Spin precession and modulation in ballistic cylindrical nanowires due to the Rashba effect

3.8 MB·English

by  A. Bringer

Spin 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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