Table Of ContentSpin precession and modulation in ballistic cylindrical nanowires due to the Rashba
effect
A. Bringer
Peter Gru¨nberg Institute (PGI-1) and JARA-Fundamentals of Future Information Technology,
Forschungszentrum Ju¨lich GmbH, 52425 Ju¨lich, Germany
Th. Sch¨apers∗
Peter Gru¨nberg Institute (PGI-9) and JARA-Fundamentals of Future Information Technology,
Forschungszentrum Ju¨lich GmbH, 52425 Ju¨lich, Germany
(Dated: January 17, 2011)
1
ThespinprecessioninacylindricalsemiconductornanowireduetoRashbaspin-orbitcouplinghas
1
beeninvestigatedtheoreticallyusinganInAsnanowirecontainingasurfacetwo-dimensionalelectron
0
2 gas as a model. The eigenstates, energy-momentum dispersion, and the energy-magnetic field dis-
persionrelationaredeterminedbysolvingtheSchro¨dingerequationinacylindricalsymmetry. The
n
combination of states with the same total angular momentum but opposite spin orientation results
a
in a periodic modulation of the axial spin component along the axis of the wire. Spin-precession
J
aboutthewiresaxisisachievedbyinterferenceoftwostateswithdifferenttotalangularmomentum.
4 Because a superposition state with exact opposite spin precession exists at zero magnetic field, an
1
oscillation of the spin orientation can be obtained. If an axially oriented magnetic field is applied,
the spin gains an additional precessing component.
]
l
l
a
h I. INTRODUCTION energy spectrum and spin precession in these structures
- are governed by the interplay between confinement and
s energy splitting due to spin-orbit coupling.20,21 Only a
e Semiconductor nanowires are almost ideal objects
m for studying quantum effects and electron interference few theoretical investigations have dealt with the effect
of spin-orbit coupling in cylindrical conductors on the
. phenomena. The use of the bottom-up approach for
t electronic states and on the quantum transport.4,9,22,23
a nanowire growth simplifies the preparation substantially
m and allows us to create novel confinement schemes, such The spin-dynamics in curved two-dimensional electron
- as axial or radial heterostructures.1,2 The large surface- gases was discussed by Trushin and Schliemann24 while
d to-volume ratio of nanowires means that surface prop- the weak antilocalization effect in cylindrical wires was
n studiedbyWenkandKettemann.25Thepresenceofspin-
erties are crucial for discussions of transport properties,
o orbitcouplingwasconfirmedforInN26andInAssemicon-
so that low band-gap semiconductors, e.g. InAs, InN,
c ductornanowiresbymeasuringtheweakantilocalization
[ or InSb, are particularly interesting. In these systems,
the Fermi level at the surface is pinned inside the con- effect.27–30
2 duction band,3 and an accumulation layer is formed. The various possibilities of spin control in two-
v Thisguaranteesthattheconductanceissufficientlylarge dimensional electron gases and planar wire structures
7
even at low nanowire radius. The presence of the sur- opened up by the Rashba effect have inspired us to an-
5
5 face accumulation layer means that a tubular conduct- alyze theoretically the spin dynamics in tubular conduc-
4 ing channel is formed, and this shape of the conductor tors. We have used a cylindrical InAs nanowire with a
. has important implications for the magnetoconductance surface two-dimensional electron gas as a model system,
1
1 of the nanowires. An example is the theoretical predic- but our findings also apply to other systems, e.g. InN
0 tionandexperimentalconfirmationofflux-periodicoscil- or InSb nanowires. In Sect. II we analyze the electronic
1 lations in nanowires with a magnetic field applied along states, focusing on spin properties, and we discuss the
: thewireaxis.4,5Theelectronicstatesofacylindricaltwo- conditionsunderwhichaspinprecessioncanbeobserved
v
i dimensional electron gas in a transverse magnetic field intubularnanowiresatzeromagneticfield(Sect.III)and
X werecalculatedbyFerrarietal.,6,7whileMagarilletal.8,9 inanaxialmagneticfield(Sect.IV).InSect.V,wecom-
r discussedthekineticsofelectronsinatubularconductor. ment on the suitability of tubular conductors for spin
a electronic devices.
Many concepts have been developed for planar semi-
conductor layer systems that make use of the spin de-
gree of freedom for device structures. The best-known
example is the spin field-effect transistor,10–12 which II. ELECTRONS IN CYLINDRICAL WIRES
uses the gate-controlled spin-precession induced by the
Rashba effect.13–15 The Rashba spin-orbit coupling orig- Electrons confined in a cylinder move along the axis
inates from a macroscopic electric field in an asymmet- withalinearmomentum(cid:126)k (k real)andaroundtheaxis
ric quantum well.16 Meanwhile, research activities have withanangularmomentum(cid:126)l(linteger). Aslongasthe
been extended to planar quasi one-dimensional struc- translational and rotational symmetries of the cylinder
tures, which promise a superior spin control.17–19 The are not perturbed these momenta are conserved quanti-
2
ties. The wave function of an electron
ψ = exp(ıkz)exp(ılφ)f(r)
is a product of exponential functions in z,φ, the coor-
dinate along the axis and the azimuthal angle around
the axis respectively, and a radial distribution function
f(r). The distribution is determined by internal forces
producedbythecylindermaterial. Inourcase,wetooka
planar 2-dimensional electron gas (2DEG) at the surface
of InAs as a reference,31,32 i.e. assuming a surface state
charge density of N =1.27×1011 cm−2, a background
S
p-doping of n = 2.8×1017 cm−3 and an effective elec-
d
tron mass of m∗ =0.026m . The calculations were done
e
for a cylinder radius r = 50 nm. A schematic illustra-
0
tion of the nanowire is depicted in Fig. 1 (upper inset).
Electrons of atoms at the surface may find energetically FIG. 1: Squared amplitude of the wave function |ψ|2, the
more favorable states in the conduction band. Due to spinorcomponentsf andhandpotentialprofileV asafunc-
the Coulomb attraction between the electrons and the tion of the normalized radius r/r . The upper inset shows a
0
ions remaining at the surface the electrons get trapped schematicillustrationofthenanowires,includingtherelevant
in a layer close to the surface forming a 2DEG.33 The electric and magnetic fields. The lower inset shows the spin
potential V resulting from the charge density of occu- orientation along the circumference for j =1/2.
pied electron states ψ , of ions at the surface and of
l,σ,k
dopants ρ
BG
matricesforσ actingona2-component(spinor)wave
occ x,y,z
ρ = e(cid:88)|ψl,σ,k|2 + ρBG /(cid:15)r (1) function (ψ↑,ψ↓). The off-diagonal terms in HSO raise
(lower)thevalueoftheorbitalangularmomentumL of
z
l,σ,k ψ (ψ ) by (cid:126). The stationary states are eigenstates of
↑ ↓
isshowninFig.1. eistheelementarycharge, σ thespin thetotalangularmomentumJz =Lz+Sz (Sz =(cid:126)σz/2)
index. (cid:15) =14.6isthebulkdielectricconstantofInAs.34 with eigenvalues j =l±1/2. The spinor is of the form:
r
It takes the polarization charges of the medium into ac-
(cid:18) (cid:19) (cid:18) (cid:19)
count. The potential profile is determined by Poisson’s ψ↑ =eıkzeılφ f(r) , (4)
equation which is solved in cylindrical symmetry analyt- ψ↓ ıeıφh(r)
ically
where f,h are real functions and solve the differential
(cid:90) r r(cid:48) equations
V = 4π(cid:15) e r(cid:48)dr(cid:48)ρ(r(cid:48))ln . (2)
0 r
0 (cid:126)2 (cid:18) 1 (cid:19) (cid:16) (cid:17)
− f(cid:48)(cid:48)+ f(cid:48) + Vˆ −(cid:15)ˆ f = kγV(cid:48) h,
Equations (1) and (2) are solved self-consistently. Start- 2m∗ r l,+
ing from the potential of a homogeneous distribution of (cid:126)2 (cid:18) 1 (cid:19) (cid:16) (cid:17)
electrons in the cylinder the distribution is recalculated − h(cid:48)(cid:48)+ h(cid:48) + Vˆ −(cid:15)ˆ h = kγV(cid:48) f .(5)
2m∗ r l+1,−
using the Schr¨odinger equation given below [see Eq. (5)]
andEq.(1). Theiterationprocedureconvergesmonoton-
ically. Weassumedaninterfacebarrierofinfiniteheight. Here, Vˆl,± = ((cid:126)l)2/(2m∗r2)+V ±γV(cid:48)l/r contains the
contributions of the centrifugal force and the diagonal
DuetotheelectricfieldE(cid:126) =−∇V/eacrossthesurface spin-orbit term, (cid:15)ˆ=(cid:15)−((cid:126)k)2/(2m∗) is the energy with-
ofthecylinderthespin(cid:126)σ oftheelectroniscoupledtoits out the axial kinetic energy. At the wire boundary we
orbital motion assumed a barrier of infinite height.35 The influence of
an external magnetic field B is not included yet.
(cid:104) (cid:105)γ
H = (cid:126)σ· p(cid:126)×eE(cid:126) In Fig. 2 the energy (cid:15)ˆis plotted for several j-bands at
SO (cid:126)
B = 0. The parabola indicates the axial kinetic energy
= γV(cid:48)(cid:20)(cid:18) 0 ı e−ıφ (cid:19) ∂ leftout. ItcrossesthebandsattheFermi-momentumkF,
−ı eıφ 0 ı∂z i.e. states with energy below the parabola are occupied.
(cid:18)1 0 (cid:19) ∂ (cid:21) At k = 0 the coupling between l and l+1 vanishes [cf.
+ . (3)
0 −1 rı∂φ Eq. (3) ]. Classification with respect to l is possible.
The splitting between the second and third band (l =
Thecoupling-strengthγ isdeterminedbythebandstruc- ±1) is caused by the diagonal part of H and increases
SO
ture of the cylinder material (1.17 nm2 for InAs).34 The proportionaltol forthehigherstates. Duetothemirror
second part of Eq. (3) expresses H in terms of Pauli symmetry z ↔ −z states with angular momentum and
SO
3
FIG. 2: Energy vs. k dispersion at B = 0. The dashed line
indicates the axial kinetic energy left out, which crosses the
FIG. 3: Spin density (s ,s ) of the lower energy states for
bands at k . The pair of dots represent states forming the T z
F total angular momenta j = 1/2,3/2,5/2 and 7/2. The spin
superpositions states ψ and ψ , while the square
−5/2,⊥ +7/2,⊥ is oriented only tangentially and along the z-axis.
indicates the state ψ . The dotted line illustrates the
+1/2,⊥
Gaussian wave packet of width δk=1/r .
0
from (0,ψ ), the solutions at k = 0, with j = l +
l+1
spin reversed have the same energy. Therefore, all bands 1/2. Theyareorthogonaltoeachotherandhaveopposite
are twofold degenerate. spin direction (±). They have different energies (cid:15)ˆ and
Thesolution(f,h)ofEq.(5)forj =1/2atkF isshown therefo√redifferentkF. Theirsuperpositionψj,(cid:107) =(ψj,++
in Fig. 1. The spin-orbit coupling increases linearly with ψ )/ 2 yields
j,−
k, i.e. at k with l = 0 there is the strongest spin-orbit
F
coupling. The spin density attains a sizable tangential (cid:104)σ (cid:105) = (cid:0)(cid:104)f2 −h2 (cid:105)+(cid:104)f2 −h2 (cid:105)(cid:1)/2
z (cid:107) j,+ j,+ j,− j,−
component
+ (cid:104)f f −h h (cid:105)cos(k −k )z .
j,+ j,− j,+ j,− F,+ F,−
(cid:18)ψ∗ (cid:19)(cid:18) 0 −ı e−ıφ (cid:19)(cid:18)ψ (cid:19)
s = ↑ ↑ =2fh. The contributions of the basis states (±) almost cancel
T ψ∗ ı eıφ 0 ψ
↓ ↓ eachotherandareneglectedfurtheron. Theinterference
between the states is constructive due to orthogonality
The component along the wire axis is
and leads to
(cid:18)ψ∗ (cid:19)(cid:18)1 0 (cid:19)(cid:18)ψ (cid:19)
s = ↑ ↑ =f2−h2. (6)
z ψ↓∗ 0 −1 ψ↓ (cid:104)σz(cid:105)(cid:107) ≈2(cid:104)fj,+fj,−(cid:105)cos(kF,+−kF,−)z , (7)
The radial component is zero. The spin orientation an oscillation of the average spin along the cylinder axis
around the cylinder for j = 1/2 is illustrated in Fig. 1 withawavelengthλ =2π/|k −k |. Thespincom-
(cid:107) F,+ F,−
(lower inset). According to Eq. (6) the spin turns to the ponents (cid:104)σ (cid:105) , (cid:104)σ (cid:105) in the cylinder plane are both zero.
x (cid:107) y (cid:107)
axial direction. This is shown for different values of j in Superpositions of eigenstates with different j’s form
the plot of the spin densities s and s in Fig. 3. As can states with a non-zero average spin component in the
T z √
beseenhere, thespinisorientedexclusivelytangentially cylinder plane, e.g. ψ = (ψ + ψ )/ 2. As
j,⊥ j,+ j−1,−
and along the axial direction. When averaged over the one can easily retrace, these states originate from states
cylinder plane (cid:104)···(cid:105) for each state ψ the spin compo- with the same angular momentum l. The interference
j
nents (cid:104)σ (cid:105) and (cid:104)σ (cid:105) are zero, while a finite contribution term gives the only φ-independent contribution to the
x y
(cid:104)σ (cid:105) remains along the z-direction. densities of σ ,σ . With
z x y
(cid:18) (cid:19)
f(r)
ψ = exp(ızk )exp(ılφ) ,
III. SUPERPOSITION STATES AND SPIN j,+ F,+ ıeıφh(r)
PRECESSION
(cid:16) (cid:17) (cid:18) f˜(r) (cid:19)
ψ = exp ızk˜ exp[ı(l−1)φ] ,
j−1,− F,− ıeıφh˜(r)
For each k and j there are two solutions of Eq. (5)
ψ . The(+)-stateoriginatesfrom(ψ ,0),the(−)-state (8)
j,± l
4
the averages are
(cid:16) (cid:17)
(cid:104)σ (cid:105) = (cid:104)hf˜(cid:105)sin k −k˜ z ,
x ⊥ F,+ F,−
(cid:16) (cid:17)
(cid:104)σ (cid:105) = (cid:104)hf˜(cid:105)cos k −k˜ z . (9)
y ⊥ F,+ F,−
The (cid:104)σ (cid:105) contribution is small and does not depend on
z ⊥
z. Inparticular,forthesuperpositionψ ofthelowest
1/2,⊥
two states (cid:104)σ (cid:105) is zero.
z ⊥
Forψ ,thesuperpositionofψ andψ ,
−5/2,⊥ −5/2,+ −7/2,−
the spin precesses counterclockwise in the cylinder plane
along the cylinder axis, as illustrated in Fig. 4(a). Here,
we assumed an initial spin orientation along the −y di-
rection, which in practice can be realized by spin injec-
tion from a spin-polarized electrode. For ψ con-
+7/2,⊥
stituted of the opposite states ψ and ψ the
+7/2,+ +5/2,−
spin precession is clockwise. Both precessions have the
same period of λ = 2π/|k − k˜ |. Their energy
⊥ F,+ F,−
is degenerate. Due to their exactly inverse precession
sense the combination of these states results in an oscil-
latorybehaviorofthenetspinorientation,asdepictedin
Fig.4(b). Foraninitialspinorientationalongthe−y di-
rection the spin oscillates in the yz-plane. Superposition
of the respective opposite states restores the left-right
symmetry and eliminates spin precession. The oscilla-
tionperiodλ ofψ dependsonj. Forsmaller|j|, e.g.
⊥ j,⊥
ψ the corresponding difference in k and k˜
−3/2,⊥ F,+ F,−
becomes smaller so that the period λ is enlarged, as
⊥
one can infer from Fig. 4(c) as compared to Fig. 4(b).
The superposition state ψ constituted of the two
+1/2,⊥
lowest lying energy states ψ (cf. Fig. 2, square)
±1/2,±
shows no precession at all, because here k and k˜
F,+ F,−
are identical. Figure 4(d) shows the spin variation for
a Gaussian wave packet of width δk = 1/r centered
0
FIG. 4: (a) Counter-clockwise spin precession of electrons in
between the k ’s of the states ψ and ψ . In
F −3/2,⊥ +5/2,⊥ the superposition state ψ at the Fermi energy consti-
positionspacethiscorrespondstoadistributionofwidth −5/2,⊥
tuted of the states ψ and ψ for a propagation
2r . The oscillation deviates from a purely harmonic os- −5/2,+ −7/2,−
0 alongthewireaxisfromz/r =0 to30. (b)Spinorientation
0
cillation, as shown in Fig. 4(c), due to the contributions
of the sum of the contribution shown in (a) and the corre-
oftheotherstatesattheFermienergy. Thiseffectisalso sponding clockwise contribution ψ being a superposi-
+7/2,⊥
increasingwithdecreasing|j|whenthekF’sgetcloserto tion of ψ+7/2,+ and ψ+5/2,−. (c) Spin oscillations resulting
each other. from the combinations of the two lower energy superposition
statesψ andψ . (d)SpinvariationforaGaussian
The electron spin is usually injected from a spin- −3/2,⊥ +5/2,⊥
wave-packet of width 1/r centered between the k ’s of the
0 F
polarized electrode in all states at the Fermi energy E
F states ψ and ψ (cf. Fig. 2).
−3/2,⊥ +5/2,⊥
having the correct spin direction. Thus, if only the di-
rectionofthespin isfixedbytheelectrode, allstatesare
likely to transport electrons through the cylinder and a
definite precession will not be observed. The total spin
will only vary in the plane which is defined by the ini-
tial spin orientation and the z-axis, similar to the situ- IV. SPIN PRECESSION IN A MAGNETIC
ation illustrated in Fig. 4. In order to observe spin pre- FIELD
cession about the cylinder axis, a selection mechanism
whichbreakstheleft-rightsymmetryofthesystemmust
be adopted. As it will be discussed in the next section, The vector potential A(cid:126) = (−By/2,Bx/2,0) of a
thisisachievedbyapplyingalongitudinalmagneticfield longitudinal magnetic field introduces a paramagnetic
B(cid:126) =(0,0,B). (Zeeman-)anddiamagnetic(Landau-)termintoEq.(5).
5
FIG. 5: Energy vs. B dispersion (left panel) at k = 0 and
energy vs. k dispersion (right panel) at B = 0.13 T. The
dashed line indicates the axial kinetic energy left out, which
crossesthebandsatk . Thepairsofdotsindicatethestates
F
forming the superposition states ψ and ψ at k
−5/2,⊥ +7/2,⊥ F
withanetspininthecylinderplane. Thetwostatesψ
+5/2,+
and ψ with j =+5/2 are marked by triangles.
+5/2,−
Vˆ is extended to
l,±
(cid:126)e (cid:18) gm∗(cid:19) e2B2
V˜ =Vˆ + B l± + r2 ,
l,± l,± 2m∗ 2m 8m∗
e
with g the gyromagnetic-factor of the electron spin
(−14.9 for InAs34). The paramagnetic (second) term
in V˜ raises (cid:15)ˆ for states with j(or l) > 0 and lowers
l,±
(cid:15)ˆfor states with j(or l) < 0. The energy difference in-
creases ∝ lB for B (cid:28) l(cid:126)/(er2) (cf. Fig. 5). For larger
0
B (cid:15)ˆincrease ∝ B2 due to the diamagnetic (third) term. FIG. 6: (a) Spin precession of electrons at the Fermi energy
In the linear range the influence of the r-dependence of propagating along the wire axis for the superposition state
the third term is negligible. The densities do not change ψ−5/2,⊥ at B = 0.13 T. (b) Corresponding spin precession
for the state ψ . (c) Spin orientation and magnitude
significantly. +7/2,⊥
of the sum of the contributions shown in (a) and (b) for a
The main effect of B is the energetic separation of
propagation from z/r = 0 to 30. The arrow indicates the
the ±j-states. It opens possibilities of observing spin 0
direction of the initially injected spin.
dynamics in electronic transport. This will be demon-
strated in the following at B = 0.13 T. Figure 5 shows
the B-dependence at k =0 up to B =0.13 T and the k-
dependence at B =0.13 T of (cid:15)ˆfor states from j =±1/2
a spin precession can be achieved.
to±9/2. Again,theparabolamarkstheFermiedge. Su-
perpositions with spin in the cylinder plane according to In the previous section, we already pointed out that
Eq.(9),ψj,⊥aremarkedaspairsinFig.5. Thelowerpair thesuperpositionstateψj,(cid:107) withequaltotalangularmo-
corresponds to ψ depicted in Fig. 4(a). As illus- mentum but opposite spin orientation result in an os-
−5/2,⊥
trated in Fig. 6(a), it shows the same counter-clockwise cillation of the average spin along the cylinder axis. In
precession. Incontrasttothezerofieldcase, nowthesu- Fig. 7(a) and (b) these oscillations of (cid:104)σz(cid:105)(cid:107) are shown
perposition state ψ has a larger k -difference, i.e. for different values of j at B = 0.13 T. One finds that
+7/2,⊥ F
a shorter precession length [cf. Fig. 6(b)]. Consequently, for larger total angular momentum values the oscillation
the precessions of ψ and ψ are not exactly period is shorter owing to the larger difference of Fermi
−5/2,⊥ +7/2,⊥
opposite. In contrast to the case at B = 0, the spin wavevectors. InFig.5thestatescontributingtoψ+5/2,(cid:107)
stillrotatesfollowingthestatewiththefasterprecession, aremarkedbytriangles. Comparedtothepreviouslydis-
when both states are superposed. This is illustrated in cussed ψj,⊥ states, here the difference in the Fermi vec-
Fig. 6(c), where one finds that in addition to the oscilla- torsisrelativelylarge,leadingtoafasteroscillationcom-
tion of the spin amplitude its orientation is also changed pared to the spin precession period shown in Fig. 6(b).
during propagation. Thus, by applying a magnetic field Once again the application of an axial magnetic field
6
j is occupied. As we observed, for each superposition
state ψ different oscillation periods are found. This
j,(cid:107)
leads to a rather complex modulation of the spin along
the axial direction. An obvious strategy for simplifica-
tion is to reduce the number of occupied states, i.e. by
depleting the channel by means of a gate. Another pos-
sibility might be to only occupy certain states by means
of k-selective filters. This might be realized by means
of an injection through a single or a resonant tunneling
barrier. As pointed out in Sect. III, one possible way
to model this situation is to assume the formation of a
state with a Gaussian distribution around the average
momentum.
In addition to a spin injection and detection along the
wire axis it is also possible to inject spins in transversal
FIG. 7: (a) Spin orientation (cid:104)σ (cid:105) along the wire axis for the
z
superposition states ψ with j = +1/2,+3/2 and +5/2 at direction. Here, the spins are carried by superposition
j,(cid:107)
B = 0.13 T. (b) Illustration of the oscillation of the average states ψj,⊥ constituted of states with different total an-
spin along the wire axis for the superposition state ψ . gularmomentaj. Aslongasnomagneticfieldisapplied
+5/2,(cid:107)
(c)Modulationofthetotalspinorientation(cid:104)σz(cid:105)(dashedline) the spin is exclusively modulated in the plane spanned
resultingfromacombinationoftheψ−5/2,(cid:107)andψ+5/2,(cid:107)states by the injection orientation and the wire axis. Here, the
at B =0.13 T. output signal in a spin field-effect transistor is gained by
gate-modulating the spin orientation along or opposite
to a detector electrode, which is polarized parallel or an-
breaks the symmetry of the ψ states. As can be tiparalleltotheinjector. Byapplyinganaxiallyoriented
±j,(cid:107)
inferred from Fig. 7(c), a different oscillation period is magneticfield,spinprecessionaboutthewireaxiscanbe
found for the ψ and ψ states. Thus, when achieved. Thisadditionalfeaturemightbeaninteresting
+5/2,(cid:107) −5/2,(cid:107)
these states are combined a beating in the oscillation of optiontoimplementmorecomplexfunctionalitiesinspin
the average spin appears. electronic devices.
In conclusion, we have shown that semiconductor
nanowires affected by Rashba spin-orbit coupling are
V. CONCLUSIONS promisingcandidatesforfuturenanowire-basedspinelec-
tronic devices. The complex spin dynamics in these
Intheprevioustwosectionswelearnedthataninjected cylindrically-shaped conductors provide many opportu-
spin is strongly modulated while propagating through nities to tailor the device functionality.
a cylindrical nanowire. For a spin injection along the
wire axis, e.g. by a ferromagnetic electrode, the spin
is carried by superposition states with equal total angu-
lar momenta. In analogy to the spin field-effect tran- Acknowledgments
sistor based on a planar 2DEG,10 a transistor structure
can be realized by placing a second magnetic electrode We thank N. Demarina (Forschungszentrum Ju¨lich)
at the opposite terminal of the nanowire as a spin de- forfruitfuldiscussionsregardingtheSchr¨odinger-Poisson
tector. Control of the spin orientation can achieved by solverincylindricalsystemsandU.Zu¨licke(MasseyUni-
manipulatingthestrengthoftheRashbaeffectbymeans versity, New Zealand) and R. Winkler (Northern Illi-
of a gate electrode. By applying a bias voltage to the nois University, USA) on the Rashba effect at semi-
gate, the strength of the electric field E(cid:126) in the surface conductor interfaces. Furthermore, we acknowledge the
2DEG is adjusted. In order to obtain a uniform control support of or work by S. Blu¨gel and D. Gru¨tzmacher
within the channel, a so-called wrap-around gate should (Forschungszentrum Ju¨lich). This work was supported
be preferred.36 Usually, in a realistic situation a larger by the Deutsche Forschungsgemeinschaft through FOR
number of states with different total angular momenta 912.
∗ Electronic address: [email protected] (2006).
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