect on the one-dimensional conductance. We develop a simple formalism to capture interfer-
Among the principal practical questions are how the ence between electrons injected into different transverse
MC traces can be used to extract information about modes. Wave-function decomposition at the injecting
electronic states in the transverse direction (such as th leadisdeterminedbytheangulardistributionaroundthe
e total number of modes and charge distribution, e.g., wire circumference of the tunneling amplitude formedat
surface-confined or nearly uniform etc.) and about var- the contact. (In a specific model discussed below, we
ious scattering mechanisms (involving in general elastic, take the tunneling strength to be uniform within an an-
phase-relaxation, and spin-orbit processes). While giv- gle2θ determinedbytheleaddeposition[seeFig.1(b)].)
0
ingcomprehensiveanswerstothesequestionsliesbeyond For simplicity, we assume that the contacts are narrow
the scope of this paper, our analysis can be used to con- along the wire in comparison to their separation L and
sider them in some simple scenarios as well as to lay the to the electron wavelength, and we focus on MC oscilla-
grounds for more systematic studies. tionsduetoafinitelengthL. Similarly,theelectronsare
This work was motivated, in large part, by mag- detected by the drain lead whose coupling to different
netoconductance studies6 of cylindrical conducting transverse modes is determined by the deposition. We
nanowires,9 which provide a fascinating medium for will focus on the MC oscillations due to the interference
mesoscopic physics4,10,11 as well as show a great of phases accumulated during quasiballistic propagation
promise for applications as building blocks of nanowire between the leads. In particular, we will disregard any
networks.9,12 After a general theoretical discussion in systematiccontributionsduetoweak-localization(oran-
Sec. II, in Sec. III we treat a specific case of trans- tilocalization) physics, restricting our attention to oscil-
port in a surface charge-accumulation layer of cylin- lation features on scales determined by relative phases
drical InAs nanowires as those studied in Ref. 6, us- accumulated for multimode ballistic propagation along
ing realistic parameters extracted from photoelectron the length L. This physical picture is made mathemati-
spectroscopy13 and magnetotransport measurements14 cally concrete in the following.
on two-dimensional electron gases at InAs surfaces. Our
approach may, however, also be useful for other types
of conducting nanowires, such as state-of-the-art semi- A. General considerations
conductoraxialheterostructureswithradiallymodulated
compositionanddoping.15 Concludingremarksaregiven
A schematic of our model is shown in Fig. 1. A long
in Sec. IV.
wire of length L and radius r is contacted via tunnel
w 0
barriers by two metallic leads 1 and 2 separated by dis-
tance L and deposited on top of the wire [see Fig. 1(b)].
II. THEORY The total capacitance of the wire and/or the attached
reservoirs is assumed large enough so that Coulomb-
Consideralong1Dwireconnectedthroughtunnelbar- blockade effects can be neglected. However, L may be
riersto two narrowmetallic sourceanddrainleadssepa- smaller than the coherencelength, and we are interested
rated by distance L, as shown in Fig. 1. In addition, the in interference effects for electron propagation between
wire may be attached to two sinks (reservoirs)A and B, contacts 1 and 2.
attheends,whosepurposewillbecomeclearlater. Disre- Electronflowin the wire is carriedby a surfacecharge
garding electron-electron interactions, the wire supports layer. The disorder (due to impurities or wire-shape im-
N transverse modes, the value of its two-terminal con- perfections) is assumed to be weak enough, so that the
ductance being therefore bounded by Ne2/h, at a fixed mean free path λ > 2πr and it makes sense to define
0
magnetic field (supposing the sinks at the ends are elec- quasi-1D bands along the wire. If, furthermore, the lead
tricallyfloating,forthemoment). Atafinitevoltagebias, separation L is not much longer than λ, some electrons
electrons are injected from the left lead 1 into the wire can propagate ballistically between the leads, giving rise
where they decompose into different transverse modes, to a magnetoresistancestructure discussedin the follow-
are carried along the wire, and are detected by the right ing.
3
(b) 1, 2 T ), where
2θ li
0
(cid:0)(cid:1)r
(cid:0)(cid:1) 0 v =∂ ǫ (k) (2)
(a) (cid:0)(cid:1) gi k i
(cid:0)(cid:1)
A 1 2 (cid:0)(cid:1) B
L is the Fermi-point group velocity corresponding to the
ith mode dispersion ǫ (k) (recall that the DOS is pro-
i
r0 portionalto the inversegroupvelocity1/vgi in 1D). The
tunneling conductance thus exhibits van Hove singulari-
(cid:0)(cid:0)(cid:1)(cid:1)Bx tieswhentheFermilevelcrossesabandedge. Ifelectrons
(cid:0)(cid:1)
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) Bz arenarrowlyconfinednearthe surfaceandarerestricted
(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) to the lowestband in the radialdirection, tunneling am-
plitudes will be approximately the same for different 1D
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)L(cid:0)(cid:1)(cid:0)(cid:1)w(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) bands and thus
FIG. 1: Schematic of the model [view parallel to the wire G = T 2 v−1. (3)
(a) and of its cross section (b)]: A long wire of length L is l | l| gi
w
i
contactedviatunnelbarrierbytwoleads1and2separatedby X
distanceL. Inaddition,thewireisØhmicallyattachedatthe In this simple model, the two-terminal conductance G
12
ends to two “sinks” A and B. Leads 1 and 2 can be voltage
given by Eq. (1) thus exhibits van Hove singularities
biasedandAandBcanbeeithergroundedorleftelectrically
whosemagneticfingerprintscouldbeausefultoolforin-
floating. Sinks A and B are assumed to form good contacts
vestigatingtheelectronicstructureofthenanowires. G
withthewireandareusedtoeitherdecohere(whenfloating) L
can also depend on the magnetic field, due to the weak-
or sink (when grounded) incoming electrons, as explained in
thetext. localizationcorrectionto semiclassicaldiffusion, but this
willbedisregardedassumingthewireissufficientlyclean.
When the lead separation is decreased and becomes
We can distinguish between several regimes based on comparable to the mean-free path λ, a mesoscopic cor-
therelationbetweenthecontactseparationL,mean-free rectiontotheconductanceδG ,whichishigherorderin
12
path λ, andthe dephasing lengthlφ. Let us assume that tunneling, may also contribute to the magnetoresistance
lφ ≫λ, which can be achievedat low temperatures, and traces for not too opaque tunnel barriers. δG12 is deter-
suppose the sinks A and B are floating. If L lφ minedbyacoherenttunneling betweenthe leads,soitis
≫ ≫
λ, the two-terminal conductance G12 is determined by fourth order in tunneling amplitude at the contacts and
summingthreeresistancesinseries(unaffectedbyAand quadratic in the electron propagator in the wire,
B),
2
G−1 =G−1+G−1+G−1, (1)
12 1 2 L δG M∗ M (L,E ) , (4)
12 ∝ (cid:12) 2,iµ 1,jνGij F (cid:12)
whereGl isthetunnelingconductanceofthelthleadand Xµν (cid:12)(cid:12)Xij (cid:12)(cid:12)
G 1/Listhediffusiveconductanceofthewiresection (cid:12) (cid:12)
beLtw∝een the leads. G can be experimentally obtained in terms of the reta(cid:12)(cid:12)rded Green’s function (cid:12)(cid:12)
1
by measuring the two-terminal conductance between 1 ∞
and A (while keeping the other contacts floating) and (L,E )= dteiEFt Ψ (L,t),Ψ†(0,0) (5)
Gij F h{ i j }i
G2—between 2 and B, for example, if the conductances Z0
of the sinks are large, G ,G G ,G . The diffusive
1 2 A B for quasi-1D electron propagation between leads 1
≪
contributionsfromthewirecanbeaccountedforbymak-
(source) and 2 (drain) at the Fermi energy E . (We
F
ingadditionaltwo-terminalmeasurementsbetween1and set ~ = 1.) M is the tunneling matrix element into
l,iµ
Band2andA,iftheintercontactseparationsareknown.
the ith channel in the wire from the lead-l state labeled
(Weare,however,interestedinthecleanlimit,wherethe
by µ at the Fermi energy. If the interband scattering in
diffusive resistance of the wire is small.) If l L λ,
φ the wire is disregarded, Eq. (4) becomes
≫ ≫
Eq.(1)isstillvalidbutthediffusiveconductanceG now
L
acquires mesoscopic fluctuations that can be modulated 2
by a nearby gate or the magnetic field. We will focus δG M∗ M (L,E ) . (6)
toinonthbeetrwegeiemnetlhφe≫conλta&ctLs,iswphheraesethceoheelercetnrtonanpdronpeaagraly- 12 ∝Xµν (cid:12)(cid:12)(cid:12)Xi 2,iµ 1,iνGi F (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
ballistic. Atlowbiasandvani(cid:12)shingtemperature,allqu(cid:12)antitiesen-
Tothelowestorderintunneling,theleadconductances teringEqs.(4)and(6)areevaluatedattheFermienergy
G ,l=1,2,aredeterminedbythedensity-of-statesofthe since we are considering the elastic contribution to the
l
quasi-1D bands in the wire and the respective tunneling tunneling. Although we willassume this inthe following
amplitudes T . For the ith 1D mode in the wire, lead-l forsimplicity,thegeneralizationtothefinitetemperature
li
tunneling conductance is G = T 2/v (with all the and bias is straightforward for noninteracting electrons.
li li gi
| |
trivial prefactors including the lead DOS lumped into (The temperature must, however,be at leastlargerthan
4
the level spacing in the wire determined by L , since pression
w
we do not intend to analyze individual levels.) An addi-
e2
tionalinelasticcontributiontothehigher-ordertunneling G = t 2 (8)
w ij
(when one electron is injected into the wire but a differ- h | |
ij
entoneisextracted,leavingbehindanelectron-holepair X
excitation) may become important at finite temperature for the conductance of a wire portion connecting two
and/or bias,16 but is disregarded here since it does not semi-infinite clean 1D reservoirs, using Kubo formula.
contribute to the discussed interference structure. For Heretij isthetransmission-scatteringamplitudebetween
simplicity, we have assumed narrow leads, so that the modes i and j. The transmission coefficient through a
propagator (5) is evaluated for a fixed distance L. The clean region ti vgi i, for example, does not show any
∝ G
presence of sinks A and B at the wire ends decoheres DOS singularities; in particular, t2 = 1 independently
| |
electronsreflectedfromtheendsandwecanneglecttheir of the density for propagating modes, giving the cele-
contribution to the propagator (5). brateduniversalcontributiontotheconductanceofe2/h
Iftheincoherentcontributionto tunnelingdetermined per mode. The two-terminal conductance we are calcu-
by Eq. (1) is much larger than the correction (4) given lating cannot, in principle, exceed the conductance (8)
byhigher-ordertunnelingprocesses,themagnetoconduc- of the wire itself. The van Hove singularitiesobtained in
tance will be dominated by the van Hove singularities, the tunneling approximationare, in fact, artifacts of the
accordingtoEq.(3)fortheDOS-dependentcontactcon- perturbative expansion in the tunneling amplitude, and
ductances. It is however possible to suppress the inco- mustsaturateatavaluebelowthewireconductance(8).
herent contribution to the signal by grounding sinks A NoteincidentallythatinthediffuselimitGw corresponds
andB atthe ends ofthe wireandmeasuringthe current to GL introduced in Eq. (1).
in a grounded contact 1 or 2, while the other contact is
voltage biased. The incoherent part of the conductance
B. Cylindrical nanowires
correspondingtothemeasuredcurrentwillthenbegiven
by
As a specific example, let us now consider a ballistic
G G G G cylindrical wire with radius r . Because of the transla-
1 2 1 2 0
G = , (7)
12 G +G +G +G ≈ G +G tional invariance and cylindrical symmetry of the wire,
1 2 A B A B
itisconvenienttoexpandtheone-electronwavefunction
neglecting for simplicity the diffusive contribution of the as
wire and thus treating it as a disordered low-resistance
einθ
node connecting four terminals. The approximation in ϕˆ(r,θ,z)=eikz ϕˆ (r), (9)
n
√r
the second equality is made assuming the leads 1 and n
X
2 to be much more resistive than the sinks: G ,G
G ,G . The coherent contribution (4), on th1e o2th≪er withafixedmomentumk alongthe wireaxisz. ϕˆn(r) is
A B the radial-position-dependentspinorwithagivenorbital
hand, is little affected by grounding the sinks since it is
angular momentum n along the wire. Assuming a uni-
governed by the coherent electron propagation between
form magnetic field, we can express the translationally
the leads 1 and 2. Note that in the approximation of
invariant Hamiltonian H according to
Eq. (7), G is proportional to the product G G which
12 1 2
is of the same order in the tunneling amplitude as the einθ
coherent correction δG12 given by Eq. (4). Hϕˆ=eikz √r Hˆn,n′ϕˆn′, (10)
Inthe restofthepaper,wewillmainlyfocusoncalcu- Xn Xn′
latingthe mesoscopicconductance(6), whichis sensitive
to electron interference on traversing the distance L be- ionrbtietramlsanogfutlhaer2m×om2emnatatrinceasnHdˆnn,n′.′ iFnosrpainqsupaasci-e1mDixwiinrge
tweentheleads. Tothatend,wewillstudytheelectronic
with the electron radial-confinement potential V(r), we
structure in the long 1D wire as a function of parallel or
obtain in a convenient gauge (A = B r/2 and A =
perpendicular magnetic field for certain simple models. θ z z
B y)
In a clean 1D wire, (L,E) v−1, the inverse group x
Gi ∝ gi
velocity, so that the conductance (6) also appears sen- 2mr2Hˆn,n′ =[−r2∂r2+(kr)2+(n+φz)2−1/4+U(r)
sitive to van Hove singularities. As we discuss in the +g(m/me)σˆ φ]δn,n′ +2ikrφx(δn+1,n′ δn−1,n′)
following, however, disorder contributes to an exponen- · −
tialdecayofthepropagatorGi(L,E)nearthebandedge, −φ2x(δn+2,n′+δn−2,n′ −2δnn′)−mrαR∂rV(r)
which suppresses DOS singularities in the conductance [ikr(σˆ+δn+1,n′ σˆ−δn−1,n′)+2(n+φz)σˆzδn,n′
× −
(4), making the interference effects more pronounced in
+2φxσˆxδn,n′ φx(σˆ+δn+2,n′ +σˆ−δn−2,n′)] , (11)
the magnetoconductance traces. −
It is interesting to compare Eq. (4) to the result ob- where U(r)=2mr2V(r) is a dimensionless confining po-
tained for a different physical situation by Fisher and tential, φ (r) = eB r2/2 is the respective magnetic-
x,z x,z
Lee.17 These authors derive the Landauer-Bu¨ttiker ex- field flux (in units of h/e) per area πr2 [thus note the
5
implied r dependence of φ in Eq.(11)], e is the elec- 15
x,z
−
tron charge, m is the free-electron mass, m is the effec-
e ρ(r)r
tive electron mass, g is the g-factor, σˆ = (σˆx,σˆy,σˆz) is 10
a vector of Pauli matrices, σˆ = σˆ iσˆ , and B is
± x y z(x) n=±1
±
themagnetic-fieldcomponentparallel(perpendicular)to
5
the wire. Notice that the perpendicular field B mixes
x n=0
angular-momentum states. The term proportional to
α in Eq. (11) is due to the Rashba spin-orbit cou- 0
R
pling in cylindrically-symmetric confinement U(r) gov-
V(r)
erned by the material-specific coefficient α . Rashba in-
R -5
teraction is believed to dominate spin-orbit coupling in
InAs surfacecharge-accumulationlayersdue to a narrow r₀
gapandstrongconfinementfields.14,18,19,20Theeffective- -10
mass approximation for the one-electron Hamiltonian 0 0.2 0.4 0.6 0.8 1 1.2 1.4
r/r
e
(11) breaks down near the wire surface, where the elec-
tronconfinementU(r)issharp. Consequently,anappro-
priate boundary condition for the wave function has to FIG. 2: Electron-density distribution in the radial direction
be appliedatthe surface(here, weassume ϕˆ =0atthe at zero magnetic field and vanishing spin-orbit interaction.
n
boundary), while ∂ U(r) defining the spin-orbit interac- The solid line is the total density ρ(r) multiplied by r (in
r
tion is smooth within the electron-gas confinement. arbitraryunits)andthedashedlinesarerespectivecontribu-
tions from then=0 and n=±1 orbital angular-momentum
Depending on the confining potential V(r) and the
states. The Fermi energy E is chosen slightly below the
wireradius,theradialelectron-densitydistributioninthe F
n = ±2 band edge. The confining potential V(r) ∝ −r3/2
wire can be either peaked near the boundary or spread
is flat at the origin and has the slope β near the hard-wall
over the entire wire cross section. Figure 2 shows the
cutoff at r=r0. The arrow shows the radius containing half
zero-field electron-density profile within the wire, calcu-
of the electrons. The specific choice of the parameters E ,
F
lated using parameters relevant to InAs nanowires (but β, and r0 is discussed in the beginning of Sec. III. re is a
for the purpose of demonstration setting αR = 0). The somewhatarbitrarilychosen“effectiveradius”oftheelectron
n = 0 mode has some finite density ρ at the center of distribution (see Sec.III for more details).
the wire [note that we are plotting ρ(r)r in the figure],
while the finite-n modes do not contribute at the center
and are pushed towards the boundary by the n2/2mr2 Fermi velocity v of the two-dimensional (2D) electron
F
potential in the cylindrical coordinates [see Eq. (11]. gas corresponding to the surface charge layer.
The disorder-averaged Green’s function for the ith
Since we base the rest of our discussion on the form
mode isreadilyobtainedafterwecalculateits dispersion
(12) of the disorder-averagedretarded Green’s function,
ǫ (k) from Eq. (11) (assuming weak disorder),
i whichdeterminestheconductanceδG throughEq.(6),
12
∞ dkeikL it is important to summarize our physical assumptions
i(L,EF)= underlying these approximations. In Eq. (6), we only
G E ǫ (k)+iΓ
Z−∞ F − i keep a coherent ballistic contribution that decays expo-
1
= eikiL, (12) nentially with intra- and interband scattering (thus ne-
vgi(ki) glecting vertex corrections). In particular, we disregard
Im(Xki)>0
terms that are affected by multiple scattering between
wherethe sumisoversimple polesof[EF ǫi(k)+iΓ]−1 disordered regions. This is a major limitation if, e.g.,
−
intheupperimaginaryhalfplane[assumingnoothersin- there are strong-scattering regions at the contacts in-
gularities under the analytic continuation of ǫi(k) in the duced by the lead-deposition procedure. We have as-
imaginary direction] and vgi is the corresponding group sumed that injected electrons that move to the left and
velocity [see Eq. (2)]. The disorder is accounted for in those electrons that have passed under the right lead do
Eq. (12) by an imaginary self-energy Γ = 1/2τ charac- not contribute to δG . Finally, cross-sectional shape
12
terized by the scattering rate 1/τ. For weak disorder, and size variation of the wire could affect the interfer-
we only take into account poles along the real axis with ence pattern due to the magnetic field. For example,
groupvelocityvgi atǫi(k)=EF,theFermienergy,anda a strong radial variation can deteriorate the periodicity
correspondingimaginarycontributiontothewavevector and strength of the magneto-oscillations in the parallel
of field if the electrons are confined at the surface.
In summary, we study the conductance of translation-
δk iΓ/v . (13)
i gi
≈ allyinvariantnanowires,focusingonMCoscillationsdue
In particular, we see that close to van Hove singularities to the ballistic paths between the leads. It is assumed
where v 0, the propagator (12) is strongly damped thatthepathsthatinvolvescatteringfromdisorderresult
gi
→
due todisorderscattering. We willcharacterizethescat- in rapidly oscillating contributions, which average out
tering strength by the mean free path λ = v τ using giving a nearly structureless background for the MC os-
F
6
cillations relatedto the ballistic interference (apart from terms
the density-of-states singularities of the incoherent con-
otruisbustuibonsecGt1io2nt)o. tIhfethcoisndisucntoatntcheedicsacsues,sethdeinnotuhre cparlecvui-- G(nn′) =ReGnGn∗′ sin[(nn−nn′′)θ0] 2 (16)
(cid:12) − (cid:12)
lations would apply to the mean conductance for a set (cid:12) (cid:12)
betweenmodesnandn′arisin(cid:12)ginthisidealiz(cid:12)edsituation
of nominally identical samples. Electron-electron inter- (contributing to the conduct(cid:12)ance δG (cid:12) G(nn′)).
actions are not included, but are briefly commented on 12 ∝ nn′
The last factor in Eq. (16) describes the suppression of
in Sec. IID. P
theinterferencebetweenmodeswithdifferentorbitalan-
gular momenta due to the finite injection and detec-
tion half-angle θ . In particular, if θ = π, all the
0 0
C. Cylindrical-shell model crosstermsvanishandG(nn′) = n 2δnn′,reminiscentof
the result17 for the Kubo condu|cGtan|ce of a finite-length
As a further simplification, let us for a moment con- wire (8). Otherwise, we have to consider the oscilla-
sider the shell model where electrons are confined in the tory part of the cross terms proportional to Re n n′:
radial direction to a narrow cylindrical well at radius G(nn′) cos[(kn kn′)L]exp[ ΓL(1/vn+1/vn′)]/GvnGvn′,
∝ − −
re. Summing over the leads’ degrees of freedom µ, ν where kn is the Fermi wavevector andvn groupvelocity
in Eq. (6) and assuming the leads are metallic with high forthenthmodealongthewire. Thecorrespondingrate
electrondensity,wearriveatthefollowingexpressionfor ofG(nn′) phaseaccumulationinmagneticfluxφz isgiven
the coherent conductance: by
δG12 ∝ i,j GiGj∗(L,EF)Z−ππdθ|λs(θ)|2ϕˆ†i(θ)ϕˆj(θ) ωnn′ = 2Lπ∂(kn∂−φzkn′). (17)
X
π For the shell model with weak scattering, k2 =
×Z−πdθ′|λd(θ′)|2ϕˆ†j(θ′)ϕˆi(θ′). (14) 2m(vE/Fv−)((nL/+2φπzr)2),/rwe2haenred,mfivxin=g(EnF+,φ(L)//2rπ)a∂nkdn/m∂nφvz ==
θ n e θ z e n
−
Here, ϕˆ (θ) is the angular component of the electron kn. ωnn′ is thus equal to the relative winding number of
i
the classical trajectories corresponding to modes n and
spinorwavefunctionattheFermienergyintheithmode
n′, on traversing the wire of length L around the cylin-
ofthewireandλ (θ)isthetunnelingamplitudeatthe
s(d)
source (drain) lead, which is assumed to be independent der of radius re. Since ωnn′ is itself field dependent, it
doesnotcorrespondtoFouriercomponentsofG(nn′)(φ ),
of the electronic state in the leads but angle-dependent z
however. Inaddition,thelatterismodulatedbythemag-
around the wire circumference. In the following, we set
thetunnelingamplitudetobeλs,d(θ)=1for θ <θ0and neticfieldviatheprefactorexp[−ΓL(1/vn+1/vn′)]/vnvn′
| | which has sharp features near the van Hove singularities
zerootherwise,forbothcontacts. Finally,expressingthe
in clean systems. Note that the fastest phase accumula-
wavefunctionsintermsoftheangular-momentumeigen-
states, ϕˆ (θ)= cˆ einθ, we obtain tionratesωnn′ correspondtosmallvelocitiesvnalongthe
i n in wire and are thus more sensitive to disorder scattering.
P 2 We now consider the effect of a perpendicular field Bx
δG12 ∝Xi,j GiGj∗(L,EF)(cid:12)(cid:12)Xnn′ cˆ†incˆjn′sin[((nn−−nn′′))θ0](cid:12)(cid:12) . olwanextcohouenrteinnaesusreugtymospaptseisocuntmruoemftahaazntedrthot-uetnhcnoicenklfiinnnegesmsDesOhnStel.ilsWmsueoffidshecali,elnlbtruleyt-
(cid:12) (cid:12)(15)
(cid:12) (cid:12) strong that at most one radial state is occupied for any
If B = 0 and the spi(cid:12)n-orbit coupling is neglec(cid:12)ted,
x givenangularmomentumn. Turningontheperpendicu-
α = 0, the z component of the angular momen-
R lar field B perturbatively gives the following correction
tum commutes with the Hamiltonian (11), and there- x
to the energyeigenstate withangularmomentumn (dis-
fore cˆ = s δ , where s is the spin-s state along the
in | i in | i regarding spin-orbit interaction):
z axis and δ is the Kronecker delta. If we also ne-
in
gthleectcotnhdeucZteaenmceanofcoouuprlsinhge,ll-tmhoendeblywigraeumgeusitnvbaeriaanpcee-, ∆ǫ (k) (eBx)2 r2 + k2 |hrinn′|2 ,
n nn
riodic function of the∞flux φz = eBzre2/2, so we may ≈ 4m h i m n′X=n±1 ǫn−ǫn′ !
write δG (φ ) = g exp(i2πmφ ). However, (18)
12 z m=−∞ m z
there may be considerable weight in the higher harmon- which is readily obtained by second-order perturbation
P
ics of the fundamental frequency, so that g may not theory using Eq. (11) [note that the lowest-order energy
m
| |
be largest for small values of m. Also, the magnitude correction is quadratic in B since the term linear in
x
of δG may oscillate multiple times within the period B in the Hamiltonian (11) mixes angular-momentum
12 x
1/2<φ <1/2, and it is natural to ask whether there states that have different energies]. Here, r2 is the
z nn
− h i
maybeacharacteristicfrequencyofsuchoscillations. Al- expectation value of the radial position squared in the
though there is no simple answer to this question, it is nth state and r nn′ is the radial-position matrix el-
h i
helpfultolookatthe physicalcontentofthe interference ement between radial wave functions of states n and
7
n′. For example, applying this result to the shell model lous compressibility of interacting electrons in 1D. Some
(hr2inn =|hrinn′|2 =re2)gives(omittingZeemanenergy) qualitative features of the ballistic magneto-oscillation
patterns, however, may remain similar to the noninter-
(n+φ )2+2φ2 k2 2φ2 acting electrons. For example, the gauge invariance still
ǫ (k) z x + 1+ x ,
n ≈ 2mr2 2m (n+φ )2 1/4 dictatesthe h/eperiodicityinthe parallelmagneticfield
e (cid:20) z − (1(cid:21)9) for the narrow-confinementmodel, if we neglectthe Zee-
where φ = eB r2/2 is the flux that the perpendicular manenergy. The spectralcompositionofthe oscillations
x x e
field B would produce in an area πr2, in units of the is, however, in general richer than discussed for nonin-
x e
flux quantum. We see that for a wire of circular cross teracting electrons in Sec. IIC: In addition to the in-
section, a weak magnetic field in the perpendicular di- terference between one-electron trajectories between the
rectionhasless effectonthe energiesthanthe samefield contacts, there will also be contributions involving other
along the wire, since the former produces energy shifts electronsin the wire excitedby the tunneling, which can
proportional to φ2 while the latter produces shifts lin- similarly collect phase due to the applied magnetic field.
x
ear in φ , for n = 0. The linear dependence on φ is a Furthermore,thespin-chargeseparationcharacteristicof
z z
6
consequence of degeneracy between states of n and n, 1D electron liquids may profoundly effect the interfer-
whichwillingeneralbeliftedbyanydeviationsfrom−cir- ence at a finite voltage bias.8 These topics are, however,
cular symmetry. The energy dependence on φ is then beyond the scope of the present paper.
z
quadratic for φ 0, but the curvature will be large if
z
→
the zero-field splitting between n and n is small, and
−
the dependence on φz becomes linear again when the III. NUMERICAL RESULTS
| |
shift due to φ is larger than the zero-field splitting.
z
As anapplicationofthe aboveformalism,we will here
calculate the magnetoconductance in a model approxi-
D. Electron-electron interactions mating InAs nanowires. InAs surface (or interfaces with
other materials) exhibits unusual properties, which are
Therecanbeseveraleffectsduetotheelectron-electron not manifested by most other semiconductors. InAs in-
interactionsaffectingmagnetoconductanceoscillationsin terfaceswithmetalstypicallydonotformSchottkybarri-
quasi-1D wires. In both disordered and clean wires, in- ers,makingitpossibleto achieveØhmiccontacts(which
teractions lead to the inelastic-scattering rate growing was utilized, e.g., in Ref. 10 for using InAs nanowires
with temperature. In the most naive form, this may be as tunable supercurrent links between superconducting
accounted for by adding a contribution to the scatter- leads). Related to this is the property of InAs sur-
ing rate Γ characterizing the single-electron propagator faces to support two-dimensional quantum-well states
(12). Afinitetemperaturecanalsoleadtoaqualitatively whosepropertiescanbetunedbythebulkdopinglevel.19
different effect, even in the absence of inelastic scatter- Owing to this surface-charge layer, 1D nanowires fab-
ing: TheMCoscillationamplitudeissuppressedasT−1/2 ricated out of InAs can be expected to carry a finite
when k T > E , the Thouless energy, i.e., the inverse number of 1D quantum channels even when the bulk is
B Th
time for electron transfer between the contacts.2 insulating.4,6Forexample,takingthe2Delectrondensity
For a wire that is connected to the outside world only to be n = 1012 cm−2,13,14 we find that a 2D electron
2D
through weak tunnel junctions, if the wire is short so gas with radius r 10 nm can support as many as 10
e
∼
that its Coulomb charging energy E = e2/2C is larger quantum channels.
C
than the temperature, the usual Coulomb-blockade res- In specific calculations, we will take the parameters14
onantstructureforthe incoherent(sequential-tunneling) measured for a 2D inversion layer at the surface of a
conductance G will set in as a function of source-drain p-doped InAs bulk: Effective mass m = 0.027m , g-
12 e
and gate voltages. Away from the resonances, and at factorg = 15,Rashbacoefficientα =1nm2,disorder-
R
−
sufficientlylowtemperatures,higher-orderquantumtun- scattering mean-free path λ = 100 nm, and triangular-
neling of the electric charge (i.e., the cotunneling) will well confinement with the slope ∂ V = 15 meV/nm,
r
dominate the current, see, e.g., Ref. 16. For free elec- which corresponds to the classical turning point of the
trons, the elastic component of the cotunneling signal ground state at the depth of about 10 nm. We will take
reduces to our coherent conductance δG , Eq. (4). The the injection/detection half angle to be θ = π/5, lead
12 0
Coulomb-blockaded regime in short wires thus opens an separation L = 300 nm, and radius r = 15 nm which,
0
interesting venue for studying quantum-interference ef- as explained below, corresponds to an effective radius of
fects, which are higher-order in tunneling. r 10 nm for electron distribution if using a quasi-2D
e
∼
At low temperatures and vanishing disorder, another shell model discussed in Sec. IIC (assuming that only
interesting regime may be achieved characteristic of the the lowest quantum-well state is occupied). The char-
nanowire geometry: that of the Luttinger-liquid state of acteristic spin-orbit splitting 2α ∂ Vk is of the order
R r F
the1Dmodes.21Inthatcase,thetunnelingamplitudeac- of 10 meV. For simplicity, we will disregard the excited
quiresapower-lawsuppressionateachcontactwithlow- well states (which is reasonable for the chosen parame-
eringtemperatureandbias,22 determinedbythe anoma- ters), keeping in mind that the profile of the charge dis-
8
50
].s0.2
0.8 tin
u
40 bra0.1
])² rme30 ]stinu0.6 [ Gδ₂₁ 0
2(/1[ )k( εn20 .bra[ Gδ₂₁0.4 0 0.5 ϕ1$ 1.5 2
0.2
10
0 0
-6 -4 -2 0 2 4 6 0 0.5 1 1.5 2
k [1/r ] ϕ [h/e]
e z
FIG.3: Bandstructureofthe1Dwireinalargeperpendicular FIG. 4: The main panel shows the interference MC traces in
magneticfield,φ =3(φ =0),fortheshellmodelwithr = the parallel magnetic field for an idealized shell model with
x z e
10 nm. Following a complicated nonmonotonic behavior of r = 10 nm, g = 0, and α = 0 (solid line), a shell model
e R
ǫ (k)atsmall k andn,theenergygrows nearlylinearly with including Zeeman splitting and spin-orbit coupling (dashed
n
katlargerk,andeventuallywiththeusual∝k2dispersionat line), and a triangular-well model with outer wire radius
k→∞. Thesedifferentregimescorrespondtothedominance r0 = 15 nm (dotted line). All other parameters are listed
of the terms on the right-hand side of Eq. (11) having the in the text (in particular, L = 300 nm). The inset shows a
respective scaling with k. The dashed line shows the Fermi- calculation similar to the dotted line in the main panel but
level position. fortheperpendicularmagneticfield. Thequantum-wellstates
aretreated intheWKBapproximation(seeSec.IIIBforde-
tails).
tribution can still be nontrivial on the scale of the wire
radius. Wefixthe1Delectrondensity,regardlessofmag-
netic field,to approximatelycorrespondto a2Delectron tions for corresponding modes are then evaluated using
gas with density n = 1012 cm−2 residing at radius r Eq. (12) and the conductance is finally obtained accord-
2D e
(more specifically, the density is chosenso that the zero- ingtoEq.(15),whichrequiresknowingwavefunctionsas
fieldlowest-bandFermiwavevectoralongthe wire isthe well as dispersions [both given by the diagonalization of
same as that of the 2D electron gas with density n ). Eq. (11)]. Examples of the magnetoconductance traces
2D
are shown in Fig. 4 for parallel and perpendicular field
orientations, taking into account both Zeeman splitting
A. Narrow-well confinement (“shell model”) and spin-orbit coupling. In the case of the parallel field,
the conductance is in general aperiodic in φ , although
z
Letusfirstconsideranidealizedsituationofanarrow- some reminiscence of the h/e periodicity is still present.
well confinement of the electron distribution at radius In the Fig. 4 inset, there are considerable MC oscilla-
r , which could be relevant for wider nanowires. As be- tions at lowperpendicular fields, before the conductance
e
fore, we define the (dimensionless) magnetic flux to be is suppressed at larger fields. The latter can be qualita-
φ = eB r2/2. In the absence of the transverse tively understood to be due to the onset of the Landau-
x(z) x(z) e
field, φ =0,andneglectingthe Zeemansplitting, gauge levelformationintheperpendicularmagneticfield,which
x
invariance dictates that the conductance is periodic in flattens the dispersions (see Fig. 3), lowers group veloc-
φ with the period of unity. The periodicity is, how- ity along the wire, and therefore, according to Eq. (13),
z
ever,brokenbyafiniteg-factorandbyanextendedcon- enhancesdisorderscatteringonthepropagationbetween
finement in the radial direction. According to the time- the contacts. It is worthwhile noting in this regard one
reversal symmetry, the linear conductance is symmetric general feature near the van Hove singularities: Fixing
under magnetic-field inversion, G( B) = G(B), so that the1Delectrondensity,theFermileveltendstobesome-
−
we need to consider positive fields only. what attracted to the van Hove singularities before 1D
The 1D dispersion is given by Eq. (19) at low perpen- modes disappear or after they start getting populated,
dicularfields,φ 1,andvanishingspin-orbitcoupling, due to the diverging compressibility. The corresponding
x
≪
α = 0. In general, we can calculate the dispersion nu- features can be seen in Fig. 5 in the Fermi-level position
R
merically by diagonalizing Hamiltonian (11) (see Fig. 3 close to where it crosses transverse energy levels in the
for φ = 3). In order to find the conductance at a finite parallel magnetic field.
x
magnetic field, we first calculate the Fermi level keeping AsdiscussedinSec.IIA,seeEqs.(1)and(3),thetwo-
electron density fixed and then find the crossing points terminal source-drain magnetoconductance is expected
of ǫ (k) = E with positive slopes. The Green’s func- to be dominatedby the density-of-statessingularities. It
n F
9
10 0.4
8
0.3
])² rme 6 ]stinu
2(/1 .bra0.2
[ )0( εn 4 ₂₁[ Gδ
0.1
2
0 0
0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1
ϕz [h/e] ϕz [h/e]
FIG. 5: k = 0 energies for quasi-1D bands calculated by di- FIG. 7: Shell-model calculation of the interference magneto-
agonalizing Hamiltonian (11) with the magnetic field along conductancefor 2D electron densities between 1 and 3 in 0.1
the wire axis for a shell model with r = 10 nm. Dotted increments (with lighter line shades corresponding to higher
e
lines show spin-degenerate energies after setting g = 0 and densities),inunitsof1012 cm−2. Thedashedlineistheaver-
α = 0. Solid lines are calculated taking into account the age. Here,gandα aresettozero,sothatthe1×1012 cm−2
R R
Zeeman and spin-orbit energies. Zero-field spin degeneracy trace is thesame as thesolid line in Fig. 4.
is lifted by the latter while Zeeman splitting dominates at
φ ≫ 1. The dashed line is the Fermi energy correspond-
z
ing to the solid-line energies, keeping the total 1D electron means, as a function of magnetic fields B and B , this
z x
density fixed. would, in principle, allow us to obtain a wealth of infor-
mation about the band structure and the electron dis-
60 tribution in the radial direction [by using relations like
Eq.(18)obtainedforlowperpendicularfieldsB ]. Ade-
x
tailedprocedurewouldrequireafurthertheoreticalanal-
50
ysis.
]s40 Finally, before closing this subsection, we would like
tin to note an interesting peculiarity of the magnetocon-
u
.bra30 delueccttaronncedaetnsliotwy, pwaeralfilenldmtahgantettihcerfieeldiss.mVoraeryoinftgenthae
[ SO20 maximum of δG12(φz) at φz = 0, rather than a min-
D imum (there must be one of the two due to the time-
reversal symmetry), over a wide range of parameters.
10
This ballistic-interference maximum at low fields per-
sists over density averaging, but it is different from the
00 0.5 1 1.5 2 weak-antilocalization peak in the presence of a strong
ϕ spin-orbit interaction in diffuse wires. See Fig. 7 for the
shell-modelcalculationwithzerospin-orbitcouplingand
Zeemansplitting,wherethe2Delectrondensityisvaried
FIG. 6: The total density of states as a function of the par-
between1and3in0.1increments,inunitsof1012 cm−2.
allel(blackanddark-grayhistograms) orperpendicularmag-
netic field (light-gray histogram), which governs theincoher- We, however, expect the opposite statistical property,
ent two-terminal double-tunnel-barrier conductance (1) ac- i.e.,alocalminimumatzerofieldforthe incoherentcon-
cordingtoEq.(3). Theblackhistogram wascalculatedusing tribution to the conductance G .
12
the same parameters as the solid line in the main panel of
Fig. 4, the dark-gray histogram—the dashed line, and the
light-gray histogram—the dotted line.
B. Triangular-well confinement
Consider now a triangular confinement potential
is plotted in Fig. 6. By comparing to Fig. 4, we see
that the sharp DOS features are fully suppressed in the V(r)=β(r r)+ Θ(r r ), (20)
0 0
− ∞ −
coherent-tunneling contribution to the conductance due
tothesignificantdisorderscattering. Wenotethatifone where β is the confinement steepness and Θ is the Heav-
was able to identify the positions of these van Hove sin- iside step function defining the sharp cutoff at the outer
gularities in the two-terminal measurements or by other radiusr . For the magneticfield alongthe wire axis,the
0
10
20 approximation, which is not expected to be very accu-
4
-6 rate for quantum-well ground-state energies in any case)
if precise energy levels are desired. The zero-field radial
15 -5
3 electron distribution is shown in Fig. 2 for a somewhat
])² rme10 -3 -4 moroigrienraeanldistniocrcmoanlfiizneedmteonthpavoetesnltoipael,βwhaitchthies flbaotunadtatrhye.
2
(/1 2 The electron distribution is indeed rather broad within
₀[ ε 5 -2 the wire. For a quantitative analysis, it would be nec-
- εn 1 -1 ceossnasriystetnotldyewteirtmhitnhee ethleectcroonnfidniisntgribpuottioennt,iaals aVf(urn)csteiolfn-
0 of the magnetic field.
0
-5
0 0.5 1 1.5 2 IV. DISCUSSION
ϕ [h/e]
z
We theoretically studied the magnetoconductance
FIG. 8: Solid lines are the ground-state energies (relative to (MC)ofcylindricalquasi-one-dimensionalnanowirescar-
the n = 0 energy) for transverse triangular-well states with ryingseveralquantumchannelsconfinednearthesurface.
fixed angular momenta n (labeled in the plot) as a function We focused on the MC oscillations arising due to the
of theparallel magnetic field, setting g=0and α =0. The
R ballistic interference of electrons propagating via several
wireradiusr0=15nmandotherparametersarelistedinthe
1D modes along a finite-length wire section between a
beginningofSec.III. Dottedlinesarecalculated fortheshell
metallic source and drain leads. Van Hove singularities
modelwithr =10nm,whichisacharacteristicradiusforthe
e
inthe1Ddensityofstatesareexpectedtodominatemost
radial-distribution center of mass at small angular momenta
n. SeeFig.2forasimilar confiningpotentialV(r)(although prominent MC features in a two-terminal configuration,
not strictly triangular as here). φ =eB r2/2. and we have therefore suggested to use additional reser-
z z e
voirsatthe endsofthe wiretoabsorbinjectedelectrons,
which diffused away from the leads, in order to enhance
totaleffectivepotentialintheradialdirectiondependson the more interesting ballistic-interference features.
the angular-momentumstate, the highermomenta (with In order to calculate mesoscopic MC interference pat-
respect to the flux φ ) being pushed more outward, as terns, we have studied the electronic structure of 1D
z
−
can be seen from Eq.(11). We calculate the correspond- modes in the long wires for an arbitrary uniform mag-
ing transverseground-stateenergiesfordifferentangular netic field, taking into account Rashba spin-orbit cou-
momenta using the WKB approximation,in the absence pling due to the surface-confinement potential. In the
of the spin-orbit interactionand perpendicular magnetic case of the parallel magnetic field along the wire axis z,
field. For simplicity, we use Eq. (16) for suppression of the MC is periodic inthe magnetic fluxφ with aperiod
z
the cross terms describing the interference between dif- of h/e only in the case of a narrow surface confinement
ferent angular-momentum states, which was derived for and vanishing Zeemansplitting. Relaxing these assump-
the shell model. The spin-orbit coupling and, if present, tions, the conductance oscillationsbecome aperiodic but
the perpendicular magnetic field are also included sim- some remnants of periodicity are still visible for realis-
ilarly to the shell model, after we choose an appropri- tic parameters, as seen in Fig. 4. A large perpendicular
ate effective radius r [by comparing triangular-welland magnetic field leads to the Landau-level formation and
e
shell-modelenergies(seeFig.8)]. Thecalculatedmagne- suppressionofthe discussedmesoscopicmagnetoconduc-
toconductance for the parallel magnetic field is plotted tance. The amplitude of the conductance fluctuations is
in Fig. 4 by the dotted line. reducedatafinite voltagebiasandtemperatureorupon
In Fig. 8, we plot the parallel-field dependence of the gate-voltageaveragingmodulatingelectrondensity. Cer-
triangular-wellground-stateenergiesatfixedangularmo- tain interference features, such as the local maximum in
menta n, which were used in calculating the magneto- δG atzeroparallelmagneticfield,however,cansurvive
12
conductance. InordertocompareWKBandshell-model energy averaging (see Fig. 7).
calculations, the lowest energy is set to zero. For larger
n (with respect to φ ), the radial potential in Eq. (11)
z
−
pushes electrons outwards so that the electrons become Acknowledgments
confinedtothe narrowwellnearthesurfaceatr =r ,as
0
n , decreasing the period of magneto-oscillations. Thediscussionpresentedherewasmotivatedbytheex-
→ ∞
This trend can be seen in Fig. 8. In particular, this perimentsonthemagnetoconductanceofInAsnanowires
means that it is rather crude to characterize different performed by A. Greytak, K. Kim, and C. M. Lieber at
angular-momentum states by a single effective radius re HarvardUniversity. This work was supportedin partby
and one would have to solve Eq. (11) for a fully three- NSF Grants No. PHY01-17795and No. DMR 05-41988
dimensionalelectrondistribution(andbeyondthe WKB and by the Harvard Society of Fellows.